Contents
1 Introduction
1.1 About This Manual
1.2 Chapter Summaries
1.3 Contact Information
1.3.1 Customer Support
1.4 Q-Chem, Inc.
1.5 Company Mission
1.6 Q-Chem Features
1.6.1 New Features in Q-Chem 4.1
1.6.2 New Features in Q-Chem 4.0.1
1.6.3 New Features in Q-Chem 4.0
1.6.4 New Features in Q-Chem 3.2
1.6.5 New Features in Q-Chem 3.1
1.6.6 New Features in Q-Chem 3.0
1.6.7 Summary of Features Prior to Q-Chem 3.0
1.7 Current Development and Future Releases
1.8 Citing Q-Chem
2 Installation
2.1 Q-Chem Installation Requirements
2.1.1 Execution Environment
2.1.2 Hardware Platforms and Operating Systems
2.1.3 Memory and Hard Disk
2.2 Installing Q-Chem
2.3 Q-Chem Auxiliary files ($QCAUX)
2.4 Q-Chem Runtime Environment Variables
2.5 User Account Adjustments
2.6 Further Customization
2.6.1 .qchemrc and Preferences File Format
2.6.2 Recommendations
2.7 Running Q-Chem
2.7.1 Running Q-Chem in parallel
2.8 IQmol Installation Requirements
2.9 Testing and Exploring Q-Chem
3 Q-Chem Inputs
3.1 IQmol
3.2 General Form
3.3 Molecular Coordinate Input ($molecule)
3.3.1 Reading Molecular Coordinates From a Previous Calculation
3.3.2 Reading Molecular Coordinates from Another File
3.4 Cartesian Coordinates
3.4.1 Examples
3.5 Z-matrix Coordinates
3.5.1 Dummy Atoms
3.6 Job Specification: The $rem Array Concept
3.7 $rem Array Format in Q-Chem Input
3.8 Minimum $rem Array Requirements
3.9 User-Defined Basis Sets ($basis and $aux_basis)
3.10 Comments ($comment)
3.11 User-Defined Pseudopotentials ($ecp)
3.12 User-defined Parameters for DFT Dispersion Correction ($empirical_dispersion)
3.13 Addition of External Charges ($external_charges)
3.14 Intracules ($intracule)
3.15 Isotopic Substitutions ($isotopes)
3.16 Applying a Multipole Field ($multipole_field)
3.17 Natural Bond Orbital Package ($nbo)
3.18 User-Defined Occupied Guess Orbitals ($occupied and $swap_occupied_virtual)
3.19 Geometry Optimization with General Constraints ($opt)
3.20 Polarizable Continuum Solvation Models ($pcm)
3.21 Effective Fragment Potential calculations ($efp_fragments and $efp_params)
3.22 SS(V)PE Solvation Modeling ($svp and $svpirf)
3.23 Orbitals, Densities and ESPs on a Mesh ($plots)
3.24 User-Defined van der Waals Radii ($van_der_waals)
3.25 User-Defined Exchange-Correlation Density Functionals ($xc_functional)
3.26 Multiple Jobs in a Single File: Q-Chem Batch Job Files
3.27 Q-Chem Output File
3.28 Q-Chem Scratch Files
4 Self-Consistent Field Ground State Methods
4.1 Introduction
4.1.1 Overview of Chapter
4.1.2 Theoretical Background
4.2 Hartree-Fock Calculations
4.2.1 The Hartree-Fock Equations
4.2.2 Wavefunction Stability Analysis
4.2.3 Basic Hartree-Fock Job Control
4.2.4 Additional Hartree-Fock Job Control Options
4.2.5 Examples
4.2.6 Symmetry
4.3 Density Functional Theory
4.3.1 Introduction
4.3.2 Kohn-Sham Density Functional Theory
4.3.3 Exchange-Correlation Functionals
4.3.4 Long-Range-Corrected DFT
4.3.4.1 LRC-DFT with the μB88, μPBE, and ωPBE exchange functionals
4.3.4.2 LRC-DFT with the BNL Functional
4.3.4.3 LRC-DFT with ωB97, ωB97X, ωB97X-D, and ωB97X-2 Functionals
4.3.4.4 LRC-DFT with the M11 Family of Functionals
4.3.5 Nonlocal Correlation Functionals
4.3.6 DFT-D Methods
4.3.6.1 Empirical dispersion correction from Grimme
4.3.6.2 Empirical dispersion correction from Chai and Head-Gordon
4.3.7 XDM DFT Model of Dispersion
4.3.8 DFT-D3 Methods
4.3.9 Double-Hybrid Density Functional Theory
4.3.10 Asymptotically Corrected Exchange-Correlation Potentials
4.3.11 DFT Numerical Quadrature
4.3.12 Angular Grids
4.3.13 Standard Quadrature Grids
4.3.14 Consistency Check and Cutoffs for Numerical Integration
4.3.15 Basic DFT Job Control
4.3.16 Example
4.3.17 User-Defined Density Functionals
4.4 Large Molecules and Linear Scaling Methods
4.4.1 Introduction
4.4.2 Continuous Fast Multipole Method (CFMM)
4.4.3 Linear Scaling Exchange (LinK) Matrix Evaluation
4.4.4 Incremental and Variable Thresh Fock Matrix Building
4.4.5 Incremental DFT
4.4.6 Fourier Transform Coulomb Method
4.4.7 Multiresolution Exchange-Correlation (mrXC) Method
4.4.8 Resolution-of-the-Identity Fock Matrix Methods
4.4.9 Examples
4.5 SCF Initial Guess
4.5.1 Introduction
4.5.2 Simple Initial Guesses
4.5.3 Reading MOs from Disk
4.5.4 Modifying the Occupied Molecular Orbitals
4.5.5 Basis Set Projection
4.5.6 Examples
4.6 Converging SCF Calculations
4.6.1 Introduction
4.6.2 Basic Convergence Control Options
4.6.3 Direct Inversion in the Iterative Subspace (DIIS)
4.6.4 Geometric Direct Minimization (GDM)
4.6.5 Direct Minimization (DM)
4.6.6 Maximum Overlap Method (MOM)
4.6.7 Relaxed Constraint Algorithm (RCA)
4.6.8 Examples
4.7 Dual-Basis Self-Consistent Field Calculations
4.7.1 Dual-Basis MP2
4.7.2 Basis Set Pairings
4.7.3 Job Control
4.7.4 Examples
4.7.5 Dual-Basis Dynamics
4.8 Hartree-Fock and Density-Functional Perturbative Corrections
4.8.1 Hartree-Fock Perturbative Correction
4.8.2 Density Functional Perturbative Correction (Density Functional "Triple Jumping")
4.8.3 Job Control
4.8.4 Examples
4.9 Constrained Density Functional Theory (CDFT)
4.10 Configuration Interaction with Constrained Density Functional Theory (CDFT-CI)
4.11 Unconventional SCF Calculations
4.11.1 CASE Approximation
4.11.2 Polarized Atomic Orbital (PAO) Calculations
4.12 SCF Metadynamics
4.13 Ground State Method Summary
5 Wavefunction-Based Correlation Methods
5.1 Introduction
5.2 Møller-Plesset Perturbation Theory
5.2.1 Introduction
5.2.2 Theoretical Background
5.3 Exact MP2 Methods
5.3.1 Algorithm
5.3.2 The Definition of Core Electron
5.3.3 Algorithm Control and Customization
5.3.4 Example
5.4 Local MP2 Methods
5.4.1 Local Triatomics in Molecules (TRIM) Model
5.4.2 EPAO Evaluation Options
5.4.3 Algorithm Control and Customization
5.4.4 Examples
5.5 Auxiliary Basis Set (Resolution-of-Identity) MP2 Methods
5.5.1 RI-MP2 Energies and Gradients.
5.5.2 Example
5.5.3 OpenMP Implementation of RI-MP2
5.5.4 GPU Implementation of RI-MP2
5.5.4.1 Requirements
5.5.4.2 Options
5.5.4.3 Input examples
5.5.5 Opposite-Spin (SOS-MP2, MOS-MP2, and O2) Energies and Gradients
5.5.6 Examples
5.5.7 RI-TRIM MP2 Energies
5.5.8 Dual-Basis MP2
5.6 Short-Range Correlation Methods
5.6.1 Attenuated MP2
5.6.2 Examples
5.7 Coupled-Cluster Methods
5.7.1 Coupled Cluster Singles and Doubles (CCSD)
5.7.2 Quadratic Configuration Interaction (QCISD)
5.7.3 Optimized Orbital Coupled Cluster Doubles (OD)
5.7.4 Quadratic Coupled Cluster Doubles (QCCD)
5.7.5 Resolution-of-identity with CC (RI-CC)
5.7.6 Cholesky decomposition with CC (CD-CC)
5.7.7 Job Control Options
5.7.8 Examples
5.8 Non-iterative Corrections to Coupled Cluster Energies
5.8.1 (T) Triples Corrections
5.8.2 (2) Triples and Quadruples Corrections
5.8.3 (dT) and (fT) corrections
5.8.4 Job Control Options
5.8.5 Example
5.9 Coupled Cluster Active Space Methods
5.9.1 Introduction
5.9.2 VOD and VOD(2) Methods
5.9.3 VQCCD
5.9.4 Local Pair Models for Valence Correlations Beyond Doubles
5.9.5 Convergence Strategies and More Advanced Options
5.9.6 Examples
5.10 Frozen Natural Orbitals in CCD, CCSD, OD, QCCD, and QCISD Calculations
5.10.1 Job Control Options
5.10.2 Example
5.11 Non-Hartree-Fock Orbitals in Correlated Calculations
5.11.1 Example
5.12 Analytic Gradients and Properties for Coupled-Cluster Methods
5.12.1 Job Control Options
5.12.2 Examples
5.13 Memory Options and Parallelization of Coupled-Cluster Calculations
5.14 Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active Space
5.14.1 Perfect pairing (PP)
5.14.2 Coupled Cluster Valence Bond (CCVB)
5.14.3 Second order correction to perfect pairing: PP(2)
5.14.4 Other GVBMAN methods and options
5.15 Geminal Models
5.15.1 Reference wavefunction
5.15.2 Perturbative corrections
6 Open-Shell and Excited-State Methods
6.1 General Excited-State Features
6.2 Non-Correlated Wavefunction Methods
6.2.1 Single Excitation Configuration Interaction (CIS)
6.2.2 Random Phase Approximation (RPA)
6.2.3 Extended CIS (XCIS)
6.2.4 Spin-Flip Extended CIS (SF-XCIS)
6.2.5 Basic Job Control Options
6.2.6 Customization
6.2.7 CIS Analytical Derivatives
6.2.8 Examples
6.2.9 Non-Orthogonal Configuration Interaction
6.3 Time-Dependent Density Functional Theory (TDDFT)
6.3.1 Brief Introduction to TDDFT
6.3.2 TDDFT within a Reduced Single-Excitation Space
6.3.3 Job Control for TDDFT
6.3.4 TDDFT coupled with C-PCM for excitation energies and properties calculations
6.3.5 Analytical Excited-State Hessian in TDDFT
6.3.6 Various TDDFT-Based Examples
6.4 Correlated Excited State Methods: the CIS(D) Family
6.4.1 CIS(D) Theory
6.4.2 Resolution of the Identity CIS(D) Methods
6.4.3 SOS-CIS(D) Model
6.4.4 SOS-CIS(D0) Model
6.4.5 CIS(D) Job Control and Examples
6.4.6 RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0): Job Control
6.4.7 Examples
6.5 Maximum Overlap Method (MOM) for SCF Excited States
6.6 Coupled-Cluster Excited-State and Open-Shell Methods
6.6.1 Excited States via EOM-EE-CCSD and EOM-EE-OD
6.6.2 EOM-XX-CCSD and CI Suite of Methods
6.6.3 Spin-Flip Methods for Di- and Triradicals
6.6.4 EOM-DIP-CCSD
6.6.5 Charge Stabilization for EOM-DIP and Other Methods
6.6.6 Frozen Natural Orbitals in CC and IP-CC Calculations
6.6.7 Equation-of-Motion Coupled-Cluster Job Control
6.6.8 Examples
6.6.9 Non-Hartree-Fock Orbitals in EOM Calculations
6.6.10 Analytic Gradients and Properties for the CCSD and EOM-XX-CCSD Methods
6.6.11 Equation-of-Motion Coupled-Cluster Optimization and Properties Job Control
6.6.12 Examples
6.6.13 EOM(2,3) Methods for Higher-Accuracy and Problematic Situations
6.6.14 Active-Space EOM-CC(2,3): Tricks of the Trade
6.6.15 Job Control for EOM-CC(2,3)
6.6.16 Examples
6.6.17 Non-Iterative Triples Corrections to EOM-CCSD and CCSD
6.6.18 Job Control for Non-Iterative Triples Corrections
6.6.19 Examples
6.6.20 Potential Energy Surface Crossing Minimization
6.6.20.1 Job Control Options
6.6.20.2 Examples
6.6.21 Dyson Orbitals for Ionization from Ground and Excited States within EOM-CCSD Formalism
6.6.21.1 Dyson Orbitals Job Control
6.6.21.2 Examples
6.6.22 Interpretation of EOM / CI Wavefunction and Orbital Numbering
6.7 Correlated Excited State Methods: ADC(n) Family
6.7.1 The Algebraic Diagrammatic Construction (ADC) Scheme
6.7.2 ADC Job Control
6.7.3 Examples
6.8 Restricted active space spin-flip (RAS-SF) and configuration interaction (RAS-CI) methods
6.8.1 The Restricted Active Space (RAS) Scheme
6.8.2 Job Control for the RASCI1 implementation
6.8.3 Job control options for the RASCI2 implementation
6.8.4 Examples
6.9 How to Compute Ionization Energies of Core Electrons and Excited States Involving Excitations of Core Electrons
6.9.1 Calculations of States Involving Core Electron Excitation/Ionization with DFT and TDDFT
6.10 Visualization of Excited States
6.10.1 Attachment / Detachment Density Analysis
6.10.2 Natural Transition Orbitals
7 Basis Sets
7.1 Introduction
7.2 Built-In Basis Sets
7.3 Basis Set Symbolic Representation
7.3.1 Customization
7.4 User-Defined Basis Sets ($basis)
7.4.1 Introduction
7.4.2 Job Control
7.4.3 Format for User-Defined Basis Sets
7.4.4 Example
7.5 Mixed Basis Sets
7.5.1 Examples
7.6 Dual basis sets
7.7 Auxiliary basis sets for RI / density fitting
7.8 Basis Set Superposition Error (BSSE)
8 Effective Core Potentials
8.1 Introduction
8.2 Built-In Pseudopotentials
8.2.1 Overview
8.2.2 Combining Pseudopotentials
8.2.3 Examples
8.3 User-Defined Pseudopotentials
8.3.1 Job Control for User-Defined ECPs
8.3.2 Example
8.4 Pseudopotentials and Density Functional Theory
8.4.1 Example
8.5 Pseudopotentials and Electron Correlation
8.5.1 Example
8.6 Pseudopotentials, Forces and Vibrational Frequencies
8.6.1 Example
8.6.2 A Brief Guide to Q-Chem's Built-In ECPs
8.6.3 The HWMB Pseudopotential at a Glance
8.6.4 The LANL2DZ Pseudopotential at a Glance
8.6.5 The SBKJC Pseudopotential at a Glance
8.6.6 The CRENBS Pseudopotential at a Glance
8.6.7 The CRENBL Pseudopotential at a Glance
8.6.8 The SRLC Pseudopotential at a Glance
8.6.9 The SRSC Pseudopotential at a Glance
9 Molecular Geometry Critical Points, ab Initio Molecular Dynamics, and QM/MM
9.1 Equilibrium Geometries and Transition Structures
9.2 User-Controllable Parameters
9.2.1 Features
9.2.2 Job Control
9.2.3 Customization
9.2.4 Example
9.3 Constrained Optimization
9.3.1 Introduction
9.3.2 Geometry Optimization with General Constraints
9.3.3 Frozen Atoms
9.3.4 Dummy Atoms
9.3.5 Dummy Atom Placement in Dihedral Constraints
9.3.6 Additional Atom Connectivity
9.3.7 Example
9.3.8 Summary
9.4 Potential Energy Scans
9.5 Intrinsic Reaction Coordinates
9.5.1 Job Control
9.5.2 Example
9.6 Freezing String Method
9.7 Hessian-Free Transition State Search
9.8 Improved Dimer Method
9.9 Ab initio Molecular Dynamics
9.9.1 Examples
9.9.2 AIMD with Correlated Wavefunctions
9.9.3 Vibrational Spectra
9.9.4 Quasi-Classical Molecular Dynamics
9.10 Ab initio Path Integrals
9.10.1 Classical Sampling
9.10.2 Quantum Sampling
9.10.3 Examples
9.11 Q-CHEM / CHARMM Interface
9.12 Stand-Alone QM / MM calculations
9.12.1 Available QM / MM Methods and Features
9.12.2 Using the Stand-Alone QM / MM Features
9.12.2.1 $molecule section
9.12.2.2 $force_field_params section
9.12.2.3 User-defined force fields
9.12.2.4 $qm_atoms and $forceman sections
9.12.3 Additional Job Control Variables
9.12.4 QM / MM Examples
10 Molecular Properties and Analysis
10.1 Introduction
10.2 Chemical Solvent Models
10.2.1 Kirkwood-Onsager Model
10.2.2 Polarizable Continuum Models
10.2.3 PCM Job Control
10.2.3.1 $rem section
10.2.3.2 $pcm section
10.2.3.3 $pcm_solvent section
10.2.4 Linear-Scaling QM / MM / PCM Calculations
10.2.5 Iso-Density Implementation of SS(V)PE
10.2.5.1 The $svp input section
10.2.6 Langevin Dipoles Solvation Model
10.2.6.1 Overview
10.2.6.2 Customizing Langevin dipoles solvation calculations
10.2.7 The SM8 Model
10.2.8 COSMO
10.3 Wavefunction Analysis
10.3.1 Population Analysis
10.3.2 Multipole Moments
10.3.3 Symmetry Decomposition
10.3.4 Localized Orbital Bonding Analysis
10.3.5 Excited-State Analysis
10.4 Intracules
10.4.1 Position Intracules
10.4.2 Momentum Intracules
10.4.3 Wigner Intracules
10.4.4 Intracule Job Control
10.4.5 Format for the $intracule Section
10.5 Vibrational Analysis
10.5.1 Job Control
10.6 Anharmonic Vibrational Frequencies
10.6.1 Partial Hessian Vibrational Analysis
10.6.2 Vibration Configuration Interaction Theory
10.6.3 Vibrational Perturbation Theory
10.6.4 Transition-Optimized Shifted Hermite Theory
10.6.5 Job Control
10.6.6 Isotopic Substitutions
10.7 Interface to the NBO Package
10.8 Orbital Localization
10.9 Visualizing and Plotting Orbitals and Densities
10.9.1 Visualizing Orbitals Using MolDen and MacMolPlt
10.9.2 Visualization of Natural Transition Orbitals
10.9.3 Generation of Volumetric Data Using $plots
10.9.4 Direct Generation of "Cube" Files
10.9.5 NCI Plots
10.10 Electrostatic Potentials
10.11 Spin and Charge Densities at the Nuclei
10.12 NMR Shielding Tensors
10.12.1 Job Control
10.12.2 Using NMR Shielding Constants as an Efficient Probe of Aromaticity
10.13 Linear-Scaling NMR Chemical Shifts: GIAO-HF and GIAO-DFT
10.14 Linear-Scaling Computation of Electric Properties
10.14.1 Examples for Section $fdpfreq
10.14.2 Features of Mopropman
10.14.3 Job Control
10.15 Atoms in Molecules
10.16 Distributed Multipole Analysis
10.17 Electronic Couplings for Electron Transfer and Energy Transfer
10.17.1 Eigenstate-Based Methods
10.17.1.1 Two-state approximation
10.17.1.2 Multi-state treatments
10.17.2 Diabatic-State-Based Methods
10.17.2.1 Electronic coupling in charge transfer
10.17.2.2 Corresponding orbital transformation
10.17.2.3 Generalized density matrix
10.17.2.4 Direct coupling method for electronic coupling
10.18 Calculating the Population of Effectively Unpaired ("odd") Electrons with DFT
10.19 Quantum Transport Properties via the Landauer Approximation
11 Effective Fragment Potential Method
11.1 Theoretical Background
11.2 Excited-State Calculations with EFP
11.3 Extension to Macromolecules: Fragmented EFP Scheme
11.4 EFP Fragment Library
11.5 EFP Job Control
11.6 Examples
11.7 Calculation of User-Defined EFP Potentials
11.7.1 Generating EFP Parameters in GAMESS
11.7.2 Converting EFP Parameters to the Q-Chem Library Format
11.7.3 Converting EFP Parameters to the Q-Chem Input Format
11.8 Converting PDB Coordinates into Q-Chem EFP Input Format
11.9 fEFP Input Structure
11.10 Advanced EFP options
12 Methods Based on Absolutely-Localized Molecular Orbitals
12.1 Introduction
12.2 Specifying Fragments in the $molecule Section
12.3 911 plus3 minus4 plus2 plus2 minus2 plus2 minus plus2 minus4 plus2 minus plus2 minus2 plus minus FRAGMO Initial Guess for SCF Methods
12.4 Locally-Projected SCF Methods
12.4.1 Locally-Projected SCF Methods with Single Roothaan-Step Correction
12.4.2 Roothaan-Step Corrections to the 911 plus3 minus4 plus2 plus2 minus2 plus2 minus plus2 minus4 plus2 minus plus2 minus2 plus minus FRAGMO Initial Guess
12.4.3 Automated Evaluation of the Basis-Set Superposition Error
12.5 Energy Decomposition and Charge-Transfer Analysis
12.5.1 Energy Decomposition Analysis
12.5.2 Analysis of Charge-Transfer Based on Complementary Occupied / Virtual Pairs
12.6 Job Control for Locally-Projected SCF Methods
12.7 The Explicit Polarization (XPol) Method
12.7.1 Supplementing XPol with Empirical Potentials
12.7.2 Job Control Variables for XPol
12.8 Symmetry-Adapted Perturbation Theory (SAPT)
12.8.1 Theory
12.8.2 Job Control for SAPT Calculations
12.9 The XPol+SAPT Method
A Geometry Optimization with Q-Chem
A.1 Introduction
A.2 Theoretical Background
A.3 Eigenvector-Following (EF) Algorithm
A.4 Delocalized Internal Coordinates
A.5 Constrained Optimization
A.6 Delocalized Internal Coordinates
A.7 GDIIS
B AOINTS
B.1 Introduction
B.2 Historical Perspective
B.3 AOINTS: Calculating ERIs with Q-Chem
B.4 Shell-Pair Data
B.5 Shell-Quartets and Integral Classes
B.6 Fundamental ERI
B.7 Angular Momentum Problem
B.8 Contraction Problem
B.9 Quadratic Scaling
B.10 Algorithm Selection
B.11 More Efficient Hartree-Fock Gradient and Hessian Evaluations
B.12 User-Controllable Variables
C Q-Chem Quick Reference
C.1 Q-Chem Text Input Summary
C.1.1 Keyword: $molecule
C.1.2 Keyword: $rem
C.1.3 Keyword: $basis
C.1.4 Keyword: $comment
C.1.5 Keyword: $ecp
C.1.6 Keyword: $empirical_dispersion
C.1.7 Keyword: $external_charges
C.1.8 Keyword: $intracule
C.1.9 Keyword: $isotopes
C.1.10 Keyword: $multipole_field
C.1.11 Keyword: $nbo
C.1.12 Keyword: $occupied
C.1.13 Keyword: $opt
C.1.14 Keyword: $svp
C.1.15 Keyword: $svpirf
C.1.16 Keyword: $plots
C.1.17 Keyword: $localized_diabatization
C.1.18 Keyword $van_der_waals
C.1.19 Keyword: $xc_functional
C.2 Geometry Optimization with General Constraints
C.2.1 Frozen Atoms
C.3 $rem Variable List
C.3.1 General
C.3.2 SCF Control
C.3.3 DFT Options
C.3.4 Large Molecules
C.3.5 Correlated Methods
C.3.6 Correlated Methods Handled by CCMAN and CCMAN2
C.3.7 Perfect pairing, Coupled cluster valence bond, and related methods
C.3.8 Excited States: CIS, TDDFT, SF-XCIS and SOS-CIS(D)
C.3.9 Excited States: EOM-CC and CI Methods
C.3.10 Geometry Optimizations
C.3.11 Vibrational Analysis
C.3.12 Reaction Coordinate Following
C.3.13 NMR Calculations
C.3.14 Wavefunction Analysis and Molecular Properties
C.3.15 Symmetry
C.3.16 Printing Options
C.3.17 Resource Control
C.3.18 Alphabetical Listing
Chapter 1 Introduction
1.1 About This Manual
This manual is intended as a general-purpose user's guide for Q-Chem, a
modern electronic structure program. The manual contains background
information that describes Q-Chem methods and user-selected parameters.
It is assumed that the user has some familiarity with the UNIX environment,
an ASCII file editor and a basic understanding of quantum chemistry.
The manual is divided into 12 chapters and 3 appendices, which are briefly
summarized below. After installing Q-Chem, and making necessary adjustments
to your user account, it is recommended that particular attention be given to
Chapters 3 and 4. The latter chapter has been
formatted so that advanced users can quickly find the information they require,
while supplying new users with a moderate level of important background
information. This format has been maintained throughout the manual, and every
attempt has been made to guide the user forward and backward to other relevant
information so that a logical progression through this manual, while
recommended, is not necessary.
1.2 Chapter Summaries
Chapter 1: | General overview of the Q-Chem program, its features and
capabilities, the people behind it, and contact information. |
Chapter 2: | Procedures to install, test, and run Q-Chem on your machine. |
Chapter 3: | Basic attributes of the Q-Chem command line input. |
Chapter 4: | Running self-consistent field ground state calculations. |
Chapter 5: | Running wavefunction-based correlation methods for ground states. |
Chapter 6: | Running calculations for excited states and open-shell species. |
Chapter 7: | Using Q-Chem's built-in basis sets and running user-defined
basis sets. |
Chapter 8: | Using Q-Chem's effective core potential capabilities. |
Chapter 9: | Options available for exploring potential energy surfaces,
such as determining critical points (transition states and local minima) as well
as molecular dynamics options. |
Chapter 10: | Techniques available for computing molecular properties and
performing wavefunction analysis. |
Chapter 11: | Techniques for molecules in environments (e.g., bulk solution)
and intermolecular interactions; Effective Fragment Potential method. |
Chapter 12: | Methods based on absolutely-localized molecular orbitals. |
Appendix A: | Optimize package used in Q-Chem for determining molecular
geometry critical points. |
Appendix B: | Q-Chem's AOINTS library, which contains some of the fastest
two-electron integral code currently available. |
Appendix C: | Quick reference section. |
1.3 Contact Information
For general information regarding broad aspects and features of the Q-Chem
program, see Q-Chem's home page (http://www.q-chem.com).
1.3.1 Customer Support
Full customer support is promptly provided though telephone or email for those
customers who have purchased Q-Chem's maintenance contract. The maintenance
contract offers free customer support and discounts on future releases and
updates. For details of the maintenance contract please see Q-Chem's home page
(http://www.q-chem.com).
1.4 Q-Chem, Inc.
Q-Chem, Inc. was founded in 1993 and
was based in Pittsburgh, USA for many years,
but will relocate to California in 2013.
Q-Chem's scientific contributors include leading quantum chemists
around the world. The company is governed by the Board of Directors
which currently consists of Peter Gill (Canberra), Anna Krylov (USC),
John Herbert (OSU), and Hilary Pople. Fritz Schaefer (Georgia) is a
Board Member Emeritus. Martin Head-Gordon is a Scientific
Advisor to the Board.
The close coupling between leading university research groups and Q-Chem
Inc. ensures that the methods and algorithms available in
Q-Chem are state-of-the-art.
In order to create this technology, the founders of Q-Chem, Inc. built
entirely new methodologies from the ground up, using the latest algorithms and
modern programming techniques. Since 1993, well over 300 person-years have been
devoted to the development of the Q-Chem program. The author list of the
program shows the full list of contributors to the current version.
The current group of developers consist of more than 100 people in 9 countries.
A brief history of Q-Chem is given in a recent Software Focus article[1], "Q-Chem: An Engine for Innovation".
1.5 Company Mission
The mission of Q-Chem, Inc. is to develop, distribute and support innovative
quantum chemistry software for industrial, government and academic researchers
in the chemical, petrochemical, biochemical, pharmaceutical and material
sciences.
1.6 Q-Chem Features
Quantum chemistry methods have proven invaluable for studying chemical and
physical properties of molecules. The Q-Chem system brings together a variety
of advanced computational methods and tools in an integrated ab initio
software package, greatly improving the speed and accuracy of calculations
being performed. In addition, Q-Chem will accommodate far large molecular
structures than previously possible and with no loss in accuracy, thereby
bringing the power of quantum chemistry to critical research projects for which
this tool was previously unavailable.
1.6.1 New Features in Q-Chem 4.1
Q-Chem 4.1 provides several bug fixes, performance enhancements, and
the following new features:
- Fundamental algorithms
- Improved parallel performance at all levels including new OpenMP capabilities for SCF/DFT, MP2, integral transformation and coupled cluster theory (Zhengting Gan, Evgeny Epifanovsky, Matt Goldey, Yihan Shao;
see Section 2.7.1).
- Significantly enhanced ECP capabilities, including availability of gradients and frequencies in all basis sets for which the energy can be evaluated (Yihan Shao and Martin Head-Gordon; see Chapter 8).
- Self-Consistent Field and Density Functional Theory capabilities
- Numerous DFT enhancements and new features:
TD-DFT energy with M06, M08, and M11-series of
functionals; XYGJ-OS analytical energy gradient;
- TD-DFT/C-PCM excitation energies, gradient, and Hessian
(Jie Liu, W. Liang;
Section 6.3.4).
- Additional features in the Maximum Overlap Method (MOM) approach for converging difficult DFT and SCF calculations (Nick Besley;
Section 4.6.6).
- Wave function correlation capabilities
- Resolution-of-identity/Cholesky decomposition implementation
of all coupled-cluster and equation-of-motion methods enabling
applications to larger systems, reducing disk/memory requirements, and
improving performance (see Sections 5.7.5 and 5.7.6;
Evgeny Epifanovsky,
Xintian Feng, Dmitri Zuev, Yihan Shao, Anna Krylov).
- Attenuated MP2 theory in the aug-cc-pVDZ and aug-cc-pVTZ basis sets, which truncate two-electron integrals to cancel basis set superposition error, yielding results for intermolecular interactions that are much more accurate than standard MP2 in the same basis sets (Matt Goldey and Martin Head-Gordon, Section 5.6.1).
- Extended RAS-nSF methodology for ground and excited states involving strong non-dynamical correlations
(see Section 6.8, Paul Zimmerman, David Casanova, Martin Head-Gordon).
- Coupled cluster valence bond (CCVB) method for describing molecules with strong spin correlations (David Small and
Martin Head-Gordon; see Section 5.14.2).
- Walking on potential energy surfaces
- Potential energy surface scans (Yihan Shao, Section 9.4).
- Improvements in automatic transition structure search algorithms by the freezing string method,
including the ability to perform such calculations without a Hessian calculation
(Shaama Sharada, Martin Head-Gordon, Section 9.7).
- Enhancements to partial Hessian vibrational analysis (Nick Besley;
Section 10.6.1).
- Calculating and characterizing inter- and intra-molecular interactions
- Extension of EFP to macromolecules: fEFP approach
(Adele Laurent, Debashree Ghosh, Anna Krylov, Lyudmila Slipchenko,
see Section 11.3).
- Symmetry-adapted perturbation theory level at the "SAPT0" level,
for intermolecular interaction energy decomposition analysis into
physically-meaningful components such as electrostatics, induction, dispersion,
and exchange. (Leif Jacobson,
John Herbert; Section 12.8). Q-Chem features an efficient resolution-of-identity (RI) version of the "SAPT0" approximation,
based on 2nd-order perturbation theory for the intermolecular interaction.
- The "explicit polarization" (XPol) monomer-based SCF calculations to
compute many-body polarization
effects in linear-scaling time via charge embedding (Section 12.7), which
can be combined either with empirical potentials (e.g., Lennard Jones) for
the non-polarization parts of the intermolecular interactions, or better yet, with
SAPT for an ab initio approach called XSAPT that extends SAPT to systems containing more that two monomers (Leif Jacobson, John Herbert;
Section 12.9).
- Extension of the absolutely-localized-molecular-orbital-based energy
decomposition analysis to unrestricted cases
(Section 12.5, Paul Horn and Martin Head-Gordon)
- Calculations of populations of effectively
unpaired electrons in low-spin state within DFT,
a new method of evaluating localized atomic magnetic moments within Kohn-Sham without symmetry breaking,
and Mayer-type bond order analysis with inclusion of static correlation effects
(Emil Proynov; Section 10.18).
- Calculations of quantum transport including electron transmission
functions and electron tunneling currents under applied bias voltage
(Section 10.19, Barry Dunietz, Nicolai Sergueev).
- Searchable online version of our pdf manual.
1.6.2 New Features in Q-Chem 4.0.1
Q-Chem 4.0.1 provides several bug fixes, performance enhancements, and
the following new features:
- Remote submission capability in IQmol (Andrew Gilbert).
- Scaled nuclear charge and charged cage stabilization capabilities
(Tomasz Ku\'s, Anna Krylov, Section 6.6.5).
- Calculations of excited state properties including transition dipole
moments between different excited states in CIS and TDDFT as well as
couplings for electron and energy transfer (see Section 10.17).
1.6.3 New Features in Q-Chem 4.0
Q-Chem 4.0 provides the following new features and upgrades:
- Exchange-Correlation Functionals
- Density functional dispersion with implementation of the efficient Becke and Johnson's XDM
model in the analytic form
(Zhengting Gan, Emil Proynov, Jing Kong; Section 4.3.7).
- Implementation of the van der Waals density functionals vdW-DF-04 and vdW-DF-10
of Langreth and co-workers (Oleg Vydrov; Section 4.3.5).
- VV09 and VV10, new analytic dispersion functionals (Oleg Vydrov,
Troy Van Voorhis; Section 4.3.5).
- Implementation of DFT-D3 Methods for improved noncovalent interactions
(Shan-Ping Mao, Jeng-Da Chai; Section 4.3.8).
- ωB97X-2, a double-hybrid functional based on long range corrected B97
functional with improved account for medium and long range interactions
(Jeng-Da Chai, Martin Head-Gordon; Section 4.3.9).
- XYGJ-OS, a double-hybrid functional for predictions of nonbonding
interactions and thermochemistry at nearly chemical accuracy (Igor Zhang,
Xin Xu, William A. Goddard, III, Yousung Jung; Section 4.3.9).
- Calculation of near-edge X-ray absorption with short-range corrected DFT
(Nick Besley).
- Improved TDDFT prediction with implementation of asymptotically corrected
exchange-correlation potential (TDDFT / TDA with LB94) (Yu-Chuan Su, Jeng-Da
Chai; Section 4.3.10).
- Nondynamic correlation in DFT with efficient RI implementation of Becke-05 model in
fully analytic formulation (Emil Proynov, Yihan Shao, Fenglai Liu, Jing Kong;
Section 4.3.3).
- Implementation of meta-GGA functionals TPSS and its hybrid version TPSSh (Fenglai Liu)
and the rPW86 GGA functional (Oleg Vydrov).
- Implementation of double hybrid functional B2PLYP-D (Jeng-Da Chai).
- Implementation of Mori-Sánchez-Cohen-Yang (MCY2) hyper-GGA functional
(Fenglai Liu).
- SOGGA, SOGGA11 and SOGGA11-X family of GGA functionals (Roberto Peverati, Yan Zhao, Don Truhlar).
- M08-HX and M08-SO suites of high HF exchange meta-GGA functionals (Yan Zhao, Don Truhlar).
- M11-L and M11 suites of meta-GGA functionals (Roberto Peverati, Yan Zhao, Don Truhlar).
- DFT Algorithms
- Fast numerical integration of exchange-correlation with mrXC
(multiresolution exchange-correlation) (Shawn Brown, Laszlo Fusti-Molnar,
Nicholas J. Russ, Chun-Min Chang, Jing Kong; Section 4.4.7).
- Efficient computation of the exchange-correlation part of the dual basis DFT
(Zhengting Gan, Jing Kong; Section 4.5.5).
- Fast DFT calculation with "triple jumps" between different sizes of basis
set and grid and different levels of functional (Jia Deng, Andrew Gilbert,
Peter M. W. Gill; Section 4.8).
- Faster DFT and HF calculation with atomic resolution of the identity
(ARI) algorithms (Alex Sodt, Martin Head-Gordon).
- POST-HF: Coupled Cluster, Equation of Motion, Configuration Interaction,
and Algebraic Diagrammatic Construction Methods
- Significantly enhanced coupled-cluster code rewritten for better performance
and multicore systems for many modules (energy and gradient for CCSD,
EOM-EE / SF / IP / EA-CCSD, CCSD(T) energy) (Evgeny Epifanovsky, Michael Wormit,
Tomasz Kus, Arik Landau, Dmitri Zuev, Kirill Khistyaev, Ilya Kaliman, Anna
Krylov, Andreas Dreuw; Chapters 5 and 6).
This new code is named CCMAN2.
- Fast and accurate coupled-cluster calculations with frozen natural orbitals
(Arik Landau, Dmitri Zuev, Anna Krylov; Section 5.10).
- Correlated excited states with the perturbation-theory based, size
consistent ADC scheme of second order (Michael Wormit, Andreas Dreuw;
Section 6.7).
- Restricted active space spin flip method for multiconfigurational ground states
and multi-electron excited states (Paul Zimmerman, Franziska Bell, David Casanova,
Martin Head-Gordon, Section 6.2.4).
- POST-HF: Strong Correlation
- Perfect Quadruples and Perfect Hextuples methods for strong
correlation problems (John Parkhill, Martin Head-Gordon, Section 5.9.4).
- Coupled Cluster Valence Bond (CCVB) and related methods for multiple
bond breaking (David Small, Keith Lawler, Martin Head-Gordon, Section 5.14).
- DFT Excited States and Charge Transfer
- Nuclear gradients of excited states with TDDFT (Yihan Shao, Fenglai Liu,
Zhengting Gan, Chao-Ping Hsu, Andreas Dreuw, Martin Head-Gordon, Jing Kong;
Section 6.3.1).
- Direct coupling of charged states for study of charge transfer
reactions (Zhi-Qiang You, Chao-Ping Hsu, Section 10.17.2.
- Analytical excited-state Hessian in TDDFT within Tamm-Dancoff approximation
(Jie Liu, Wanzhen Liang; Section 6.3.5).
- Obtaining an excited state self-consistently with MOM, the Maximum Overlap Method
(Andrew Gilbert, Nick Besley, Peter M. W. Gill; Section 6.5).
- Calculation of reactions with configuration interactions of
charge-constrained states with constrained DFT (Qin Wu, Benjamin Kaduk, Troy
Van Voorhis; Section 4.9).
- Overlap analysis of the charge transfer in a excited state with TDDFT (Nick
Besley; Section 6.3.2).
- Localizing diabatic states with Boys or Edmiston-Ruedenberg localization
scheme for charge or energy transfer (Joe Subotnik, Ryan Steele, Neil
Shenvi, Alex Sodt; Section 10.17.1.2).
- Implementation of non-collinear formulation extends SF-TDDFT to a broader
set of functionals and improves its accuracy (Yihan Shao, Yves Bernard,
Anna Krylov; Section 6.3).
- Solvation and Condensed Phase
- Smooth solvation energy surface with switching/Gaussian polarizable
continuum medium (PCM) solvation models for QM and QM / MM calculations
(Adrian Lange, John Herbert; Sections 10.2.2 and 10.2.4).
- The original COSMO solvation model by Klamt and Schüürmann with DFT
energy and gradient (ported by Yihan Shao; Section 10.2.8).
- Accurate and fast energy computation for large systems including polarizable
explicit solvation for ground and excited states with effective fragment
potential using DFT / TDDFT, CCSD / EOM-CCSD, as well as CIS and CIS(D); library
of effective fragments for common solvents; energy gradient for EFP-EFP
systems (Vitalii Vanovschi, Debashree Ghosh, Ilya Kaliman, Dmytro Kosenkov,
Chris Williams, John Herbert, Mark Gordon, Michael Schmidt, Yihan Shao,
Lyudmila Slipchenko, Anna Krylov; Chapter 11).
- Optimizations, Vibrations, and Dynamics
- Freezing and Growing String Methods for efficient automatic reaction path
finding (Andrew Behn, Paul Zimmerman, Alex Bell, Martin Head-Gordon,
Section 9.6).
- Improved robustness of IRC code (intrinsic reaction coordinate following) (Martin
Head-Gordon).
- Exact, quantum mechanical treatment of nuclear motions at equilibrium with
path integral methods (Ryan Steele; Section 9.10).
- Calculation of local vibrational modes of interest with partial Hessian
vibrational analysis (Nick Besley; Section 10.6.1).
- Ab initio dynamics with extrapolated Z-vector techniques for MP2 and / or
dual-basis methods (Ryan Steele; Section 4.7.5).
- Quasiclassical ab initio molecular dynamics (Daniel Lambrecht, Martin Head-Gordon,
Section 9.9.4).
- Properties and Wavefunction Analysis
- Analysis of metal oxidation states via localized orbital bonding
analysis (Alex Thom, Eric Sundstrom, Martin Head-Gordon; Section 10.3.4).
- Hirshfeld population analysis (Sina Yeganeh; Section 10.3.1).
- Visualization of noncovalent bonding using Johnson and Yang's algorithm
(Yihan Shao; Section 10.9.5).
- ESP on a grid for transition density (Yihan Shao;
Section 10.10).
- Support for Modern Computing Platforms
- Efficient multicore parallel CC/EOM/ADC methods.
- Better performance for multicore systems with shared-memory parallel DFT/HF
(Zhengting Gan, Yihan Shao, Jing Kong) and RI-MP2 (Matthew Goldey, Martin
Head-Gordon; Section 5.13).
- Accelerating RI-MP2 calculation with GPU (graphic processing unit) (Roberto
Olivares-Amaya, Mark Watson, Richard Edgar, Leslie Vogt, Yihan Shao,
Alan Aspuru-Guzik; Section 5.5.4).
- Graphic User Interface
- Input file generation, Q-Chem job submission, and visualization
is supported by IQmol, a fully integrated GUI
developed by Andrew Gilbert from the Australian
National University. IQmol is a free software and does not require purchasing
a license. See http://www.iqmol.org for details and installation instructions.
- Other graphic interfaces are also available.
1.6.4 New Features in Q-Chem 3.2
Q-Chem 3.2 provides the following important upgrades:
- Several new DFT options:
- Long-ranged corrected (LRC) functionals (Dr. Rohrdanz, Prof. Herbert)
- Baer-Neuhauser-Livshits (BNL) functional (Prof. Baer, Prof. Neuhauser, Dr. Livshits)
- Variations of ωB97 Functional (Dr. Chai, Prof. Head-Gordon)
- Constrained DFT (CDFT) (Dr. Wu, Cheng, Prof. Van Voorhis)
- Grimme's empirical dispersion correction (Prof. Sherrill)
- Default XC grid for DFT:
- Default xc grid is now SG-1. It used to be SG-0 before this release.
- Solvation models:
- SM8 model (energy and analytical gradient) for water and organic solvents
(Dr. Marenich, Dr. Olson, Dr. Kelly, Prof. Cramer, Prof. Truhlar)
- Updates to Onsager reaction-field model
(Mr. Cheng, Prof. Van Voorhis, Dr. Thanthiriwatte, Prof. Gwaltney)
- Intermolecular interaction analysis (Dr. Khaliullin, Prof. Head-Gordon):
- SCF with absolutely localized molecular orbitals for molecule interaction (SCF-MI)
- Roothaan-step (RS) correction following SCF-MI
- Energy decomposition analysis (EDA)
- Complimentary occupied-virtual pair (COVP) analysis for charge transfer
- Automated basis-set superposition error (BSSE) calculation
- Electron transfer analysis (Dr. You, Prof. Hsu)
- Relaxed constraint algorithm (RCA) for converging SCF (Mr. Cheng, Prof. Van Voorhis)
- G3Large basis set for transition metals (Prof. Rassolov)
- New MP2 options:
- dual-basis RIMP2 energy and analytical gradient (Dr. Steele, Mr. DiStasio Jr., Prof. Head-Gordon)
- O2 energy and gradient (Dr. Lochan, Prof. Head-Gordon)
- New wavefunction-based methods for efficiently calculating excited state properties (Dr. Casanova, Prof. Rhee, Prof. Head-Gordon):
- SOS-CIS(D) energy for excited states
- SOS-CIS(D0) energy and gradient for excited states
- Coupled-cluster methods (Dr. Pieniazek, Dr. Epifanovsky, Prof. Krylov):
- IP-CISD energy and gradient
- EOM-IP-CCSD energy and gradient
- OpenMP for parallel coupled-cluster calculations
- QM/MM methods (Dr. Woodcock, Ghysels, Dr. Shao, Dr. Kong, Dr. Brooks)
- QM/MM full Hessian evaluation
- QM/MM mobile-block Hessian (MBH) evaluation
- Description for MM atoms with Gaussian-delocalized charges
- Partial Hessian method for vibrational analysis (Dr. Besley)
- Wavefunction analysis tools:
- Improved algorithms for computing localized orbitals
(Dr. Subotnik, Prof. Rhee, Dr. Thom, Mr. Kurlancheek, Prof. Head-Gordon)
- Distributed multipole analysis
(Dr. Vanovschi, Prof. Krylov, Dr. Williams, Prof. Herbert)
- Analytical Wigner intracule
(Dr. Crittenden, Prof. Gill)
1.6.5 New Features in Q-Chem 3.1
Q-Chem 3.1 provides the following important upgrades:
- Several new DFT functional options:
- The nonempirical GGA functional PBE (from the open DF Repository
distributed by the QCG CCLRC Daresbury Lab., implemented in Q-Chem 3.1 by Dr. E. Proynov).
- M05 and M06 suites of meta-GGA functionals for more
accurate predictions of various types of reactions and systems
(Dr. Yan Zhao, Dr. Nathan E. Schultz, Prof. Don Truhlar).
- A faster correlated excited state method:
RI-CIS(D) (Dr. Young Min Rhee).
- Potential energy surface crossing minimization with CCSD and EOM-CCSD
methods (Dr. Evgeny Epifanovsky).
- Dyson orbitals for ionization from the ground and excited states
within CCSD and EOM-CCSD methods (Dr. Melania Oana).
1.6.6 New Features in Q-Chem 3.0
Q-Chem 3.0 includes many new features, along with many enhancements in
performance and robustness over previous versions. Below is a list of some
of the main additions, and who is primarily to thank for implementing them.
Further details and references can be found in the official citation for
Q-Chem (see Section 1.8).
- Improved two-electron integrals package (Dr. Yihan Shao):
- Code for the Head-Gordon-Pople algorithm rewritten to avoid
cache misses and to take advantage of modern computer
architectures.
- Overall increased in performance, especially for computing
derivatives.
- Fourier Transform Coulomb method (Dr. Laszlo Fusti-Molnar):
- Highly efficient implementation for the calculation of
Coulomb matrices and forces for DFT calculations.
- Linear scaling regime is attained earlier than previous
linear algorithms.
- Present implementation works well for basis sets with
high angular momentum and diffuse functions.
- Improved DFT quadrature evaluation:
- Incremental DFT method avoids calculating negligible contributions
from grid points in later SCF cycles (Dr. Shawn Brown).
- Highly efficient SG-0 quadrature grid with approximately half the
accuracy and number of grid points as the SG-1 grid (Siu-Hung Chien).
- Dual basis self-consistent field calculations (Dr. Jing Kong, Ryan Steele):
- Two stage SCF calculations can reduce computational cost by an
order of magnitude.
- Customized basis subsets designed for optimal projection into larger
bases.
- Auxiliary basis expansions for MP2 calculations:
- RI-MP2 and SOS-MP2 energies (Dr. Yousung Jung) and gradients (Robert A. DiStasio Jr.).
- RI-TRIM MP2 energies (Robert A. DiStasio Jr.).
- Scaled opposite spin energies and gradients (Rohini Lochan).
- Enhancements to the correlation package including:
- Most extensive range of EOM-CCSD methods available including
EOM-SF-CCSD, EOM-EE-CCSD, EOM-DIP-CCSD, EOM-IP/EA-CCSD (Prof. Anna Krylov).
- Available for RHF/UHF/ROHF references.
- Analytic gradients and properties calculations (permanent and
transition dipoles etc.).
- Full use of abelian point-group symmetry.
- Singlet strongly orthogonal geminal (SSG) methods (Dr. Vitaly Rassolov).
- Coupled-cluster perfect-paring methods (Prof. Martin Head-Gordon):
- Perfect pairing (PP), imperfect pairing (IP) and restricted pairing
(RP) models.
- PP(2) Corrects for some of the worst failures of MP2 theory.
- Useful in the study of singlet molecules with diradicaloid
character.
- Applicable to systems with more than 100 active electrons.
- Hybrid quantum mechanics / molecular mechanics (QM / MM) methods:
- Fixed point-charge model based on the AMBER force field.
- Two-layer ONIOM model (Dr. Yihan Shao).
- Integration with the Molaris simulation package (Dr. Edina Rosta).
- Q-Chem/ CHARMM interface (Dr. Lee Woodcock)
- New continuum solvation models (Dr. Shawn Brown):
- Surface and Simulation of Volume Polarization for Electrostatics
[SS(V)PE] model.
- Available for HF and DFT calculations.
- New transition structure search algorithms (Andreas Heyden and Dr. Baron
Peters):
- Growing string method for finding transition states.
- Dimer Method which does not use the Hessian and is therefore
useful for large systems.
- Ab Initio Molecular dynamics (Dr. John Herbert):
- Available for SCF wavefunctions (HF, DFT).
- Direct Born-Oppenheimer molecular dynamics (BOMD).
- Extended Lagrangian ab initio molecular dynamics (ELMD).
- Linear scaling properties for large systems (Jörg Kussmann and
Prof. Christian Ochsenfeld):
- NMR chemical shifts.
- Static and dynamic polarizabilities.
- Static hyperpolarizabilities, optical rectification and
electro-optical Pockels effect.
- Anharmonic frequencies (Dr. Ching Yeh Lin):
- Efficient implementation of high-order derivatives
- Corrections via perturbation theory (VPT) or configuration
interaction (VCI).
- New transition optimized shifted Hermite (TOSH) method.
- Wavefunction analysis tools:
- Spin densities at the nuclei (Dr. Vitaly Rassolov).
- Efficient calculation of localized orbitals.
- Optimal atomic point-charge models for densities (Andrew Simmonett).
- Calculation of position, momentum and Wigner intracules (Dr Nick
Besley and Dr. Darragh O'Neill).
- Graphical user interface options:
- IQmol, a fully integrated GUI. IQmol includes
input file generator and contextual help, molecular builder,
job submission tool, and visualization kit
(molecular orbital and density viewer, frequencies, etc).
For the latest version and download/installation instructions, please
see the IQmol homepage (www.iqmol.org).
- Support for the public domain version of WebMO
(see www.webmo.net).
- Seamless integration with the Spartan package
(see www.wavefun.com).
- Support for the public domain version of Avogadro (see:
http: / / avogadro.openmolecules.net / wiki / Get_Avogadro).
- Support the MolDen molecular orbital viewer
(see www.cmbi.ru.nl/molden).
- Support the JMol package
(see http://sourceforge.net/project/showfiles.php?group_id= 23629&release_id=66897).
1.6.7 Summary of Features Prior to Q-Chem 3.0
- Efficient algorithms for large-molecule density functional calculations:
- CFMM for linear scaling Coulomb interactions (energies and gradients) (Dr. Chris White).
- Second-generation J-engine and J-force engine (Dr. Yihan Shao).
- LinK for exchange energies and forces (Dr. Christian Ochsenfeld and Dr. Chris White).
- Linear scaling DFT exchange-correlation quadrature.
- Local, gradient-corrected, and hybrid DFT functionals:
- Slater, Becke, GGA91 and Gill `96 exchange functionals.
- VWN, PZ81, Wigner, Perdew86, LYP and GGA91 correlation functionals.
- EDF1 exchange-correlation functional (Dr. Ross Adamson).
- B3LYP, B3P and user-definable hybrid functionals.
- Analytical gradients and analytical frequencies.
- SG-0 standard quadrature grid (Siu-Hung Chien).
- Lebedev grids up to 5294 points (Dr. Shawn Brown).
- High level wavefunction-based electron correlation methods (Chapter
5):
- Efficient semi-direct MP2 energies and gradients.
- MP3, MP4, QCISD, CCSD energies.
- OD and QCCD energies and analytical gradients.
- Triples corrections (QCISD(T), CCSD(T) and OD(T) energies).
- CCSD(2) and OD(2) energies.
- Active space coupled cluster methods: VOD, VQCCD, VOD(2).
- Local second order Møller-Plesset (MP2) methods (DIM and TRIM).
- Improved definitions of core electrons for post-HF correlation (Dr. Vitaly Rassolov).
- Extensive excited state capabilities:
- CIS energies, analytical gradients and analytical frequencies.
- CIS(D) energies.
- Time-dependent density functional theory energies (TDDFT).
- Coupled cluster excited state energies, OD and VOD (Prof. Anna Krylov).
- Coupled-cluster excited-state geometry optimizations.
- Coupled-cluster property calculations (dipoles, transition dipoles).
- Spin-flip calculations for CCSD and TDDFT excited states (Prof. Anna
Krylov and Dr. Yihan Shao).
- High performance geometry and transition structure optimization (Jon Baker):
- Optimizes in Cartesian, Z-matrix or delocalized internal coordinates.
- Impose bond angle, dihedral angle (torsion) or out-of-plane bend constraints.
- Freezes atoms in Cartesian coordinates.
- Constraints do not need to be satisfied in the starting structure.
- Geometry optimization in the presence of fixed point charges.
- Intrinsic reaction coordinate (IRC) following code.
- Evaluation and visualization of molecular properties
- Onsager, SS(V)PE and Langevin dipoles solvation models.
- Evaluate densities, electrostatic potentials, orbitals over cubes for plotting.
- Natural Bond Orbital (NBO) analysis.
- Attachment / detachment densities for excited states via CIS, TDDFT.
- Vibrational analysis after evaluation of the nuclear coordinate Hessian.
- Isotopic substitution for frequency calculations (Robert Doerksen).
- NMR chemical shifts (Joerg Kussmann).
- Atoms in Molecules (AIMPAC) support (Jim Ritchie).
- Stability analysis of SCF wavefunctions (Yihan Shao).
- Calculation of position and momentum molecular intracules (Aaron
Lee, Nick Besley and Darragh O'Neill).
- Flexible basis set and effective core potential (ECP) functionality:
(Ross Adamson and Peter Gill)
- Wide range of built-in basis sets and ECPs.
- Basis set superposition error correction.
- Support for mixed and user-defined basis sets.
- Effective core potentials for energies and gradients.
- Highly efficient PRISM-based algorithms to evaluate ECP matrix elements.
- Faster and more accurate ECP second derivatives for frequencies.
1.7 Current Development and Future Releases
All details of functionality currently under development, information relating
to future releases, and patch information are regularly updated on the Q-Chem
web page (http://www.q-chem.com). Users are referred to this page for updates on developments,
release information and further information on ordering and licenses. For any
additional information, please contact Q-Chem, Inc. headquarters.
1.8 Citing Q-Chem
The most recent official citation for Q-Chem is a journal article
that has been written describing the main technical features of Q-Chem3.0
version of the program.
The full citation for this article is:
"Advances in quantum chemical methods and algorithms in the Q-Chem 3.0 program
package",
Yihan Shao, Laszlo Fusti-Molnar, Yousung Jung, Jürg Kussmann, Christian
Ochsenfeld, Shawn T. Brown, Andrew T.B. Gilbert, Lyudmila V. Slipchenko,
Sergey V. Levchenko, Darragh P. O'Neill, Robert A. DiStasio Jr., Rohini C.
Lochan, Tao Wang, Gregory J.O. Beran, Nicholas A. Besley, John M. Herbert,
Ching Yeh Lin, Troy Van Voorhis, Siu Hung Chien, Alex Sodt, Ryan P. Steele,
Vitaly A. Rassolov, Paul E. Maslen, Prakashan P. Korambath, Ross D. Adamson,
Brian Austin, Jon Baker, Edward F. C. Byrd, Holger Daschel, Robert J. Doerksen,
Andreas Dreuw, Barry D. Dunietz, Anthony D. Dutoi, Thomas R. Furlani, Steven
R. Gwaltney, Andreas Heyden, So Hirata, Chao-Ping Hsu, Gary Kedziora, Rustam Z.
Khaliullin, Phil Klunzinger, Aaron M. Lee, Michael S. Lee, WanZhen Liang, Itay
Lotan, Nikhil Nair, Baron Peters, Emil I. Proynov, Piotr A. Pieniazek, Young
Min Rhee, Jim Ritchie, Edina Rosta, C. David Sherrill, Andrew C. Simmonett,
Joseph E. Subotnik, H. Lee Woodcock III, Weimin Zhang, Alexis T. Bell, Arup
K. Chakraborty, Daniel M. Chipman, Frerich J. Keil, Arieh Warshel, Warren J.
Hehre, Henry F. Schaefer III, Jing Kong, Anna I. Krylov, Peter M.W. Gill and
Martin Head-Gordon. Phys. Chem. Chem. Phys., 8, 3172 (2006).
This citation will soon be replaced by an official Q-Chem 4 paper, which is
in preparation. The full list of current authors can be found on
Q-Chem's website.
Meanwhile, one can acknowledge using Q-Chem 4.0 by additionally citing
a more recent feature article describing Q-Chem:
"Q-Chem: An engine for innovation",
A.I. Krylov and P.M.W. Gill.
WIREs Comput. Mol. Sci., 3, 317-326 (2013).
Chapter 2 Installation
2.1 Q-Chem Installation Requirements
2.1.1 Execution Environment
Q-Chem is shipped as a single executable along with several scripts for
the computer system you will run Q-Chem on. No compilation is required. Once
the package is installed, it is ready to run.
Please refer to the installation notes for your particular platform which
are distributed with the software.
The system software required to run Q-Chem on your platform is
minimal, and includes:
- A suitable operating system.
- Run-time libraries (usually provided with your operating system).
- Perl, version 5.
- Vendor implementation of MPI or MPICH libraries (MPI-parallel
version only).
Please check the Q-Chem web site, or contact Q-Chem support (email:
support@q-chem.com) if further details are required.
2.1.2 Hardware Platforms and Operating Systems
Q-Chem runs on a wide varieties of computer systems, ranging from Intel and AMD microprocessor
based PCs and workstations to high performance server nodes used in clusters and supercomputers.
Currently Q-Chem support Linux, Mac, Windows and IBM AIX systems.
For the availability of a specific platform/operating system, please contact
support@q-chem.com for details.
2.1.3 Memory and Hard Disk
Memory
Q-Chem, Inc. has endeavored to minimize memory requirements and maximize the
efficiency of its use. Still, the larger the structure or the higher the level
of theory, the more random access memory (RAM) is needed. Although Q-Chem can
be run with very small memory, we recommend 1 GB as a minimum. Q-Chem also offers
the ability for user control of important memory intensive aspects of the
program, an important consideration for non-batch constrained multi-user
systems. In general, the more memory your system has, the larger the
calculation you will be able to perform.
Q-Chem uses two types of memory: a chunk of static memory that is used by
multiple data sets and managed by the code, and dynamic memory which is allocated
using system calls. The size of the static memory is specified by the user
through the $rem word MEM_STATIC and has a default value of 64 MB.
The $rem word MEM_TOTAL specifies the limit of the total memory the
user's job can use. The default value is sufficiently large that on most
machines it will allow Q-Chem to use all the available memory. This value
should be reduced on machines where this is undesirable (for example if the
machine is used by multiple users). The limit for the dynamic memory
allocation is given by (MEM_TOTAL − MEM_STATIC). The amount
of MEM_STATIC needed depends on the size of the user's particular
job. Please note that one should not specify an excessively large value for
MEM_STATIC, otherwise it will reduce the available memory for dynamic
allocation. Memory settings in CC/EOM/ADC
calculations are described in Section 5.13. The use of $rem
words will be discussed in the next Chapter.
Disk
The Q-Chem executables, shell scripts, auxiliary files, samples and
documentation require between 360 to 400 MB of disk space, depending on the
platform. The default Q-Chem output, which is printed to the designated
output file, is usually only a few KBs. This will be exceeded, of course, in
difficult geometry optimizations, and in cases where users invoke non-default
print options. In order to maximize the capabilities of your copy of Q-Chem,
additional disk space is required for scratch files created during execution,
and these are automatically deleted on normal termination of a job. The amount of
disk space required for scratch files depends critically on the type of job,
the size of the molecule and the basis set chosen.
Q-Chem uses direct methods for Hartree-Fock and density functional theory
calculations, which do not require large amount of scratch disk space.
Wavefunction-based correlation methods, such as MP2 and coupled-cluster theory
require substantial amounts of temporary (scratch) disk storage, and the faster
the access speeds, the better these jobs will perform. With the low cost of
disk drives, it is feasible to have between 100 and 1000 Gb of scratch space
available as a dedicated file system for these large temporary job files. The
more you have available, the larger the jobs that will be feasible and in the
case of some jobs, like MP2, the jobs will also run faster as two-electron
integrals are computed less often.
Although the size of any one of the Q-Chem temporary files will not exceed
2 Gb, a user's job will not be limited by this. Q-Chem writes large temporary
data sets to multiple files so that it is not bounded by the 2 Gb file size
limitation on some operating systems.
2.2 Installing Q-Chem
Users are referred to the detailed installation instructions distributed
with your copy of Q-Chem.
An encrypted license file, qchem.license.dat, must be obtained from your
vendor before you will be able to use Q-Chem. This file should be placed in
the directory $QCAUX/license and must be able to be read by all users
of the software. This file is node-locked, i.e., it will only operate
correctly on the machine for which it was generated. Further details about
obtaining this file, can be found in the installation instructions.
Do not alter the license file unless directed by Q-Chem, Inc.
2.3 Q-Chem Auxiliary files ($QCAUX)
The $QCAUX environment variable determines the directory where Q-Chem
searches for auxiliary files and the machine license. If not set explicitly,
it defaults to $QC/qcaux.
The $QCAUX directory contains files required to run Q-Chem
calculations, including basis set and ECP specifications, SAD
guess (see Chapter 4),
library of standard effective fragments (see Chapter 11),
and instructions for the AOINTS package for generating two-electron integrals
efficiently.
2.4 Q-Chem Runtime Environment Variables
Q-Chem requires the following shell environment variables setup prior to running any
calculations:
QC | Defines the location of the Q-Chem directory structure. The
qchem.install shell script determines this
automatically. |
QCAUX | Defines the location of the auxiliary information required by
Q-Chem, which includes the license required to run Q-Chem.
If not explicitly set by the user, this defaults to
$QC/qcaux. |
QCSCRATCH | Defines the directory in which Q-Chem will store temporary
files. Q-Chem will usually remove these files on successful
completion of the job, but they can be saved, if so wished.
Therefore, $QCSCRATCH should not reside in a directory that
will be automatically removed at the end of a job, if the files
are to be kept for further calculations.
Note that many of these files can be very large, and it should be
ensured that the volume that contains this directory has
sufficient disk space available. The $QCSCRATCH directory
should be periodically checked for scratch files remaining from
abnormally terminated jobs. $QCSCRATCH defaults to the
working directory if not explicitly set. Please see section
2.7 for details on saving temporary files and
consult your systems administrator. |
QCLOCALSCR | On certain platforms, such as Linux clusters, it is sometimes
preferable to write the temporary files to a disk local to the
node. $QCLOCALSCR specifies this directory. The temporary
files will be copied to $QCSCRATCH at the end of the job,
unless the job is terminated abnormally. In such cases Q-Chem
will attempt to remove the files in $QCLOCALSCR, but may
not be able to due to access restrictions. Please specify this
variable only if required.
|
2.5 User Account Adjustments
In order for individual users to run Q-Chem, User file access permissions
must be set correctly so that the user can read, write
and execute the necessary Q-Chem files. It may be advantageous to
create a qchem user group on your machine and recursively
change the group ownership of the Q-Chem directory to qchem group.
The Q-Chem runtime environment need to be initiated prior to running any Q-Chem
calculations, which is done by sourcing the environment setup script qcenv.sh [for bash] or
qcenv.csh [for tcsh/csh] placed in your Q-Chem top directory after a successful
installation. It might be more convenient for user to include the Q-Chem environment setup
in their shell startup script, e.g., .cshrc/.tcshrc for csh/tcsh or .bashrc for bash.
For users using the csh shell (or equivalent), add the following lines to their
home directory .cshrc file:
#
setenv QC qchem_root_directory_name
setenv QCSCRATCH scratch_directory_name
source $QC/qcenv.csh
#
For users using the Bourne-again shell (bash), add the following lines to
their home directory .bashrc file:
#
export QC=qchem_root_directory_name
export QCSCRATCH=scratch_directory_name
. $QC/qcenv.sh
#
2.6 Further Customization
Q-Chem has developed a simple mechanism for users to set user-defined
long-term defaults to override the built-in program defaults. Such defaults
may be most suited to machine specific features such as memory allocation,
as the total available memory will vary from machine to machine depending on
specific hardware and accounting configurations. However, users may identify
other important uses for this customization feature.
Q-Chem obtains input initialization variables from four sources:
- User input file
- $HOME/.qchemrc file
- $QC/config/preferences file
- Program defaults
The order of preference of initialization is as above, where the higher placed
input mechanism overrides the lower.
Details of the requirements for the Q-Chem input file are discussed in detail
in this manual. In reviewing the $rem variables and their defaults, users
may identify some variable defaults that they find too limiting or variables
which they find repeatedly need to be set within their input files to make the
most of Q-Chem's features. Rather than having to remember to place such
variables into the Q-Chem input file, users are able to set long-term
defaults which are read each time the user runs a Q-Chem job. This is done by
placing these defaults into the file .qchemrc stored in the users home
directory. Additionally, system administrators can override Q-Chem defaults
with an additional preferences file in the $QC/config directory
achieving a hierarchy of input as illustrated above.
Note:
The .qchemrc and preferences files are not requisites for
running Q-Chem and currently only support $rem keywords. |
2.6.1 .qchemrc and Preferences File Format
The format of the .qchemrc and preferences files is similar to
that for the input file, except that only a $rem keyword section may be
entered, terminated with the usual $end keyword. Any other keyword sections
will be ignored. So that jobs may easily be reproduced, a copy of the
.qchemrc file (if present) is now included near the top of the job
output file.
It is important that the .qchemrc and preferences files have
appropriate file permissions so that they are readable by the user invoking
Q-Chem. The format of both of these files is as follows:
$rem
rem_variable option comment
rem_variable option comment
...
$end
Example 2.0 An example of a .qchemrc file to apply program default
override $rem settings to all of the user's Q-Chem jobs.
$rem
INCORE_INTS_BUFFER 4000000 More integrals in memory
DIIS_SUBSPACE_SIZE 5 Modify max DIIS subspace size
THRESH 10
$end
2.6.2 Recommendations
As mentioned, the customization files are specifically suited for placing
long-term machine specific defaults as clearly some of the defaults placed by
Q-Chem will not be optimal on large or very small machines. The following
$rem variables are examples of those which should be considered, but the user
is free to include as few or as many as desired:
AO2MO_DISK
INCORE_INTS_BUFFER
MEM_STATIC
SCF_CONVERGENCE
THRESH
NBO
Q-Chem will print a warning message to advise the user if a $rem
keyword section has been detected in either .qchemrc or
preferences.
2.7 Running Q-Chem
Once installation is complete, and any necessary adjustments are made to the
user account, the user is now able to run Q-Chem. There are several ways to
invoke
Q-Chem:
- IQmol offers a fully integrated graphical interface for the Q-Chem
package and includes a sophisticated input generator with contextual help
which is able to guide you through the many Q-Chem options available. It
also provides a molecular builder, job submission and monitoring tools, and
is able to visualize molecular orbitals, densities and vibrational frequencies.
For the latest version and download/installation instructions, please
see the IQmol homepage (www.iqmol.org).
- qchem command line shell script.
The simple format for command line execution
is given below. The remainder of this manual covers the creation of input
files in detail.
- Via a third-party GUI. The two most popular ones are:
- A general web-based interface for electronic structure software, WebMO
(see www.webmo.net).
- Wavefunction's Spartan user interface on some platforms. Contact
Wavefunction (www.wavefun.com) or Q-Chem for full details of
current availability.
Using the Q-Chem command line shell script (qchem) is straightforward
provided Q-Chem has been correctly installed on your machine and the
necessary environment variables have been set in your .cshrc,
.profile, or equivalent login file. If done correctly, the necessary
changes will have been made to the $PATH variable automatically on
login so that Q-Chem can be invoked from your working directory.
The qchem shell script can be used in either of the following ways:
qchem infile outfile
qchem infile outfile savename
qchem --save infile outfile savename
where infile is the name of a suitably formatted Q-Chem input file
(detailed in Chapter 3, and the remainder of this manual), and
the outfile is the name of the file to which Q-Chem will place the job
output information.
Note:
If the outfile already exists in the working directory, it will be
overwritten. |
The use of the savename command line variable allows the saving of a few
key scratch files between runs, and is necessary when instructing Q-Chem to
read information from previous jobs. If the savename argument is not
given, Q-Chem deletes all temporary scratch files at the end of a run. The
saved files are in $QCSCRATCH/savename/, and include files with the
current molecular geometry, the current molecular orbitals and density matrix
and the current force constants (if available). The -save option in
conjunction with savename means that all temporary files are saved,
rather than just the few essential files described above. Normally this is not
required. When $QCLOCALSCR has been specified, the temporary files
will be stored there and copied to $QCSCRATCH/savename/ at the end of
normal termination.
The name of the input parameters infile, outfile and save
can be chosen at the discretion of the user (usual UNIX file and directory name
restrictions apply). It maybe helpful to use the same job name for
infile and outfile, but with varying suffixes. For example:
localhost-1> qchem water.in water.out &
invokes Q-Chem where the input is taken from water.in and the output
is placed into water.out. The & places the job into the
background so that you may continue to work in the current shell.
localhost-2> qchem water.com water.log water &
invokes Q-Chem where the input is assumed to reside in water.com, the
output is placed into water.log and the key scratch files are saved in a
directory $QCSCRATCH/water/.
Note:
A checkpoint file can be requested by setting GUI=2 in the
$rem section of the input. The checkpoint file name is determined by the
GUIFILE environment variable which by default is set to ${input}.fchk |
2.7.1 Running Q-Chem in parallel
The parallel execution of Q-Chem can be based on either OpenMP
multi-threading on a single node or MPI protocol using multiple cores
or multiple nodes. In the current release (version 4.1)
hybrid MPI+OpenMP parallelization is not supported.
This restriction will be lifted in our future releases.
As of the 4.1 release OpenMP parallelization is fully supported only by CC, EOM-CC,
and ADC methods. Experimental OpenMP code is available for parallel
SCF, DFT, and MP2 calculations. The MPI parallel capability is
available for SCF, DFT, and MP2 methods.
Table 2.1 summarizes the parallel capabilities of Q-Chem 4.1.
Method | OpenMP | MPI | |
HF energy & gradient | noa | yes |
DFT energy & gradient | noa | yes |
MP2 energy and gradient | yesb | yes |
Integral transformation | yes | no |
CCMAN & CCMAN2 methods | yes | no |
ADC methods | yes | no |
CIS | no | no |
TDDFT | no | no | |
Table 2.1: Parallel capabilities of Q-Chem 4.0.1. a Experimental code in version 4.0.1.
b To invoke an experimental OpenMP RI-MP2 code (RHF energies only),
use CORR=primp2.
To run Q-Chem calculation with OpenMP threads specify the number of threads ( nthreads )
using qchem command option -nt.
Since each thread uses one CPU core, you should not specify more threads than
the total number of available CPU cores for performance reason.
When unspecified, the number of threads defaults to 1 (serial calculation).
qchem -nt nthreads infile outfile
qchem -nt nthreads infile outfile save
qchem -save -nt nthreads infile outfile save
Similarly, to run parallel calculations with MPI use the option -np to specify
the number of MPI processes to be spawned.
qchem -np n infile outfile
qchem -np n infile outfile savename
qchem -save -np n infile outfile savename
where n is the number of processors to use. If the -np switch is not
given, Q-Chem will default to running locally on a single node.
When the additional argument savename is specified, the temporary files
for MPI-parallel Q-Chem are stored in $QCSCRATCH/savename.0 At the start
of a job, any existing files will be copied into this directory, and on
successful completion of the job, be copied to $QCSCRATCH/savename/ for
future use. If the job terminates abnormally, the files will not be copied.
To run parallel Q-Chem using a batch scheduler such as PBS, users may need to
set QCMPIRUN environment variable to point to the mpirun command used in
the system. For further details users should read the
$QC/README.Parallel file, and contact Q-Chem if any
problems are encountered (email: support@q-chem.com).
2.8 IQmol Installation Requirements
IQmol provides a fully integrated molecular builder and viewer for the
Q-Chem package. It is available for the Windows, Linux, and Mac OS X
platforms and instructions for downloading and installing the latest version
can be found at www.iqmol.org/downloads.html.
IQmol can be run as a stand-alone package which is able to open existing
Q-Chem input/output files, but it can also be used as a fully functional
front end which is able to submit and monitor Q-Chem jobs, and analyze the
resulting output. Before Q-Chem can be launched from IQmol an appropriate
server must be configured. First, ensure Q-Chem has been correctly installed
on the target machine and can be run from the command line. Second, open IQmol
and carry out the following steps:
- Select the Calculation→Edit Servers menu option. A dialog
will appear with a list of configured servers (which will initially be
empty).
- Click the Add New Server button with the `+' icon. This opens a dialog
which allows the new server to be configured. The server is the machine
which has your Q-Chem installation.
- Give the server a name (this is simply used to identify the current
server configuration and does not have to match the actual machine name)
and select if the machine is local (i.e. the same machine as IQmol is
running on) or remote.
- If there is PBS software running on the server, select the PBS `Type'
option, otherwise in most cases the Basic option should be sufficient.
Please note that the server must be Linux based and cannot be a Windows
server.
- If required, the server can be further configured using the Configure
button. Details on this can be found in the embedded IQmol help which
can be accessed via the Help→Show Help menu option.
- For non-PBS servers the number of concurrent Q-Chem jobs can be limited
using a simple inbuilt queuing system. The maximum number of jobs is set
by the Job Limit control. If the Job Limit is set to zero the queue is
disabled and any number of jobs can be run concurrently. Please note
that this limit applies to the current IQmol session and does not
account for jobs submitted by other users or by other IQmol sessions.
- The $QC environment variable should be entered in the given box.
- For remote servers the address of the machine and your user name are also
required. IQmol uses SSH2 to connect to remote machines and the most
convenient way to set this up is by using authorized keys (see
http://www.debian.org/devel/passwordlessssh for details on how these
can be set up). IQmol can then connect via the SSH Agent and will
not have to prompt you for your password. If you are not able to use
an SSH Agent, several other authentication methods are offered:
- Public Key This requires you to enter your SSH passphrase (if
any) to unlock your private key file. The passphrase is stored in
memory, not disk, so you will need to re-enter this each time
IQmol is run.
- Password Vault This allows a single password (the vault key)
to be used to unlock the passwords for all the configured servers.
The server passwords are salted with 64 random bits and encrypted
using the AES algorithm before being stored on disk. The vault key
is not stored permanently and must be re-entered each time IQmol
is run.
- Password Prompt This requires each server password to be
entered each time IQmol is run. Once the connection has been
established the memory used to hold the password is overwritten to
reduce the risk of recovery from a core dump.
Further configuration of SSH options should not be required unless your
public/private keys are stored in a non-standard location.
It is recommended that you test the server configuration to ensure everything
is working before attempting to submit a job. Multiple servers can be
configured if you have access to more than one copy of Q-Chem or have
different account configurations. In this case the default server is the first
on the list and if you want to change this you should use the arrow buttons in
the Server List dialog. The list of configured servers will be displayed when
submitting Q-Chem jobs and you will be able to select the desired server for
each job.
Please note that while Q-Chem is file-based, as of version 2.1 IQmol uses
a directory to keep the various files from a calculation.
2.9 Testing and Exploring Q-Chem
Q-Chem is shipped with a small number of test jobs which are located in the
$QC/samples directory. If you wish to test your version of
Q-Chem, run the test jobs in the samples directory and compare the output
files with the reference files (suffixed .out) of the same name.
These test jobs are not an exhaustive quality control test (a small subset of
the test suite used at Q-Chem, Inc.), but they should all run correctly on
your platform. If any fault is identified in these, or any output files created
by your version, do not hesitate to contact customer service immediately.
These jobs are also an excellent way to begin learning about Q-Chem's
text-based input and output formats in detail. In many cases you can use these
inputs as starting points for building your own input files, if you wish to
avoid reading the rest of this manual!
Please check the Q-Chem web page (http://www.q-chem.com) and the
README files in the $QC/bin directory for updated information.
Chapter 3 Q-Chem Inputs
3.1 IQmol
The easiest way to run Q-Chem is by using the IQmol interface which can be
downloaded for free from www.iqmol.org. Before submitting a Q-Chem job
from you will need to configure a Q-Chem server and details on how to do this
are given in Section 2.8 of this manual.
IQmol provides a free-form molecular builder and a comprehensive interface
for setting up the input for Q-Chem jobs. Additionally calculations can be
submitted to either the local or a remote machine and monitored using the built
in job monitor. The output can also be analyzed allowing visualization of
molecular orbitals and densities, and animation of vibrational modes and
reaction pathways. A more complete list of features can be found at
www.iqmol.org/features.html.
The IQmol program comes with a built-in help system that details how to set
up and submit Q-Chem calculations. This help can be accessed via the
Help→Show Help menu option.
3.2 General Form
IQmol (or another graphical interface) is the simplest way to control
Q-Chem. However, the low level command line interface is available to enable
maximum customization and allow the user to exploit all Q-Chem's features.
The command line interface requires a Q-Chem input file which is simply an
ASCII text file. This input file can be created using your favorite editor
(e.g., vi, emacs, jot, etc.) following the basic steps outlined in the next
few chapters.
Q-Chem's input mechanism uses a series of keywords to signal user
input sections of the input file. As required, the Q-Chem program searches
the input file for supported keywords. When Q-Chem finds a keyword, it then
reads the section of the input file beginning at the keyword until that keyword
section is terminated the $end keyword. A short description of all
Q-Chem keywords is provided in Table 3.2 and the following
sections. The user
must understand the function and format of the $molecule (Section
3.3) and $rem (Section 3.6) keywords, as these
keyword sections are where the user places the molecular geometry information
and job specification details.
The keywords $rem and $molecule are requisites of Q-Chem input
files
As each keyword has a different function, the format required for specific
keywords varies somewhat, to account for these differences (format requirements
are summarized in Appendix C). However, because each keyword in
the input file is sought out independently by the program, the overall format
requirements of Q-Chem input files are much less stringent. For example, the
$molecule section does not have to occur at the very start of the input file.
Keyword | Description | |
$molecule | Contains the molecular coordinate input (input file requisite). |
$rem | Job specification and customization parameters (input file requisite). |
$end | Terminates each keyword section. |
$basis | User-defined basis set information (see Chapter 7). |
$comment | User comments for inclusion into output file. |
$ecp | User-defined effective core potentials (see Chapter 8). |
$empirical_dispersion | User-defined van der Waals parameters for DFT dispersion |
| correction. |
$external_charges | External charges and their positions. |
$force_field_params | Force field parameters for QM / MM calculations
(see Section 9.12). |
$intracule | Intracule parameters (see Chapter 10). |
$isotopes | Isotopic substitutions for vibrational calculations (see
Chapter 10). |
$localized_diabatization | Information for mixing together multiple adiabatic states into |
| diabatic states (see Chapter 10). |
$multipole_field | Details of a multipole field to apply. |
$nbo | Natural Bond Orbital package. |
$occupied | Guess orbitals to be occupied. |
$swap_occupied_virtual | Guess orbitals to be swapped. |
$opt | Constraint definitions for geometry optimizations. |
$pcm | Special parameters for polarizable continuum models (see Section |
| 10.2.3). |
$pcm_solvent | Special parameters for polarizable continuum models (see Section |
| 10.2.3). |
$plots | Generate plotting information over a grid of points (see |
| Chapter 10). |
$qm_atoms | Specify the QM region for QM / MM calculations
(see Section 9.12). |
$svp | Special parameters for the SS(V)PE module
(see Section 10.2.5). |
$svpirf | Initial guess for SS(V)PE module. |
$van_der_waals | User-defined atomic radii for Langevin dipoles solvation (see |
| Chapter 10). |
$xc_functional | Details of user-defined DFT exchange-correlation functionals. |
$cdft | Options for the constrained DFT method
(see Section 4.9). |
$efp_fragments | Specifies labels and positions of EFP fragments
(see Chapter 11). |
$efp_params | Contains user-defined parameters for effective fragments
(see Chapter 11). |
Table 3.1: Q-Chem user input section keywords. See the $QC/samples
directory with your release for specific examples of Q-Chem input using these
keywords.
Note:
(1) Users are able to enter keyword sections in any order.
(2) Each keyword section must be terminated with the $end keyword.
(3) The $rem and $molecule sections must be included.
(4) It is not necessary to have all keywords in an input file.
(5) Each keyword section is described in Appendix C.
(6) The entire Q-Chem input is case-insensitive. |
The second general aspect of Q-Chem input is that there are effectively four
input sources:
- User input file (required)
- .qchemrc file in $HOME (optional)
- preferences file in $QC/config (optional)
- Internal program defaults and calculation results (built-in)
The order of preference is as shown, i.e., the input mechanism offers a program
default override for all users, default override for individual
users and, of course, the input file provided by the user overrides all
defaults. Refer to Section 2.6 for details of
.qchemrc and preferences. Currently, Q-Chem only supports the
$rem keyword in .qchemrc and preferences files.
In general, users will need to enter variables for the $molecule and $rem
keyword section and are encouraged to add a $comment for future reference.
The necessity of other keyword input will become apparent throughout the
manual.
3.3 Molecular Coordinate Input ($molecule)
The $molecule section communicates to the program the charge, spin
multiplicity, and geometry of the molecule being considered. The molecular
coordinates input begins with two integers: the net charge and the spin
multiplicity of the molecule. The net charge must be between -50 and 50,
inclusive (0 for neutral molecules, 1 for cations, -1 for anions, etc.).
The multiplicity must be between 1 and 10, inclusive (1 for a singlet, 2 for
a doublet, 3 for a triplet, etc.). Each subsequent line of the molecular
coordinate input corresponds to a single atom in the molecule (or dummy atom),
irrespective of whether using Z-matrix internal coordinates or Cartesian
coordinates.
Note:
The coordinate system used for declaring an initial molecular geometry by
default does not affect that used in a geometry optimization procedure. See
Appendix A which discusses the OPTIMIZE package in further detail. |
Q-Chem begins all calculations by rotating and translating the user-defined
molecular geometry into a Standard Nuclear Orientation whereby the center of
nuclear charge is placed at the origin. This is a standard feature of most
quantum chemistry programs. This action can be turned off by
using SYM_IGNORE=TRUE.
Note:
SYM_IGNORE=TRUE will also turn off
determining and using of the point group symmetry. |
Note:
Q-Chem ignores commas and equal signs, and requires all distances,
positions and angles to be entered as Angstroms and degrees unless the
INPUT_BOHR $rem variable is set to TRUE, in which case all
lengths are assumed to be in bohr. |
Example 3.0 A molecule in Z-matrix coordinates. Note that the $molecule
input begins with the charge and multiplicity.
$molecule
0 1
O
H1 O distance
H2 O distance H1 theta
distance = 1.0
theta = 104.5
$end
3.3.1 Reading Molecular Coordinates From a Previous Calculation
Often users wish to perform several calculations in quick succession, whereby
the later calculations rely on results obtained from the previous ones. For
example, a geometry optimization at a low level of theory, followed by a
vibrational analysis and then, perhaps, single-point energy at a higher level.
Rather than having the user manually transfer the coordinates from the output
of the optimization to the input file of a vibrational analysis or single point
energy calculation, Q-Chem can transfer them directly from job to job.
To achieve this requires that:
- The READ variable is entered into the molecular coordinate input
- Scratch files from a previous calculation have been saved. These may be
obtained explicitly by using the save option across multiple job
runs as described below and in Chapter 2, or
implicitly when running multiple calculations in one input file,
as described later in this Chapter.
Example 3.0 Reading a geometry from a prior calculation.
localhost-1> qchem job1.in job1.out job1
localhost-2> qchem job2.in job2.out job1
In this example, the job1 scratch files are saved in a directory
$QCSCRATCH/job1 and are then made available to the job2
calculation.
Note:
The program must be instructed to read specific scratch files
by the input of job2. |
Users are also able to use the READ function for molecular coordinate input
using Q-Chem's batch job file (see later in this Chapter).
3.3.2 Reading Molecular Coordinates from Another File
Users are able to use the READ function to read molecular coordinates
from a second input file. The format for the coordinates in the second file
follows that for standard Q-Chem input, and must be delimited with the
$molecule and $end keywords.
Example 3.0 Reading molecular coordinates from another file.
filename may be given either as the full file path, or path relative to
the working directory.
$molecule
READ filename
$end
3.4 Cartesian Coordinates
Q-Chem can accept a list of N atoms and their 3N Cartesian coordinates.
The atoms can be entered either as atomic numbers or atomic symbols where each
line corresponds to a single atom. The Q-Chem format for declaring a
molecular geometry using Cartesian coordinates (in Angstroms) is:
atom x-coordinate y-coordinate z-coordinate
Note:
The geometry can by specified in bohr; to do so, set the
INPUT_BOHR $rem variable to TRUE. |
3.4.1 Examples
Example 3.0 Atomic number Cartesian coordinate input for H2O.
$molecule
0 1
8 0.000000 0.000000 -0.212195
1 1.370265 0.000000 0.848778
1 -1.370265 0.000000 0.848778
$end
Example 3.0 Atomic symbol Cartesian coordinate input for H2O.
$molecule
0 1
O 0.000000 0.000000 -0.212195
H 1.370265 0.000000 0.848778
H -1.370265 0.000000 0.848778
$end
Note:
(1) Atoms can be declared by either atomic number or symbol.
(2) Coordinates can be entered either as variables/parameters or real numbers.
(3) Variables/parameters can be declared in any order.
(4) A single blank line separates parameters from the atom declaration. |
Once all the molecular Cartesian coordinates have been entered, terminate
the molecular coordinate input with the $end keyword.
3.5 Z-matrix Coordinates
Z-matrix notation is one of the most common molecular coordinate input forms.
The Z-matrix defines the positions of atoms relative to previously defined
atoms using a length, an angle and a dihedral angle. Again, note that all bond
lengths and angles must be in Angstroms and degrees.
Note:
As with the Cartesian coordinate input method, Q-Chem begins a
calculation by taking the user-defined coordinates and translating and
rotating them into a Standard Nuclear Orientation. |
The first three atom entries of a Z-matrix are different from the subsequent
entries. The first Z-matrix line declares a single atom. The second line of
the Z-matrix input declares a second atom, refers to the first atom and gives
the distance between them. The third line declares the third atom, refers to
either the first or second atom, gives the distance between them, refers to the
remaining atom and gives the angle between them. All subsequent entries begin
with an atom declaration, a reference atom and a distance, a second reference
atom and an angle, a third reference atom and a dihedral angle. This can be
summarized as:
- First atom.
- Second atom, reference atom, distance.
- Third atom, reference atom A, distance between A and the third atom,
reference atom B, angle defined by atoms A, B and the third atom.
- Fourth atom, reference atom A, distance, reference atom B, angle,
reference atom C, dihedral angle (A, B, C and the fourth atom).
- All subsequent atoms follow the same basic form as (4)
Example 3.0 Z-matrix for hydrogen peroxide
O1
O2 O1 oo
H1 O1 ho O2 hoo
H2 O2 ho O1 hoo H1 hooh
Line 1 declares an oxygen atom (O1). Line 2 declares the second oxygen atom
(O2), followed by a reference to the first atom (O1) and a distance between
them denoted oo. Line 3 declares the first hydrogen atom (H1), indicates
it is separated from the first oxygen atom (O1) by a distance HO and makes an
angle with the second oxygen atom (O2) of hoo. Line 4 declares the
fourth atom and the second hydrogen atom (H2), indicates it is separated from
the second oxygen atom (O2) by a distance HO and makes an angle with the first
oxygen atom (O1) of hoo and makes a dihedral angle with the first
hydrogen atom (H1) of hooh.
Some further points to note are:
- Atoms can be declared by either atomic number or symbol.
- If declared by atomic number, connectivity needs to be indicated
by Z-matrix line number.
- If declared by atomic symbol either number similar atoms (e.g.,
H1, H2, O1, O2 etc.) and refer connectivity using this symbol, or
indicate connectivity by the line number of the referred atom.
- Bond lengths and angles can be entered either as variables/parameters or
real numbers.
- Variables/parameters can be declared in any order.
- A single blank line separates parameters from the Z-matrix.
All the following examples are equivalent in the information forwarded to the
Q-Chem program.
Example 3.0 Using parameters to define bond lengths and angles, and
using numbered symbols to define atoms and indicate connectivity.
$molecule
0 1
O1
O2 O1 oo
H1 O1 ho O2 hoo
H2 O2 ho O1 hoo H1 hooh
oo = 1.5
oh = 1.0
hoo = 120.0
hooh = 180.0
$end
Example 3.0 Not using parameters to define bond lengths and angles,
and using numbered symbols to define atoms and indicate connectivity.
$molecule
0 1
O1
O2 O1 1.5
H1 O1 1.0 O2 120.0
H2 O2 1.0 O1 120.0 H1 180.0
$end
Example 3.0 Using parameters to define bond lengths and angles, and
referring to atom connectivities by line number.
$molecule
0 1
8
8 1 oo
1 1 ho 2 hoo
1 2 ho 1 hoo 3 hooh
oo = 1.5
oh = 1.0
hoo = 120.0
hooh = 180.0
$end
Example 3.0 Referring to atom connectivities by line number, and
entering bond length and angles directly.
$molecule
0 1
8
8 1 1.5
1 1 1.0 2 120.0
1 2 1.0 1 120.0 3 180.0
$end
Obviously, a number of the formats outlined above are less appealing to the
eye and more difficult for us to interpret than the others, but each
communicates exactly the same Z-matrix to the Q-Chem program.
3.5.1 Dummy Atoms
Dummy atoms are indicated by the identifier X and followed, if necessary, by
an integer. (e.g., X1, X2. Dummy atoms are often useful for molecules
where symmetry axes and planes are not centered on a real atom, and have also
been useful in the past for choosing variables for structure optimization and
introducing symmetry constraints.
Note:
Dummy atoms play no role in the quantum mechanical calculation, and are
used merely for convenience in specifying other atomic positions or geometric
variables. |
3.6 Job Specification: The $rem Array Concept
The $rem array is the means by which users convey to Q-Chem the type of
calculation they wish to perform (level of theory, basis set, convergence
criteria, etc.). The keyword $rem signals the beginning of the overall job
specification. Within the $rem section the user inserts $rem
variables (one per line) which define the essential details of the
calculation. The format for entering $rem variables within the $rem keyword
section of the input is shown in the following example shown in the following
example:
Example 3.0 Format for declaring $rem variables in the $rem keyword
section of the Q-Chem input file. Note, Q-Chem only reads the first two
arguments on each line of $rem. All other text is ignored and can be used for
placing short user comments.
REM_VARIABLE VALUE [comment]
The $rem array stores all details required to perform the calculation, and
details of output requirements. It provides the flexibility to customize a
calculation to specific user requirements. If a default $rem variable setting
is indicated in this manual, the user does not have to declare the variable in
order for the default to be initiated (e.g., the default JOBTYPE is a
single point energy, SP). Thus, to perform a single point energy
calculation, the user does not need to set the $rem variable
JOBTYPE to SP. However, to perform an optimization, for
example, it is necessary to override the program default by setting
JOBTYPE to OPT.
A number of the $rem variables have been set aside for internal program use,
as they represent variables automatically determined by Q-Chem (e.g., the
number of atoms, the number of basis functions). These need not concern the
user.
User communication to the internal program $rem array comes in two general
forms: (1) long term, machine-specific customization via the
.qchemrc and preferences files (Section 2.6)
and, (2) the Q-Chem input deck. There are many defaults already set within
the Q-Chem program many of which can be overridden by the user. Checks are
made to ensure that the user specifications are permissible (e.g. integral
accuracy is confined to 10−12 and adjusted, if necessary. If adjustment is
not possible, an error message is returned. Details of these checks and
defaults will be given as they arise.
The user need not know all elements, options and details of the $rem array in
order to fully exploit the Q-Chem program. Many of the necessary elements and
options are determined automatically by the program, or the optimized default
parameters, supplied according to the user's basic requirements, available disk
and memory, and the operating system and platform.
3.7 $rem Array Format in Q-Chem Input
All data between the $rem keyword and the next appearance of $end is
assumed to be user $rem array input. On a single line for each $rem
variable, the user declares the $rem variable, followed by a blank space (tab
stop inclusive) and then the $rem variable option. It is recommended that a
comment be placed following a space after the $rem variable option. $rem
variables are case insensitive and a full listing is supplied in Appendix C.
Depending on the particular $rem variable, $rem options are entered either
as a case-insensitive keyword, an integer value or logical identifier
(true/false). The format for describing each $rem variable in this manual is
as follows:
REM_VARIABLE
A short description of what the variable controls. |
TYPE:
The type of variable, i.e. either INTEGER, LOGICAL or STRING |
DEFAULT:
The default value, if any. |
OPTIONS:
A list of the options available to the user. |
RECOMMENDATION:
A quick recommendation, where appropriate. |
|
Example 3.0 General format of the $rem section of the text input file.
$rem
REM_VARIABLE value [ user_comment ]
REM_VARIABLE value [ user_comment ]
...
$end
Note:
(1) Erroneous lines will terminate the calculation.
(2) Tab stops can be used to format input.
(3) A line prefixed with an exclamation mark `!' is treated as a comment and
will be ignored by the program. |
3.8 Minimum $rem Array Requirements
Although Q-Chem provides defaults for most $rem variables, the user will
always have to stipulate a few others. For example, in a single point energy
calculation, the minimum requirements will be BASIS (defining the
basis set), EXCHANGE (defining the level of theory to treat exchange)
and CORRELATION (defining the level of theory to treat correlation, if
required). If a wavefunction-based correlation treatment (such as
MP2) is used, HF is taken as the default for exchange.
Example 3.0 Example of minimum $rem requirements to run an MP2/6-31G*
energy calculation.
$rem
BASIS 6-31G* Just a small basis set
CORRELATION mp2 MP2 energy
$end
3.9 User-Defined Basis Sets ($basis and $aux_basis)
The $rem variable BASIS allows the user to indicate that
the basis set is being user-defined. The user-defined basis set is entered in
the $basis section of the input. For further details of entering a
user-defined basis set, see Chapter 7. Similarly, a user-defined
auxiliary basis set may be entered in a $aux_basis section of the input if
the $rem list includes AUX_BASIS = GEN.
3.10 Comments ($comment)
Users are able to add comments to the input file outside keyword input
sections, which will be ignored by the program. This can be useful as
reminders to the user, or perhaps, when teaching another user to set up inputs.
Comments can also be provided in a $comment block, although currently the
entire input deck is copied to the output file, rendering this redundant.
3.11 User-Defined Pseudopotentials ($ecp)
The $rem variable ECP allows the user to indicate that
pseudopotentials (effective core potentials) are being user-defined. The
user-defined effective core potential is entered in the $ecp section of the
input. For further details, see Chapter 8.
3.12 User-defined Parameters for DFT Dispersion Correction ($empirical_dispersion)
If a user wants to change from the default values recommended by Grimme,
the user-defined dispersion parameters can be entered in the
$empirical_dispersion section of the input. For further details, see Section 4.3.6.
3.13 Addition of External Charges ($external_charges)
If the $external_charges keyword is present, Q-Chem scans for
a set of external charges to be incorporated into a calculation. The format for
a set of external charges is the Cartesian coordinates, followed by the charge
size, one charge per line. Charges are in atomic units, and coordinates are in
angstroms (unless atomic units are specifically selected, see INPUT_BOHR).
The external charges are rotated with the molecule into the standard nuclear orientation.
Example 3.0 General format for incorporating a set of external charges.
$external_charges
x-coord1 y-coord1 z-coord1 charge1
x-coord2 y-coord2 z-coord2 charge2
x-coord3 y-coord3 z-coord3 charge3
$end
In addition, the user can request to add a charged cage around the molecule
by using ADD_CHARGED_CAGE keyword
The cage parameters are controlled by
CAGE_RADIUS, CAGE_POINTS, and CAGE_CHARGE.
More details are given in Section 6.6.5.
3.14 Intracules ($intracule)
The $intracule section allows the user to enter options to customize the
calculation of molecular intracules. The INTRACULE $rem variable
must also be set to TRUE before this section takes effect. For further
details see Section 10.4.
3.15 Isotopic Substitutions ($isotopes)
By default Q-Chem uses atomic masses that correspond to the most abundant
naturally occurring isotopes. Alternative masses for any or all of the atoms in
a molecule can be specified using the $isotopes keyword. The
ISOTOPES $rem variable must be set to TRUE for this section to take
effect. See Section 10.6.6 for details.
3.16 Applying a Multipole Field ($multipole_field)
Q-Chem has the capability to apply a multipole field to the molecule under
investigation. Q-Chem scans the input deck for the $multipole_field keyword,
and reads each line (up to the terminator keyword, $end) as a single
component of the applied field.
Example 3.0 General format for imposing a multipole field.
$multipole_field
field_component_1 value_1
field_component_2 value_2
$end
The field_component is simply stipulated using the Cartesian
representation e.g. X, Y, Z, (dipole), XX, XY, YY (quadrupole) XXX,
etc., and the value or size of the imposed field is in atomic units.
3.17 Natural Bond Orbital Package ($nbo)
The default action in Q-Chem is not to run the NBO package. To turn the NBO
package on, set the $rem variable NBO to ON. To access
further features of NBO, place standard NBO package parameters into a keyword
section in the input file headed with the $nbo keyword. Terminate the section
with the termination string $end.
3.18 User-Defined Occupied Guess Orbitals ($occupied and $swap_occupied_virtual)
It is sometimes useful for the occupied guess orbitals to be other than the
lowest Nα (or Nβ) orbitals. Q-Chem allows the occupied
guess orbitals to be defined using the $occupied keyword. The user defines
occupied guess orbitals by listing the alpha orbitals to be occupied on the
first line, and beta on the second. Alternatively, orbital choice can be controlled
by the $swap_occupied_virtualkeyword. See Section 4.5.4.
3.19 Geometry Optimization with General Constraints ($opt)
When a user defines the JOBTYPE to be a molecular geometry
optimization, Q-Chem scans the input deck for the $opt keyword. Distance,
angle, dihedral and out-of-plane bend constraints imposed on any atom declared
by the user in this section, are then imposed on the optimization procedure.
See Chapter 9 for details.
3.20 Polarizable Continuum Solvation Models ($pcm)
The $pcm section is available to provide special parameters for polarizable
continuum models (PCMs). These include the C-PCM and IEF-PCM models, which
share a common set of parameters. Details are provided in Section 10.2.2.
3.21 Effective Fragment Potential calculations ($efp_fragments and
$efp_params)
These keywords are used to specify positions and parameters for effective fragments
in EFP calculations. Details are provided in Chapter 11.
3.22 SS(V)PE Solvation Modeling ($svp and $svpirf)
The $svp section is available to specify special
parameters to the solvation module such as cavity grid parameters and
modifications to the numerical integration procedure. The $svpirf section
allows the user to specify an initial guess for the solution of the cavity
charges. As discussed in section 10.2.5, the $svp and $svpirf
input sections are used to specify parameters for the iso-density implementation
of SS(V)PE. An alternative implementation of the SS(V)PE mode, based on a more
empirical definition of the solute cavity, is available within the PCM code
(Section 10.2.2).
3.23 Orbitals, Densities and ESPs on a Mesh ($plots)
The $plots part of the input permits the evaluation of molecular orbitals,
densities, electrostatic potentials, transition densities, electron attachment
and detachment densities on a user-defined mesh of points. For more details,
see Section 10.9.
3.24 User-Defined van der Waals Radii ($van_der_waals)
The $van_der_waals section of the input enables the user to customize the Van der Waals
radii that are important parameters in the Langevin dipoles solvation model.
For more details, see Section 10.2.
3.25 User-Defined Exchange-Correlation Density Functionals ($xc_functional)
The EXCHANGE and CORRELATION $rem variables
(Chapter 4) allow the user to indicate that the exchange-correlation
density functional will be user-defined. The user defined exchange-correlation
is to be entered in the $xc_functional part of the input. The format is:
$xc_functional
X exchange_symbol coefficient
X exchange_symbol coefficient
...
C correlation_symbol coefficient
C correlation_symbol coefficient
...
K coefficient
$end
Note:
Coefficients are real numbers. |
3.26 Multiple Jobs in a Single File: Q-Chem Batch Job Files
It is sometimes useful to place a series of jobs into a single ASCII file.
This feature is supported by Q-Chem and is invoked by separating jobs with
the string @@@ on a single line. All output is subsequently appended
to the same output file for each job within the file.
Note:
The first job will overwrite any existing output file of the same name in
the working directory. Restarting the job will also overwrite any existing
file. |
In general, multiple jobs are placed in a single file for two reasons:
- To use information from a prior job in a later job
- To keep projects together in a single file
The @@@ feature allows these objectives to be met, but the following points
should be noted:
- Q-Chem reads all the jobs from the input file on initiation and stores
them. The user cannot make changes to the details of jobs which have not
been run post command line initiation.
- If any single job fails, Q-Chem proceeds to the next job in the batch
file.
- No check is made to ensure that dependencies are satisfied, or that
information is consistent (e.g. an optimization job followed by a
frequency job; reading in the new geometry from the optimization for the
frequency). No check is made to ensure that the optimization was
successful. Similarly, it is assumed that both jobs use the same basis
set when reading in MO coefficients from a previous job.
- Scratch files are saved between multi-job / single files runs (i.e.,
using a batch file with @@@ separators), but are deleted on
completion unless a third qchem command line argument is
supplied (see Chapter 2).
Using batch files with the @@@ separator is clearly most useful for cases
relating to point 1 above. The alternative would be to cut and paste output,
and/or use a third command line argument to save scratch files between separate
runs.
For example, the following input file will optimize the geometry of H2 at
HF/6-31G*, calculate vibrational frequencies at HF/6-31G* using the optimized
geometry and the self-consistent MO coefficients from the optimization and,
finally, perform a single point energy using the optimized geometry at the
MP2/6-311G(d,p) level of theory. Each job will use the same scratch area,
reading files from previous runs as instructed.
Example 3.0 Example of using information from previous jobs in a single
input file.
$comment
Optimize H-H at HF/6-31G*
$end
$molecule
0 1
H
H 1 r
r = 1.1
$end
$rem
JOBTYPE opt Optimize the bond length
EXCHANGE hf
CORRELATION none
BASIS 6-31G*
$end
@@@
$comment
Now calculate the frequency of H-H at the same level of theory.
$end
$molecule
read
$end
$rem
JOBTYPE freq Calculate vibrational frequency
EXCHANGE hf
CORRELATION none
BASIS 6-31G*
SCF_GUESS read Read the MOs from disk
$end
@@@
$comment
Now a single point calculation at at MP2/6-311G(d,p)//HF/6-31G*
$end
$molecule
read
$end
$rem
EXCHANGE hf
CORRELATION mp2
BASIS 6-311G(d,p)
$end
Note:
(1) Output is concatenated into the same output file.
(2) Only two arguments are necessarily supplied to the command line interface. |
3.27 Q-Chem Output File
The Q-Chem output file is the file to which details of the job invoked by
the user are printed. The type of information printed to this files depends
on the type of job (single point energy, geometry optimization etc.) and the
$rem variable print levels. The general and default form is as follows:
- Q-Chem citation
- User input
- Molecular geometry in Cartesian coordinates
- Molecular point group, nuclear repulsion energy, number of alpha and beta electrons
- Basis set information (number of functions, shells and function pairs)
- SCF details (method, guess, optimization procedure)
- SCF iterations (for each iteration, energy and DIIS error is reported)
- {depends on job type}
- Molecular orbital symmetries
- Mulliken population analysis
- Cartesian multipole moments
- Job completion
Note:
Q-Chem overwrites any existing output files in the working directory
when it is invoked with an existing file as the output file parameter. |
3.28 Q-Chem Scratch Files
The directory set by the environment variable $QCSCRATCH is the
location Q-Chem places scratch files it creates on execution. Users may wish
to use the information created for subsequent calculations. See Chapter 2
for information on saving files.
The 32-bit architecture on some platforms means there can be problems
associated with files larger than about 2 Gb. Q-Chem handles this issue by
splitting scratch files that are larger than this into several files, each of
which is smaller than the 2 Gb limit. The maximum number of these files (which
in turn limits the maximum total file size) is determined by the following
$rem variable:
MAX_SUB_FILE_NUM
|
Sets the maximum number of sub files allowed. |
TYPE:
DEFAULT:
16 Corresponding to a total of 32Gb for a given file. |
OPTIONS:
n | User-defined number of gigabytes. |
RECOMMENDATION:
Leave as default, or adjust according to your system limits.
|
Chapter 4 Self-Consistent Field Ground State Methods
4.1 Introduction
4.1.1 Overview of Chapter
Theoretical chemical models [6] involve two principal
approximations. One must specify the type of atomic orbital basis set used (see
Chapters 7 and 8), and one must specify the way in
which the instantaneous interactions (or correlations) between electrons are
treated. Self-consistent field (SCF) methods are the simplest and most widely
used electron correlation treatments, and contain as special cases all
Kohn-Sham density functional methods and the Hartree-Fock method. This
Chapter summarizes Q-Chem's SCF capabilities, while the next Chapter
discusses more complex (and computationally expensive!) wavefunction-based
methods for describing electron correlation. If you are new to quantum
chemistry, we recommend that you also purchase an introductory textbook on the
physical content and practical performance of standard
methods [6,[7,[8].
This Chapter is organized so that the earlier sections provide a mixture of
basic theoretical background, and a description of the minimum number of
program input options that must be specified to run SCF jobs. Specifically,
this includes the sections on:
- Hartree-Fock theory
- Density functional theory. Note that all basic input options described in
the Hartree-Fock also apply to density functional calculations.
Later sections introduce more specialized options that can be consulted as
needed:
- Large molecules and linear scaling methods. A short overview of the ideas
behind methods for very large systems and the options that control them.
- Initial guesses for SCF calculations. Changing the default initial guess
is sometimes important for SCF calculations that do not converge.
- Converging the SCF calculation. This section describes the iterative
methods available to control SCF calculations in Q-Chem. Altering the
standard options is essential for SCF jobs that have failed to converge
with the default options.
- Unconventional SCF calculations. Some nonstandard SCF methods
with novel physical and mathematical features. Explore further if you are
interested!
- SCF Metadynamics. This can be used to locate multiple solutions to the SCF
equations and help check that your solution is the lowest minimum.
4.1.2 Theoretical Background
In 1926, Schrödinger [9] combined the wave nature of the
electron with the statistical knowledge of the electron viz.
Heisenberg's Uncertainty Principle [10] to formulate an
eigenvalue equation for the total energy of a molecular system. If we focus on
stationary states and ignore the effects of relativity, we have the
time-independent, non-relativistic equation
where the coordinates R and r refer to nuclei and electron position
vectors respectively and H is the Hamiltonian operator. In atomic units,
H=− |
1
2
|
|
N ∑
i=1
|
∇i2 − |
1
2
|
|
M ∑
A=1
|
|
1
MA
|
∇A2 − |
N ∑
i=1
|
|
M ∑
A=1
|
|
ZA
riA
|
+ |
N ∑
i=1
|
|
N ∑
j > i
|
|
1
rij
|
+ |
M ∑
A=1
|
|
M ∑
B > A
|
|
ZA ZB
RAB
|
|
| (4.2) |
where ∇2 is the Laplacian operator,
∇2 ≡ |
∂2
∂x2
|
+ |
∂2
∂y2
|
+ |
∂2
∂z2
|
|
| (4.3) |
In Eq. ,
Z is the nuclear charge, MA is the ratio of the mass of nucleus A to
the mass of an electron, RAB = |RA − RB| is the distance between
the Ath and Bth nucleus, rij = |ri − rj|
is the distance between the ith and jth electrons, riA = | ri − RA| is the distance between the ith electron and the Ath
nucleus, M is the number of nuclei and N is the number of
electrons. E is an eigenvalue of H, equal to the total energy, and the wave
function Ψ, is an eigenfunction of H.
Separating the motions of the electrons from that of the nuclei, an idea
originally due to Born and Oppenheimer [11], yields the electronic
Hamiltonian operator:
Helec = − |
1
2
|
|
N ∑
i=1
|
∇i2 − |
N ∑
i=1
|
|
M ∑
A=1
|
|
ZA
riA
|
+ |
N ∑
i=1
|
|
N ∑
j > i
|
|
1
rij
|
|
| (4.4) |
The solution of the corresponding electronic Schrödinger equation,
Helec Ψelec = Eelec Ψelec |
| (4.5) |
gives the total electronic energy, Eelec, and electronic wave
function, Ψelec, which describes the motion of the electrons for a
fixed nuclear position. The total energy is obtained by simply adding the
nuclear-nuclear repulsion energy [the fifth term in Eq. (4.2)] to the
total electronic energy:
Solving the eigenvalue problem in Eq. (4.5) yields a set of eigenfunctions
(Ψ0, Ψ1, Ψ2 …) with corresponding eigenvalues
(E0, E1, E2…) where E0 ≤ E1 ≤ E2 ≤ ….
Our interest lies in determining the lowest eigenvalue and associated
eigenfunction which correspond to the ground state energy and wavefunction
of the molecule. However, solving Eq. (4.5) for other than the most trivial
systems is extremely difficult and the best we can do in practice is to find
approximate solutions.
The first approximation used to solve Eq. (4.5) is that electrons move
independently within molecular orbitals (MO), each of which describes the
probability distribution of a single electron. Each MO is determined by
considering the electron as moving within an average field of all the other
electrons. Ensuring that the wavefunction is antisymmetric upon electron
interchange, yields the well known Slater-determinant wavefunction [12,[13],
where χi, a spin orbital, is the product of a molecular orbital ψi
and a spin function (α or β).
One obtains the optimum set of MOs by variationally minimizing the energy in
what is called a "self-consistent field" or SCF approximation to the
many-electron problem. The archetypal SCF method is the Hartree-Fock
approximation, but these SCF methods also include Kohn-Sham Density Functional
Theories (see Section 4.3). All SCF methods lead to equations of the form
where the Fock operator f(i) can be written
Here xi are spin and spatial coordinates of the ith electron,
χ are the spin orbitals and υeff is the effective potential
"seen" by the ith electron which depends on the spin orbitals of the
other electrons. The nature of the effective potential υeff depends on
the SCF methodology and will be elaborated on in further sections.
The second approximation usually introduced when solving Eq. (4.5), is the
introduction of an Atomic Orbital (AO) basis. AOs (ϕμ) are usually
combined linearly to approximate the true MOs. There are many standardized,
atom-centered basis sets and details of these are discussed in Chapter 7.
After eliminating the spin components in Eq. (4.8) and introducing a finite basis,
Eq. (4.8) reduces to the Roothaan-Hall matrix equation,
where F is the Fock matrix, C is a square matrix of molecular
orbital coefficients, S is the overlap matrix with elements
and ε is a diagonal matrix of the orbital energies.
Generalizing to an unrestricted formalism by introducing separate spatial
orbitals for α and β spin in Eq. (4.7) yields the
Pople-Nesbet [14] equations
Solving Eq. (4.11) or Eq. (4.13) yields the restricted or unrestricted
finite basis Hartree-Fock approximation. This approximation inherently
neglects the instantaneous electron-electron correlations which are averaged
out by the SCF procedure, and while the chemistry resulting from HF
calculations often offers valuable qualitative insight, quantitative energetics
are often poor. In principle, the DFT SCF methodologies are able to capture
all the correlation energy (the difference in energy between the HF energy and
the true energy). In practice, the best currently available density functionals
perform well, but not perfectly and conventional HF-based approaches to
calculating the correlation energy are still often required. They are discussed
separately in the following Chapter.
In self-consistent field methods, an initial guess is calculated for the MOs
and, from this, an average field seen by each electron can be calculated. A new
set of MOs can be obtained by solving the Roothaan-Hall or Pople-Nesbet
eigenvalue equations. This procedure is repeated until the new MOs differ
negligibly from those of the previous iteration.
Because they often yield acceptably accurate chemical predictions at a
reasonable computational cost, self-consistent field methods are the corner
stone of most quantum chemical programs and calculations. The formal costs of
many SCF algorithms is O(N4), that is, they grow with the fourth power of
the size, N, of the system. This is slower than the growth of the cheapest
conventional correlated methods but recent work by Q-Chem, Inc. and its
collaborators has dramatically reduced it to O(N), an improvement that now
allows SCF methods to be applied to molecules previously considered beyond the
scope of ab initio treatment.
In order to carry out an SCF calculation using Q-Chem, three $rem variables
need to be set:
BASIS | to specify the basis set (see Chapter 7). |
EXCHANGE | method for treating Exchange. |
CORRELATION | method for treating Correlation (defaults to NONE)
|
Types of ground state energy calculations currently available in Q-Chem
are summarized in Table 4.1.
Calculation | $rem Variable JOBTYPE | |
Single point energy (default) | SINGLE_POINT, SP |
Force | FORCE |
Equilibrium Structure Search | OPTIMIZATION, OPT |
Transition Structure Search | TS |
Intrinsic reaction pathway | RPATH |
Frequency | FREQUENCY, FREQ |
NMR Chemical Shift | NMR |
Table 4.1: The type of calculation to be run by Q-Chem is controlled by the
$rem variable JOBTYPE.
4.2 Hartree-Fock Calculations
4.2.1 The Hartree-Fock Equations
As with much of the theory underlying modern quantum chemistry, the
Hartree-Fock approximation was developed shortly after publication of the
Schrödinger equation, but remained a qualitative theory until the advent of
the computer. Although the HF approximation tends to yield qualitative chemical
accuracy, rather than quantitative information, and is generally inferior to
many of the DFT approaches available, it remains as a useful tool in the
quantum chemist's toolkit. In particular, for organic chemistry, HF predictions
of molecular structure are very useful.
Consider once more the Roothaan-Hall equations, Eq. (4.11), or the
Pople-Nesbet equations, Eq. (4.13), which can be traced back to the
integro-differential Eq. (4.8) in which the effective potential
υeff depends on the SCF methodology. In a restricted HF (RHF)
formalism, the effective potential can be written as
υeff= |
N/2 ∑
a
|
[ 2Ja (1)−Ka (1) ] − |
M ∑
A=1
|
|
ZA
r1A
|
|
| (4.14) |
where the Coulomb and exchange operators are defined as
Ja (1)= | ⌠ ⌡
|
ψa∗ (2) |
1
r12
|
ψa (2)dr 2 |
| (4.15) |
and
Ka (1)ψi (1)= | ⎡ ⎣
| ⌠ ⌡
|
ψa∗ (2) |
1
r12
|
ψi (2)dr 2 | ⎤ ⎦
|
ψa (1) |
| (4.16) |
respectively. By introducing an atomic orbital basis, we obtain Fock matrix elements
where the core Hamiltonian matrix elements
consist of kinetic energy elements
Tμν = | ⌠ ⌡
|
ϕμ (r) | ⎡ ⎣
|
− |
1
2
|
∇2 | ⎤ ⎦
|
ϕν (r)dr |
| (4.19) |
and nuclear attraction elements
Vμν = | ⌠ ⌡
|
ϕμ (r) | ⎡ ⎣
|
− |
∑
A
|
|
ZA
| R A −r |
| ⎤ ⎦
|
ϕν (r)dr |
| (4.20) |
The Coulomb and Exchange elements are given by
and
Kμν = |
1
2
|
|
∑
λσ
|
Pλσ ( μλ|νσ ) |
| (4.22) |
respectively, where the density matrix elements are
and the two electron integrals are
( μν|λσ ) = | ⌠ ⌡
|
| ⌠ ⌡
|
ϕμ (r 1 )ϕν (r 1 ) | ⎡ ⎣
|
1
r12
| ⎤ ⎦
|
ϕλ (r 2 )ϕσ (r2 )dr 1 dr 2 |
| (4.24) |
Note:
The formation and utilization of two-electron integrals is a topic
central to the overall performance of SCF methodologies. The performance of the
SCF methods in new quantum chemistry software programs can be quickly estimated
simply by considering the quality of their atomic orbital integrals packages.
See Appendix B for details of Q-Chem's AOINTS package. |
Substituting the matrix element in Eq. (4.17) back into the Roothaan-Hall
equations, Eq. (4.11), and iterating until self-consistency is achieved will
yield the Restricted Hartree-Fock (RHF) energy and wavefunction.
Alternatively, one could have adopted the unrestricted form of the wavefunction
by defining an alpha and beta density matrix:
The total electron density matrix PT is simply the sum of the alpha
and beta density matrices. The unrestricted alpha Fock matrix,
Fμνα = Hμνcore +Jμν −Kμνα |
| (4.26) |
differs from the restricted one only in the exchange contributions where the
alpha exchange matrix elements are given by
Kμνα = |
N ∑
λ
|
|
N ∑
σ
|
Pλσα ( μλ|νσ ) |
| (4.27) |
4.2.2 Wavefunction Stability Analysis
At convergence, the SCF energy will be at a stationary point with respect to
changes in the MO coefficients. However, this stationary point is not
guaranteed to be an energy minimum, and in cases where it is not, the
wavefunction is said to be unstable. Even if the wavefunction is at a minimum,
this minimum may be an artifact of the constraints placed on the form of the
wavefunction. For example, an unrestricted calculation will usually give a
lower energy than the corresponding restricted calculation, and this can give
rise to a RHF→UHF instability.
To understand what instabilities can occur, it is useful to consider the
most general form possible for the spin orbitals:
χi (r,ζ)=ψiα (r)α(ζ)+ψiβ (r)β(ζ) |
| (4.28) |
Here, the ψ's are complex functions of the Cartesian coordinates
r, and α and β are spin eigenfunctions of the
spin-variable ζ. The first constraint that is almost universally
applied is to assume the spin orbitals depend only on one or other of the
spin-functions α or β. Thus, the spin-functions take the form
χi(r,ζ)=ψiα(r)α(ζ) or χi(r,ζ)=ψiβ (r)β(ζ) |
| (4.29) |
where the ψ's are still complex functions. Most SCF packages,
including Q-Chem's, deal only with real functions, and this places an
additional constraint on the form of the wavefunction. If there exists a
complex solution to the SCF equations that has a lower energy, the
wavefunction will exhibit either a RHF → CRHF or a UHF → CUHF
instability. The final constraint that is commonly placed on the
spin-functions is that ψiα = ψiβ, i.e., the spatial
parts of the spin-up and spin-down orbitals are the same. This gives the
familiar restricted formalism and can lead to a RHF→ UHF instability as
mentioned above. Further details about the possible instabilities can be
found in Ref. .
Wavefunction instabilities can arise for several reasons, but frequently occur if
- There exists a singlet diradical at a lower energy then the closed-shell
singlet state.
- There exists a triplet state at a lower energy than the lowest singlet
state.
- There are multiple solutions to the SCF equations, and the calculation
has not found the lowest energy solution.
If a wavefunction exhibits an instability, the seriousness of it can be judged
from the magnitude of the negative eigenvalues of the stability matrices. These
matrices and eigenvalues are computed by Q-Chem's Stability Analysis package,
which was implemented by Dr Yihan Shao. The package is invoked by setting the
STABILITY_ANALYSIS $rem variable is set to TRUE. In order
to compute these stability matrices Q-Chem must first perform a CIS
calculation. This will be performed automatically, and does not require any
further input from the user. By default Q-Chem computes only the lowest
eigenvalue of the stability matrix. This is usually sufficient to determine if
there is a negative eigenvalue, and therefore an instability. Users wishing to
calculate additional eigenvalues can do so by setting the
CIS_N_ROOTS $rem variable to a number larger than 1.
Q-Chem's Stability Analysis package also seeks to correct internal
instabilities (RHF→RHF or UHF→UHF). Then, if such an instability is
detected, Q-Chem automatically performs a unitary transformation of the
molecular orbitals following the directions of the lowest eigenvector, and
writes a new set of MOs to disk. One can read in these MOs as an initial guess
in a second SCF calculation (set the SCF_GUESS $rem variable to
READ), it might also be desirable to set the SCF_ALGORITHM
to GDM. In cases where the lowest-energy SCF solution breaks the
molecular point-group symmetry, the SYM_IGNORE $rem should be set
to TRUE.
Note:
The stability analysis package can be used to analyze both DFT and HF
wavefunctions. |
4.2.3 Basic Hartree-Fock Job Control
In brief, Q-Chem supports the three main variants of the Hartree-Fock
method. They are:
- Restricted Hartree-Fock (RHF) for closed shell molecules. It is typically
appropriate for closed shell molecules at their equilibrium geometry,
where electrons occupy orbitals in pairs.
- Unrestricted Hartree-Fock (UHF) for open shell molecules. Appropriate for
radicals with an odd number of electrons, and also for molecules with
even numbers of electrons where not all electrons are paired (for example
stretched bonds and diradicaloids).
- Restricted open shell Hartree-Fock (ROHF) for open shell molecules, where
the alpha and beta orbitals are constrained to be identical.
Only two $rem variables are required in order to run Hartree-Fock (HF)
calculations:
-
EXCHANGE must be set to HF.
- A valid keyword for BASIS must be specified (see Chapter
7).
In slightly more detail, here is a list of basic $rem variables associated
with running Hartree-Fock calculations. See Chapter 7 for
further detail on basis sets available and Chapter 8 for
specifying effective core potentials.
JOBTYPE
Specifies the type of calculation. |
TYPE:
DEFAULT:
OPTIONS:
SP | Single point energy. |
OPT | Geometry Minimization. |
TS | Transition Structure Search. |
FREQ | Frequency Calculation. |
FORCE | Analytical Force calculation. |
RPATH | Intrinsic Reaction Coordinate calculation. |
NMR | NMR chemical shift calculation. |
BSSE | BSSE calculation. |
EDA | Energy decomposition analysis. |
RECOMMENDATION:
|
| EXCHANGE
Specifies the exchange level of theory. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use HF for Hartree-Fock calculations. |
|
|
|
BASIS
Specifies the basis sets to be used. |
TYPE:
DEFAULT:
OPTIONS:
General, Gen | User defined ($basis keyword required). |
Symbol | Use standard basis sets as per Chapter 7. |
Mixed | Use a mixture of basis sets (see Chapter 7). |
RECOMMENDATION:
Consult literature and reviews to aid your selection. |
|
| PRINT_ORBITALS
Prints orbital coefficients with atom labels in analysis part of output. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not print any orbitals. |
TRUE | Prints occupied orbitals plus 5 virtuals. |
NVIRT | Number of virtuals to print. |
RECOMMENDATION:
Use TRUE unless more virtuals are desired. |
|
|
|
THRESH
Cutoff for neglect of two electron integrals. 10−THRESH (THRESH
≤ 14). |
TYPE:
DEFAULT:
8 | For single point energies. |
10 | For optimizations and frequency calculations. |
14 | For coupled-cluster calculations. |
OPTIONS:
n | for a threshold of 10−n. |
RECOMMENDATION:
Should be at least three greater than SCF_CONVERGENCE. Increase for
more significant figures, at greater computational cost. |
|
| SCF_CONVERGENCE
SCF is considered converged when the wavefunction error is less that
10−SCF_CONVERGENCE. Adjust the value of THRESH at the same
time. Note that in Q-Chem 3.0 the DIIS error is measured by the maximum error
rather than the RMS error as in previous versions. |
TYPE:
DEFAULT:
5 | For single point energy calculations. |
7 | For geometry optimizations and vibrational analysis. |
8 | For SSG calculations, see Chapter 5. |
OPTIONS:
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger
values provide more significant figures, at greater computational cost. |
|
|
|
UNRESTRICTED
Controls the use of restricted or unrestricted orbitals. |
TYPE:
DEFAULT:
FALSE | (Restricted) Closed-shell systems. |
TRUE | (Unrestricted) Open-shell systems. |
OPTIONS:
TRUE | (Unrestricted) Open-shell systems. |
FALSE | Restricted open-shell HF (ROHF). |
RECOMMENDATION:
Use default unless ROHF is desired. Note that for unrestricted calculations on
systems with an even number of electrons it is usually necessary to break
alpha / beta symmetry in the initial guess, by using SCF_GUESS_MIX or
providing $occupied information (see Section 4.5 on
initial guesses). |
|
4.2.4 Additional Hartree-Fock Job Control Options
Listed below are a number of useful options to customize a Hartree-Fock
calculation. This is only a short summary of the function of these $rem
variables. A full list of all SCF-related variables is provided in
Appendix C.
A number of other specialized topics (large molecules, customizing initial
guesses, and converging the calculation) are discussed separately in
Sections 4.4, 4.5, and 4.6, respectively.
INTEGRALS_BUFFER
Controls the size of in-core integral storage buffer. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default, or consult your systems administrator for hardware limits. |
|
| DIRECT_SCF
TYPE:
DEFAULT:
OPTIONS:
TRUE | Forces direct SCF. |
FALSE | Do not use direct SCF. |
RECOMMENDATION:
Use default; direct SCF switches off in-core integrals. |
|
|
|
METECO
Sets the threshold criteria for discarding shell-pairs. |
TYPE:
DEFAULT:
2 | Discard shell-pairs below 10−THRESH. |
OPTIONS:
1 | Discard shell-pairs four orders of magnitude below machine precision. |
2 | Discard shell-pairs below 10−THRESH. |
RECOMMENDATION:
|
| STABILITY_ANALYSIS
Performs stability analysis for a HF or DFT solution. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform stability analysis. |
FALSE | Do not perform stability analysis. |
RECOMMENDATION:
Set to TRUE when a HF or DFT solution is suspected to be unstable. |
|
|
|
SCF_PRINT
Controls level of output from SCF procedure to Q-Chem output file. |
TYPE:
DEFAULT:
0 | Minimal, concise, useful and necessary output. |
OPTIONS:
0 | Minimal, concise, useful and necessary output. |
1 | Level 0 plus component breakdown of SCF electronic energy. |
2 | Level 1 plus density, Fock and MO matrices on each cycle. |
3 | Level 2 plus two-electron Fock matrix components (Coulomb, HF exchange |
| and DFT exchange-correlation matrices) on each cycle. |
RECOMMENDATION:
Proceed with care; can result in extremely large output files at level 2 or higher.
These levels are primarily for program debugging. |
|
| SCF_FINAL_PRINT
Controls level of output from SCF procedure to Q-Chem output file at the
end of the SCF. |
TYPE:
DEFAULT:
OPTIONS:
0 | No extra print out. |
1 | Orbital energies and break-down of SCF energy. |
2 | Level 1 plus MOs and density matrices. |
3 | Level 2 plus Fock and density matrices. |
RECOMMENDATION:
The break-down of energies is often useful (level 1). |
|
|
|
DIIS_SEPARATE_ERRVEC
Control optimization of DIIS error vector in unrestricted calculations. |
TYPE:
DEFAULT:
FALSE | Use a combined alpha and beta error vector. |
OPTIONS:
FALSE | Use a combined alpha and beta error vector. |
TRUE | Use separate error vectors for the alpha and beta spaces. |
RECOMMENDATION:
When using DIIS in Q-Chem a convenient optimization for unrestricted calculations is to sum
the alpha and beta error vectors into a single vector which is used for extrapolation. This
is often extremely effective, but in some pathological systems with symmetry breaking, can lead
to false solutions being detected, where the alpha and beta components of the error vector
cancel exactly giving a zero DIIS error. While an extremely uncommon occurrence, if it is suspected,
set DIIS_SEPARATE_ERRVEC to TRUE to check. |
|
4.2.5 Examples
Provided below are examples of Q-Chem input files to run ground state,
Hartree-Fock single point energy calculations.
Example 4.0 Example Q-Chem input for a single point energy calculation on
water. Note that the declaration of the single point $rem variable and level
of theory to treat correlation are redundant because they are the same as the
Q-Chem defaults.
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.2
hoh = 120.0
$end
$rem
JOBTYPE sp Single Point energy
EXCHANGE hf Exact HF exchange
CORRELATION none No correlation
BASIS sto-3g Basis set
$end
$comment
HF/STO-3G water single point calculation
$end
Example 4.0 UHF/6-311G calculation on the Lithium atom. Note that
correlation and the job type were not indicated because Q-Chem defaults
automatically to no correlation and single point energies. Note also that,
since the number of alpha and beta electron differ, MOs default to an
unrestricted formalism.
$molecule
0,2
3
$end
$rem
EXCHANGE HF Hartree-Fock
BASIS 6-311G Basis set
$end
Example 4.0 ROHF/6-311G calculation on the Lithium atom. Note again that
correlation and the job type need not be indicated.
$molecule
0,2
3
$end
$rem
EXCHANGE hf Hartree-Fock
UNRESTRICTED false Restricted MOs
BASIS 6-311G Basis set
$end
Example 4.0 RHF/6-31G stability analysis calculation on the singlet state of
the oxygen molecule. The wavefunction is RHF→UHF unstable.
$molecule
0 1
O
O 1 1.165
$end
$rem
EXCHANGE hf Hartree-Fock
UNRESTRICTED false Restricted MOs
BASIS 6-31G(d) Basis set
STABILITY_ANALYSIS true Perform a stability analysis
$end
4.2.6 Symmetry
Symmetry is a powerful branch of mathematics and is often exploited in quantum
chemistry, both to reduce the computational workload and to classify the final
results obtained [16,[17,[18]. Q-Chem is able
to determine the point group symmetry of the molecular nuclei and, on
completion of the SCF procedure, classify the symmetry of molecular orbitals,
and provide symmetry decomposition of kinetic and nuclear attraction energy
(see Chapter 10).
Molecular systems possessing point group symmetry offer the possibility of
large savings of computational time, by avoiding calculations of integrals
which are equivalent i.e., those integrals which can be mapped on to one
another under one of the symmetry operations of the molecular point group.
The Q-Chem default is to use symmetry to reduce computational time, when
possible.
There are several keywords that are related to symmetry, which causes frequent
confusion. SYM_IGNORE controls symmetry throughout all modules.
The default is FALSE. In some cases it may be desirable to turn off
symmetry altogether, for example if you do not want Q-Chem to reorient the
molecule into the standard nuclear orientation, or if you want to turn it off for
finite difference calculations. If the SYM_IGNORE $rem is set to
TRUE then the coordinates will not be altered from the input, and the
point group will be set to C1.
The SYMMETRY (an alias for ISYM_RQ) keyword controls symmetry in
some integral routines.
It is set to FALSE by default.
Note that setting it to FALSE
does not turn point group symmetry off, and does not disable
symmetry in the coupled-cluster suite (CCMAN and CCMAN2), which is controlled
by CC_SYMMETRY (see Chapters 5
and 6), although we noticed that sometimes it may mess up
the determination of orbital symmetries, possibly due to numeric noise.
In some cases, SYMMETRY=TRUE
can cause problems (poor convergence and crazy SCF energies) and
turning it off can help.
Note:
The user should be aware about different conventions for defining
symmetry
elements.
The arbitrariness affects, for example, C2v point group.
The specific choice affects how the irreps in the affected groups are labeled.
For example, b1 and b2 irreps in C2v are flipped when using different
conventions. Q-Chem uses non-Mulliken symmetry convention.
See http://iopenshell.usc.edu/howto/symmetry
for detailed explanations. |
| SYMMETRY
Controls the efficiency through the use of point group symmetry for
calculating integrals. |
TYPE:
DEFAULT:
TRUE | Use symmetry for computing integrals. |
OPTIONS:
TRUE | Use symmetry when available. |
FALSE | Do not use symmetry. This is
always the case for RIMP2 jobs |
RECOMMENDATION:
Use default unless benchmarking.
Note that symmetry usage is disabled for RIMP2, FFT, and QM/MM jobs. |
|
|
|
SYM_IGNORE
Controls whether or not Q-Chem determines the point group of the molecule and reorients the molecule to the standard orientation. |
TYPE:
DEFAULT:
FALSE | Do determine the point group (disabled for RIMP2 jobs). |
OPTIONS:
RECOMMENDATION:
Use default unless you do not want the molecule to be reoriented.
Note that symmetry usage is disabled for RIMP2 jobs. |
|
| SYM_TOL
Controls the tolerance for determining point group symmetry. Differences in
atom locations less than 10−SYM_TOL are treated as zero. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default unless the molecule has high symmetry which is not being
correctly identified. Note that relaxing this tolerance too much may introduce
errors into the calculation. |
|
|
|
4.3 Density Functional Theory
4.3.1 Introduction
In recent years, Density Functional Theory [19,[20,[21,[22]
has emerged as an accurate alternative first-principles approach to quantum
mechanical molecular investigations. DFT currently accounts for approximately
90% of all quantum chemical calculations being performed, not only because
of its proven chemical accuracy, but also because of its relatively cheap
computational expense. These two features suggest that DFT is likely to remain
a leading method in the quantum chemist's toolkit well into the future.
Q-Chem contains fast, efficient and accurate algorithms for all popular
density functional theories, which make calculations on quite large molecules
possible and practical.
DFT is primarily a theory of electronic ground state structures based on the
electron density, ρ(r), as opposed to the many-electron wavefunction
Ψ(r1,…,rN) There are a number of distinct similarities and
differences to traditional wavefunction approaches and modern DFT
methodologies. Firstly, the essential building blocks of the many electron
wavefunction are single-electron orbitals are directly analogous to the
Kohn-Sham (see below) orbitals in the current DFT framework. Secondly, both the
electron density and the many-electron wavefunction tend to be constructed
via a SCF approach that requires the construction of matrix elements
which are remarkably and conveniently very similar.
However, traditional approaches using the many electron wavefunction as a
foundation must resort to a post-SCF calculation (Chapter 5)
to incorporate correlation effects, whereas DFT
approaches do not. Post-SCF methods, such as perturbation theory or coupled
cluster theory are extremely expensive relative to the SCF procedure. On the
other hand, the DFT approach is, in principle, exact, but in practice relies on
modeling the unknown exact exchange correlation energy functional. While more
accurate forms of such functionals are constantly being developed, there is no
systematic way to improve the functional to achieve an arbitrary level of
accuracy. Thus, the traditional approaches offer the possibility of achieving
an arbitrary level of accuracy, but can be computationally demanding, whereas
DFT approaches offer a practical route but the theory is currently incomplete.
4.3.2 Kohn-Sham Density Functional Theory
The Density Functional Theory by Hohenberg, Kohn and Sham [23,[24]
stems from the original work of
Dirac [25], who found that the exchange energy of a uniform electron gas
may be calculated exactly, knowing only the charge density. However, while the
more traditional DFT constitutes a direct approach and the necessary equations
contain only the electron density, difficulties associated with the kinetic
energy functional obstructed the extension of DFT to anything more than a crude
level of approximation. Kohn and Sham developed an indirect approach to the
kinetic energy functional which transformed DFT into a practical tool for
quantum chemical calculations.
Within the Kohn-Sham formalism [24], the ground state
electronic energy, E, can be written as
where ET is the kinetic energy, EV is the electron-nuclear
interaction energy, EJ is the Coulomb self-interaction of the
electron density ρ(r) and EXC is the exchange-correlation
energy. Adopting an unrestricted format, the alpha and beta total electron
densities can be written as
where nα and nβ are the number of alpha and beta electron
respectively and, ψi are the Kohn-Sham orbitals. Thus, the total
electron density is
Within a finite basis set [26], the density is represented by
ρ(r) = |
∑
μν
|
PμνT ϕμ (r) ϕν (r) |
| (4.33) |
The components of Eq. (4.28) can now be written as
| |
|
|
nα ∑
i=1
|
|
|
ψiα | ⎢ ⎢
|
− |
1
2
|
∇2 | ⎢ ⎢
|
ψiα |
|
+ |
nβ ∑
i=1
|
|
|
ψiβ | ⎢ ⎢
|
− |
1
2
|
∇2 | ⎢ ⎢
|
ψiβ |
|
|
| |
| |
|
|
∑
μν
|
PμνT |
|
ϕμ(r) | ⎢ ⎢
|
− |
1
2
|
∇2 | ⎢ ⎢
|
ϕν(r) |
|
|
| | (4.34) |
| |
|
− |
M ∑
A=1
|
ZA |
ρ(r)
|r−RA|
|
dr |
| |
| |
|
− |
∑
μν
|
PμνT |
∑
A
|
|
|
ϕμ(r) | ⎢ ⎢
|
ZA
|r−RA|
| ⎢ ⎢
|
ϕν(r) |
|
|
| | (4.35) |
| |
|
|
1
2
|
|
|
ρ(r1) | ⎢ ⎢
|
1
|r1 − r2|
| ⎢ ⎢
|
ρ(r2) |
|
|
| |
| |
|
|
1
2
|
|
∑
μν
|
|
∑
λσ
|
PμνT PλσT (μν|λσ) |
| | (4.36) |
| |
|
| | (4.37) |
|
Minimizing E with respect to the unknown Kohn-Sham orbital coefficients
yields a set of matrix equations exactly analogous to the UHF case
where the Fock matrix elements are generalized to
|
Fμνα = Hμνcore + Jμν − FμνXCα |
| | (4.40) |
| Fμνβ = Hμνcore + Jμν − FμνXCβ |
| | (4.41) |
|
where FμνXCα and FμνXCβ are the
exchange-correlation parts of the Fock matrices dependent on the
exchange-correlation functional used. The Pople-Nesbet equations are obtained
simply by allowing
and similarly for the beta equation. Thus, the density and energy are obtained
in a manner analogous to that for the Hartree-Fock method. Initial guesses are
made for the MO coefficients and an iterative process applied until self
consistency is obtained.
4.3.3 Exchange-Correlation Functionals
There are an increasing number of exchange and correlation functionals and
hybrid DFT methods available to the quantum chemist, many of which are very
effective. In short, there are nowadays five basic working types of functionals
(five rungs on the Perdew's "Jacob`s Ladder"): those based on
the local spin density approximation (LSDA) are on the first rung,
those based on generalized gradient approximations (GGA) are on the second rung.
Functionals that include not only density gradient
corrections (as in the GGA functionals), but also a dependence on
the electron kinetic energy density and / or
the Laplacian of the electron density, occupy
the third rung of the Jacob`s Ladder and are known as "meta-GGA"
functionals. The latter lead to a systematic,
and often substantial improvement over GGA for thermochemistry
and reaction kinetics. Among the meta-GGA functionals, a particular attention
deserve the VSXC functional [27],
the functional of Becke and Roussel for exchange [28],
and for correlation [29] (the BR89B94 meta-GGA
combination [29]). The latter functional did not receive enough
popularity until recently, mainly because it was not representable
in an analytic form. In Q-Chem, BR89B94 is implemented now self-consistently
in a fully analytic form, based on the recent work [30].
The one and only non-empirical meta-GGA functional,
the TPSS functional [31],
was also implemented recently in Q-Chem [32].
Each of the above mentioned "pure" functionals can be
combined with a fraction of exact (Hartree-Fock) non-local
exchange energy replacing a similar fraction
from the DFT local exchange energy.
When a nonzero amount of Hartree-Fock exchange
is used (less than a 100%), the corresponding functional is a hybrid extension
(a global hybrid) of the parent "pure" functional.
In most cases a hybrid functional would have one or more
(up to 21 so far) linear mixing parameters that are fitted
to experimental data. An exception is the hybrid extension
of the TPSS meta-GGA functional, the non-empirical TPSSh scheme, which is
also implemented now in Q-Chem [32].
The forth rung of functionals ("hyper-GGA" functionals)
involve occupied Kohn-Sham orbitals as additional
non-local variables [33,[34,[35,[36].
This helps tremendously in describing cases of strong inhomogeneity
and strong non-dynamic correlation, that are evasive for global
hybrids at GGA and meta-GGA levels of the theory. The success is
mainly due to one novel feature of these functionals:
they incorporate a 100% of exact (or HF) exchange
combined with a hyper-GGA model correlation.
Employing a 100% of exact exchange has been a long standing dream in DFT,
but most previous attempts were unsuccessful. The correlation models
used in the hyper-GGA schemes B05 [33] and
PSTS [36], properly compensate the spuriously
high non-locality of the exact exchange hole,
so that cases of strong non-dynamic correlation become treatable.
In addition to some GGA and meta-GGA variables, the B05 scheme employs a new
functional variable, namely, the exact-exchange energy density:
eHFX(r) = − |
1
2
|
| ⌠ ⌡
|
dr′ |
|n(r,r′)|2
|r−r′|
|
, |
| (4.43) |
where
n(r,r′) = |
1
ρ(r)
|
|
occ ∑
i
|
φiks(r)φiks(r′) . |
| (4.44) |
This new variable enters the correlation energy component in a
rather sophisticated nonlinear manner [33]:
This presents a huge challenge for the practical implementation
of such functionals, since they require a Hartree-Fock
type of calculation at each grid point, which renders the
task impractical. Significant progress in implementing
efficiently the B05 functional was reported only
recently [37,[38]. This new implementation
achieves a speed-up of the B05 calculations by a factor of 100
based on resolution-of-identity (RI) technique (the RI-B05 scheme) and
analytical interpolations. Using this methodology, the PSTS
hyper-GGA was also implemented in Q-Chem more recently [32].
For the time being only single-point SCF calculations are
available for RI-B05 and RI-PSTS (the energy gradient will be available soon).
In contrast to B05 and PSTS, the forth-rung functional MCY employs a
100% global exact exchange, not only as a separate energy component of the functional,
but also as a non-linear variable used the MCY correlation energy
expression [34,[35]. Since this variable is
the same at each grid point, it has to be calculated
only once per SCF iteration. The form of the MCY correlation functional
is deduced from known adiabatic connection and coordinate
scaling relationships which, together with a few fitting
parameters, provides a good correlation match to the exact exchange.
The MCY functional [34] in its MCY2 version [35] is now
implemented in Q-Chem, as described in Ref. [32].
The fifth-rung functionals include not only occupied Kohn-Sham orbitals, but
also unoccupied orbitals, which improves further the quality of the
exchange-correlation energy. The practical application so far of these
consists of adding empirically a small fraction of correlation energy obtained
from MP2-like post-SCF calculation [39,[40].
Such functionals are known as "double-hybrids".
A more detailed
description of some these as implemented in Q-Chem is
given in Subsections 4.3.9 and 4.3.4.3.
Finally, the so-called range-separated (or long-range corrected, LRC)
functionals that employ exact
exchange for the long-range part of the functional are showing excellent
performance and considerable
promise (see Section 4.3.4). In addition, many of the functionals
can be augmented by an empirical dispersion correction, "-D" (see Section 4.3.6).
In summary, Q-Chem includes the following exchange and correlation functionals:
LSDA functionals:
- Slater-Dirac (Exchange) [25]
- Vokso-Wilk-Nusair (Correlation) [41]
- Perdew-Zunger (Correlation) [42]
- Wigner (Correlation) [43]
- Perdew-Wang 92 (Correlation) [44]
- Proynov-Kong 2009 (Correlation) [45]
GGA functionals:
- Becke86 (Exchange) [46]
- Becke88 (Exchange) [47]
- PW86 (Exchange) [48]
- refit PW86 (Exchange) [49]
- Gill96 (Exchange) [50]
- Gilbert-Gill99 (Exchange [51]
- Lee-Yang-Parr (Correlation) [52]
- Perdew86 (Correlation) [53]
- GGA91 (Exchange and correlation) [54]
- mPW1PW91 (Exchange and Correlation) [55]
- mPW1PBE (Exchange and Correlation)
- mPW1LYP (Exchange and Correlation)
- PBE (Exchange and correlation) [56,[57]
- revPBE (Exchange) [58]
- PBE0 (25% Hartree-Fock exchange + 75% PBE exchange + 100% PBE correlation) [59]
- PBE50 (50% Hartree-Fock exchange + 50% PBE exchange + 100% PBE correlation)
- B3LYP (Exchange and correlation within a hybrid scheme) [60]
- B3PW91 (B3 Exchange + PW91 correlation)
- B3P86 (B3 Exchange + PW86 correlation)
- B5050LYP (50% Hartree-Fock exchange + 5% Slater exchange + 42% Becke exchange + 100% LYP correlation) [61]
- BHHLYP (50% Hartree-Fock exchange + 50% Becke exchange + 100% LYP correlation) [60]
- O3LYP (Exchange and correlation) [62]
- X3LYP (Exchange and correlation) [63]
- CAM-B3LYP (Range separated exchange and LYP correlation) [64]
- Becke97 (Exchange and correlation within a hybrid scheme) [65,[57]
- Becke97-1 (Exchange and correlation within a hybrid scheme) [66,[57]
- Becke97-2 (Exchange and correlation within a hybrid scheme) [67,[57]
- B97-D (Exchange and correlation and empirical dispersion correction) [68]
- HCTH (Exchange- correlation within a hybrid scheme) [66,[57]
- HCTH-120 (Exchange- correlation within a hybrid scheme) [69,[57]
- HCTH-147 (Exchange- correlation within a hybrid scheme) [69,[57]
- HCTH-407 (Exchange- correlation within a hybrid scheme) [70,[57]
- The ωB97X functionals developed by Chai and Gordon [71]
(Exchange and correlation within a hybrid scheme, with long-range correction,
see further in this manual for details)
- BNL (Exchange GGA functional) [72,[73]
- BOP (Becke88 exchange plus the "one-parameter progressive" correlation
functional, OP) [74]
- PBEOP (PBE Exchange plus the OP correlation functional) [74]
- SOGGA (Exchange plus the PBE correlation functional) [75]
- SOGGA11 (Exchange and Correlation) [76]
- SOGGA11-X (Exchange and Correlation within a hybrid scheme, with re-optimized SOGGA11 parameters) [77]
- LRC-wPBEPBE (Long-range corrected PBE exchange and PBE correlation) [78]
- LRC-wPBEhPBE (Long-range corrected hybrid PBE exchange and PBE correlation) [79]
Note:
The OP correlation functional used in BOP has been parameterized for use with
Becke88 exchange, whereas in the PBEOP functional, the same correlation ansatz
is re-parameterized for use with PBE exchange. These two versions of OP correlation
are available as the correlation functionals (B88)OP and (PBE)OP. The BOP
functional, for example, consists of (B88)OP correlation combined with Becke88 exchange. |
Meta-GGA functionals involving the kinetic energy
density (τ), and or the Laplacian of the electron density:
- VSXC (Exchange and Correlation) [27]
- TPSS (Exchange and Correlation in a single non-empirical
scheme) [31,[32]
- TPSSh (Exchange and Correlation within a non-empirical hybrid scheme) [80]
- BMK (Exchange and Correlation within a hybrid scheme) [81]
- M05 (Exchange and Correlation within a hybrid scheme) [82,]
- M05-2X (Exchange and Correlation within a hybrid scheme) [84,[83]
- M06-L (Exchange and Correlation) [85,[83]
- M06-HF (Exchange and Correlation within a hybrid scheme) [86,[83]
- M06 (Exchange and Correlation within a hybrid scheme) [87,[83]
- M06-2X (Exchange and Correlation within a hybrid scheme) [87,[83]
- M08-HX (Exchange and Correlation within a hybrid scheme) [88]
- M08-SO (Exchange and Correlation within a hybrid scheme) [88]
- M11-L (Exchange and Correlation) [89]
- M11 (Exchange and Correlation within a hybrid scheme, with long-range correction) [90]
- BR89 (Exchange) [28,[30]
- B94 (Correlation) [29,[30]
- B95 (Correlation) [91]
- B1B95 (Exchange and Correlation) [91]
- PK06 (Correlation) [92]
Hyper-GGA functionals:
- B05 (A full exact-exchange Kohn-Sham scheme of Becke that accounts for static corrrelation
via real-space corrections) [33,[37,[38]
- mB05 (Modified B05 method that has simpler functional form and SCF potential) [93]
- PSTS (Hyper-GGA functional of Perdew-Staroverov-Tao-Scuseria) [36]
- MCY2 (The adiabatic connection-based MCY2 functional) [34,[35,[32]
Fifth-rung, double-hybrid (DH) functionals:
- ωB97X-2 (Exchange and Correlation within a DH generalization of the
LC corrected ωB97X scheme) [40]
- B2PLYP (another DH scheme proposed by Grimme, based on GGA exchange and correlation
functionals) [68]
- XYG3 and XYGJ-OS (an efficient DH scheme based on generalization of B3LYP) [94]
In addition to the above functional types, Q-Chem contains the
Empirical Density Functional 1 (EDF1), developed by Adamson, Gill and
Pople [95]. EDF1 is a combined exchange and correlation
functional that is specifically adapted to yield good results with the
relatively modest-sized 6-31+G* basis set, by direct fitting to thermochemical
data. It has the interesting feature that exact exchange mixing was not found
to be helpful with a basis set of this size. Furthermore, for this basis set,
the performance substantially exceeded the popular B3LYP functional,
while the cost of the calculations is considerably lower because there is no
need to evaluate exact (non-local) exchange. We recommend consideration of
EDF1 instead of either B3LYP or BLYP for density functional calculations on
large molecules, for which basis sets larger than 6-31+G* may be too
computationally demanding.
EDF2, another Empirical Density Functional, was developed by Ching Yeh Lin and
Peter Gill [96] in a similar vein to EDF1, but is specially designed for
harmonic frequency calculations. It was optimized using the cc-pVTZ basis
set by fitting into experimental harmonic frequencies and is designed
to describe the potential energy curvature well. Fortuitously, it also
performs better than B3LYP for thermochemical properties.
A few more words deserve the hybrid functionals [60], where several
different exchange and correlation functionals can be combined linearly to form a
hybrid functional. These have proven successful in a number of reported applications.
However, since the hybrid functionals contain HF exchange they are more expensive
that pure DFT functionals. Q-Chem has incorporated two of the most popular
hybrid functionals, B3LYP [97] and B3PW91 [28],
with the additional option for users to define their own hybrid functionals via
the $xc_functional keyword (see user-defined functionals in
Section 4.3.17, below). Among the latter, a recent new hybrid combination
available in Q-Chem is the 'B3tLap' functional, based on Becke's B88
GGA exchange and the 'tLap' (or 'PK06') meta-GGA
correlation [92,[98]. This
hybrid combination is on average more accurate than B3LYP, BMK, and M06
functionals for thermochemistry and better than B3LYP for reaction
barriers, while involving only five fitting parameters.
Another hybrid functional in Q-Chem that deserves attention
is the hybrid extension of the BR89B94 meta-GGA
functional [29,[98].
This hybrid functional yields a very good thermochemistry results, yet has
only three fitting parameters.
In addition, Q-Chem now includes the M05 and M06 suites of density
functionals. These are designed to be used only with
certain definite percentages of Hartree-Fock exchange. In
particular, M06-L [85] is designed to be used with no Hartree-Fock
exchange (which reduces the cost for large molecules), and M05 [82],
M05-2X [84], M06, and M06-2X [87] are designed to be used with
28%, 56%, 27%, and 54% Hartree-Fock exchange. M06-HF [86] is designed to
be used with 100% Hartree-Fock exchange, but it still contains some
local DFT exchange because the 100% non-local Hartree-Fock
exchange replaces only some of the local exchange.
Note:
The hybrid functionals are not simply a pairing of an exchange
and correlation functional, but are a combined exchange-correlation functional
(i.e., B-LYP and B3LYP vary in the correlation contribution in addition to the
exchange part). |
4.3.4 Long-Range-Corrected DFT
As pointed out in Ref. and elsewhere, the description of
charge-transfer excited states within density functional theory
(or more precisely, time-dependent DFT, which is discussed in
Section 6.3) requires full (100%)
non-local Hartree-Fock exchange, at least in the limit of large
donor-acceptor distance.
Hybrid functionals such as B3LYP [97] and
PBE0 [59] that are well-established and in widespread use,
however, employ only 20%
and 25% Hartree-Fock exchange, respectively. While these functionals
provide excellent results for many ground-state properties, they cannot
correctly describe the distance dependence of charge-transfer excitation
energies, which are enormously underestimated by most common density
functionals. This is a serious problem in any case, but it is a
catastrophic
problem in large molecules and in clusters, where TDDFT often predicts a
near-continuum of of spurious, low-lying charge transfer states [100,[101].
The problems with TDDFT's description of
charge transfer are not limited to large donor-acceptor distances, but have
been observed at ∼ 2 Å separation, in systems as small as
uracil-(H2O)4 [100]. Rydberg excitation energies
also tend to be substantially underestimated by standard TDDFT.
One possible avenue by which to correct such problems is to parameterize
functionals
that contain 100% Hartree-Fock exchange. To date, few such functionals exist,
and those that do (such as M06-HF) contain a very large number of
empirical adjustable parameters. An alternative option is to
attempt to preserve the form of common GGAs and hybrid functionals at
short range (i.e., keep the 25% Hartree-Fock exchange in PBE0)
while incorporating 100% Hartree-Fock exchange
at long range. Functionals along these lines are known variously
as "Coulomb-attenuated" functionals, "range-separated"
functionals, or (our preferred designation) "long-range-corrected"
(LRC) density functionals. Whatever the nomenclature, these functionals
are all based upon
a partition of the electron-electron Coulomb potential into long- and
short-range components, using the error function (erf):
|
1
r12
|
≡ |
1−erf(ωr12)
r12
|
+ |
erf(ωr12)
r12
|
|
| (4.45) |
The first term on the right in Eq. (4.45)
is singular but short-range, and decays
to zero on a length scale of ∼ 1/ω, while the second term
constitutes a non-singular, long-range background. The basic idea of LRC-DFT
is to utilize the short-range component of the Coulomb operator in
conjunction with standard DFT exchange (including any component of
Hartree-Fock exchange, if the functional is a hybrid), while at the same
time incorporating full Hartree-Fock exchange using the long-range
part of the Coulomb operator. This provides a
rigorously correct description of the long-range distance dependence of
charge-transfer excitation energies, but aims to avoid contaminating short-range
exchange-correlation effects with extra Hartree-Fock exchange.
Consider an exchange-correlation functional of the form
EXC = EC + EXGGA + CHF EXHF |
| (4.46) |
in which EC is the correlation energy, EXGGA
is the (local) GGA exchange energy, and EXHF is the (non-local)
Hartree-Fock exchange energy. The constant CHF denotes
the fraction of Hartree-Fock exchange in the functional, therefore
CHF = 0 for GGAs, CHF = 0.20 for B3LYP,
CHF = 0.25 for PBE0, etc..
The LRC version of the generic functional in Eq. (4.46) is
EXCLRC = EC + EXGGA, SR + CHF EXHF, SR + EXHF, LR |
| (4.47) |
in which the designations "SR" and "LR" in the various exchange
energies indicate that these components
of the functional are evaluated using either the short-range (SR) or the
long-range (LR) component of the Coulomb operator. (The correlation
energy EC is evaluated using the full Coulomb operator.)
The LRC functional in
Eq. (4.47) incorporates full Hartree-Fock exchange in the
asymptotic limit via the final term, EXHF, LR. To fully
specify the LRC functional, one must choose a value for the range
separation parameter ω in Eq. (4.45);
in the limit ω→ 0, the LRC functional in
Eq. (4.47) reduces to the original functional in
Eq. (4.46), while the ω→∞ limit
corresponds to a new functional, EXC = EC + EXHF. It is well known that full Hartree-Fock exchange
is inappropriate for use with most contemporary GGA correlation functionals,
so the latter limit is expected to perform quite poorly. Values of
ω > 1.0 bohr−1 are probably not worth considering [102,[78].
Evaluation of the short- and long-range Hartree-Fock exchange energies
is straightforward [103], so the crux of LRC-DFT rests upon
the form of the short-range GGA exchange energy. Several different
short-range GGA exchange functionals are available in Q-Chem, including
short-range variants of B88 and PBE exchange described by Hirao and
co-workers [104,[105], an alternative formulation of
short-range PBE exchange proposed by Scuseria and co-workers [106],
and several short-range variants of B97 introduced by
Chai and Head-Gordon [71,[107,[108,[40].
The reader is referred to these papers for additional methodological details.
These LRC-DFT functionals have been shown to remove
the near-continuum of spurious charge-transfer excited states that
appear in large-scale TDDFT calculations [102].
However, certain results depend sensitively upon the range-separation
parameter ω [101,[102,[78,[79], and the results
of LRC-DFT calculations must therefore be interpreted with caution,
and probably for a range of ω values. In two recent benchmark
studies of several LRC density functionals,
Rohrdanz and Herbert [78,[79]
have considered the errors engendered, as a function of ω, in
both ground-state properties and also TDDFT vertical excitation energies.
In Ref. , the sensitivity of valence excitations versus charge-transfer
excitation energies in TDDFT was considered, again as a function of ω.
A careful reading of these references is suggested prior to
performing any LRC-DFT calculations.
Within Q-Chem 3.2, there are three ways to perform LRC-DFT calculations.
4.3.4.1 LRC-DFT with the μB88, μPBE, and ωPBE exchange functionals
The form of EXGGA, SR is different for each different
GGA exchange functional, and short-range versions of B88
and PBE exchange are available in Q-Chem through the efforts of
the Herbert group. Versions of B88 and PBE, in which the Coulomb attenuation
is performed according to the procedure of Hirao [105], are denoted
as μB88 and μPBE, respectively (since μ, rather than ω, is
the Hirao group's notation for the range-separation parameter). Alternatively,
a short-range version of PBE exchange called ωPBE is available, which is
constructed according to the prescription of Scuseria and co-workers [106].
These short-range exchange functionals can be used in the absence of long-range
Hartree-Fock exchange, and using a combination of
ωPBE exchange and PBE correlation, a user could, for example, employ the
short-range hybrid functional recently described by Heyd, Scuseria, and
Ernzerhof [109]. Short-range hybrids appear to be most appropriate for
extended systems, however. Thus, within Q-Chem, short-range GGAs should
be used with long-range Hartree-Fock exchange, as in Eq. 4.47.
Long-range Hartree-Fock exchange is requested by setting LRC_DFT to
TRUE.
LRC-DFT is thus available
for any functional whose exchange component consists of some combination
of Hartree-Fock, B88, and PBE exchange (e.g., BLYP, PBE, PBE0,
BOP, PBEOP, and various user-specified combinations, but not B3LYP or other
functionals whose exchange components are more involved). Having
specified such a functional via the EXCHANGE and
CORRELATION variables, a user may request the
corresponding LRC functional by setting
LRC_DFT to TRUE. Long-range-corrected variants of PBE0,
BOP, and PBEOP must be obtained through the appropriate user-specified
combination of exchange and correlation functionals (as demonstrated in the
example below). In any case,
the value of ω must also be
specified by the user. Analytic energy gradients are available but
analytic Hessians are not. TDDFT vertical excitation energies are also
available.
LRC_DFT
Controls the application of long-range-corrected DFT |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply long-range correction. |
TRUE | (or 1) Use the long-range-corrected version of the requested functional.
|
RECOMMENDATION:
Long-range correction is available for any combination of Hartree-Fock,
B88, and PBE exchange (along with any stand-alone correlation functional).
|
|
| OMEGA
Sets the Coulomb attenuation parameter ω. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
|
|
Example 4.0 Application of LRC-BOP to (H2O)2−.
$comment
To obtain LRC-BOP, a short-range version of BOP must be specified,
using muB88 short-range exchange plus (B88)OP correlation, which is
the version of OP parameterized for use with B88.
$end
$molecule
-1 2
O 1.347338 -.017773 -.071860
H 1.824285 .813088 .117645
H 1.805176 -.695567 .461913
O -1.523051 -.002159 -.090765
H -.544777 -.024370 -.165445
H -1.682218 .174228 .849364
$end
$rem
EXCHANGE GEN
BASIS 6-31(1+,3+)G*
LRC_DFT TRUE
OMEGA 330 ! = 0.330 a.u.
$end
$xc_functional
C (B88)OP 1.0
X muB88 1.0
$end
Regarding the choice of functionals and ω values, it has been found that
the Hirao and Scuseria ansatz afford virtually identical TDDFT
excitation energies, for all values of ω [79]. Thus,
functionals based on μPBE versus ωPBE provide the same excitation
energies, as a function of ω. However, the ωPBE functional appears
to be somewhat superior in the sense that it can provide accurate TDDFT
excitation energies and accurate ground-state properties using the
same value of ω [79], whereas this does not seem
to be the case for functionals based on μB88 or μPBE [78].
Recently, Rohrdanz et al. [79] have published a thorough
benchmark study of both ground- and excited-state properties, using the
"LRC-ωPBEh" functional, a hybrid (hence the "h")
that contains a fraction of short-range
Hartree-Fock exchange in addition to full long-range Hartree-Fock exchange:
EXC(LRC−ωPBEh) = EC(PBE) + EXSR(ωPBE) + CHF EXSR(HF) + EXLR(HF) |
| (4.48) |
The statistically-optimal parameter set, consider both ground-state properties and
TDDFT excitation energies together, was found to be CHF = 0.2 and
ω = 0.2 bohr−1 [79]. With these parameters, the
LRC-ωPBEh functional outperforms the traditional hybrid functional PBE0
for ground-state atomization energies and barrier heights. For TDDFT excitation
energies corresponding to localized excitations, TD-PBE0 and TD-LRC-ωPBEh
show similar statistical errors of ∼ 0.3 eV, but the latter functional also
exhibits only ∼ 0.3 eV errors for charge-transfer excitation energies, whereas
the statistical error for TD-PBE0 charge-transfer excitation energies is 3.0 eV!
Caution is definitely warranted in the case of charge-transfer excited states,
however, as these excitation energies are very sensitive to the precise value of
ω [101,[79]. It was later found that the parameter
set (CHF = 0, ω = 0.3 bohr−1) provides similar
(statistical) performance to that described above, although the predictions for
specific charge-transfer excited states can be somewhat different as compared to
the original parameter set [101].
Example 4.0 Application of LRC-ωPBEh to the C2H4-C2F4
hetero-dimer at 5 Å separation.
$comment
This example uses the "optimal" parameter set discussed above.
It can also be run by setting "EXCHANGE LRC-WPBEhPBE".
$end
$molecule
0 1
C 0.670604 0.000000 0.000000
C -0.670604 0.000000 0.000000
H 1.249222 0.929447 0.000000
H 1.249222 -0.929447 0.000000
H -1.249222 0.929447 0.000000
H -1.249222 -0.929447 0.000000
C 0.669726 0.000000 5.000000
C -0.669726 0.000000 5.000000
F 1.401152 1.122634 5.000000
F 1.401152 -1.122634 5.000000
F -1.401152 -1.122634 5.000000
F -1.401152 1.122634 5.000000
$end
$rem
EXCHANGE GEN
BASIS 6-31+G*
LRC_DFT TRUE
OMEGA 200 ! = 0.2 a.u.
CIS_N_ROOTS 4
CIS_TRIPLETS FALSE
$end
$xc_functional
C PBE 1.00
X wPBE 0.80
X HF 0.20
$end
4.3.4.2 LRC-DFT with the BNL Functional
The Baer-Neuhauser-Livshits (BNL) functional [72,[73]
is also based on the range separation of the Coulomb
operator in Eq. 4.45.
Its functional form resembles Eq. 4.47:
EXC = EC + CGGA,X EXGGA, SR + EXHF, LR |
| (4.49) |
where the recommended GGA correlation functional is LYP.
The recommended GGA exchange functional is BNL,
which is described by a local functional [110].
For ground state properties, the optimized value for
CGGA,X (scaling factor for the BNL exchange functional) was found to be 0.9.
The value of ω in BNL calculations can be
chosen in several different ways.
For example, one can use the optimized value ω=0.5 bohr−1.
For calculation of excited states and properties related to orbital energies,
it is strongly recommend to tune ω as described
below[111,[112].
System-specific optimization of
ω is based on Koopmans conditions that would be satisfied for
the exact functional[111], that is, ω is varied
until the Koopmans IE/EA for the
HOMO/LUMO is equal to ∆E IE/EA.
Based on published
benchmarks [73,[113], this system-specific
approach yields the most accurate values of IEs and excitation
energies.
The script that optimizes ω is called
.pl@ and is located in the $QC/bin directory.
The script optimizes ω in the range 0.1-0.8 (100-800).
See the script for the instructions how
to modify the script to optimize in a broader range.
To execute the script, you need to create three inputs for a BNL
job using the same geometry and basis set for a neutral molecule (.in@),
anion (.in@), and cation (.in@), and then type
'OptOmegaIPEA.pl >& optomega'.
The script will run creating outputs for each step (_*@, _*@,
_*@) writing the optimization output into .
A similar script, .pl@, will optimize ω to satisfy
the Koopmans condition for the IP only. This script minimizes
J=(IP+ϵHOMO)2, not the absolute values.
Note:
(i) If the system does not have positive EA, then the tuning should be done
according to the IP condition only. The IPEA script will
yield a wrong value of ω in such cases.
(ii) In order for the scripts to work, one must
specify SCF_FINAL_PRINT=1 in the inputs.
The scripts look for specific regular expressions and will not work correctly
without this keyword.
(iii) When tuning omega we recommend taking the amount of X BNL in the
XC part as 1.0 and not 0.9. |
The $xc_functional keyword for a BNL calculation reads:
$xc_functional
X HF 1.0
X BNL 0.9
C LYP 1.0
$end
and the $rem keyword reads
$rem
EXCHANGE GENERAL
SEPARATE_JK TRUE
OMEGA 500 != 0.5 Bohr$^{-1}$
DERSCREEN FALSE !if performing unrestricted calcn
IDERIV 0 !if performing unrestricted Hessian evaluation
$end
4.3.4.3 LRC-DFT with ωB97, ωB97X, ωB97X-D, and ωB97X-2 Functionals
Also available in Q-Chem are the ωB97 [71], ωB97X [71],
ωB97X-D [107], and ωB97X-2 [40] functionals,
recently developed by Chai and Head-Gordon. These authors have proposed a very simple ansatz to extend any EXGGA to
EXGGA,SR, as long as the SR operator has considerable spatial
extent [71,[108]. With the use of flexible GGAs, such
as Becke97 functional [65], their new LRC hybrid
functionals [71,[107,[108]
outperform the corresponding global hybrid
functionals (i.e., B97) and popular hybrid functionals (e.g., B3LYP) in thermochemistry, kinetics, and non-covalent interactions, which has not been easily
achieved by the previous LRC hybrid functionals. In addition, the qualitative failures of the commonly used hybrid density functionals in
some "difficult problems", such as dissociation of symmetric radical cations and long-range charge-transfer excitations, are significantly reduced
by these new functionals [71,[107,[108]. Analytical gradients and analytical Hessians are available for ωB97, ωB97X, and ωB97X-D.
Example 4.0 Application of ωB97 functional to nitrogen dimer.
$comment
Geometry optimization, followed by a TDDFT calculation.
$end
$molecule
0 1
N1
N2 N1 1.1
$end
$rem
jobtype opt
exchange omegaB97
basis 6-31G*
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97
basis 6-31G*
scf_guess READ
cis_n_roots 10
rpa true
$end
Example 4.0 Application of ωB97X functional to nitrogen dimer.
$comment
Frequency calculation (with analytical Hessian methods).
$end
$molecule
0 1
N1
N2 N1 1.1
$end
$rem
jobtype freq
exchange omegaB97X
basis 6-31G*
$end
Among these new LRC hybrid functionals, ωB97X-D is a DFT-D
(density functional theory with empirical dispersion corrections) functional, where
the total energy is computed as the sum of a DFT part and an empirical
atomic-pairwise dispersion correction. Relative to ωB97 and ωB97X,
ωB97X-D is significantly superior for non-bonded interactions,
and very similar in performance for bonded interactions. However, it should be noted
that the remaining short-range self-interaction error is somewhat
larger for ωB97X-D than for ωB97X than for ωB97.
A careful reading of Refs.
is suggested prior to performing any DFT and TDDFT calculations based on variations
of ωB97 functional. ωB97X-D functional automatically involves two keywords for the
dispersion correction, DFT_D and DFT_D_A, which are described in Section 4.3.6.
Example 4.0 Application of ωB97X-D functional to methane dimer.
$comment
Geometry optimization.
$end
$molecule
0 1
C 0.000000 -0.000323 1.755803
H -0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 -1.024959 1.393014
H 0.000000 0.001084 2.842908
C 0.000000 0.000323 -1.755803
H 0.000000 -0.001084 -2.842908
H -0.887097 -0.510784 -1.390695
H 0.887097 -0.510784 -1.390695
H 0.000000 1.024959 -1.393014
$end
$rem
jobtype opt
exchange omegaB97X-D
basis 6-31G*
$end
Similar to the existing double-hybrid density functional theory
(DH-DFT) [39,[114,[115,[116,[94],
which is described in Section 4.3.9,
LRC-DFT can be extended to include non-local orbital correlation energy from second-order
Møller-Plesset perturbation theory (MP2) [117], that includes a same-spin (ss) component
Ecss, and an opposite-spin (os) component Ecos of PT2 correlation energy. The two scaling
parameters, css and cos, are introduced to avoid double-counting
correlation with the LRC hybrid functional.
Etotal = ELRC−DFT + css Ecss + cos Ecos |
| (4.50) |
Among the ωB97 series, ωB97X-2 [40]
is a long-range corrected double-hybrid (DH) functional, which can
greatly reduce the self-interaction errors (due to its
high fraction of Hartree-Fock exchange), and has been shown
significantly superior for systems with bonded and non-bonded interactions. Due to the sensitivity of PT2
correlation energy with respect to the choices of basis sets, ωB97X-2
was parameterized with two different basis sets. ωB97X-2(LP) was parameterized with
the 6-311++G(3df,3pd) basis set (the large Pople type basis set), while
ωB97X-2(TQZ) was parameterized with the TQ extrapolation to the basis set limit.
A careful reading of Ref. is thus highly advised.
ωB97X-2(LP) and ωB97X-2(TQZ) automatically
involve three keywords for the PT2 correlation energy, DH, DH_SS and DH_OS,
which are described in Section 4.3.9. The PT2 correlation
energy can also be computed with the efficient resolution-of-identity
(RI) methods (see Section 5.5).
Example 4.0 Application of ωB97X-2(LP) functional to LiH molecules.
$comment
Geometry optimization and frequency calculation on LiH, followed by
single-point calculations with non-RI and RI approaches.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype opt
exchange omegaB97X-2(LP)
correlation mp2
basis 6-311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype freq
exchange omegaB97X-2(LP)
correlation mp2
basis 6-311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X-2(LP)
correlation mp2
basis 6-311++G(3df,3pd)
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X-2(LP)
correlation rimp2
basis 6-311++G(3df,3pd)
aux_basis rimp2-aug-cc-pvtz
$end
Example 4.0 Application of ωB97X-2(TQZ) functional to LiH molecules.
$comment
Single-point calculations on LiH.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype sp
exchange omegaB97X-2(TQZ)
correlation mp2
basis cc-pvqz
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange omegaB97X-2(TQZ)
correlation rimp2
basis cc-pvqz
aux_basis rimp2-cc-pvqz
$end
4.3.4.4 LRC-DFT with the M11 Family of Functionals
The Minnesota family of functional by Truhlar's group has been recently updated
by adding two new functionals: M11-L [89] and M11 [90].
The M11 functional is a long-range corrected meta-GGA,
obtained by using the LRC scheme of Chai and Head-Gordon (see above),
with the successful parameterization of the Minnesota meta-GGA functionals:
EM11xc = | ⎛ ⎝
|
X
100
| ⎞ ⎠
|
ESR−HFx + | ⎛ ⎝
|
1 − |
X
100
| ⎞ ⎠
|
ESR−M11x + ELR−HFx + EM11c |
| (4.51) |
with the percentage of Hartree-Fock exchange at short range X being 42.8.
An extension of the LRC scheme to local functional (no HF exchange) was
introduced in the M11-L functional by means of the dual-range exchange:
EM11−Lxc = ESR−M11x + ELR−M11x + EM11−Lc |
| (4.52) |
The correct long-range scheme is selected automatically with the input keywords.
A careful reading of the references [89,[90]
is suggested prior to performing any calculations with the M11 functionals.
Example 4.0 Application of M11 functional to water molecule
$comment
Optimization of H2O with M11
$end
$molecule
0 1
O 0.000000 0.000000 0.000000
H 0.000000 0.000000 0.956914
H 0.926363 0.000000 -0.239868
$end
$rem
jobtype opt
exchange m11
basis 6-31+G(d,p)
$end
4.3.5 Nonlocal Correlation Functionals
Q-Chem includes four nonlocal correlation functionals that describe long-range
dispersion (i.e. van der Waals) interactions:
- vdW-DF-04, developed by Langreth, Lundqvist, and coworkers [118,[119]
and implemented as described in Ref. [120];
- vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization [121]
of vdW-DF-04, implemented in the same way as its precursor [120];
- VV09, developed [122] and implemented [123] by Vydrov and Van Voorhis;
- VV10 by Vydrov and Van Voorhis [124].
All these functionals are implemented self-consistently and analytic gradients with respect
to nuclear displacements are available [120,[123,[124]. The nonlocal
correlation is governed by the $rem variable NL_CORRELATION, which can be set to
one of the four values: vdW-DF-04, vdW-DF-10, VV09, or VV10.
Note that vdW-DF-04, vdW-DF-10, and VV09 functionals are used in combination with LSDA correlation,
which must be specified explicitly. For instance, vdW-DF-10 is invoked by the following keyword
combination:
CORRELATION PW92
NL_CORRELATION vdW-DF-10
VV10 is used in combination with PBE correlation, which must be added explicitly.
In addition, the values of two parameters, C and b must be specified for VV10.
These parameters are controlled by the $rem variables NL_VV_C and
NL_VV_B, respectively. For instance, to invoke VV10 with C = 0.0093
and b = 5.9, the following input is used:
CORRELATION PBE
NL_CORRELATION VV10
NL_VV_C 93
NL_VV_B 590
The variable NL_VV_C may also be specified for VV09, where it has the same
meaning. By default, C = 0.0089 is used in VV09 (i.e. NL_VV_C is set to
89). However, in VV10 neither C nor b are assigned a default value and must
always be provided in the input.
As opposed to local (LSDA) and semilocal (GGA and meta-GGA) functionals, evaluated as a single 3D
integral over space [see Eq. (4.37)], non-local functionals require double integration over the spatial
variables:
Ecnl = | ⌠ ⌡
|
f(r,r′) dr dr′. |
| (4.53) |
In practice, this double integration is performed numerically on a quadrature
grid [120,[123,[124].
By default, the SG-1 quadrature (described in Section 4.3.13 below) is used
to evaluate Ecnl, but a different grid can be requested via the $rem variable
NL_GRID. The non-local energy is rather insensitive to the fineness of the grid, so that
SG-1 or even SG-0 grids can be used in most cases. However, a finer grid may be required for the (semi)local
parts of the functional, as controlled by the XC_GRID variable.
Example 4.0 Geometry optimization of the methane dimer using VV10 with rPW86 exchange.
$molecule
0 1
C 0.000000 -0.000140 1.859161
H -0.888551 0.513060 1.494685
H 0.888551 0.513060 1.494685
H 0.000000 -1.026339 1.494868
H 0.000000 0.000089 2.948284
C 0.000000 0.000140 -1.859161
H 0.000000 -0.000089 -2.948284
H -0.888551 -0.513060 -1.494685
H 0.888551 -0.513060 -1.494685
H 0.000000 1.026339 -1.494868
$end
$rem
JobType Opt
BASIS aug-cc-pVTZ
EXCHANGE rPW86
CORRELATION PBE
XC_GRID 75000302
NL_CORRELATION VV10
NL_GRID 1
NL_VV_C 93
NL_VV_B 590
$end
In the above example, an EML-(75,302) grid is used to evaluate the rPW86 exchange and PBE
correlation, but a coarser SG-1 grid is used for the non-local part of VV10.
4.3.6 DFT-D Methods
4.3.6.1 Empirical dispersion correction from Grimme
Thanks to the efforts of the Sherrill group, the popular empirical
dispersion corrections due to Grimme [68] are
now available in Q-Chem. Energies, analytic gradients, and analytic
second derivatives are available. Grimme's empirical dispersion
corrections can be added to any of the density functionals
available in Q-Chem.
DFT-D methods add an extra term,
| |
|
−s6 |
∑
A
|
|
∑
B < A
|
|
C6AB
RAB6
|
fdmp(RAB) |
| | (4.54) |
| |
|
| | (4.55) |
| |
|
| | (4.56) |
|
where s6 is a global scaling parameter (near unity), fdmp is a
damping parameter meant to help avoid double-counting correlation effects at
short range, d is a global scaling parameter for the damping function,
and RAB0 is the sum of the van der Waals radii of atoms A and B.
DFT-D using Grimme's parameters may be turned on using
DFT_D EMPIRICAL_GRIMME
Grimme has suggested scaling factors s6 of 0.75 for PBE, 1.2 for BLYP,
1.05 for BP86, and 1.05 for B3LYP; these are the default values of s6 when
those functionals are used. Otherwise, the default value of s6 is 1.0.
It is possible to specify different values of s6, d, the atomic C6
coefficients, or the van der Waals radii by using the
$empirical_dispersion keyword; for example:
$empirical_dispersion
S6 1.1
D 10.0
C6 Ar 4.60 Ne 0.60
VDW_RADII Ar 1.60 Ne 1.20
$end
Any values not specified explicitly will default to the values in Grimme's
model.
4.3.6.2 Empirical dispersion correction from Chai and Head-Gordon
The empirical dispersion correction from Chai and Head-Gordon [107]
employs a different damping function and can be activated by using
DFT_D EMPIRICAL_CHG
It uses another keyword DFT_D_A to control the strength of dispersion corrections.
DFT_D
Controls the application of DFT-D or DFT-D3 scheme. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the DFT-D or DFT-D3 scheme |
EMPIRICAL_GRIMME | dispersion correction from Grimme |
EMPIRICAL_CHG | dispersion correction from Chai and Head-Gordon |
EMPIRICAL_GRIMME3 | dispersion correction from Grimme's DFT-D3 method |
| (see Section 4.3.8)
|
RECOMMENDATION:
|
| DFT_D_A
Controls the strength of dispersion corrections in the Chai-Head-Gordon DFT-D scheme in Eq.(3) of Ref. . |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a = n/100. |
RECOMMENDATION:
|
|
|
4.3.7 XDM DFT Model of Dispersion
While standard DFT functionals describe chemical bonds relatively
well, one major deficiency is their inability
to cope with dispersion interactions, i.e., van der Waals (vdW) interactions.
Becke and Johnson have proposed a conceptually simple yet accurate
dispersion model called the exchange-dipole model (XDM) [33,[125].
In this model the dispersion attraction emerges from the interaction between the instant
dipole moment of the exchange hole in one molecule and the induced
dipole moment in another. It is a conceptually simple but powerful approach
that has been shown to yield very accurate dispersion coefficients without
fitting parameters. This allows the calculation of both intermolecular
and intramolecular dispersion interactions within a single DFT framework.
The implementation and validation of this method in the Q-Chem code
is described in Ref. .
Fundamental to the XDM model is the calculation of the norm of the dipole
moment of the exchange hole at a given point:
dσ(r)=− | ⌠ ⌡
|
hσ(r,r′)r′d3r′−r |
| (4.57) |
where σ labels the spin and hσ(r,r′) is
the exchange-hole function. The XDM version that is implemented
in Q-Chem employs the Becke-Roussel (BR) model exchange-hole function.
It was not given in an analytical form and one had to determine
its value at each grid point numerically.
Q-Chem has developed for the first time an analytical
expression for this function based on non-linear interpolation and spline
techniques, which greatly improves efficiency as well as the numerical
stability [28].
There are two different damping functions used in the XDM model of
Becke and Johnson. One of them uses only the intermolecular C6 dispersion
coefficient. In its Q-Chem implementation it is denoted as "XDM6". In this version
the dispersion energy is computed as
EvdW= |
∑
| EvdW,ij=− |
∑
i > j
|
|
C6,ij
Rij6+kC6,ij/(EiC+EjC)
|
|
| (4.58) |
where k is a universal parameter, Rij is the distance
between atoms i and j, and EijC is the sum of the absolute values of
the correlation energy of free atoms i and j. The dispersion
coefficients C6,ij is computed as
C6,ij= |
〈dX2〉i〈dX2〉jαiαj
〈dX2〉iαj+〈dX2〉jαi
|
|
| (4.59) |
where 〈dX2〉i is the exchange hole dipole moment of
the atom, and αi is the effective polarizability of the atom i in
the molecule.
The XDM6 scheme is further generalized to include higher-order dispersion
coefficients, which leads to the "XDM10" model in Q-Chem implementation. The
dispersion energy damping function used in XDM10 is
EvdW=− |
∑
i > j
|
| ⎛ ⎝
|
C6,ij
RvdW,ij6+Rij6
|
+ |
C8,ij
RvdW,ij8+Rij8
|
+ |
C10,ij
RvdW,ij10+Rij10
| ⎞ ⎠
|
|
| (4.60) |
where C6,ij, C8,ij and C10,ij are dispersion coefficients
computed at higher-order multipole (including dipole, quadrupole and octopole)
moments of the exchange hole [127]. In above,
RvdW,ij is the sum of the effective
vdW radii of atoms i and j, which is a linear function of the
so called critical distance RC,ij between atoms i and j:
The critical distance, RC,ij, is computed by averaging these three distances:
RC,ij = |
1
3
|
| ⎡ ⎣
| ⎛ ⎝
|
C8,ij
C6,ij
| ⎞ ⎠
|
1/2
|
+ | ⎛ ⎝
|
C10,ij
C6,ij
| ⎞ ⎠
|
1/4
|
+ | ⎛ ⎝
|
C10,ij
C8,ij
| ⎞ ⎠
|
1/2
| ⎤ ⎦
|
|
| (4.62) |
In the XDM10 scheme there are two universal parameters, a1 and a2. Their
default values of 0.83 and 1.35, respectively, are
due to Johnson and Becke [125], determined by
least square fitting to the binding energies of a set
of intermolecular complexes. Please keep in mind that these values are not
the only possible optimal set to use with XDM.
Becke's group has suggested later on several
other XC functional combinations with XDM
that employ different a1 and a2
values. The user is advised to consult their recent papers for more details
(e.g., Refs. ).
The computed vdW energy is added as a post-SCF correction. In addition, Q-Chem
also has implemented the first and second nuclear derivatives of vdW energy
correction in both the XDM6 and XDM10 schemes.
Listed below are a number of useful options to customize the vdW calculation
based on the XDM DFT approach.
DFTVDW_JOBNUMBER
TYPE:
DEFAULT:
OPTIONS:
0 | Do not apply the XDM scheme. |
1 | add vdW gradient correction to SCF. |
2 | add VDW as a DFT functional and do full SCF. |
RECOMMENDATION:
This option only works with C6 XDM formula |
|
| DFTVDW_METHOD
Choose the damping function used in XDM |
TYPE:
DEFAULT:
OPTIONS:
1 | use Becke's damping function including C6 term only. |
2 | use Becke's damping function with higher-order (C8,C10) terms. |
RECOMMENDATION:
|
|
|
DFTVDW_MOL1NATOMS
The number of atoms in the first monomer in dimer calculation |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| DFTVDW_KAI
Damping factor K for C6 only damping function |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DFTVDW_ALPHA1
Parameter in XDM calculation with higher-order terms |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| DFTVDW_ALPHA2
Parameter in XDM calculation with higher-order terms. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DFTVDW_USE_ELE_DRV
Specify whether to add the gradient correction to the XDM energy.
only valid with Becke's C6 damping function
using the interpolated BR89 model. |
TYPE:
DEFAULT:
OPTIONS:
1 | use density correction when applicable (default). |
0 | do not use this correction (for debugging purpose) |
RECOMMENDATION:
|
| DFTVDW_PRINT
Printing control for VDW code |
TYPE:
DEFAULT:
OPTIONS:
0 no printing. |
1 | minimum printing (default) |
2 | debug printing |
RECOMMENDATION:
|
|
|
Example 4.0 Below is a sample input
illustrating a frequency calculation
of a vdW complex consisted of He atom and N2 molecule.
$molecule
0 1
He .0 .0 3.8
N .000000 .000000 0.546986
N .000000 .000000 -0.546986
$end
$rem
JOBTYPE FREQ
IDERIV 2
EXCHANGE B3LYP
!default SCF setting
INCDFT 0
SCF_CONVERGENCE 8
BASIS 6-31G*
XC_GRID 1
SCF_GUESS SAD
!vdw parameters setting
DFTVDW_JOBNUMBER 1
DFTVDW_METHOD 1
DFTVDW_PRINT 0
DFTVDW_KAI 800
DFTVDW_USE_ELE_DRV 0
$end
One should note that the XDM option can be used in
conjunction with different GGA, meta-GGA pure or hybrid functionals,
even though the original implementation of Becke and Johnson
was in combination with Hartree-Fock exchange, or with a specific
meta-GGA exchange and correlation (the BR89 exchange and the BR94
correlation described in previous sections above). For example,
encouraging results were obtained using the XDM option with the
popular B3LYP [126]. Becke has found more recently
that this model can be efficiently combined with the old GGA
exchange of Perdew 86 (the P86 exchange option in Q-Chem), and
with his hyper-GGA functional B05. Using XDM together
with PBE exchange plus LYP correlation, or PBE exchange plus
BR94 correlation has been also found fruitful.
4.3.8 DFT-D3 Methods
Recently, Grimme proposed DFT-D3 method [130] to
improve his previous DFT-D method [68] (see Section 4.3.6).
Energies and analytic gradients of DFT-D3 methods are available in Q-Chem.
Grimme's DFT-D3 method can be combined with any of the
density functionals available in Q-Chem.
The total DFT-D3 energy is given by
EDFT−D3 = EKS−DFT + Edisp |
| (4.63) |
where EKS-DFT is the total energy from KS-DFT and Edisp is the dispersion correction
as a sum of two- and three-body energies,
DFT-D3 method can be turned on by five keywords, DFT_D, DFT_D3_S6, DFT_D3_RS6, DFT_D3_S8 and DFT_D3_3BODY.
DFT_D
Controls the application of DFT-D3 or DFT-D scheme. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the DFT-D3 or DFT-D scheme |
EMPIRICAL_GRIMME3 | dispersion correction from Grimme's DFT-D3 method |
EMPIRICAL_GRIMME | dispersion correction from Grimme (see Section 4.3.6) |
EMPIRICAL_CHG | dispersion correction from Chai and Head-Gordon (see Section 4.3.6)
|
RECOMMENDATION:
|
Grimme suggested three scaling factors s6, sr,6 and s8 that were optimized for several functionals
(see Table IV in Ref. ).
For example, sr,6 of 1.217 and s8 of 0.722 for PBE, 1.094 and 1.682 for BLYP, 1.261 and 1.703 for B3LYP, 1.532 and 0.862 for PW6B95,
0.892 and 0.909 for BECKE97, and 1.287 and 0.928 for PBE0; these are the Q-Chem
default values of sr,6 and s8. Otherwise, the default values
of s6, sr,6 and s8 are 1.0.
DFT_D3_S6
Controls the strength of dispersion corrections, s6, in Grimme's DFT-D3 method (see Table IV in
Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to s6 = n/1000. |
RECOMMENDATION:
|
| DFT_D3_RS6
Controls the strength of dispersion corrections, sr6, in the Grimme's DFT-D3 method (see Table IV in Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to sr6 = n/1000. |
RECOMMENDATION:
|
|
|
DFT_D3_S8
Controls the strength of dispersion corrections, s8, in Grimme's DFT-D3 method (see Table IV in
Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to s8 = n/1000. |
RECOMMENDATION:
|
The three-body interaction term, mentioned in Ref. , can also be turned on, if needed.
DFT_D3_3BODY
Controls whether the three-body interaction in Grimme's DFT-D3 method should be applied
(see Eq. (14) in Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the three-body interaction term |
TRUE | Apply the three-body interaction term
|
RECOMMENDATION:
|
Example 4.0 Applications of B3LYP-D3 to a methane dimer.
$comment
Geometry optimization, followed by single-point calculations
using a larger basis set.
$end
$molecule
0 1
C 0.000000 -0.000323 1.755803
H -0.887097 0.510784 1.390695
H 0.887097 0.510784 1.390695
H 0.000000 -1.024959 1.393014
H 0.000000 0.001084 2.842908
C 0.000000 0.000323 -1.755803
H 0.000000 -0.001084 -2.842908
H -0.887097 -0.510784 -1.390695
H 0.887097 -0.510784 -1.390695
H 0.000000 1.024959 -1.393014
$end
$rem
jobtype opt
exchange B3LYP
basis 6-31G*
DFT_D EMPIRICAL_GRIMME3
DFT_D3_S6 1000
DFT_D3_RS6 1261
DFT_D3_S8 1703
DFT_D3_3BODY FALSE
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange B3LYP
basis 6-311++G**
DFT_D EMPIRICAL_GRIMME3
DFT_D3_S6 1000
DFT_D3_RS6 1261
DFT_D3_S8 1703
DFT_D3_3BODY FALSE
$end
4.3.9 Double-Hybrid Density Functional Theory
The recent advance in double-hybrid density functional theory
(DH-DFT) [39,[114,[115,[116,[94],
has demonstrated its great potential for approaching the chemical accuracy with a
computational cost comparable to the second-order Møller-Plesset perturbation theory (MP2).
In a DH-DFT, a Kohn-Sham (KS) DFT
calculation is performed first, followed by a treatment of
non-local orbital correlation energy at the level of second-order Møller-Plesset perturbation
theory (MP2) [117]. This MP2 correlation correction includes a
a same-spin (ss) component, Ecss, as well as an
opposite-spin (os) component, Ecos, which are added to the total energy obtained from
the KS-DFT calculation. Two scaling parameters, css and cos, are introduced
in order to avoid double-counting correlation:
EDH−DFT = EKS−DFT + css Ecss + cos Ecos |
| (4.65) |
Among DH functionals, ωB97X-2 [40], a long-range corrected
DH functional, is described in Section 4.3.4.3.
There are three keywords for turning on DH-DFT as below.
DH
Controls the application of DH-DFT scheme. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the DH-DFT scheme |
TRUE | (or 1) Apply DH-DFT scheme |
RECOMMENDATION:
|
| DH_SS
Controls the strength of the same-spin component of PT2 correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to css = n/1000000 in Eq. (4.65). |
RECOMMENDATION:
|
|
|
DH_OS
Controls the strength of the opposite-spin component of PT2 correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to cos = n/1000000 in Eq. (4.65). |
RECOMMENDATION:
|
For example, B2PLYP [68], which involves 53% Hartree-Fock exchange,
47% Becke 88 GGA exchange, 73% LYP GGA correlation and 27% PT2
orbital correlation, can be called with the following input file.
The PT2 correlation energy can also be computed with the
efficient resolution-of-identity (RI) methods
(see Section 5.5).
Example 4.0 Applications of B2PLYP functional to LiH molecule.
$comment
Geometry optimization and frequency calculation on LiH, followed by
single-point calculations with non-RI and RI approaches.
$end
$molecule
0 1
H
Li H 1.6
$end
$rem
jobtype opt
exchange general
correlation mp2
basis cc-pvtz
DH 1
DH_SS 270000 !0.27 = 270000/1000000
DH_OS 270000 !0.27 = 270000/1000000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype freq
exchange general
correlation mp2
basis cc-pvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange general
correlation mp2
basis cc-pvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
@@@
$molecule
READ
$end
$rem
jobtype sp
exchange general
correlation rimp2
basis cc-pvtz
aux_basis rimp2-cc-pvtz
DH 1
DH_SS 270000
DH_OS 270000
$end
$XC_Functional
X HF 0.53
X B 0.47
C LYP 0.73
$end
A more detailed gist of one particular class of DH functionals, the
XYG3 & XYGJ-OS functionals follows below thanks to Dr Yousung Jung
who implemented these functionals in Q-Chem.
A starting point of these DH functionals is the adiabatic connection formula
which provides a rigorous way to define them. One considers an adiabatic
path between the fictitious noninteracting Kohn-Sham system (λ= 0)
and the real physical system (λ= 1) while holding the electron
density fixed at its physical state for all λ:
EXC [ρ]= | ⌠ ⌡
|
1
0
|
UXC,λ [ρ]dλ , |
| (4.66) |
where UXC,λ is the exchange correlation potential energy at
a coupling strength λ. If one assumes a linear model of the latter:
one obtains the popular hybrid functional that includes the Hartree-Fock exchange
(or occupied orbitals) such as B3LYP. If one further uses the Gorling-Levy's
perturbation theory (GL2) to define the initial slope at λ= 0,
one obtains the doubly hybrid functional (see Eq. 4.65) that includes
MP2 type perturbative terms (PT2) involving virtual Kohn-Sham orbitals:
UXC,λ = |
∂UXC,λ
λ
| ⎢ ⎢
|
λ = 0
|
=2ECGL2 . |
| (4.68) |
In the DH functional XYG3, as implemented in Q-Chem, the B3LYP orbitals
are used to generate the density and evaluate the PT2 terms. This is different
from P2PLYP, an earlier doubly hybrid functional by Grimme. P2PLYP
uses truncated Kohn-Sham orbitals while XYG3 uses converged KS orbitals
to evaluate the PT2 terms. The performance of XYG3 is not only comparable
to that of the G3 or G2 theory for thermochemistry, but barrier heights
and non-covalent interactions, including stacking interactions,
are also very well described by XYG3 [94].
The recommended basis set for XYG3 is 6-311+G(3df,2p).
Due to the inclusion of PT2 terms, XYG3 or all other forms of doubly
hybrid functionals formally scale as the 5th power of system size as
in conventional MP2, thereby not applicable to large systems and
partly losing DFT's cost advantages. With the success of SOS-MP2 and
local SOS-MP2 algorithms developed in Q-Chem, the natural extension
of XYG3 is to include only opposite-spin correlation contributions,
ignoring the same-spin parts. The resulting DH functional is XYGJ-OS
also implemented in Q-Chem. It has 4 parameters that are optimized
using thermochemistry data. This new functional is both accurate
(comparable or even slightly better than XYG3) and faster. If the local
algorithm is applied, the formal scaling of XYGJ-OS is cubic, without
the locality, it has still 4th order scaling.
Recently, XYGJ-OS becomes the only DH functional with analytical gradient [131].
Example 1: XYG3 calculation of N2. XYG3 invokes automatically
the B3LYP calculation first, and use the resulting orbitals
for evaluating the MP2-type correction terms. One can also
use XYG3 combined with RI approximation for the PT2 terms; use
EXCHANGE = XYG3RI to do so, along with an appropriate
choice of auxiliary basis set.
Example 4.0 XYG3 calculation of N2
$molecule
0 1
N 0.00000000 0.00000000 0.54777500
N 0.00000000 0.00000000 -0.54777500
$end
$rem
exchange xyg3
basis 6-311+G(3df,2p)
$end
Example 2: XYGJ-OS calculation of N2. Since it uses
the RI approximation by default, one must define the auxiliary basis.
Example 4.0 XYGJ-OS calculation of N2
$molecule
0 1
N 0.00000000 0.00000000 0.54777500
N 0.00000000 0.00000000 -0.54777500
$end
$rem
exchange xygjos
basis 6-311+G(3df,2p)
aux_basis rimp2-cc-pVtZ
purecart 1111
time_mp2 true
$end
Example 3: Local XYGJ-OS calculation of N2. The same as XYGJ-OS,
except for the use of the attenuated Coulomb metric to solve
the RI coefficients. Omega determines the locality of the metric.
Example 4.0 Local XYGJ-OS calculation of N2
$molecule
0 1
N 0.000 0.000 0.54777500
N 0.000 0.000 -0.54777500
$end
$rem
exchange lxygjos
omega 200
basis 6-311+G(3df,2p)
aux_basis rimp2-cc-pVtZ
purecart 1111
$end
4.3.10 Asymptotically Corrected Exchange-Correlation Potentials
It is well known that no gradient-corrected exchange functional can simultaneously produce the correct
contribution to the exchange energy density and exchange potential in the
asymptotic region of molecular systems [132]. Existing GGA exchange-correlation
(xc) potentials decay much faster than the correct −1/r xc potential in the asymptotic
region [133]. High-lying occupied orbitals and low-lying virtual orbitals are therefore
too loosely bounded from these GGA functionals, and the minus HOMO energy becomes
much less than the exact ionization potential (as required by the exact
DFT) [134,[135].
Moreover, response properties could be poorly predicted from TDDFT calculations
with GGA functionals [135]. Long-range corrected hybrid DFT (LRC-DFT), described in
Section 4.3.4, has greatly remedied this situation. However, due to the
use of long-range HF exchange, LRC-DFT is computationally more expensive than KS-DFT with GGA functionals.
To circumvent this, van Leeuwen and Baerends proposed an asymptotically corrected (AC) exchange potential [132]:
vxLB = −β |
x2
1+3 βsinh−1(x)
|
|
| (4.69) |
that will reduce to −1/r, for an exponentially decaying density, in the
asymptotic region of molecular systems, where x = [( |∇ρ|)/(ρ4/3)] is the reduced
density gradient. The LB94 xc potential is formed by a linear combination of LDA
xc potential and the LB exchange potential [132]:
The parameter β was determined by fitting the LB94 xc potential to the beryllium atom.
As mentioned in Ref. , for TDDFT and TDDFT/TDA calculations,
it is sufficient to include the AC xc potential for ground-state calculations followed by TDDFT
calculations with an adiabatic LDA xc kernel. The implementation of LB94 xc potential
in Q-Chem thus follows this; using LB94 xc potential for ground-state calculations, followed
by TDDFT calculations with an adiabatic LDA xc kernel. This TDLDA / LB94 approach
has been widely applied to study excited-state properties of large molecules in literature.
Since the LB exchange potential does not come from the functional derivative of some exchange
functional, we use the Levy-Perdew virial relation [138] (implemented in Q-Chem)
to obtain its exchange energy:
ExLB = − | ⌠ ⌡
|
vxLB([ρ],r)[3ρ(r)+r∇ρ(r)]dr |
| (4.71) |
LB94_BETA
Set the β parameter of LB94 xc potential |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to β = n/10000. |
RECOMMENDATION:
Use default, i.e., β = 0.05 |
|
Example 4.0 Applications of LB94 xc potential to N2 molecule.
$comment
TDLDA/LB94 calculation is performed for excitation energies.
$end
$molecule
0 1
N 0.0000 0.0000 0.0000
N 1.0977 0.0000 0.0000
$end
$rem
jobtype = sp
exchange = lb94
basis = 6-311(2+,2+)G**
cis_n_roots = 30
rpa = true
$end
4.3.11 DFT Numerical Quadrature
In practical DFT calculations, the forms of the approximate
exchange-correlation functionals used are quite complicated, such that the
required integrals involving the functionals generally cannot be evaluated
analytically. Q-Chem evaluates these integrals through numerical quadrature
directly applied to the exchange-correlation integrand (i.e., no fitting of
the XC potential in an auxiliary basis is done). Q-Chem provides a standard
quadrature grid by default which is sufficient for most purposes.
The quadrature approach in Q-Chem is generally similar to that found in many
DFT programs. The multi-center XC integrals are first partitioned into
"atomic" contributions using a nuclear weight function. Q-Chem uses the
nuclear partitioning of Becke [139], though without the atomic
size adjustments". The atomic integrals are then evaluated through standard
one-center numerical techniques.
Thus, the exchange-correlation energy EXC is obtained as
EXC = |
∑
A
|
|
∑
i
|
wAi f( rAi ) |
| (4.72) |
where the first summation is over the atoms and the second is over the
numerical quadrature grid points for the current atom. The f function is the
exchange-correlation functional. The wAi are the quadrature weights, and
the grid points rAi are given by
where RA is the position of nucleus A, with the
ri defining a suitable one-center integration grid, which is
independent of the nuclear configuration.
The single-center integrations are further separated into radial and angular
integrations. Within Q-Chem, the radial part is usually treated by the
Euler-Maclaurin scheme proposed by Murry et al. [140]. This scheme
maps the semi-infinite domain [0,∞)→ [0,1) and applies the extended
trapezoidal rule to the transformed integrand. Recently Gill and
Chien [141] proposed a radial scheme based on a Gaussian quadrature on
the interval [0,1] with weight function ln2x. This scheme is exact for
integrands that are a linear combination of a geometric sequence of exponential
functions, and is therefore well suited to evaluating atomic integrals. The
authors refer to this scheme as MultiExp.
4.3.12 Angular Grids
Angular quadrature rules may be characterized by their degree, which is the
highest degree of spherical harmonics for which the formula is exact, and
their efficiency, which is the number of spherical harmonics exactly
integrated per degree of freedom in the formula. Q-Chem supports the
following types of angular grids:
Lebedev
These are specially constructed grids for quadrature on the surface of a
sphere [142,[143,[144] based on the octahedral group.
Lebedev grids of the following degrees are available:
Degree | 3rd | 5th | 7th | 9th | 11th | 15th | 17th | 19th | 23rd | 29th |
Points | 6 | 18 | 26 | 38 | 50 | 86 | 110 | 146 | 194 | 302 |
Additional grids with the following number of points are also available: 74,
170, 230, 266, 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702,
3074, 3470, 3890, 4334, 4802, 5294. Lebedev grids typically have
efficiencies near one, with efficiencies greater than one in some cases.
Gauss-Legendre
These are spherical product rules separating the two angular dimensions
θ and ϕ. Integration in the θ dimension is carried out
with a Gaussian quadrature rule derived from the Legendre polynomials
(orthogonal on [−1,1] with weight function unity), while the ϕ
integration is done with equally spaced points.
A Gauss-Legendre grid is selected by specifying the total number of points,
2N2, to be used for the integration. This gives a grid with
2Nϕ ϕ-points, Nθ θ-points, and a degree of 2N−1.
In contrast with Lebedev grids, Gauss-Legendre grids have efficiency of only
2/3 (hence more Gauss-Legendre points are required to attain the same
accuracy as Lebedev). However, since Gauss-Legendre grids of general degree
are available, this is a convenient mechanism for achieving arbitrary
accuracy in the angular integration if desired.
Combining these radial and angular schemes yields an intimidating selection
of three-dimensional quadratures. In practice, is it useful to standardize
the grids used in order to facilitate the comparison of calculations at
different levels of theory.
4.3.13 Standard Quadrature Grids
Both the SG-0 [145] and SG-1 [146] standard quadrature
grids were designed to yield the performance of a large, accurate quadrature
grid, but with as few points as possible for the sake of computational
efficiency. This is accomplished by reducing the number of angular points in
regions where sophisticated angular quadrature is not necessary, such as near
the nuclei where the charge density is nearly spherically symmetric, while
retaining large numbers of angular points in the valence region where angular
accuracy is critical.
The SG-0 grid was derived in this fashion from a MultiExp-Lebedev-(23,170),
(i.e., 23 radial points and 170 angular points per radial point). This grid
was pruned whilst ensuring the error in the computed exchange energies for the
atoms and a selection of small molecules was not larger than the corresponding
error associated with SG-1. In our evaluation, the RMS error associated with
the atomization energies for the molecules in the G1 data set is 72
microhartrees. While relative energies are expected to be reproduced well by
this scheme, if absolute energies are being sought, a larger grid is
recommended.
The SG-0 grid is implemented in Q-Chem from H to micro Hartrees, excepted He and Na; in
this scheme, each atom has around 1400-point, and SG-1 is used for those
their SG-0 grids have not been defined. It should be noted that, since
the SG-0 grid used for H has been re-optimized in this version of Q-Chem
(version 3.0), quantities calculated in this scheme may not reproduce those
generated by the last version (version 2.1).
The SG-1 grid is derived from a Euler-Maclaurin-Lebedev-(50,194) grid
(i.e., 50 radial points, and 194 angular points per radial point). This grid
has been found to give numerical integration errors of the order of 0.2 kcal/mol
for medium-sized molecules, including particularly demanding test
cases such as isomerization energies of alkanes. This error is deemed
acceptable since it is significantly smaller than the accuracy typically
achieved by quantum chemical methods. In SG-1 the total number of points is
reduced to approximately 1/4 of that of the original EML-(50,194) grid, with
SG-1 generally giving the same total energies as EML-(50,194) to within a few
microhartrees (0.01 kcal/mol). Therefore, the SG-1 grid is relatively
efficient while still maintaining the numerical accuracy necessary for chemical
reliability in the majority of applications.
Both the SG-0 and SG-1 grids were optimized so that the error in the energy
when using the grid did not exceed a target threshold. For single point
calculations this criterion is appropriate. However, derivatives of the energy
can be more sensitive to the quality of the integration grid, and it is
recommended that a larger grid be used when calculating these. Special care
is required when performing DFT vibrational calculations as imaginary
frequencies can be reported if the grid is inadequate. This is more of a
problem with low-frequency vibrations. If imaginary frequencies are found, or
if there is some doubt about the frequencies reported by Q-Chem, the
recommended procedure is to perform the calculation again with a larger grid
and check for convergence of the frequencies. Of course the geometry must be
re-optimized, but if the existing geometry is used as an initial guess, the
geometry optimization should converge in only a few cycles.
4.3.14 Consistency Check and Cutoffs for Numerical Integration
Whenever Q-Chem calculates numerical density functional integrals, the
electron density itself is also integrated numerically as a test on the
quality of the quadrature formula used. The deviation of the numerical
result from the number of electrons in the system is an indication of the
accuracy of the other numerical integrals. If the relative error in the
numerical electron count reaches 0.01%, a warning is printed; this is an
indication that the numerical XC results may not be reliable. If the warning
appears at the first SCF cycle, it is probably not serious, because the
initial-guess density matrix is sometimes not idempotent, as is the case
with the SAD guess and the density matrix taken from a different geometry in
a geometry optimization. If that is the case, the problem will be corrected
as the idempotency is restored in later cycles. On the other hand, if the
warning is persistent to the end of SCF iterations, then either a finer grid
is needed, or choose an alternative method for generating the initial guess.
Users should be aware, however, of the potential flaws that have been
discovered in some of the grids currently in use. Jarecki and Davidson [147],
for example, have recently shown that correctly
integrating the density is a necessary, but not sufficient, test of grid
quality.
By default, Q-Chem will estimate the magnitude of various XC contributions on
the grid and eliminate those determined to be numerically insignificant.
Q-Chem uses specially developed cutoff procedures which permits evaluation of
the XC energy and potential in only O(N) work for large molecules, where N
is the size of the system. This is a significant improvement over the formal
O(N3) scaling of the XC cost, and is critical in enabling DFT calculations
to be carried out on very large systems. In very rare cases, however, the
default cutoff scheme can be too aggressive, eliminating contributions that
should be retained; this is almost always signaled by an inaccurate numerical
density integral. An example of when this could occur is in calculating anions
with multiple sets of diffuse functions in the basis. As mentioned above, when
an inaccurate electron count is obtained, it maybe possible to remedy the
problem by increasing the size of the quadrature grid.
Finally we note that early implementations of quadrature-based Kohn-Sham DFT
employing standard basis sets were plagued by lack of rotational invariance.
That is, rotation of the system yielded a significantly energy change.
Clearly, such behavior is highly undesirable. Johnson et al. rectified the
problem of rotational invariance by completing the specification of the grid
procedure [148] to ensure that the computed XC energy is the same
for any orientation of the molecule in any Cartesian coordinate system.
4.3.15 Basic DFT Job Control
Three $rem variables are required to run a DFT job: EXCHANGE,
CORRELATION and BASIS. In addition, all of the basic input
options discussed for Hartree-Fock calculations in Section
4.2.3, and the extended options discussed in Section
4.2.4 are all valid for DFT calculations. Below we list
only the basic DFT-specific options (keywords).
| EXCHANGE
Specifies the exchange functional or exchange-correlation functional for hybrid. |
TYPE:
DEFAULT:
No default exchange functional |
OPTIONS:
NAME | Use EXCHANGE = NAME, where NAME is |
| one of the exchange functionals listed in Table 4.2. |
RECOMMENDATION:
Consult the literature to guide your selection. |
|
|
|
CORRELATION
Specifies the correlation functional. |
TYPE:
DEFAULT:
OPTIONS:
None | No correlation |
VWN | Vosko-Wilk-Nusair parameterization #5 |
LYP | Lee-Yang-Parr (LYP) |
PW91, PW | GGA91 (Perdew-Wang) |
PW92 | LSDA 92 (Perdew and Wang) [44] |
PK09 | LSDA (Proynov-Kong) [45] |
LYP(EDF1) | LYP(EDF1) parameterization |
Perdew86, P86 | Perdew 1986 |
PZ81, PZ | Perdew-Zunger 1981 |
PBE | Perdew-Burke-Ernzerhof 1996 |
TPSS | The correlation component of the TPSS functional |
B94 | Becke 1994 correlation in fully analytic form |
B95 | Becke 1995 correlation |
B94hyb | Becke 1994 correlation as above, but re-adjusted for use only within |
| the hybrid scheme BR89B94hyb |
PK06 | Proynov-Kong 2006 correlation (known also as "tLap" |
(B88)OP | OP correlation [74], optimized for use with B88 exchange |
(PBE)OP | OP correlation [74], optimized for use with PBE exchange |
Wigner | Wigner |
RECOMMENDATION:
Consult the literature to guide your selection. |
|
|
EXCHANGE = | Description | |
|
HF | Fock exchange |
Slater, S | Slater (Dirac 1930) |
Becke86, B86 | Becke 1986 |
Becke, B, B88 | Becke 1988 |
muB88 | Short-range Becke exchange, as formulated by Song et al. [105] |
Gill96, Gill | Gill 1996 |
GG99 | Gilbert and Gill, 1999 |
Becke(EDF1), B(EDF1) | Becke (uses EDF1 parameters) |
PW86, | Perdew-Wang 1986 |
rPW86, | Re-fitted PW86 [49] |
PW91, PW | Perdew-Wang 1991 |
mPW1PW91 | modified PW91 |
mPW1LYP | modified PW91 exchange + LYP correlation |
mPW1PBE | modified PW91 exchange + PBE correlation |
mPW1B95 | modified PW91 exchange + B97 correlation |
mPWB1K | mPWB1K |
PBE | Perdew-Burke-Ernzerhof 1996 |
TPSS | The nonempirical exchange-correlation scheme of Tao, |
| Perdew, Staroverov, and Scuseria (requires also that the user |
| specify "TPSS" for correlation) |
TPSSH | The hybrid version of TPSS (with no input line for correlation) |
PBE0, PBE1PBE | PBE hybrid with 25% HF exchange |
PBE50 | PBE hybrid with 50% HF exchange |
revPBE | revised PBE exchange [58] |
PBEOP | PBE exchange + one-parameter progressive correlation |
wPBE | Short-range ωPBE exchange, as formulated by
Henderson et al. [106] |
muPBE | Short-range μPBE exchange, as formulated by
Song et al. [105] |
LRC-wPBEPBE | long-range corrected pure PBE |
LRC-wPBEhPBE | long-range corrected hybrid PBE |
B1B95 | Becke hybrid functional with B97 correlation |
B97 | Becke97 XC hybrid |
B97-1 | Becke97 re-optimized by Hamprecht et al. (1998) |
B97-2 | Becke97-1 optimized further by Wilson et al. (2001) |
B97-D | Grimme's modified B97 with empirical dispersion |
B3PW91, Becke3PW91, B3P | B3PW91 hybrid |
B3LYP, Becke3LYP | B3LYP hybrid |
B3LYP5 | B3LYP based on correlation functional #5 of Vosko, Wilk, |
| and Nusair (rather than their functional #3) |
CAM-B3LYP | Coulomb-attenuated B3LYP |
HC-O3LYP | O3LYP from Handy |
X3LYP | X3LYP from Xu and Goddard |
B5050LYP | modified B3LYP functional with 50% Hartree-Fock exchange |
BHHLYP | modified BLYP functional with 50% Hartree-Fock exchange |
B3P86 | B3 exchange and Perdew 86 correlation |
B3PW91 | B3 exchange and GGA91 correlation |
B3tLAP | Hybrid Becke exchange and PK06 correlation |
HCTH | HCTH hybrid |
HCTH-120 | HCTH-120 hybrid |
HCTH-147 | HCTH-147 hybrid |
HCTH-407 | HCTH-407 hybrid |
BOP | B88 exchange + one-parameter progressive correlation |
EDF1 | EDF1 |
EDF2 | EDF2 |
VSXC | VSXC meta-GGA, not a hybrid |
BMK | BMK hybrid |
M05 | M05 hybrid |
M052X | M05-2X hybrid |
M06L | M06-L hybrid |
M06HF | M06-HF hybrid |
M06 | M06 hybrid |
M062X | M06-2X hybrid |
M08HX | M08-HX hybrid |
M08SO | M08-SO hybrid |
M11L | M11-L hybrid |
M11 | M11 long-range corrected hybrid |
SOGGA | SOGGA hybrid |
SOGGA11 | SOGGA11 hybrid |
SOGGA11X | SOGGA11-X hybrid |
BR89 | Becke-Roussel 1989 represented in analytic form |
BR89B94h | Hybrid BR89 exchange and B94hyb correlation |
omegaB97 | ωB97 long-range corrected hybrid |
omegaB97X | ωB97X long-range corrected hybrid |
omegaB97X-D | ωB97X-D long-range corrected hybrid with dispersion |
| corrections |
omegaB97X-2(LP) | ωB97X-2(LP) long-range corrected double-hybrid |
omegaB97X-2(TQZ) | ωB97X-2(TQZ) long-range corrected double-hybrid |
MCY2 | The MCY2 hyper-GGA exchange-correlation (with no |
| input line for correlation) |
B05 | Full exact-exchange hyper-GGA functional of Becke 05 with |
| RI approximation for the exact-exchange energy density |
BM05 | Modified B05 hyper-GGA scheme with RI approximation for |
| the exact-exchange energy density used as a variable. |
XYG3 | XYG3 double-hybrid functional |
XYGJOS | XYGJ-OS double-hybrid functional |
LXYGJOS | LXYGJ-OS double-hybrid functional with localized MP2 |
General, Gen | User defined combination of K, X and C (refer to the next |
| section) |
|
Table 4.2: DFT exchange functionals available within Q-Chem. |
|
NL_CORRELATION
Specifies a non-local correlation functional that includes non-empirical dispersion. |
TYPE:
DEFAULT:
None | No non-local correlation. |
OPTIONS:
None | No non-local correlation |
vdW-DF-04 | the non-local part of vdW-DF-04 |
vdW-DF-10 | the nonlocal part of vdW-DF-10 (aka vdW-DF2) |
VV09 | the nonlocal part of VV09 |
VV10 | the nonlocal part of VV10 |
RECOMMENDATION:
Do not forget to add the LSDA correlation (PW92 is recommended) when
using vdW-DF-04, vdW-DF-10, or VV09. VV10 should be used with PBE
correlation. Choose exchange functionals carefully: HF, rPW86, revPBE, and
some of the LRC exchange functionals are among the recommended choices. |
|
| NL_VV_C
Sets the parameter C in VV09 and VV10. This parameter is fitted to asymptotic
van der Waals C6 coefficients. |
TYPE:
DEFAULT:
89 | for VV09 |
No default | for VV10 |
OPTIONS:
n | Corresponding to C = n/10000 |
RECOMMENDATION:
C = 0.0093 is recommended when a semilocal exchange functional is used.
C = 0.0089 is recommended when a long-range corrected (LRC) hybrid functional is used.
See further details in Ref. [124]. |
|
|
|
NL_VV_B
Sets the parameter b in VV10. This parameter controls the short range behavior
of the nonlocal correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to b = n/100 |
RECOMMENDATION:
The optimal value depends strongly on the exchange functional used.
b = 5.9 is recommended for rPW86. See further details in Ref. [124]. |
|
| FAST_XC
Controls direct variable thresholds to accelerate exchange correlation (XC) in
DFT. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Turn FAST_XC on. |
FALSE | Do not use FAST_XC. |
RECOMMENDATION:
Caution: FAST_XC improves the speed of a DFT calculation, but
may occasionally cause the SCF calculation to diverge. |
|
|
|
XC_GRID
Specifies the type of grid to use for DFT calculations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use SG-0 for H, C, N, and O, SG-1 for all other atoms. |
1 | Use SG-1 for all atoms. |
2 | Low Quality. |
mn | The first six integers correspond to m radial points and the second six |
| integers correspond to n angular points where possible numbers of Lebedev |
| angular points are listed in section 4.3.11. |
−mn | The first six integers correspond to m radial points and the second six |
| integers correspond to n angular points where the number of Gauss-Legendre |
| angular points n = 2N2. |
RECOMMENDATION:
Use default unless numerical integration problems arise. Larger grids may be
required for optimization and frequency calculations. |
|
| XC_SMART_GRID
Uses SG-0 (where available) for early SCF cycles, and switches to the
(larger) grid specified by XC_GRID (which defaults to SG-1, if not
otherwise specified) for final cycles of the SCF. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The use of the smart grid can save some time on initial SCF cycles. |
|
|
|
NL_GRID
Specifies the grid to use for non-local correlation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless computational cost becomes prohibitive, in which case SG-0 may be used.
XC_GRID should generally be finer than NL_GRID. |
|
4.3.16 Example
Example 4.0 Q-Chem input for a DFT single point energy calculation on
water.
$comment
B-LYP/STO-3G water single point calculation
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.2
hoh = 120.0
$end.
$rem
EXCHANGE Becke Becke88 exchange
CORRELATION lyp LYP correlation
BASIS sto-3g Basis set
$end
4.3.17 User-Defined Density Functionals
The format for entering user-defined exchange-correlation density functionals
is one line for each component of the functional. Each line requires three
variables: the first defines whether the component is an exchange or
correlation functional by declaring an X or C, respectively. The second
variable is the symbolic representation of the functional as used for the
EXCHANGE and CORRELATION $rem variables. The final variable
is a real number corresponding to the contribution of the component to the
functional. Hartree-Fock exchange contributions (required for hybrid density
functionals) can be entered using only two variables (K, for HF exchange)
followed by a real number.
$xc_functional
X exchange_symbol coefficient
X exchange_symbol coefficient
...
C correlation_symbol coefficient
C correlation_symbol coefficient
...
K coefficient
$end
Note:
(1) Coefficients are real.
(2) A user-defined functional does not require all X, C and K components. |
Examples of user-defined XCs: these are XC options that
for the time being can only be invoked via the user defined XC input section:
Example 4.0 Q-Chem input of water with B3tLap.
$comment
water with B3tLap
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 0.97
hoh = 120.0
$end
$rem
EXCHANGE gen
CORRELATION none
XC_GRID 000120000194 ! recommended for high accuracy
BASIS G3LARGE ! recommended for high accuracy
THRESH 14 ! recommended for high accuracy and better convergence
$end
$xc_functional
X Becke 0.726
X S 0.0966
C PK06 1.0
K 0.1713
$end
Example 4.0 Q-Chem input of water with BR89B94hyb.
$comment
water with BR89B94hyb
$end
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 0.97
hoh = 120.0
$end
$rem
EXCHANGE gen
CORRELATION none
XC_GRID 000120000194 ! recommended for high accuracy
BASIS G3LARGE ! recommended for high accuracy
THRESH 14 ! recommended for high accuracy and better convergence
$end
$xc_functional
X BR89 0.846
C B94hyb 1.0
K 0.154
$end
More specific is the use of the RI-B05 and RI-PSTS functionals. In this release
we offer only single-point SCF calculations with these functionals.
Both options require a converged LSD or HF solution as initial guess,
which greatly facilitates the convergence. It
also requires specifying a particular auxiliary basis set:
Example 4.0 Q-Chem input of H2 using RI-B05.
$comment
H2, example of SP RI-B05.
First do a well-converged LSD, G3LARGE is the basis of choice
for good accuracy. The input lines
purecar 222
SCF_GUESS CORE
are obligatory for the time being here.
$end
$molecule
0 1
H 0. 0. 0.0
H 0. 0. 0.7414
$end
$rem
JOBTYPE SP
SCF_GUESS CORE
EXCHANGE SLATER
CORRELATION VWN
BASIS G3LARGE
purcar 222
THRESH 14
MAX_SCF_CYCLES 80
PRINT_INPUT TRUE
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
SCF_CONVERGENCE 9
$end
@@@@
$comment
For the time being the following input lines are obligatory:
purcar 22222
AUX_BASIS riB05-cc-pvtz
dft_cutoffs 0
1415 1
MAX_SCF_CYCLES 0
JOBTYPE SP
$end
$molecule
READ
$end
$rem
JOBTYPE SP
SCF_GUESS READ
EXCHANGE B05
! EXCHANGE PSTS ! use this line for RI-PSTS
purcar 22222
BASIS G3LARGE
AUX_BASIS riB05-cc-pvtz ! The aux basis for RI-B05 and RI-PSTS
THRESH 14
PRINT_INPUT TRUE
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
MAX_SCF_CYCLES 0
dft_cutoffs 0
1415 1
$end
Besides post-LSD, the RI-B05 option can be used as post-Hartree-Fock
method as well, in which case one first does a well-converged HF
calculation and uses it as a guess read in the consecutive RI-B05
run.
4.4 Large Molecules and Linear Scaling Methods
4.4.1 Introduction
Construction of the effective Hamiltonian, or Fock matrix, has traditionally
been the rate-determining step in self-consistent field calculations, due
primarily to the cost of two-electron integral evaluation, even with the
efficient methods available in Q-Chem (see Appendix B). However, for
large enough molecules, significant speedups are possible by employing
linear-scaling methods for each of the nonlinear terms that can arise. Linear
scaling means that if the molecule size is doubled, then the computational
effort likewise only doubles. There are three computationally significant
terms:
- Electron-electron Coulomb interactions, for which Q-Chem incorporates
the Continuous Fast Multipole Method (CFMM) discussed in section
4.4.2
- Exact exchange interactions, which arise in hybrid DFT calculations and
Hartree-Fock calculations, for which Q-Chem incorporates the LinK
method discussed in section 4.4.3 below.
- Numerical integration of the exchange and correlation functionals in DFT
calculations, which we have already discussed in section 4.3.11.
Q-Chem supports energies and efficient analytical gradients for all three of
these high performance methods to permit structure optimization of large
molecules, as well as relative energy evaluation. Note that analytical second
derivatives of SCF energies do not exploit these methods at present.
For the most part, these methods are switched on automatically by the program
based on whether they offer a significant speedup for the job at hand.
Nevertheless it is useful to have a general idea of the key concepts behind
each of these algorithms, and what input options are necessary to control them.
That is the primary purpose of this section, in addition to briefly describing
two more conventional methods for reducing computer time in large calculations
in Section 4.4.4.
There is one other computationally significant step in SCF calculations, and
that is diagonalization of the Fock matrix, once it has been constructed.
This step scales with the cube of molecular size (or basis set size), with a
small pre-factor. So, for large enough SCF calculations (very roughly in the
vicinity of 2000 basis functions and larger), diagonalization becomes the
rate-determining step. The cost of cubic scaling with a small pre-factor at
this point exceeds the cost of the linear scaling Fock build, which has a
very large pre-factor, and the gap rapidly widens thereafter. This sets an
effective upper limit on the size of SCF calculation for which Q-Chem is
useful at several thousand basis functions.
4.4.2 Continuous Fast Multipole Method (CFMM)
The quantum chemical Coulomb problem, perhaps better known as the DFT
bottleneck, has been at the forefront of many research efforts throughout the
1990s. The quadratic computational scaling behavior conventionally seen in the
construction of the Coulomb matrix in DFT or HF calculations has prevented the
application of ab initio methods to molecules containing many hundreds
of atoms. Q-Chem, Inc., in collaboration with White and Head-Gordon at the
University of California at Berkeley, and Gill now at the Australian National
University, were the first to develop the generalization of Greengard's Fast
Multipole Method (FMM) [149] to Continuous charged matter
distributions in the form of the CFMM, which is the first linear scaling
algorithm for DFT calculations. This initial breakthrough has since lead to an
increasing number of linear scaling alternatives and analogies, but for Coulomb
interactions, the CFMM remains state of the art. There are two computationally
intensive contributions to the Coulomb interactions which we discuss in turn:
- Long-range interactions, which are treated by the CFMM
- Short-range interactions, corresponding to overlapping charge
distributions, which are treated by a specialized "J-matrix engine"
together with Q-Chem's state-of-the art two-electron integral methods.
The Continuous Fast Multipole Method was the first implemented linear scaling
algorithm for the construction of the J matrix. In collaboration with
Q-Chem, Inc., Dr. Chris White began the development of the CFMM by more
efficiently deriving [150] the original Fast Multipole Method
before generalizing it to the CFMM [151]. The generalization
applied by White et al. allowed the principles underlying the success of the
FMM to be applied to arbitrary (subject to constraints in evaluating the
related integrals) continuous, but localized, matter distributions. White and
co-workers further improved the underlying CFMM algorithm [152,[153]
then implemented it efficiently [154],
achieving performance that is an order of magnitude faster than some competing
implementations.
The success of the CFMM follows similarly with that of the FMM, in that the
charge system is subdivided into a hierarchy of boxes. Local charge
distributions are then systematically organized into multipole representations
so that each distribution interacts with local expansions of the potential due
to all distant charge distributions. Local and distant distributions are
distinguished by a well-separated (WS) index, which is the number of boxes that
must separate two collections of charges before they may be considered distant
and can interact through multipole expansions; near-field interactions must be
calculated directly. In the CFMM each distribution is given its own WS index
and is sorted on the basis of the WS index, and the position of their space
centers. The implementation in Q-Chem has allowed the efficiency gains of
contracted basis functions to be maintained.
The CFMM algorithm can be summarized in five steps:
- Form and translate multipoles.
- Convert multipoles to local Taylor expansions.
- Translate Taylor information to the lowest level.
- Evaluate Taylor expansions to obtain the far-field potential.
- Perform direct interactions between overlapping distributions.
Accuracy can be carefully controlled by due consideration of tree depth,
truncation of the multipole expansion and the definition of the extent of
charge distributions in accordance with a rigorous mathematical error bound.
As a rough guide, 10 poles are adequate for single point energy calculations,
while 25 poles yield sufficient accuracy for gradient calculations. Subdivision
of boxes to yield a one-dimensional length of about 8 boxes works quite well
for systems of up to about one hundred atoms. Larger molecular systems, or
ones which are extended along one dimension, will benefit from an increase in
this number. The program automatically selects an appropriate number of boxes
by default.
For the evaluation of the remaining short-range interactions, Q-Chem
incorporates efficient J-matrix engines, originated by White and Head-Gordon [155].
These are analytically exact methods that are based on
standard two-electron integral methods, but with an interesting twist. If one
knows that the two-electron integrals are going to be summed into a Coulomb
matrix, one can ask whether they are in fact the most efficient intermediates
for this specific task. Or, can one instead find a more compact and
computationally efficient set of intermediates by folding the density matrix
into the recurrence relations for the two-electron integrals. For integrals
that are not highly contracted (i.e., are not linear combinations of more than a
few Gaussians), the answer is a dramatic yes. This is the basis of the
J-matrix approach, and Q-Chem includes the latest algorithm developed by
Yihan Shao working with Martin Head-Gordon at Berkeley for this purpose.
Shao's J-engine is employed for both energies [156] and forces [157]
and gives substantial speedups relative to the use of
two-electron integrals without any approximation (roughly a factor of 10
(energies) and 30 (forces) at the level of an uncontracted dddd shell quartet,
and increasing with angular momentum). Its use is automatically selected for
integrals with low degrees of contraction, while regular integrals are employed
when the degree of contraction is high, following the state of the art PRISM
approach of Gill and co-workers [158].
The CFMM is controlled by the following input parameters:
CFMM_ORDER
Controls the order of the multipole expansions in CFMM calculation. |
TYPE:
DEFAULT:
15 | For single point SCF accuracy |
25 | For tighter convergence (optimizations) |
OPTIONS:
n | Use multipole expansions of order n |
RECOMMENDATION:
|
| GRAIN
Controls the number of lowest-level boxes in one dimension for CFMM. |
TYPE:
DEFAULT:
-1 | Program decides best value, turning on CFMM when useful |
OPTIONS:
-1 | Program decides best value, turning on CFMM when useful |
1 | Do not use CFMM |
n ≥ 8 | Use CFMM with n lowest-level boxes in one dimension |
RECOMMENDATION:
This is an expert option; either use the default, or use a value of 1 if CFMM
is not desired. |
|
|
|
4.4.3 Linear Scaling Exchange (LinK) Matrix Evaluation
Hartree-Fock calculations and the popular hybrid density functionals such as
B3LYP also require two-electron integrals to evaluate the exchange energy
associated with a single determinant. There is no useful multipole expansion
for the exchange energy, because the bra and ket of the two-electron integral
are coupled by the density matrix, which carries the effect of exchange.
Fortunately, density matrix elements decay exponentially with distance for
systems that have a HOMO-LUMO gap [159]. The better the
insulator, the more localized the electronic structure, and the faster the rate
of exponential decay. Therefore, for insulators, there are only a linear number
of numerically significant contributions to the exchange energy. With
intelligent numerical thresholding, it is possible to rigorously evaluate the
exchange matrix in linear scaling effort. For this purpose, Q-Chem contains
the linear scaling K (LinK) method [160] to evaluate both
exchange energies and their gradients [161] in linear scaling
effort (provided the density matrix is highly sparse). The LinK method
essentially reduces to the conventional direct SCF method for exchange in the
small molecule limit (by adding no significant overhead), while yielding large
speedups for (very) large systems where the density matrix is indeed highly
sparse. For full details, we refer the reader to the original
papers [160,[161]. LinK can be explicitly requested by the
following option (although Q-Chem automatically switches it on when the
program believes it is the preferable algorithm).
LIN_K
Controls whether linear scaling evaluation of exact exchange (LinK) is used. |
TYPE:
DEFAULT:
Program chooses, switching on LinK whenever CFMM is used. |
OPTIONS:
TRUE | Use LinK |
FALSE | Do not use LinK |
RECOMMENDATION:
Use for HF and hybrid DFT calculations with large numbers of atoms. |
|
4.4.4 Incremental and Variable Thresh Fock Matrix Building
The use of a variable integral threshold, operating for the first few cycles of
an SCF, is justifiable on the basis that the MO coefficients are usually of
poor quality in these cycles. In Q-Chem, the integrals in the first iteration
are calculated at a threshold of 10−6 (for an anticipated final integral
threshold greater than, or equal to 10−6 to ensure the error in the first
iteration is solely sourced from the poor MO guess. Following this, the
integral threshold used is computed as
tmp_thresh = varthresh×DIIS_error |
| (4.74) |
where the DIIS_error is that calculated from the previous cycle, varthresh
is the variable threshold set by the program (by default) and tmp_thresh is
the temporary threshold used for integral evaluation. Each cycle requires
recalculation of all integrals. The variable integral threshold procedure has
the greatest impact in early SCF cycles.
In an incremental Fock matrix build [162], F is computed
recursively as
where m is the SCF cycle, and ∆Jm and ∆Km are computed using the difference density
Using Schwartz integrals and elements of the difference density, Q-Chem is
able to determine at each iteration which ERIs are required, and if necessary,
recalculated. As the SCF nears convergence, ∆Pm becomes sparse and
the number of ERIs that need to be recalculated declines dramatically, saving
the user large amounts of computational time.
Incremental Fock matrix builds and variable thresholds are only used when the
SCF is carried out using the direct SCF algorithm and are clearly complementary
algorithms. These options are controlled by the following input parameters,
which are only used with direct SCF calculations.
INCFOCK
Iteration number after which the incremental Fock matrix algorithm is
initiated |
TYPE:
DEFAULT:
1 | Start INCFOCK after iteration number 1 |
OPTIONS:
User-defined (0 switches INCFOCK off) |
RECOMMENDATION:
May be necessary to allow several iterations before switching on INCFOCK. |
|
| VARTHRESH
Controls the temporary integral cut-off threshold. tmp_thresh = 10−VARTHRESH×DIIS_error |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
3 has been found to be a practical level, and can slightly speed up SCF
evaluation. |
|
|
|
4.4.5 Incremental DFT
Incremental DFT (IncDFT) uses the difference density and functional values to improve the
performance of the DFT quadrature procedure by providing a better screening of negligible
values. Using this option will yield improved efficiency at each successive iteration
due to more effective screening.
INCDFT
Toggles the use of the IncDFT procedure for DFT energy calculations. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use IncDFT |
TRUE | Use IncDFT |
RECOMMENDATION:
Turning this option on can lead to faster SCF calculations, particularly
towards the end of the SCF. Please note that for some systems use of this
option may lead to convergence problems. |
|
| INCDFT_DENDIFF_THRESH
Sets the threshold for screening density matrix values in the IncDFT procedure. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. |
|
|
|
INCDFT_GRIDDIFF_THRESH
Sets the threshold for screening functional values in the IncDFT procedure |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. |
|
| INCDFT_DENDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening density matrix
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_DENDIFF_THRESH is reached. This means this
value must be set lower than INCDFT_DENDIFF_THRESH. |
TYPE:
DEFAULT:
0 | Variable threshold is not used. |
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. |
|
|
|
INCDFT_GRIDDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening the functional
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_GRIDDIFF_THRESH is reached. This means that
this value must be set lower than INCDFT_GRIDDIFF_THRESH. |
TYPE:
DEFAULT:
0 | Variable threshold is not used. |
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. |
|
4.4.6 Fourier Transform Coulomb Method
The Coulomb part of the DFT calculations using `ordinary' Gaussian
representations can be sped up dramatically using plane waves as a secondary
basis set by replacing the most costly analytical electron repulsion integrals
with numerical integration techniques. The main advantages to keeping the
Gaussians as the primary basis set is that the diagonalization step is much
faster than using plane waves as the primary basis set, and all electron
calculations can be performed analytically.
The Fourier Transform Coulomb (FTC) technique [163,[164] is
precise and tunable and all results are practically identical with the
traditional analytical integral calculations. The FTC technique is at least
2-3 orders of magnitude more accurate then other popular plane wave based
methods using the same energy cutoff. It is also at least 2-3 orders of
magnitude more accurate than the density fitting (resolution of identity)
technique. Recently, an efficient way to implement the forces of the Coulomb
energy was introduced [165], and a new technique to localize
filtered core functions. Both of these features have been implemented within
Q-Chem and contribute to the efficiency of the method.
The FTC method achieves these spectacular results by replacing the analytical
integral calculations, whose computational costs scales as O(N4) (where
N is the number of basis function) with procedures that scale as only
O(N2). The asymptotic scaling of computational costs with system size is
linear versus the analytical integral evaluation which is quadratic. Research
at Q-Chem Inc. has yielded a new, general, and very efficient implementation
of the FTC method which work in tandem with the J-engine and the CFMM
(Continuous Fast Multipole Method) techniques [166].
In the current implementation the speed-ups arising from the FTC technique are
moderate when small or medium Pople basis sets are used. The reason is that the
J-matrix engine and CFMM techniques provide an already highly efficient
solution to the Coulomb problem. However, increasing the number of
polarization functions and, particularly, the number of diffuse functions
allows the FTC to come into its own and gives the most significant
improvements. For instance, using the 6-311G+(df,pd) basis set for a medium-to-large
size molecule is more affordable today then before. We found also significant
speed ups when non-Pople basis sets are used such as cc-pvTZ. The FTC energy
and gradients calculations are implemented to use up to f-type basis
functions.
FTC
Controls the overall use of the FTC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use FTC in the Coulomb part |
1 | Use FTC in the Coulomb part |
RECOMMENDATION:
Use FTC when bigger and/or diffuse basis sets are used. |
|
| FTC_SMALLMOL
Controls whether or not the operator is evaluated on a large grid and stored in
memory to speed up the calculation. |
TYPE:
DEFAULT:
OPTIONS:
1 | Use a big pre-calculated array to speed up the FTC calculations |
0 | Use this option to save some memory |
RECOMMENDATION:
Use the default if possible and use 0 (or buy some more memory) when
needed. |
|
|
|
FTC_CLASS_THRESH_ORDER
Together with FTC_CLASS_THRESH_MULT, determines the cutoff
threshold for included a shell-pair in the dd class, i.e., the class that
is expanded in terms of plane waves. |
TYPE:
DEFAULT:
5 | Logarithmic part of the FTC classification threshold. Corresponds to 10−5 |
OPTIONS:
RECOMMENDATION:
|
| FTC_CLASS_THRESH_MULT
Together with FTC_CLASS_THRESH_ORDER, determines the cutoff
threshold for included a shell-pair in the dd class, i.e., the class that
is expanded in terms of plane waves. |
TYPE:
DEFAULT:
5 | Multiplicative part of the FTC classification threshold. Together with |
| the default value of the FTC_CLASS_THRESH_ORDER this leads to |
| the 5×10−5 threshold value. |
OPTIONS:
RECOMMENDATION:
Use the default. If diffuse basis sets are used and the molecule is relatively
big then tighter FTC classification threshold has to be used. According to our
experiments using Pople-type diffuse basis sets, the default 5×10−5 value provides accurate result for an alanine5 molecule while 1×10−5 threshold value for alanine10 and 5×10−6 value for
alanine15 has to be used. |
|
|
|
4.4.7 Multiresolution Exchange-Correlation (mrXC) Method
MrXC (multiresolution exchange-correlation) [167,[168,[169]
is a new method developed by the Q-Chem development
team for the accelerating the computation of exchange-correlation
(XC) energy and matrix originated from the XC functional.
As explained in 4.4.6, the XC
functional is so complicated that the integration of it is usually
done on a numerical quadrature. There are two basic types of
quadrature. One is the atom-centered grid (ACG), a superposition of
atomic quadrature described in 4.4.6.
ACG has high density of points near the nucleus to handle the
compact core density and low density of points in the valence and
nonbonding region where the electron density is smooth. The other
type is even-spaced cubic grid (ESCG), which is typically used
together with pseudopotentials and planewave basis functions where
only the e electron density is assumed smooth. In quantum chemistry,
ACG is more often used as it can handle accurately all-electron
calculations of molecules. MrXC combines those two integration
schemes seamlessly to achieve an optimal computational efficiency
by placing the calculation of the smooth part of the density and XC
matrix onto the ESCG. The computation associated with the smooth
fraction of the electron density is the
major bottleneck of the XC part of a DFT
calculation and can be done at a much faster rate on the ESCG due
to its low resolution. Fast Fourier transform and B-spline
interpolation are employed for the accurate transformation between
the two types of grids such that the final results remain the
same as they would be on the ACG alone. Yet, a speed-up of several times for
the calculations of electron-density and XC matrix is achieved. The smooth
part of the calculation with mrXC can also be combined with FTC
(see section 4.4.6) to achieve further gain of efficiency.
MRXC
Controls the use of MRXC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use MRXC |
1 | Use MRXC in the evaluation of the XC part |
RECOMMENDATION:
MRXC is very efficient for medium and large molecules,
especially when medium and large basis sets are used. |
|
The following two keywords control the smoothness precision.
The default value is carefully selected to maintain high accuracy.
MRXC_CLASS_THRESH_MULT
Controls the of smoothness precision |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
a prefactor in the threshold for mrxc error control:
im*10.0−io |
|
| MRXC_CLASS_THRESH_ORDER
Controls the of smoothness precision |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The exponent in the threshold of the mrxc error control:
im*10.0−io |
|
|
|
The next keyword controls the order of the
B-spline interpolation:
LOCAL_INTERP_ORDER
Controls the order of the B-spline |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The default value is sufficiently accurate |
|
4.4.8 Resolution-of-the-Identity Fock Matrix Methods
Evaluation of the Fock matrix (both Coulomb, J, and exchange, K, pieces) can be sped up by an approximation known as the resolution-of-the-identity approximation (RI-JK).
Essentially, the full complexity in common basis sets required to describe chemical bonding is not necessary to describe the mean-field Coulomb and exchange interactions between electrons. That is, ρ in the left side of
( - ) = _ ( - ) _
is much less complicated than an individual λσ function pair.
The same principle applies to the FTC method in subsection 4.4.6, in which case the slowly varying piece of the electron density is replaced with a plane-wave expansion.
With the RI-JK approximation, the Coulomb interactions of the function pair ρ(r)=λσ(r) Pλσ are fit by a smaller set of atom-centered basis functions. In terms of J:
_ d^3 _1 P_ (_1) 1 _K d^3 _1 P_K K(_1) 1
The coefficients PK must be determined to accurately represent the potential.
This is done by performing a least-squared minimization of the difference between Pλσ λσ(r1) and PK K(r1), with differences measured by the Coulomb metric.
This requires a matrix inversion over the space of auxiliary basis functions, which may be done rapidly by Cholesky decomposition.
The RI method applied to the Fock matrix may be further enhanced by performing local fitting of a density or function pair element. This is the basis of the atomic-RI method (ARI), which has been developed for
both Coulomb (J) matrix [170] and exchange (K) matrix evaluation [171].
In ARI, only nearby auxiliary functions K(r) are employed to fit the target function.
This reduces the asymptotic scaling of the matrix-inversion step as well as that of many intermediate steps in the digestion of RI integrals.
Briefly, atom-centered auxiliary functions on nearby atoms are only used if they are within the "outer" radius (R1) of the fitting region.
Between R1 and the "inner" radius (R0), the amplitude of interacting auxiliary functions is smoothed by a function that goes from zero to one and has continuous derivatives.
To optimize efficiency, the van der Waals radius of the atom is included in the cutoff so that smaller atoms are dropped from the fitting radius sooner.
The values of R0 and R1 are specified as REM variables as described below.
RI_J
Toggles the use of the RI algorithm to compute J. |
TYPE:
DEFAULT:
FALSE | RI will not be used to compute J. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI. |
|
| RI_K
Toggles the use of the RI algorithm to compute K. |
TYPE:
DEFAULT:
FALSE | RI will not be used to compute K. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI. |
|
|
|
ARI
Toggles the use of the atomic resolution-of-the-identity (ARI) approximation. |
TYPE:
DEFAULT:
FALSE | ARI will not be used by default for an RI-JK calculation. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaulation time. |
|
| ARI_R0
Determines the value of the inner fitting radius (in Å ngstroms) |
TYPE:
DEFAULT:
4 | A value of 4 Å will be added to the atomic van der Waals radius. |
OPTIONS:
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable. |
|
|
|
ARI_R1
Determines the value of the outer fitting radius (in Å ngstroms) |
TYPE:
DEFAULT:
5 | A value of 5 Å will be added to the atomic van der Waals radius. |
OPTIONS:
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable. This value also determines, in part, the smoothness of the potential energy surface. |
|
4.4.9 Examples
Example 4.0 Q-Chem input for a large single point energy calculation. The
CFMM is switched on automatically when LinK is requested.
$comment
HF/3-21G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
EXCHANGE HF HF exchange
BASIS 3-21G Basis set
LIN_K TRUE Calculate K using LinK
$end
Example 4.0 Q-Chem input for a large single point energy calculation. This
would be appropriate for a medium-sized molecule, but for truly large
calculations, the CFMM and LinK algorithms are far more efficient.
$comment
HF/3-21G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
exchange hf HF exchange
basis 3-21G Basis set
incfock 5 Incremental Fock after 5 cycles
varthresh 3 1.0d-03 variable threshold
$end
4.5 SCF Initial Guess
4.5.1 Introduction
The Roothaan-Hall and Pople-Nesbet equations of SCF theory are non-linear in
the molecular orbital coefficients. Like many mathematical problems involving
non-linear equations, prior to the application of a technique to search for a
numerical solution, an initial guess for the solution must be generated. If the
guess is poor, the iterative procedure applied to determine the numerical
solutions may converge very slowly, requiring a large number of iterations, or
at worst, the procedure may diverge.
Thus, in an ab initio SCF procedure, the quality of the initial guess
is of utmost importance for (at least) two main reasons:
- To ensure that the SCF converges to an appropriate ground state. Often
SCF calculations can converge to different local minima in wavefunction
space, depending upon which part of that space the initial guess places
the system in.
- When considering jobs with many basis functions requiring the
recalculation of ERIs at each iteration, using a good initial guess that
is close to the final solution can reduce the total job time
significantly by decreasing the number of SCF iterations.
For these reasons, sooner or later most users will find it helpful to have some
understanding of the different options available for customizing the initial
guess. Q-Chem currently offers five options for the initial guess:
- Superposition of Atomic Density (SAD)
- Core Hamiltonian (CORE)
- Generalized Wolfsberg-Helmholtz (GWH)
- Reading previously obtained MOs from disk. (READ)
- Basis set projection (BASIS2)
The first three of these guesses are built-in, and are briefly described in
Section 4.5.2. The option of reading MOs from disk is described in
Section 4.5.3. The initial guess MOs can be modified, either by
mixing, or altering the order of occupation. These options are discussed in
Section 4.5.4. Finally, Q-Chem's novel basis set projection
method is discussed in Section 4.5.5.
4.5.2 Simple Initial Guesses
There are three simple initial guesses available in Q-Chem. While they are
all simple, they are by no means equal in quality, as we discuss below.
- Superposition of Atomic Densities (SAD): The SAD guess is almost
trivially constructed by summing together atomic densities that have been
spherically averaged to yield a trial density matrix. The SAD guess is far
superior to the other two options below, particularly when large basis
sets and/or large molecules are employed. There are three issues
associated with the SAD guess to be aware of:
- No molecular orbitals are obtained, which means that SCF algorithms
requiring orbitals (the direct minimization methods discussed in
Section 4.6) cannot directly use the SAD guess,
and,
- The SAD guess is not available for general (read-in) basis sets. All
internal basis sets support the SAD guess.
- The SAD guess is not idempotent and thus requires at least
two SCF iterations to ensure proper SCF convergence (idempotency of
the density).
- Generalized Wolfsberg-Helmholtz (GWH): The GWH guess
procedure [172] uses a combination of the overlap matrix elements
in Eq. (4.12), and the diagonal elements of the Core Hamiltonian matrix
in Eq. (4.18). This initial guess is most satisfactory in small basis
sets for small molecules. It is constructed according to the relation
given below, where cx is a constant.
Hμυ = cx Sμυ (Hμμ +Hυυ ) | / |
(Hμμ +Hυυ ) 2 2 |
| (4.77) |
- Core Hamiltonian: The core Hamiltonian guess simply obtains the
guess MO coefficients by diagonalizing the core Hamiltonian matrix in
Eq. (4.18). This approach works best with small basis sets, and
degrades as both the molecule size and the basis set size are increased.
The selection of these choices (or whether to read in the orbitals) is
controlled by the following $rem variables:
SCF_GUESS
Specifies the initial guess procedure to use for the SCF. |
TYPE:
DEFAULT:
SAD | Superposition of atomic density (available only with standard basis
sets) |
GWH | For ROHF where a set of orbitals are required. |
FRAGMO | For a fragment MO calculation |
OPTIONS:
CORE | Diagonalize core Hamiltonian |
SAD | Superposition of atomic density |
GWH | Apply generalized Wolfsberg-Helmholtz approximation |
READ | Read previous MOs from disk |
FRAGMO | Superimposing converged fragment MOs |
RECOMMENDATION:
SAD guess for standard basis sets. For general basis sets, it is best to use
the BASIS2 $rem. Alternatively, try the GWH or core Hamiltonian
guess. For ROHF it can be useful to READ guesses from an SCF calculation on the
corresponding cation or anion. Note that because the density is made spherical,
this may favor an undesired state for atomic systems, especially transition
metals. Use FRAGMO in a fragment MO calculation. |
|
| SCF_GUESS_ALWAYS
Switch to force the regeneration of a new initial guess for each series of
SCF iterations (for use in geometry optimization). |
TYPE:
DEFAULT:
OPTIONS:
False | Do not generate a new guess for each series of SCF iterations in an |
| optimization; use MOs from the previous SCF calculation for the guess, |
| if available. |
True | Generate a new guess for each series of SCF iterations in a geometry |
| optimization. |
RECOMMENDATION:
Use default unless SCF convergence issues arise |
|
|
|
4.5.3 Reading MOs from Disk
There are two methods by which MO coefficients can be used from a previous
job by reading them from disk:
- Running two independent jobs sequentially invoking qchem with
three command line variables:.
localhost-1> qchem job1.in job1.out save
localhost-2> qchem job2.in job2.out save
Note:
(1) The $rem variable SCF_GUESS must be set to READ
in job2.in.
(2) Scratch files remain in $QCSCRATCH/save on exit. |
- Running a batch job where two jobs are placed into a single input file
separated by the string @@@ on a single line.
Note:
(1) SCF_GUESS must be set to
READ in the second job of the batch file.
(2) A third qchem command line variable is not necessary.
(3) As for the SAD guess, Q-Chem requires at least two SCF cycles to
ensure proper SCF convergence (idempotency of the density). |
Note:
It is up to the user to make sure that the basis
sets match between the two jobs. There is no internal checking for this,
although the occupied orbitals are re-orthogonalized in the current basis after
being read in. If you want to project from a smaller basis into a larger basis,
consult section 4.5.5. |
4.5.4 Modifying the Occupied Molecular Orbitals
It is sometimes useful for the occupied guess orbitals to be other than the
lowest Nα (or Nβ) orbitals. Reasons why one may need
to do this include:
- To converge to a state of different symmetry or orbital occupation.
- To break spatial symmetry.
- To break spin symmetry, as in unrestricted calculations on molecules with
an even number of electrons.
There are two mechanisms for modifying a set of guess orbitals: either by
SCF_GUESS_MIX, or by specifying the orbitals to occupy. Q-Chem
users may define the occupied guess orbitals using the $occupied or
$swap_occupied_virtual
keywords. In the former, occupied guess orbitals are defined by listing the
alpha orbitals to be occupied on the first line and beta on the second. In the
former, only pair of orbitals that needs to be swapped is specified.
Note:
To prevent Q-Chem to change orbital occupation during SCF procedure,
MOMSTART option is often used in combination with
$occupied or $swap_occupied_virtual keywords. |
Note:
The need for orbitals renders these options incompatible with the SAD
guess. Most often, they are used with SCF_GUESS=READ. |
Example 4.0 Format for modifying occupied guess orbitals.
$occupied
1 2 3 4 ... nalpha
1 2 3 4 ... nbeta
$end
Example 4.0 Alternative format for modifying occupied guess orbitals.
$swap_occupied_virtual
<spin> <io1> <iv1>
<spin> <io2> <iv2>
$end
Example 4.0 Example of swapping guess orbitals.
$swap_occupied_virtual
alpha 5 6
beta 6 7
$end
This is identical to:
Example 4.0 Example of specifying occupied guess orbitals.
$occupied
1 2 3 4 6 5 7
1 2 3 4 5 7 6
$end
The other $rem variables related to altering the orbital occupancies are:
SCF_GUESS_PRINT
Controls printing of guess MOs, Fock and density matrices. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not print guesses. |
SAD | |
1 | Atomic density matrices and molecular matrix. |
2 | Level 1 plus density matrices. |
CORE and GWH | |
1 | No extra output. |
2 | Level 1 plus Fock and density matrices and, MO coefficients and |
| eigenvalues. |
READ | |
1 | No extra output |
2 | Level 1 plus density matrices, MO coefficients and eigenvalues. |
RECOMMENDATION:
|
| SCF_GUESS_MIX
Controls mixing of LUMO and HOMO to break symmetry in the initial guess. For
unrestricted jobs, the mixing is performed only for the alpha orbitals. |
TYPE:
DEFAULT:
0 (FALSE) | Do not mix HOMO and LUMO in SCF guess. |
OPTIONS:
0 (FALSE) | Do not mix HOMO and LUMO in SCF guess. |
1 (TRUE) | Add 10% of LUMO to HOMO to break symmetry. |
n | Add n×10% of LUMO to HOMO (0 < n < 10). |
RECOMMENDATION:
When performing unrestricted calculations on molecules with an even number of
electrons, it is often necessary to break alpha / beta symmetry in the initial
guess with this option, or by specifying input for $occupied. |
|
|
|
4.5.5 Basis Set Projection
Q-Chem also includes a novel basis set projection method developed by Dr Jing
Kong of Q-Chem Inc. It permits a calculation in a large basis set to
bootstrap itself up via a calculation in a small basis set that is
automatically spawned when the user requests this option. When basis set
projection is requested (by providing a valid small basis for BASIS2),
the program executes the following steps:
- A simple DFT calculation is performed in the small basis,
BASIS2, yielding a converged density matrix in this basis.
- The large basis set SCF calculation (with different values of
EXCHANGE and CORRELATION set by the input) begins by
constructing the DFT Fock operator in the large basis but with the
density matrix obtained from the small basis set.
- By diagonalizing this matrix, an accurate initial guess for the density
matrix in the large basis is obtained, and the target SCF calculation
commences.
Two different methods of projection are available and can be set using the
BASISPROJTYPE $rem. The OVPROJECTION option expands the MOs
from the BASIS2 calculation in the larger basis, while the
FOPPROJECTION option constructs the Fock matrix in the larger basis
using the density matrix from the initial, smaller basis set calculation. Basis
set projection is a very effective option for general basis sets, where the SAD
guess is not available. In detail, this initial guess is controlled by the
following $rem variables:
BASIS2
Sets the small basis set to use in basis set projection. |
TYPE:
DEFAULT:
No second basis set default. |
OPTIONS:
Symbol. Use standard basis sets as per Chapter 7. |
BASIS2_GEN | General BASIS2 |
BASIS2_MIXED | Mixed BASIS2 |
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using
a basis larger than a minimal basis when BASIS2 is used for initial guess purposes.
Larger, standardized BASIS2 options are available for dual-basis calculations
(see Section 4.7). |
|
| BASISPROJTYPE
Determines which method to use when projecting the density matrix of
BASIS2 |
TYPE:
DEFAULT:
FOPPROJECTION (when DUAL_BASIS_ENERGY=false) |
OVPROJECTION (when DUAL_BASIS_ENERGY=true) |
OPTIONS:
FOPPROJECTION | Construct the Fock matrix in the second basis |
OVPROJECTION | Projects MO's from BASIS2 to BASIS. |
RECOMMENDATION:
|
|
|
Note:
BASIS2 sometimes messes up post-Hartree-Fock calculations. It is recommended
to split such jobs into two subsequent one, such that in the first job a desired Hartree-Fock solution is found using BASIS2, and in the second job, which performs a
post-HF calculation, SCF_GUESS=READ is invoked. |
4.5.6 Examples
Example 4.0 Input where basis set projection is used to generate a good
initial guess for a calculation employing a general basis set, for which the
default initial guess is not available.
$molecule
0 1
O
H 1 r
H 1 r 2 a
r 0.9
a 104.0
$end
$rem
EXCHANGE hf
CORRELATION mp2
BASIS general
BASIS2 sto-3g
$end
$basis
O 0
S 3 1.000000
3.22037000E+02 5.92394000E-02
4.84308000E+01 3.51500000E-01
1.04206000E+01 7.07658000E-01
SP 2 1.000000
7.40294000E+00 -4.04453000E-01 2.44586000E-01
1.57620000E+00 1.22156000E+00 8.53955000E-01
SP 1 1.000000
3.73684000E-01 1.00000000E+00 1.00000000E+00
SP 1 1.000000
8.45000000E-02 1.00000000E+00 1.00000000E+00
****
H 0
S 2 1.000000
5.44717800E+00 1.56285000E-01
8.24547000E-01 9.04691000E-01
S 1 1.000000
1.83192000E-01 1.00000000E+00
****
$end
Example 4.0 Input for an ROHF calculation on the OH radical. One SCF cycle
is initially performed on the cation, to get reasonably good initial guess
orbitals, which are then read in as the guess for the radical. This avoids the
use of Q-Chem's default GWH guess for ROHF, which is often poor.
$comment
OH radical, part 1. Do 1 iteration of cation orbitals.
$end
$molecule
1 1
O 0.000 0.000 0.000
H 0.000 0.000 1.000
$end
$rem
BASIS = 6-311++G(2df)
EXCHANGE = hf
MAX_SCF_CYCLES = 1
THRESH = 10
$end
@@@
$comment
OH radical, part 2. Read cation orbitals, do the radical
$end
$molecule
0 2
O 0.000 0.000 0.000
H 0.000 0.000 1.000
$end
$rem
BASIS = 6-311++G(2df)
EXCHANGE = hf
UNRESTRICTED = false
SCF_ALGORITHM = dm
SCF_CONVERGENCE = 7
SCF_GUESS = read
THRESH = 10
$end
Example 4.0 Input for an unrestricted HF calculation on H2 in the
dissociation limit, showing the use of SCF_GUESS_MIX = 2
(corresponding to 20% of the alpha LUMO mixed with the alpha HOMO).
Geometric direct minimization with DIIS is used to converge the SCF, together
with MAX_DIIS_CYCLES = 1 (using the default value for
MAX_DIIS_CYCLES, the DIIS procedure just oscillates).
$molecule
0 1
H 0.000 0.000 0.0
H 0.000 0.000 -10.0
$end
$rem
UNRESTRICTED = true
EXCHANGE = hf
BASIS = 6-31g**
SCF_ALGORITHM = diis_gdm
MAX_DIIS_CYCLES = 1
SCF_GUESS = gwh
SCF_GUESS_MIX = 2
$end
4.6 Converging SCF Calculations
4.6.1 Introduction
As for any numerical optimization procedure, the rate of convergence of the SCF
procedure is dependent on the initial guess and on the algorithm used to step
towards the stationary point. Q-Chem features a number of alternative SCF
optimization algorithms, which are discussed in the following sections, along
with the $rem variables that are used to control the calculations. The main
options are discussed in sections which follow and are, in brief:
- The highly successful DIIS procedures, which are the default, except for
restricted open-shell SCF calculations.
- The new geometric direct minimization (GDM) method, which is highly
robust, and the recommended fall-back when DIIS fails. It can also be
invoked after a few initial iterations with DIIS to improve the initial
guess. GDM is the default algorithm for restricted open-shell SCF
calculations.
- The older and less robust direct minimization method (DM). As for GDM,
it can also be invoked after a few DIIS iterations (except for RO jobs).
- The maximum overlap method (MOM) which ensures that DIIS always occupies
a continuous set of orbitals and does not oscillate between different
occupancies.
- The relaxed constraint algorithm (RCA) which guarantees that the energy
goes down at every step.
4.6.2 Basic Convergence Control Options
See also more detailed options in the following sections, and note that the SCF
convergence criterion and the integral threshold must be set in a compatible
manner, (this usually means THRESH should be set to at least 3 higher
than SCF_CONVERGENCE).
MAX_SCF_CYCLES
Controls the maximum number of SCF iterations permitted. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase for slowly converging systems such as those containing transition
metals. |
|
| SCF_ALGORITHM
Algorithm used for converging the SCF. |
TYPE:
DEFAULT:
OPTIONS:
DIIS | Pulay DIIS. |
DM | Direct minimizer. |
DIIS_DM | Uses DIIS initially, switching to direct minimizer for later iterations |
| (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). |
DIIS_GDM | Use DIIS and then later switch to geometric direct minimization |
| (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). |
GDM | Geometric Direct Minimization. |
RCA | Relaxed constraint algorithm |
RCA_DIIS | Use RCA initially, switching to DIIS for later iterations (see |
| THRESH_RCA_SWITCH and MAX_RCA_CYCLES described |
| later in this chapter) |
ROOTHAAN | Roothaan repeated diagonalization. |
RECOMMENDATION:
Use DIIS unless performing a restricted open-shell calculation, in which case GDM is recommended.
If DIIS fails to find a reasonable approximate solution in the initial iterations,
RCA_DIIS is the recommended fallback option.
If DIIS approaches the correct solution but fails to finally converge,
DIIS_GDM is the recommended fallback. |
|
|
|
SCF_CONVERGENCE
SCF is considered converged when the wavefunction error is less that
10−SCF_CONVERGENCE. Adjust the value of THRESH at the same
time. Note that in Q-Chem 3.0 the DIIS error is measured by the maximum error
rather than the RMS error. |
TYPE:
DEFAULT:
5 | For single point energy calculations. |
7 | For geometry optimizations and vibrational analysis. |
8 | For SSG calculations, see Chapter 5. |
OPTIONS:
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger
values provide more significant figures, at greater computational cost. |
|
In some cases besides the total SCF energy, one needs its separate energy components, like
kinetic energy, exchange energy, correlation energy, etc. The values of these
components are printed at each SCF cycle if one specifies in the input:
SCF_PRINT 1 .
4.6.3 Direct Inversion in the Iterative Subspace (DIIS)
The SCF implementation of the Direct Inversion in the Iterative Subspace
(DIIS) method [173,[174] uses the property of
an SCF solution that requires the density matrix to commute with the Fock matrix:
During the SCF cycles, prior to achieving self-consistency, it is therefore possible
to define an error vector ei, which is non-zero except at convergence:
Here, Pi is obtained from diagonalization of ∧Fi , and
The DIIS coefficients ck, are obtained by a least-squares constrained
minimization of the error vectors, viz
Z= | ⎛ ⎝
|
∑
k
|
ck ek | ⎞ ⎠
|
· | ⎛ ⎝
|
∑
k
|
ck ek | ⎞ ⎠
|
|
| (4.81) |
where the constraint
is imposed to yield a set of linear equations, of dimension N+1:
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
= | ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
|
| (4.83) |
Convergence criteria requires the largest element of the Nth error vector
to be below a cutoff threshold, usually 10−5 for single point energies,
often increased to 10−8 for optimizations and frequency calculations.
The rate of convergence may be improved by restricting the number of previous
Fock matrices (size of the DIIS subspace, $rem variable
DIIS_SUBSPACE_SIZE) used for determining the DIIS coefficients:
|
^
F
|
k
|
= |
k−1 ∑
j=k−(L+1)
|
cj Fj |
| (4.84) |
where L is the size of the DIIS subspace. As the Fock matrix nears
self-consistency, the linear matrix equations in Eq. (4.85) tend to become
severely ill-conditioned and it is often necessary to reset the DIIS subspace
(this is automatically carried out by the program).
Finally, on a practical note, we observe that DIIS has a tendency to converge
to global minima rather than local minima when employed for SCF calculations.
This seems to be because only at convergence is the density matrix in the DIIS
iterations idempotent. On the way to convergence, one is not on the "true"
energy surface, and this seems to permit DIIS to "tunnel" through barriers in
wavefunction space. This is usually a desirable property, and is the motivation
for the options that permit initial DIIS iterations before switching to direct
minimization to converge to the minimum in difficult cases.
The following $rem variables permit some customization of the DIIS iterations:
DIIS_SUBSPACE_SIZE
Controls the size of the DIIS and/or RCA subspace during the SCF. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| DIIS_PRINT
Controls the output from DIIS SCF optimization. |
TYPE:
DEFAULT:
OPTIONS:
0 | Minimal print out. |
1 | Chosen method and DIIS coefficients and solutions. |
2 | Level 1 plus changes in multipole moments. |
3 | Level 2 plus Multipole moments. |
4 | Level 3 plus extrapolated Fock matrices. |
RECOMMENDATION:
|
|
|
Note:
In Q-Chem 3.0 the DIIS error is determined by the maximum error rather
than the RMS error. For backward compatibility the RMS error can be forced by
using the following $rem |
| DIIS_ERR_RMS
Changes the DIIS convergence metric from the maximum to the RMS error. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default, the maximum error provides a more reliable criterion. |
|
|
|
4.6.4 Geometric Direct Minimization (GDM)
Troy Van Voorhis, working at Berkeley with Martin Head-Gordon, has developed a
novel direct minimization method that is extremely robust, and at the same time
is only slightly less efficient than DIIS. This method is called geometric
direct minimization (GDM) because it takes steps in an orbital rotation space
that correspond properly to the hyper-spherical geometry of that space. In other
words, rotations are variables that describe a space which is curved like a
many-dimensional sphere. Just like the optimum flight paths for airplanes are
not straight lines but great circles, so too are the optimum steps in orbital
rotation space. GDM takes this correctly into account, which is the origin of
its efficiency and its robustness. For full details, we refer the reader to
Ref. . GDM is a good alternative to DIIS
for SCF jobs that exhibit convergence difficulties with DIIS.
Recently, Barry Dunietz, also working at Berkeley with Martin Head-Gordon, has
extended the GDM approach to restricted open-shell SCF calculations. Their
results indicate that GDM is much more efficient than the older
direct minimization method (DM).
In section 4.6.3, we discussed the fact that DIIS can efficiently
head towards the global SCF minimum in the early iterations. This can be true
even if DIIS fails to converge in later iterations. For this reason, a hybrid
scheme has been implemented which uses the DIIS minimization procedure to
achieve convergence to an intermediate cutoff threshold. Thereafter, the
geometric direct minimization algorithm is used. This scheme combines the
strengths of the two methods quite nicely: the ability of DIIS to recover from
initial guesses that may not be close to the global minimum, and the ability of
GDM to robustly converge to a local minimum, even when the local surface
topology is challenging for DIIS. This is the recommended procedure with which
to invoke GDM (i.e., setting SCF_ALGORITHM = DIIS_GDM). This
hybrid procedure is also compatible with the SAD guess, while GDM itself is
not, because it requires an initial guess set of orbitals. If one wishes to
disturb the initial guess as little as possible before switching on GDM, one
should additionally specify MAX_DIIS_CYCLES = 1 to obtain only a
single Roothaan step (which also serves up a properly orthogonalized set of
orbitals).
$rem options relevant to GDM are SCF_ALGORITHM which should be
set to either GDM or DIIS_GDM and the following:
MAX_DIIS_CYCLES
The maximum number of DIIS iterations before switching to (geometric) direct
minimization when SCF_ALGORITHM is DIIS_GDM or
DIIS_DM. See also THRESH_DIIS_SWITCH. |
TYPE:
DEFAULT:
OPTIONS:
1 | Only a single Roothaan step before switching to (G)DM |
n | n DIIS iterations before switching to (G)DM. |
RECOMMENDATION:
|
| THRESH_DIIS_SWITCH
The threshold for switching between DIIS extrapolation and direct minimization
of the SCF energy is 10−THRESH_DIIS_SWITCH when
SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See
also MAX_DIIS_CYCLES |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
4.6.5 Direct Minimization (DM)
Direct minimization (DM) is a less sophisticated forerunner of the geometric
direct minimization (GDM) method discussed in the previous section. DM does not
properly step along great circles in the hyper-spherical space of orbital
rotations, and therefore converges less rapidly and less robustly than GDM, in
general. It is retained for legacy purposes, and because it is at present the
only method available for restricted open shell (RO) SCF calculations in
Q-Chem. In general, the input options are the same as for GDM, with the
exception of the specification of SCF_ALGORITHM, which can be either
DIIS_DM (recommended) or DM.
PSEUDO_CANONICAL
When SCF_ALGORITHM = DM, this controls the way the initial
step, and steps after subspace resets are taken. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use Roothaan steps when (re)initializing |
TRUE | Use a steepest descent step when (re)initializing |
RECOMMENDATION:
The default is usually more efficient, but choosing TRUE sometimes
avoids problems with orbital reordering. |
|
4.6.6 Maximum Overlap Method (MOM)
In general, the DIIS procedure is remarkably successful. One difficulty that
is occasionally encountered is the problem of an SCF that occupies two
different sets of orbitals on alternating iterations, and therefore
oscillates and fails to converge. This can be overcome by choosing orbital
occupancies that maximize the overlap of the new occupied orbitals with the set
previously occupied. Q-Chem contains the maximum overlap method (MOM) [176],
developed by Andrew Gilbert and Peter Gill now at the
Australian National University.
MOM is therefore is a useful adjunct to DIIS in convergence problems involving
flipping of orbital occupancies. It is controlled by the $rem variable
MOM_START, which specifies the SCF iteration on which the MOM
procedure is first enabled. There are two strategies that are useful in setting
a value for MOM_START. To help maintain an initial configuration it
should be set to start on the first cycle. On the other hand, to assist
convergence it should come on later to avoid holding on to an initial
configuration that may be far from the converged one.
The MOM-related $rem variables in full are the following:.
MOM_PRINT
Switches printing on within the MOM procedure. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Printing is turned off |
TRUE | Printing is turned on. |
RECOMMENDATION:
|
| MOM_START
Determines when MOM is switched on to stabilize DIIS iterations. |
TYPE:
DEFAULT:
OPTIONS:
0 (FALSE) | MOM is not used |
n | MOM begins on cycle n. |
RECOMMENDATION:
Set to 1 if preservation of initial orbitals is desired. If MOM is to be
used to aid convergence, an SCF without MOM should be run to determine when
the SCF starts oscillating. MOM should be set to start just before the
oscillations. |
|
|
|
4.6.7 Relaxed Constraint Algorithm (RCA)
The relaxed constraint algorithm (RCA) is an ingenious and simple means of minimizing the SCF energy
that is particularly effective in cases where the initial guess is poor.
The latter is true, for example, when employing a user-specified basis
(when the Core or GWH guess must be employed) or when near-degeneracy effects imply
that the initial guess will likely occupy the wrong orbitals relative to the desired converged solution.
Briefly, RCA begins with the SCF problem as a constrained minimization of the energy
as a function of the density matrix, E(P) [177,[178].
The constraint is that the density matrix be idempotent, P ·P=P,
which basically forces the occupation numbers to be either zero or one.
The fundamental realization of RCA is that this constraint can be relaxed to
allow sub-idempotent density matrices, P ·P ≤ P.
This condition forces the occupation numbers to be between zero and one.
Physically, we expect that any state with fractional occupations can
lower its energy by moving electrons from higher energy orbitals to lower ones.
Thus, if we solve for the minimum of E(P) subject to the relaxed sub-idempotent constraint,
we expect that the ultimate solution will nonetheless be idempotent.
In fact, for Hartree-Fock this can be rigorously proven.
For density functional theory, it is possible that the minimum will have
fractional occupation numbers but these occupations have a physical interpretation
in terms of ensemble DFT.
The reason the relaxed constraint is easier to deal with is that it is easy to prove
that a linear combination of sub-idempotent matrices is also
sub-idempotent as long as the linear coefficients are between zero and one.
By exploiting this property, convergence can be accelerated in a way
that guarantees the energy will go down at every step.
The implementation of RCA in Q-Chem closely follows the "Energy DIIS" implementation of
the RCA algorithm [179].
Here, the current density matrix is written as a linear combination of the previous density matrices:
To a very good approximation (exact for Hartree-Fock) the energy for P(x)
can be written as a quadratic function of x:
E(x) = |
∑
i
|
Ei xi+ |
1
2
|
|
∑
i
|
xi(Pi− Pj) ·(Fi− Fj) xj |
| (4.86) |
At each iteration, x is chosen to minimize E(x) subject to the constraint
that all of the xi are between zero and one.
The Fock matrix for P(x) is further written as a linear combination of the previous Fock matrices,
F(x) = |
∑
i
|
xi Fi + δFxc(x) |
| (4.87) |
where δFxc(x) denotes a (usually quite small) change in the exchange-correlation part
that is computed once x has been determined.
We note that this extrapolation is very similar to that used by DIIS.
However, this procedure is guaranteed to reduce the energy E(x) at every iteration, unlike DIIS.
In practice, the RCA approach is ideally suited to difficult convergence situations
because it is immune to the erratic orbital swapping that can occur in DIIS.
On the other hand, RCA appears to perform relatively poorly near convergence,
requiring a relatively large number of steps to improve the precision of a "good" approximate solution.
It is thus advantageous in many cases to run RCA for the initial steps and
then switch to DIIS either after some specified number of iterations or
after some target convergence threshold has been reached.
Finally, note that by its nature RCA considers the energy as a function of the density matrix.
As a result, it cannot be applied to restricted open shell calculations which are explicitly orbital-based.
Note: RCA interacts poorly with INCDFT, so INCDFT is disabled by default
when an RCA or RCA_DIIS calculation is requested. To enable INCDFT with
such a calculation, set INCDFT = 2 in the $rem section. RCA may also have
poor interactions with INCFOCK; if RCA fails to converge, disabling INCFOCK
may improve convergence in some cases.
RCA options are:
RCA_PRINT
Controls the output from RCA SCF optimizations. |
TYPE:
DEFAULT:
OPTIONS:
0 | No print out |
1 | RCA summary information |
2 | Level 1 plus RCA coefficients |
3 | Level 2 plus RCA iteration details |
RECOMMENDATION:
|
| MAX_RCA_CYCLES
The maximum number of RCA iterations before switching to DIIS when SCF_ALGORITHM is RCA_DIIS. |
TYPE:
DEFAULT:
OPTIONS:
N | N RCA iterations before switching to DIIS |
RECOMMENDATION:
|
|
|
THRESH_RCA_SWITCH
The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS. |
TYPE:
DEFAULT:
OPTIONS:
N | Algorithm changes from RCA to DIIS when Error is less than 10−N. |
RECOMMENDATION:
|
Please see next section for an example using RCA.
4.6.8 Examples
Example 4.0 Input for a UHF calculation using geometric direct minimization
(GDM) on the phenyl radical, after initial iterations with DIIS. This example
fails to converge if DIIS is employed directly.
$molecule
0 2
c1
x1 c1 1.0
c2 c1 rc2 x1 90.0
x2 c2 1.0 c1 90.0 x1 0.0
c3 c1 rc3 x1 90.0 c2 tc3
c4 c1 rc3 x1 90.0 c2 -tc3
c5 c3 rc5 c1 ac5 x1 -90.0
c6 c4 rc5 c1 ac5 x1 90.0
h1 c2 rh1 x2 90.0 c1 180.0
h2 c3 rh2 c1 ah2 x1 90.0
h3 c4 rh2 c1 ah2 x1 -90.0
h4 c5 rh4 c3 ah4 c1 180.0
h5 c6 rh4 c4 ah4 c1 180.0
rc2 = 2.672986
rc3 = 1.354498
tc3 = 62.851505
rc5 = 1.372904
ac5 = 116.454370
rh1 = 1.085735
rh2 = 1.085342
ah2 = 122.157328
rh4 = 1.087216
ah4 = 119.523496
$end
$rem
BASIS = 6-31G*
EXCHANGE = hf
INTSBUFFERSIZE = 15000000
SCF_ALGORITHM = diis_gdm
SCF_CONVERGENCE = 7
THRESH = 10
$end
Example 4.0 An example showing how to converge a ROHF calculation on the 3A2
state of DMX. Note the use of reading in orbitals from a previous closed-shell
calculation and the use of MOM to maintain the orbital occupancies. The 3B1 is
obtained if MOM is not used.
$molecule
+1 1
C 0.000000 0.000000 0.990770
H 0.000000 0.000000 2.081970
C -1.233954 0.000000 0.290926
C -2.444677 0.000000 1.001437
H -2.464545 0.000000 2.089088
H -3.400657 0.000000 0.486785
C -1.175344 0.000000 -1.151599
H -2.151707 0.000000 -1.649364
C 0.000000 0.000000 -1.928130
C 1.175344 0.000000 -1.151599
H 2.151707 0.000000 -1.649364
C 1.233954 0.000000 0.290926
C 2.444677 0.000000 1.001437
H 2.464545 0.000000 2.089088
H 3.400657 0.000000 0.486785
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 6-31+G*
SCF_GUESS core
$end
@@@
$molecule
read
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 6-31+G*
SCF_GUESS read
MOM_START 1
$end
$occupied
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28
$end
@@@
$molecule
-1 3
... <as above> ...
$end
$rem
UNRESTRICTED false
EXCHANGE hf
BASIS 6-31+G*
SCF_GUESS read
$end
Example 4.0 RCA_DIIS algorithm applied a radical
$molecule
0 2
H 1.004123 -0.180454 0.000000
O -0.246002 0.596152 0.000000
O -1.312366 -0.230256 0.000000
$end
$rem
UNRESTRICTED true
EXCHANGE hf
BASIS cc-pVDZ
SCF_GUESS gwh
SCF_ALGORITHM RCA_DIIS
DIIS_SUBSPACE_SIZE 15
THRESH 9
$end
4.7 Dual-Basis Self-Consistent Field Calculations
The dual-basis approximation [180,[181,[182,[183,[184,[185]
to self-consistent field (HF or DFT) energies provides an
efficient means for obtaining large basis set effects at vastly less cost
than a full SCF calculation in a large basis set.
First, a full SCF calculation is performed in a
chosen small basis (specified by BASIS2). Second, a single SCF-like
step in the larger, target basis (specified, as usual, by BASIS) is used to
perturbatively approximate the large basis energy. This correction amounts to a
first-order approximation in the change in density matrix, after the single
large-basis step:
Etotal = Esmall basis + Tr[(∆P)·F]large basis |
| (4.88) |
where F (in the large basis) is built from the converged (small basis)
density matrix. Thus, only a single Fock build is required in the large basis
set. Currently, HF and DFT energies (SP) as well as analytic first derivatives
(FORCE or OPT) are available. [Note: As of version 4.0,
first derivatives of unrestricted dual-basis DFT energies-though correct-require a
code-efficiency fix. We do not recommend use of these derivatives until
this improvement has been made.]
Across the G3 set [186,[187,[188] of 223 molecules, using cc-pVQZ, dual-basis
errors for B3LYP are 0.04 kcal/mol (energy) and 0.03 kcal/mol (atomization
energy per bond) and are at least an order of magnitude less than using a smaller
basis set alone. These errors are obtained at roughly an order of magnitude savings
in cost, relative to the full, target-basis calculation.
4.7.1 Dual-Basis MP2
The dual-basis approximation can also be used for the reference energy
of a correlated second-order Møller-Plesset (MP2) calculation [181,[185].
When activated, the dual-basis HF energy is first calculated as described above;
subsequently, the MO coefficients and orbital energies are used to calculate
the correlation energy in the large basis. This technique is particularly
effective for RI-MP2 calculations (see Section 5.5), in which the
cost of the underlying SCF calculation often dominates.
Furthermore, efficient analytic gradients
of the DB-RI-MP2 energy have been developed [183] and added to Q-Chem.
These gradients allow for the optimization of molecular structures with RI-MP2 near the
basis set limit. Typical computational savings are on the order of 50% (aug-cc-pVDZ) to 71% (aug-cc-pVTZ).
Resulting dual-basis errors are only 0.001 Å in molecular structures and are, again,
significantly less than use of a smaller basis set alone.
4.7.2 Basis Set Pairings
We recommend using basis pairings in which the small basis set is a proper subset of the target basis
(6-31G into 6-31G*, for example).
They not only produce more accurate results; they also lead to more efficient integral
screening in both energies and gradients. Subsets for many standard basis sets (including Dunning-style
cc-pVXZ basis sets and their augmented analogs)
have been developed and thoroughly tested for these purposes. A summary of the
pairings is provided in Table 4.7.2; details of these truncations are provided in
Figure 4.1.
A new pairing for 6-31G*-type calculations is also available. The 6-4G subset (named r64G in Q-Chem)
is a subset by primitive functions and provides a smaller, faster alternative for this
basis set regime [184]. A case-dependent switch in the projection code
(still OVPROJECTION) properly handles 6-4G. For DB-HF, the calculations proceed as described
above. For DB-DFT, empirical scaling factors (see Ref. for details) are applied
to the dual-basis correction. This scaling is handled automatically by the code and prints accordingly.
As of Q-Chem version 3.2, the basis set projection code has also been adapted to properly account for
linear dependence [185], which can often be problematic for large, augmented
(aug-cc-pVTZ, etc..) basis set calculations. The same standard keyword (LINDEPTHRESH)
is utilized for linear dependence in the projection code. Because of the scheme utilized to account
for linear dependence, only proper-subset pairings are now allowed.
Like single-basis calculations, user-specified general or mixed basis sets may be
employed (see Chapter 7) with dual-basis calculations. The target basis specification
occurs in the standard $basis section. The smaller, secondary basis is placed in a similar
$basis2 section; the syntax within this section is the same as the syntax for $basis.
General and mixed small
basis sets are activated by BASIS2=BASIS2_GEN and BASIS2=BASIS2_MIXED, respectively.
BASIS | BASIS2 | |
cc-pVTZ | rcc-pVTZ |
cc-pVQZ | rcc-pVQZ |
aug-cc-pVDZ | racc-pVDZ |
aug-cc-pVTZ | racc-pVTZ |
aug-cc-pVQZ | racc-pVQZ |
6-31G* | r64G, 6-31G |
6-31G** | r64G, 6-31G |
6-31++G** | 6-31G* |
6-311++G(3df,3pd) | 6-311G*, 6-311+G* |
Table 4.3: Summary and nomenclature of recommended dual-basis pairings
#1Structure of the truncated basis set pairings for cc-pV(T,Q)Z and aug-cc-pV(D,T,Q)Z.
The most compact functions are listed at the top. Primed functions depict aug (diffuse) functions.
Dashes indicate eliminated functions, relative to the paired standard basis set. In each case, the
truncations for hydrogen and heavy atoms are shown, along with the nomenclature used in Q-Chem.
4.7.3 Job Control
Dual-Basis calculations are controlled with the following $rem.
DUAL_BASIS_ENERGY turns on the Dual-Basis approximation. Note that
use of BASIS2 without DUAL_BASIS_ENERGY only uses
basis set projection to generate the initial guess and does not invoke the
Dual-Basis approximation (see Section 4.5.5). OVPROJECTION is used as the default
projection mechanism for Dual-Basis calculations; it is not recommended that
this be changed. Specification of SCF variables
(e.g., THRESH) will apply to calculations in both basis sets.
DUAL_BASIS_ENERGY
Activates dual-basis SCF (HF or DFT) energy correction. |
TYPE:
DEFAULT:
OPTIONS:
Analytic first derivative available for HF and DFT (see JOBTYPE) |
Can be used in conjunction with MP2 or RI-MP2 |
See BASIS, BASIS2, BASISPROJTYPE |
RECOMMENDATION:
Use Dual-Basis to capture large-basis effects at smaller basis cost.
Particularly useful with RI-MP2, in which HF often dominates. Use only proper
subsets for small-basis calculation. |
|
4.7.4 Examples
Example 4.0 Input for a Dual-Basis B3LYP single-point calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE b3lyp
BASIS 6-311++G(3df,3pd)
BASIS2 6-311G*
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a Dual-Basis B3LYP single-point calculation with a minimal 6-4G small basis.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE b3lyp
BASIS 6-31G*
BASIS2 r64G
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a Dual-Basis RI-MP2 single-point calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS rimp2-cc-pVQZ
BASIS cc-pVQZ
BASIS2 rcc-pVQZ
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a Dual-Basis RI-MP2 geometry optimization.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE opt
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS rimp2-aug-cc-pVDZ
BASIS aug-cc-pVDZ
BASIS2 racc-pVDZ
DUAL_BASIS_ENERGY true
$end
Example 4.0 Input for a Dual-Basis RI-MP2 single-point calculation with mixed basis sets.
$molecule
0 1
H
O 1 1.1
H 2 1.1 1 104.5
$end
$rem
JOBTYPE opt
EXCHANGE hf
CORRELATION rimp2
AUX_BASIS aux_mixed
BASIS mixed
BASIS2 basis2_mixed
DUAL_BASIS_ENERGY true
$end
$basis
H 1
cc-pVTZ
****
O 2
aug-cc-pVTZ
****
H 3
cc-pVTZ
****
$end
$basis2
H 1
rcc-pVTZ
****
O 2
racc-pVTZ
****
H 3
rcc-pVTZ
****
$end
$aux_basis
H 1
rimp2-cc-pVTZ
****
O 2
rimp2-aug-cc-pVTZ
****
H 3
rimp2-cc-pVTZ
****
$end
4.7.5 Dual-Basis Dynamics
The ability to compute SCF and MP2 energies and forces at reduced cost makes dual-basis calculations
attractive for ab initio molecular dynamics simulations. Dual-basis BOMD has
demonstrated [189]
savings of 58%, even relative to state-of-the-art, Fock-extrapolated BOMD. Savings are further
increased to 71% for dual-basis RI-MP2 dynamics. Notably, these timings outperform estimates
of extended-Lagrangian (Car-Parrinello) dynamics, without detrimental energy conservation
artifacts that are sometimes observed in the latter [190].
Two algorithmic factors make modest but worthwhile improvements to dual-basis dynamics. First,
the iterative, small-basis calculation can benefit from Fock matrix extrapolation [190].
Second,
extrapolation of the response equations (the so-called "Z-vector" equations) for
nuclear forces further increases efficiency [191] . Both sets of keywords
are described in Section 9.9, and the code automatically
adjusts to extrapolate in the proper basis set when DUAL_BASIS_ENERGY
is activated.
4.8 Hartree-Fock and Density-Functional Perturbative Corrections
4.8.1 Hartree-Fock Perturbative Correction
An HFPC [192,[193] calculation consists of an iterative HF calculation in a small primary
basis followed by a single Fock matrix formation, diagonalization, and energy
evaluation in a larger, secondary basis. We denote a conventional HF
calculation by HF / basis, and a HFPC calculation by HFPC / primary / secondary.
Using a primary basis of n functions, the restricted HF matrix elements for a
2m-electron system are
Fμν = hμν + |
n ∑
λσ
|
Pλσ | ⎡ ⎣
|
(μν|λσ) − |
1
2
|
(μλ|νσ) | ⎤ ⎦
|
|
| (4.89) |
Solving the Roothaan-Hall equation in the primary basis results in
molecular orbitals and an associated density matrix, P. In an HFPC
calculation, P is subsequently used to build a new Fock matrix,
F[1], in a larger secondary basis of N functions
Fab[1] = hab + |
n ∑
λσ
|
Pλσ | ⎡ ⎣
|
(ab|λσ) − |
1
2
|
(aλ|bσ) | ⎤ ⎦
|
|
| (4.90) |
where λ, σ indicate primary basis functions and a, b
represent secondary basis functions. Diagonalization of F[1] yields
improved molecular orbitals and an associated density matrix P[1]. The
HFPC energy is given by
EHFPC = |
N ∑
ab
|
P[1]ab hab + |
1
2
|
|
N ∑
abcd
|
P[1]abP[1]cd [2(ab|cd) − (ac|bd)] |
| (4.91) |
where a, b, c and d represent secondary basis functions. This
differs from the DBHF energy evaluation where P P[1], rather than
P[1]P[1], is used. The inclusion of contributions that are
quadratic in P[1] is the key reason for the fact that HFPC is more
accurate than DBHF.
Unlike DBHF, HFPC does not require proper subset/superset
basis set combinations and is therefore able to jump between
any two basis sets. Benchmark study of HFPC on a large and diverse data set of
total and reaction energies show that, for a range of primary/secondary basis set
combinations the HFPC scheme can reduce the error of the primary calculation by
around two orders of magnitude at a cost of about one third that of the full
secondary calculation.
4.8.2 Density Functional Perturbative Correction (Density Functional "Triple Jumping")
Density Functional Perturbation Theory (DFPC) [194] seeks to combine the low cost of pure calculations using small bases and grids with the high accuracy of hybrid calculations using large bases and grids. Our method is motivated by the dual functional method of Nakajima and
Hirao [195] and the dual grid scheme of Tozer et al. [196]
We combine these with dual basis ideas to obtain a triple perturbation in the functional,
grid and basis directions.
4.8.3 Job Control
HFPC/DFPC calculations are controlled with the following $rem.
HFPT turns on the HFPC/DFPC approximation. Note that
HFPT_BASIS specifies the secondary basis set.
HFPT
Activates HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use Dual-Basis to capture large-basis effects at smaller basis cost. See reference for recommended basis set, functional, and grid pairings. |
|
| HFPT_BASIS
Specifies the secondary basis in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
|
|
DFPT_XC_GRID
Specifies the secondary grid in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
| DFPT_EXCHANGE
Specifies the secondary functional in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
|
|
4.8.4 Examples
Example 4.0 Input for a HFPC single-point calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE hf
BASIS cc-pVDZ !primary basis
HFPT_BASIS cc-pVQZ !secondary basis
PURECART 1111 ! set to purecart of the target basis
HFPT true
$end
Example 4.0 Input for a DFPC single-point calculation.
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE sp
EXCHANGE blyp !primary functional
DFPT_EXCHANGE b3lyp !secondary functional
DFPT_XC_GRID 00075000302 !secondary grid
XC_GRID 0 !primary grid
HFPT_BASIS 6-311++G(3df,3pd) !secondary basis
BASIS 6-311G* !primary basis
PURECART 1111
HFPT true
$end
4.9 Constrained Density Functional Theory (CDFT)
Under certain circumstances,
it is desirable to apply constraints to the electron density during a self-consistent calculation.
For example, in a transition metal complex it may be desirable to constrain the net spin density
on a particular metal atom to integrate to a value consistent with the MS value expected from ligand field theory.
Similarly, in a donor-acceptor complex one may be interested in constraining the total density
on the acceptor group so that the formal charge on the acceptor is either neutral or negatively charged,
depending as the molecule is in its neutral or charge transfer configuration.
In these situations, one is interested in controlling the average value of some density observable,
O(r), to take on a given value, N:
There are of course many states that satisfy such a constraint,
but in practice one is usually looking for the lowest energy such state.
To solve the resulting constrained minimization problem,
one introduces a Lagrange multiplier, V, and solves for the stationary point of
V[ρ, V] = E[ρ] − V( | ⌠ ⌡
|
ρ(r) O(r) d3r − N ) |
| (4.93) |
where E[ρ] is the energy of the system described using density functional theory (DFT).
At convergence, the functional W gives the density, ρ, that satisfies the constraint exactly
(i.e., it has exactly the prescribed number of electrons on the acceptor or spins on the metal center)
but has the lowest energy possible.
The resulting self-consistent procedure can be efficiently solved by ensuring at every SCF step the constraint is satisfied exactly.
The Q-Chem implementation of these equations closely parallels those in Ref. .
The first step in any constrained DFT calculation is the specification of the constraint operator, O(r).
Within Q-Chem, the user is free to specify any constraint operator
that consists of a linear combination of the Becke's atomic partitioning functions:
Here the summation runs over the atoms in the system (A) and over the electron spin
(σ = α, β).
Note that each weight function is designed to be nearly 1 near the nucleus of atom A
and rapidly fall to zero near the nucleus of any other atom in the system.
The specification of the CAσ coefficients is accomplished using
$cdft
CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...
$end
Here, each CONSTRAINT_VALUE is a real number that specifies the desired average value (N) of
the ensuing linear combination of atomic partition functions.
Each COEFFICIENT specifies the coefficient (Cα) of a partition function
or group of partition functions in the constraint operator O.
For each coefficient, all the atoms between the integers FIRST_ATOM and LAST_ATOM
contribute with the specified weight in the constraint operator.
Finally, TYPE specifies the type of constraint being applied-either "CHARGE" or "SPIN".
For a CHARGE constraint the spin up and spin down densities contribute equally
(CAα=CAβ = CA) yielding the total number of electrons on the atom A.
For a SPIN constraint, the spin up and spin down densities contribute with opposite sign
(CAα−CAβ = CA) resulting in a measure of the net spin on the atom A.
Each separate CONSTRAINT_VALUE creates a new operator whose average is to be
constrained-for instance, the example above includes several independent constraints:
X, Y, …. Q-Chem can handle an arbitrary number of constraints
and will minimize the energy subject to all of these constraints simultaneously.
In addition to the $cdft input section of the input file,
a constrained DFT calculation must also set the CDFT flag to TRUE for the calculation to run.
If an atom is not included in a particular operator,
then the coefficient of that atoms partition function is set to zero for that operator.
The TYPE specification is optional, and the default is to perform a charge constraint.
Further, note that any charge constraint is on the net atomic charge.
That is, the constraint is on the difference between the average number of electrons
on the atom and the nuclear charge.
Thus, to constrain CO to be negative, the constraint value would be 1 and not 15.
The choice of which atoms to include in different constraint regions
is left entirely to the user and in practice must be based somewhat on chemical intuition.
Thus, for example, in an electron transfer reaction the user must specify which atoms
are in the "donor" and which are in the "acceptor".
In practice, the most stable choice is typically to make the constrained region
as large as physically possible.
Thus, for the example of electron transfer again, it is best to assign every atom
in the molecule to one or the other group (donor or acceptor),
recognizing that it makes no sense to assign any atoms to both groups.
On the other end of the spectrum, constraining the formal charge on a single atom
is highly discouraged.
The problem is that while our chemical intuition tells us that the lithium atom in LiF
should have a formal charge of +1, in practice the quantum mechanical charge is much closer
to +0.5 than +1.
Only when the fragments are far enough apart do our intuitive pictures of formal charge
actually become quantitative.
Finally, we note that SCF convergence is typically more challenging in constrained DFT calculations
as compared to their unconstrained counterparts.
This effect arises because applying the constraint typically leads to a broken symmetry, diradical-like state.
As SCF convergence for these cases is known to be difficult even for unconstrained states,
it is perhaps not surprising that there are additional convergence difficulties in this case.
Please see the section on SCF convergence for ideas on how to improve convergence for constrained calculations.
[Special Note: The direct minimization methods are not available for constrained calculations.
Hence, some combination of DIIS and RCA must be used to obtain convergence.
Further, it is often necessary to break symmetry in the initial guess (using SCF_GUESS_MIX)
to ensure that the lowest energy solution is obtained.]
Analytic gradients are available for constrained DFT calculations [198].
Second derivatives are only available by finite difference of gradients.
For details on how to apply constrained DFT to compute magnetic exchange couplings,
see Ref. .
For details on using constrained DFT to compute electron transfer parameters,
see Ref. .
CDFT options are:
CDFT
Initiates a constrained DFT calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform a Constrained DFT Calculation |
FALSE | No Density Constraint |
RECOMMENDATION:
Set to TRUE if a Constrained DFT calculation is desired. |
|
| CDFT_POSTDIIS
Controls whether the constraint is enforced after DIIS extrapolation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Enforce constraint after DIIS |
FALSE | Do not enforce constraint after DIIS |
RECOMMENDATION:
Use default unless convergence problems arise,
in which case it may be beneficial to experiment with setting CDFT_POSTDIIS to FALSE.
With this option set to TRUE, energies should be variational after the first iteration. |
|
|
|
CDFT_PREDIIS
Controls whether the constraint is enforced before DIIS extrapolation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Enforce constraint before DIIS |
FALSE | Do not enforce constraint before DIIS |
RECOMMENDATION:
Use default unless convergence problems arise, in which case it may be beneficial to experiment with setting
CDFT_PREDIIS to TRUE. Note that it is possible to enforce the constraint
both before and after DIIS by setting both CDFT_PREDIIS and CDFT_POSTDIIS to TRUE. |
|
| CDFT_THRESH
Threshold that determines how tightly the constraint must be satisfied. |
TYPE:
DEFAULT:
OPTIONS:
N | Constraint is satisfied to within 10−N. |
RECOMMENDATION:
Use default unless problems occur. |
|
|
|
CDFT_CRASHONFAIL
Whether the calculation should crash or not if the constraint iterations do not converge. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Crash if constraint iterations do not converge. |
FALSE | Do not crash. |
RECOMMENDATION:
|
| CDFT_BECKE_POP
Whether the calculation should print the Becke atomic charges at convergence |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Print Populations |
FALSE | Do not print them |
RECOMMENDATION:
Use default. Note that the Mulliken populations printed at the end of an SCF run will not typically add up to the prescribed constraint value. Only the Becke populations are guaranteed to satisfy the user-specified constraints. |
|
|
|
Example 4.0 Charge separation on FAAQ
$molecule
0 1
C -0.64570736 1.37641945 -0.59867467
C 0.64047568 1.86965826 -0.50242683
C 1.73542663 1.01169939 -0.26307089
C 1.48977850 -0.39245666 -0.15200261
C 0.17444585 -0.86520769 -0.27283957
C -0.91002699 -0.02021483 -0.46970395
C 3.07770780 1.57576311 -0.14660056
C 2.57383948 -1.35303134 0.09158744
C 3.93006075 -0.78485926 0.20164558
C 4.16915637 0.61104948 0.08827557
C 5.48914671 1.09087541 0.20409492
H 5.64130588 2.16192921 0.11315072
C 6.54456054 0.22164774 0.42486947
C 6.30689287 -1.16262761 0.53756193
C 5.01647654 -1.65329553 0.42726664
H -1.45105590 2.07404495 -0.83914389
H 0.85607395 2.92830339 -0.61585218
H 0.02533661 -1.93964850 -0.19096085
H 7.55839768 0.60647405 0.51134530
H 7.13705743 -1.84392666 0.71043613
H 4.80090178 -2.71421422 0.50926027
O 2.35714021 -2.57891545 0.20103599
O 3.29128460 2.80678842 -0.23826460
C -2.29106231 -0.63197545 -0.53957285
O -2.55084900 -1.72562847 -0.95628300
N -3.24209015 0.26680616 0.03199109
H -2.81592456 1.08883943 0.45966550
C -4.58411403 0.11982669 0.15424004
C -5.28753695 1.14948617 0.86238753
C -5.30144592 -0.99369577 -0.39253179
C -6.65078185 1.06387425 1.01814801
H -4.73058059 1.98862544 1.26980479
C -6.66791492 -1.05241167 -0.21955088
H -4.76132422 -1.76584307 -0.92242502
C -7.35245187 -0.03698606 0.47966072
H -7.18656323 1.84034269 1.55377875
H -7.22179827 -1.89092743 -0.62856041
H -8.42896369 -0.10082875 0.60432214
$end
$rem
JOBTYPE FORCE
EXCHANGE B3LYP
BASIS 6-31G*
SCF_PRINT TRUE
CDFT TRUE
$end
$cdft
2
1 1 25
-1 26 38
$end
Example 4.0 Cu2-Ox High Spin
$molecule
2 3
Cu 1.4674 1.6370 1.5762
O 1.7093 0.0850 0.3825
O -0.5891 1.3402 0.9352
C 0.6487 -0.3651 -0.1716
N 1.2005 3.2680 2.7240
N 3.0386 2.6879 0.6981
N 1.3597 0.4651 3.4308
H 2.1491 -0.1464 3.4851
H 0.5184 -0.0755 3.4352
H 1.3626 1.0836 4.2166
H 1.9316 3.3202 3.4043
H 0.3168 3.2079 3.1883
H 1.2204 4.0865 2.1499
H 3.8375 2.6565 1.2987
H 3.2668 2.2722 -0.1823
H 2.7652 3.6394 0.5565
Cu -1.4674 -1.6370 -1.5762
O -1.7093 -0.0850 -0.3825
O 0.5891 -1.3402 -0.9352
C -0.6487 0.3651 0.1716
N -1.2005 -3.2680 -2.7240
N -3.0386 -2.6879 -0.6981
N -1.3597 -0.4651 -3.4308
H -2.6704 -3.4097 -0.1120
H -3.6070 -3.0961 -1.4124
H -3.5921 -2.0622 -0.1485
H -0.3622 -3.1653 -3.2595
H -1.9799 -3.3721 -3.3417
H -1.1266 -4.0773 -2.1412
H -0.5359 0.1017 -3.4196
H -2.1667 0.1211 -3.5020
H -1.3275 -1.0845 -4.2152
$end
$rem
JOBTYPE SP
EXCHANGE B3LYP
BASIS 6-31G*
SCF_PRINT TRUE
CDFT TRUE
$end
$cdft
2
1 1 3 s
-1 17 19 s
$end
4.10 Configuration Interaction with Constrained Density Functional Theory (CDFT-CI)
There are some situations in which a system is not well-described by a
DFT calculation on a single configuration. For example, transition
states are known to be poorly described by most functionals, with
the computed barrier being too low. We can, in particular, identify
homolytic dissociation of diatomic species as situations where static
correlation becomes extremely important. Existing DFT functionals
have proved to be very effective in capturing dynamic correlation,
but frequently exhibit difficulties in the presence of strong
static correlation. Configuration Interaction, well known in
wavefunction methods, is a multireference method that is quite
well-suited for capturing static correlation; the CDFT-CI technique
allows for CI calculations on top of DFT calculations, harnessing
both static and dynamic correlation methods.
Constrained DFT is used to compute densities (and Kohn-Sham wavefunctions)
for two or more diabatic-like states; these states are then used to
build a CI matrix. Diagonalizing this matrix yields energies
for the ground and excited states within the configuration space.
The coefficients of the initial diabatic states are printed, to
show the characteristics of the resultant states.
Since Density-Functional Theory only gives converged densities,
not actual wavefunctions, computing the off-diagonal coupling
elements H12 is not completely straightforward, as the physical
meaning of the Kohn-Sham wavefunction is not entirely clear.
We can, however, perform the following manipulation [201]:
| |
|
|
1
2
|
[〈1|H+VC1ωC1−VC1ωC1 |2〉+ 〈1|H+VC2ωC2−VC2ωC2|2〉] |
| |
| |
|
|
1
2
|
[(E1+VC1NC1+E2+VC2NC2) 〈1|2〉−VC1〈1|ωC1|2〉 −VC2〈1|ωC2|2〉] |
| |
|
(where the converged states |i〉 are assumed to be the
ground state of H+VCiωCi with eigenvalue Ei+VCiNCi).
This manipulation eliminates the two-electron integrals from the
expression, and experience has shown that the use of Slater determinants
of Kohn-Sham orbitals is a reasonable approximation for the
quantities 〈1|2〉 and 〈1|ωCi|2〉.
We note that since these constrained states are eigenfunctions
of different Hamiltonians (due to different constraining potentials),
they are not orthogonal states, and we must set up our
CI matrix as a generalized eigenvalue problem. Symmetric orthogonalization
is used by default, though the overlap matrix and Hamiltonian
in non-orthogonal basis are also printed at higher print levels
so that other orthogonalization schemes can be used after-the-fact.
In a limited number of cases, it is possible to find an orthogonal
basis for the CDFT-CI Hamiltonian, where a physical interpretation
can be assigned to the orthogonal states. In such cases, the
matrix representation of the Becke weight operator is diagonalized,
and the (orthogonal) eigenstates can be characterized [202].
This matrix is printed as the "CDFT-CI Population Matrix" at
increased print levels.
In order to perform a CDFT-CI calculation, the N interacting
states must be defined; this is done in a very similar fashion
to the specification for CDFT states:
$cdft
STATE_1_CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
STATE_1_CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...
---
STATE_2_CONSTRAINT_VALUE_X
COEFFICIENT1_X FIRST_ATOM1_X LAST_ATOM1_X TYPE1_X
COEFFICIENT2_X FIRST_ATOM2_X LAST_ATOM2_X TYPE2_X
...
STATE_2_CONSTRAINT_VALUE_Y
COEFFICIENT1_Y FIRST_ATOM1_Y LAST_ATOM1_Y TYPE1_Y
COEFFICIENT2_Y FIRST_ATOM2_Y LAST_ATOM2_Y TYPE2_Y
...
...
...
$end
Each state is specified with the CONSTRAINT_VALUE and the
corresponding weights on sets of atoms whose average value should
be the constraint value.
Different states are separated by a single line containing three or more
dash characters.
If it is desired to use an unconstrained state as one of the
interacting configurations, charge and spin constraints of
zero may be applied to the atom range from 0 to 0.
It is MANDATORY to specify a spin constraint corresponding
to every charge constraint (and it must be immediately following
that charge constraint in the input deck), for reasons described below.
In addition to the $cdft input section of the input file,
a CDFT-CI calculation must also set the CDFTCI flag
to TRUE for the calculation to run. Note, however, that the
CDFT flag is used internally by CDFT-CI, and should
not be set in the input deck. The variable CDFTCI_PRINT
may also be set manually to control the level of output. The default
is 0, which will print the energies and weights (in the diabatic
basis) of the N CDFT-CI states. Setting it to 1 or above will
also print the CDFT-CI overlap matrix, the CDFT-CI Hamiltonian matrix
before the change of basis, and the CDFT-CI Population matrix.
Setting it to 2 or above
will also print the eigenvectors and eigenvalues of the CDFT-CI
Population matrix. Setting it to 3 will produce more output
that is only useful during application debugging.
For convenience, if CDFTCI_PRINT is not set in the
input file, it will be set to the value of SCF_PRINT.
As mentioned in the previous section, there is a disparity
between our chemical intuition of what charges should be
and the actual quantum-mechanical charge. The example was given
of LiF, where our intuition gives the lithium atom a formal
charge of +1; we might similarly imagine performing a
CDFT-CI calculation on H2, with two ionic states
and two spin-constrained states. However, this would result
in attempting to force both electrons of H2 onto
the same nucleus, and this calculation is impossible to converge
(since by the nature of the Becke weight operators, there will
be some non-zero amount of the density that gets proportioned
onto the other atom, at moderate internuclear separations).
To remedy problems such as this, we have adopted a mechanism by
which to convert the formal charges of our chemical intuition
into reasonable quantum-mechanical charge constraints.
We use the formalism of "promolecule" densities, wherein
the molecule is divided into fragments (based on the partitioning
of constraint operators), and a DFT calculation is performed on
these fragments, completely isolated from each other [202].
(This step is why both spin and charge constraints are required, so
that the correct partitioning of electrons for each fragment
may be made.) The resulting promolecule densities, converged
for the separate fragments, are then added together, and the
value of the various weight operators as applied to this
new density, is used as a constraint for the actual CDFT
calculations on the interacting states.
The promolecule density method compensates for the effect
of nearby atoms on the actual density that will be constrained.
The comments about SCF convergence for CDFT calculations also
apply to the calculations used for CDFT-CI, with the addition
that if the SCF converges but CDFT does not, it may be necessary
to use a denser integration grid or reduce the value of CDFT_THRESH.
Analytic gradients are not available. For details on using CDFT-CI
to calculate reaction barrier heights, see Ref. .
CDFT-CI options are:
| CDFTCI
Initiates a constrained DFT-configuration interaction calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform a CDFT-CI Calculation |
FALSE | No CDFT-CI |
RECOMMENDATION:
Set to TRUE if a CDFT-CI calculation is desired. |
|
|
|
CDFTCI_PRINT
Controls level of output from CDFT-CI procedure to Q-Chem output file. |
TYPE:
DEFAULT:
OPTIONS:
0 | Only print energies and coefficients of CDFT-CI final states |
1 | Level 0 plus CDFT-CI overlap, Hamiltonian, and population matrices |
2 | Level 1 plus eigenvectors and eigenvalues of the CDFT-CI population matrix |
3 | Level 2 plus promolecule orbital coefficients and energies |
RECOMMENDATION:
Level 3 is primarily for program debugging; levels 1 and 2 may be useful
for analyzing the coupling elements |
|
| CDFT_LAMBDA_MODE
Allows CDFT potentials to be specified directly, instead of being
determined as Lagrange multipliers. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Standard CDFT calculations are used. |
TRUE | Instead of specifying target charge and spin constraints, use the values |
| from the input deck as the value of the Becke weight potential
|
RECOMMENDATION:
Should usually be set to FALSE. Setting to TRUE can be useful to
scan over different strengths of charge or spin localization, as
convergence properties are improved compared to regular CDFT(-CI) calculations. |
|
|
|
CDFTCI_SKIP_PROMOLECULES
Skips promolecule calculations and allows fractional charge and spin
constraints to be specified directly. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Standard CDFT-CI calculation is performed. |
TRUE | Use the given charge/spin constraints directly, with no
promolecule calculations. |
RECOMMENDATION:
Setting to TRUE can be useful for scanning over constraint values. |
|
Note that CDFT_LAMBDA_MODE and CDFTCI_SKIP_PROMOLECULES
are mutually incompatible.
CDFTCI_SVD_THRESH
By default, a symmetric orthogonalization is performed on the CDFT-CI
matrix before diagonalization. If the CDFT-CI overlap matrix is nearly
singular (i.e., some of the diabatic states are nearly degenerate), then
this orthogonalization can lead to numerical instability. When computing
→S−1/2, eigenvalues smaller than 10−CDFTCI_SVD_THRESH
are discarded. |
TYPE:
DEFAULT:
OPTIONS:
n | for a threshold of 10−n. |
RECOMMENDATION:
Can be decreased if numerical instabilities are encountered in the
final diagonalization. |
|
| CDFTCI_STOP
The CDFT-CI procedure involves performing independent SCF calculations
on distinct constrained states. It sometimes occurs that the same
convergence parameters are not successful for all of the states of
interest, so that a CDFT-CI calculation might converge one of these
diabatic states but not the next. This variable allows a user to
stop a CDFT-CI calculation after a certain number of states have
been converged, with the ability to restart later on the next state,
with different convergence options. |
TYPE:
DEFAULT:
OPTIONS:
n | stop after converging state n (the first state is state 1) |
0 | do not stop early |
RECOMMENDATION:
Use this setting if some diabatic states converge but others do not. |
|
|
|
CDFTCI_RESTART
To be used in conjunction with CDFTCI_STOP, this variable
causes CDFT-CI to read already-converged states from disk and begin
SCF convergence on later states. Note that the same $cdft section
must be used for the stopped calculation and the restarted calculation. |
TYPE:
DEFAULT:
OPTIONS:
n | start calculations on state n+1 |
RECOMMENDATION:
Use this setting in conjunction with CDFTCI_STOP. |
|
Many of the CDFT-related rem variables are also applicable to CDFT-CI calculations.
4.11 Unconventional SCF Calculations
4.11.1 CASE Approximation
The Coulomb Attenuated Schrödinger Equation (CASE) [204]
approximation follows from the KWIK [205] algorithm in which
the Coulomb operator is separated into two pieces using the error
function, Eq. (4.45). Whereas in Section 4.3.4
this partition of the Coulomb operator was used to incorporate long-range
Hartree-Fock exchange into DFT, within the CASE approximation it is used
to attenuate all occurrences of the Coulomb operator in
Eq. (4.2), by
neglecting the long-range portion of the identity in
Eq. (4.45). The parameter ω in
Eq. (4.45) is used to tune the level of attenuation.
Although the
total energies from Coulomb attenuated calculations are significantly different
from non-attenuated energies, it is found that relative energies, correlation
energies and, in particular, wavefunctions, are not, provided a reasonable
value of ω is chosen.
By virtue of the exponential decay of the attenuated operator, ERIs can be
neglected on a proximity basis yielding a rigorous O(N) algorithm for single
point energies. CASE may also be applied in geometry optimizations and
frequency calculations.
OMEGA
Controls the degree of attenuation of the Coulomb operator. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
| INTEGRAL_2E_OPR
Determines the two-electron operator. |
TYPE:
DEFAULT:
OPTIONS:
-1 | Apply the CASE approximation. |
-2 | Coulomb Operator. |
RECOMMENDATION:
Use default unless the CASE operator is desired. |
|
|
|
4.11.2 Polarized Atomic Orbital (PAO) Calculations
Polarized atomic orbital (PAO) calculations are an interesting unconventional
SCF method, in which the molecular orbitals and the density matrix are not
expanded directly in terms of the basis of atomic orbitals. Instead, an
intermediate molecule-optimized minimal basis of polarized atomic orbitals
(PAOs) is used [206]. The polarized atomic orbitals are defined by
an atom-blocked linear transformation from the fixed atomic orbital basis,
where the coefficients of the transformation are optimized to minimize the
energy, at the same time as the density matrix is obtained in the PAO
representation. Thus a PAO-SCF calculation is a constrained variational
method, whose energy is above that of a full SCF calculation in the same basis.
However, a molecule optimized minimal basis is a very compact and useful
representation for purposes of chemical analysis, and it also has potential
computational advantages in the context of MP2 or local MP2 calculations, as
can be done after a PAO-HF calculation is complete to obtain the PAO-MP2
energy.
PAO-SCF calculations tend to systematically underestimate binding energies
(since by definition the exact result is obtained for atoms, but not for
molecules). In tests on the G2 database, PAO-B3LYP/6-311+G(2df,p) atomization
energies deviated from full B3LYP/6-311+G(2df,p) atomization energies by
roughly 20 kcal/mol, with the error being essentially extensive with the number
of bonds. This deviation can be reduced to only 0.5 kcal/mol
with the use of a simple non-iterative second order
correction for "beyond-minimal basis" effects [207]. The second
order correction is evaluated at the end of each PAO-SCF calculation, as it
involves negligible computational cost. Analytical gradients are available
using PAOs, to permit structure optimization. For additional discussion of the
PAO-SCF method and its uses, see the references cited above.
Calculations with PAOs are determined controlled by the following $rem
variables. PAO_METHOD = PAO invokes PAO-SCF calculations, while the
algorithm used to iterate the PAO's can be controlled with
PAO_ALGORITHM.
PAO_ALGORITHM
Algorithm used to optimize polarized atomic orbitals (see PAO_METHOD) |
TYPE:
DEFAULT:
OPTIONS:
0 | Use efficient (and riskier) strategy to converge PAOs. |
1 | Use conservative (and slower) strategy to converge PAOs. |
RECOMMENDATION:
|
| PAO_METHOD
Controls evaluation of polarized atomic orbitals (PAOs). |
TYPE:
DEFAULT:
EPAO | For local MP2 calculations Otherwise no default. |
OPTIONS:
PAO | Perform PAO-SCF instead of conventional SCF. |
EPAO | Obtain EPAO's after a conventional SCF. |
RECOMMENDATION:
|
|
|
4.12 SCF Metadynamics
As the SCF equations are non-linear in the electron density, there are in
theory very many solutions (i.e., sets of orbitals where the energy is stationary
with respect to changes in the orbital subset). Most often sought is the
solution with globally minimal energy as this is a variational upper bound
to the true eigenfunction in this basis.
The SCF methods available in Q-Chem allow the user to converge upon an
SCF solution, and (using STABILITY_ANALYSIS) ensure it is a minimum,
but there is no known method of ensuring that the found solution is a global
minimum; indeed in systems with many low-lying energy levels the solution
converged upon may vary considerably with initial guess.
SCF metadynamics [208] is a technique which can be used to
locate multiple SCF solutions, and thus gain some confidence that
the calculation has converged upon the global minimum.
It works by searching out a solution to the SCF equations. Once found,
the solution is stored, and a biasing potential added so as
to avoid re-converging to the same solution.
More formally, the distance between two solutions, w and x,
can be expressed as dwx2=〈wΨ| w∧ρ− x∧ρ | wΨ〉,
where wΨ is a Slater determinant formed from the orthonormal orbitals,
wϕi, of solution w, and w∧ρ is the one-particle
density operator for wΨ. This definition is equivalent to
dwx2=N−wPμνSνσ·xPστSτμ. and is easily calculated.
dwx2 is bounded by 0 and the number of electrons, and can be taken as the
distance between two solutions. As an example, any singly excited
determinant from an SCF determinant (which will not in general be
another SCF solution), would be a distance 1 away from it.
In a manner analogous to classical metadynamics, to bias against the set of
previously located solutions, x, we create a new Lagrangian,
where 0 represents the present density.
From this we may derive a new effective Fock matrix,
| |
~
F
|
μν
|
=Fμν+ |
x ∑
x
|
Pμν Nx λx e−λx d0x2 |
| | (4.96) |
|
This may be used with very little modification within a standard DIIS procedure to locate multiple solutions.
When close to a new solution, the biasing potential is removed so the location of that solution is not affected by it.
If the calculation ends up re-converging to the same solution, Nx and λx can be modified to avert this.
Once a solution is found it is added to the list of solutions, and the orbitals mixed
to provide a new guess for locating a different solution.
This process can be customized by the REM variables below.
Both DIIS and GDM methods can be used, but it is advisable
to turn on MOM when using DIIS to maintain the orbital ordering.
Post-HF correlation methods can also be applied. By default
they will operate for the last solution located, but this can be changed with the
SCF_MINFIND_RUNCORR variable.
The solutions found through metadynamics also appear to be good approximations to diabatic surfaces
where the electronic structure does not significantly change with geometry.
In situations where there are such multiple electronic states close in energy, an adiabatic state
may be produced by diagonalizing a matrix of these states - Configuration Interaction. As they are distinct solutions of the
SCF equations, these states are non-orthogonal (one cannot be constructed as a single determinant
made out of the orbitals of another), and so the CI is a little more complicated and is a Non-Orthogonal CI.
For more information see the NOCI section in Chapter 6
SCF_SAVEMINIMA
Turn on SCF Metadynamics and specify how many solutions to locate. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use SCF Metadynamics |
n | Attempt to find n distinct SCF solutions. |
RECOMMENDATION:
Perform SCF Orbital metadynamics and attempt to locate
n different SCF solutions. Note that these may not all be minima. Many saddle points are often located.
The last one located will be the one used in any post-SCF treatments.
In systems where there are infinite point groups, this procedure
cannot currently distinguish between spatial rotations of different
densities, so will likely converge on these multiply. |
|
| SCF_READMINIMA
Read in solutions from a previous SCF Metadynamics calculation |
TYPE:
DEFAULT:
OPTIONS:
n | Read in n previous solutions and attempt to locate them all. |
−n | Read in n previous solutions, but only attempt to locate solution n. |
RECOMMENDATION:
This may not actually locate all solutions required and will probably
locate others too. The SCF will also stop when the number of
solutions specified in SCF_SAVEMINIMA are found.
Solutions from other geometries may also be read in and used as starting orbitals.
If a solution is found and matches one that is read in within
SCF_MINFIND_READDISTTHRESH, its orbitals are saved in
that position for any future calculations.
The algorithm works by restarting from the orbitals and density
of a the minimum it is attempting to find. After 10 failed
restarts (defined by SCF_MINFIND_RESTARTSTEPS), it moves
to another previous minimum and attempts to locate that instead.
If there are no minima to find, the restart does random mixing
(with 10 times the normal random mixing parameter).
|
|
|
|
SCF_MINFIND_WELLTHRESH
Specify what SCF_MINFIND believes is the basin of a solution |
TYPE:
DEFAULT:
OPTIONS:
n for a threshold of 10−n |
RECOMMENDATION:
When the DIIS error is less than 10−n, penalties are switched
off to see whether it has converged to a new solution. |
|
| SCF_MINFIND_RESTARTSTEPS
Restart with new orbitals if no minima have been found within this many steps |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If the SCF calculation spends many steps not finding a
solution, lowering this number may speed up solution-finding.
If the system converges to solutions very
slowly, then this number may need to be raised. |
|
|
|
SCF_MINFIND_INCREASEFACTOR
Controls how the height of the penalty function
changes when repeatedly trapped at the same solution |
TYPE:
DEFAULT:
OPTIONS:
abcde | corresponding to a.bcde |
RECOMMENDATION:
If the algorithm converges to a solution which corresponds
to a previously located solution, increase both the
normalization N and the width lambda of the penalty function there. Then do a restart. |
|
| SCF_MINFIND_INITLAMBDA
Control the initial width of the penalty function. |
TYPE:
DEFAULT:
OPTIONS:
abcde | corresponding to ab.cde |
RECOMMENDATION:
The initial inverse-width (i.e., the inverse-variance) of the
Gaussian to place to fill solution's well. Measured in electrons(−1).
Increasing this will repeatedly converging on the same solution. |
|
|
|
SCF_MINFIND_INITNORM
Control the initial height of the penalty function. |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The initial normalization of the Gaussian to place to fill a well. Measured in Hartrees. |
|
| SCF_MINFIND_RANDOMMIXING
Control how to choose new orbitals after locating a solution |
TYPE:
DEFAULT:
00200 meaning .02 radians |
OPTIONS:
abcde corresponding to a.bcde radians |
RECOMMENDATION:
After locating an SCF solution, the orbitals are mixed
randomly to move to a new position in orbital space. For each
occupied and virtual orbital pair picked at random and rotate
between them by a random angle between 0 and this. If this
is negative then use exactly this number, e.g., −15708 will
almost exactly swap orbitals. Any number < −15708 will cause the orbitals to be swapped exactly. |
|
|
|
SCF_MINFIND_NRANDOMMIXES
Control how many random mixes to do to generate new orbitals |
TYPE:
DEFAULT:
OPTIONS:
n | Perform n random mixes. |
RECOMMENDATION:
This is the number of occupied/virtual pairs to attempt to mix,
per separate density (i.e., for unrestricted calculations both
alpha and beta space will get this many rotations). If this
is negative then only mix the highest 25% occupied and lowest 25% virtuals. |
|
| SCF_MINFIND_READDISTTHRESH
The distance threshold at which to consider two solutions the same |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The threshold to regard a minimum as the same as a read in
minimum. Measured in electrons. If two minima are closer
together than this, reduce the threshold to distinguish them. |
|
|
|
SCF_MINFIND_MIXMETHOD
Specify how to select orbitals for random mixing |
TYPE:
DEFAULT:
OPTIONS:
0 | Random mixing: select from any orbital to any orbital. |
1 | Active mixing: select based on energy, decaying with distance from the Fermi level. |
2 | Active Alpha space mixing: select based on energy, decaying with distance from the |
| Fermi level only in the alpha space. |
RECOMMENDATION:
Random mixing will often find very high energy solutions.
If lower energy solutions are desired, use 1 or 2. |
|
| SCF_MINFIND_MIXENERGY
Specify the active energy range when doing Active mixing |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The standard deviation of the Gaussian distribution used
to select the orbitals for mixing (centered on the Fermi level). Measured in Hartree.
To find less-excited solutions, decrease this value |
|
|
|
SCF_MINFIND_RUNCORR
Run post-SCF correlated methods on multiple SCF solutions |
TYPE:
DEFAULT:
OPTIONS:
If this is set > 0, then run correlation methods for all found SCF solutions. |
RECOMMENDATION:
Post-HF correlation methods should function correctly with excited
SCF solutions, but their convergence is often much more difficult owing to intruder states. |
|
4.13 Ground State Method Summary
To summarize the main features of Q-Chem's ground state self-consistent field
capabilities, the user needs to consider:
- Input a molecular geometry ($molecule keyword)
- Cartesian
- Z-matrix
- Read from prior calculations
- Declare the job specification ($remkeyword)
-
JOBTYPE
- Single point
- Optimization
- Frequency
- See Table 4.1 for further options
- BASIS
- Refer to Chapter 7 (note: $basis keyword for user defined basis sets)
- Effective core potentials, as described in Chapter 8
- EXCHANGE
- Linear scaling algorithms for all methods
- Arsenal of exchange density functionals
- User definable functionals and hybrids
- CORRELATION
- DFT or wavefunction-based methods
- Linear scaling (CPU and memory) incorporation of correlation with DFT
- Arsenal of correlation density functionals
- User definable functionals and hybrids
- See Chapter 5 for wavefunction-based correlation methods.
- Exploit Q-Chem's special features
- CFMM, LinK large molecule options
- SCF rate of convergence increased through improved guesses and
alternative minimization algorithms
- Explore novel methods if
desired: CASE approximation, PAOs.
Chapter 5 Wavefunction-Based Correlation Methods
5.1 Introduction
The Hartree-Fock procedure, while often qualitatively correct, is frequently
quantitatively deficient. The deficiency is due to the underlying assumption of
the Hartree-Fock approximation: that electrons move independently
within molecular orbitals subject to an averaged field imposed by the remaining
electrons. The error that this introduces is called the correlation energy and
a wide variety of procedures exist for estimating its magnitude. The purpose of
this Chapter is to introduce the main wavefunction-based methods available in
Q-Chem to describe electron correlation.
Wavefunction-based electron correlation methods concentrate on the design of
corrections to the wavefunction beyond the mean-field Hartree-Fock
description. This is to be contrasted with the density functional theory
methods discussed in the previous Chapter. While density functional methods
yield a description of electronic structure that accounts for electron
correlation subject only to the limitations of present-day functionals (which,
for example, omit dispersion interactions), DFT cannot be systematically
improved if the results are deficient. Wavefunction-based approaches for
describing electron correlation [211,[212] offer this main
advantage. Their main disadvantage is relatively high computational cost,
particularly for the higher-level theories.
There are four broad classes of models for describing electron correlation that
are supported within Q-Chem. The first three directly approximate the full
time-independent Schrödinger equation. In order of increasing accuracy, and
also increasing cost, they are:
- Perturbative treatment of pair correlations between electrons, typically
capable of recovering 80% or so of the correlation energy in stable
molecules.
- Self-consistent treatment of pair correlations between electrons (most often
based on coupled-cluster theory),
capable of recovering on the order of 95% or so of the correlation
energy.
- Non-iterative corrections for higher than double substitutions, which
can account for more than 99% of the correlation energy. They are the
basis of many modern methods that are capable of yielding chemical
accuracy for ground state reaction energies, as exemplified by the
G2 [186] and G3 methods [187].
These methods are discussed in the following subsections.
There is also a fourth class of methods supported in Q-Chem, which have a
different objective. These active space methods aim to obtain a balanced
description of electron correlation in highly correlated systems, such as
diradicals, or along bond-breaking coordinates. Active space methods are
discussed in Section 5.9. Finally, equation-of-motion (EOM)
methods provide tools for describing open-shell and electronically excited
species. Selected configuration interaction (CI) models are also available.
In order to carry out a wavefunction-based electron correlation calculation
using Q-Chem, three $rem variables need to be set:
-
BASIS to specify the basis set (see Chapter 7)
- CORRELATION method for treating Correlation (defaults to
NONE)
- N_FROZEN_CORE frozen core electrons (0 default, optionally
FC, or n)
Additionally, for EOM or CI calculations the number of target states of each type
(excited, spin-flipped, ionized, attached, etc.) in each
irreducible representation (irrep) should be specified (see Section
6.6.7). The level of correlation of the target EOM states
may be different from that used for the reference, and can be specified by
EOM_CORR keyword.
Note that for wavefunction-based correlation methods, the default option for
EXCHANGE is HF (Hartree-Fock). It can therefore be omitted
from the input. If desired, correlated calculations can employ DFT
orbitals by setting EXCHANGE to a specific DFT method (see
Section 5.11).
The full range of ground state wavefunction-based correlation methods
available (i.e. the recognized options to the CORRELATION keyword)
are as follows:.
CORRELATION
Specifies the correlation level of theory, either DFT or wavefunction-based. |
TYPE:
DEFAULT:
OPTIONS:
MP2 | Sections 5.2 and 5.3 |
Local_MP2 | Section 5.4 |
RILMP2 | Section 5.5.1 |
ATTMP2 | Section 5.6.1 |
ATTRIMP2 | Section 5.6.1 |
ZAPT2 | A more efficient restricted open-shell MP2 method [213]. |
MP3 | Section 5.2 |
MP4SDQ | Section 5.2 |
MP4 | Section 5.2 |
CCD | Section 5.7 |
CCD(2) | Section 5.8 |
CCSD | Section 5.7 |
CCSD(T) | Section 5.8 |
CCSD(2) | Section 5.8 |
CCSD(fT) | Section 5.8.3 |
CCSD(dT) | Section 5.8.3 |
QCISD | Section 5.7 |
QCISD(T) | Section 5.8 |
OD | Section 5.7 |
OD(T) | Section 5.8 |
OD(2) | Section 5.8 |
VOD | Section 5.9 |
VOD(2) | Section 5.9 |
QCCD | Section 5.7 |
QCCD(T) | |
QCCD(2) | |
VQCCD | Section 5.9 |
RECOMMENDATION:
Consult the literature for guidance. |
|
5.2 Møller-Plesset Perturbation Theory
5.2.1 Introduction
Møller-Plesset Perturbation Theory [117] is a widely used
method for approximating the correlation energy of molecules. In particular,
second order Møller-Plesset perturbation theory (MP2) is one of the
simplest and most useful levels of theory beyond the Hartree-Fock
approximation. Conventional and local MP2 methods available in Q-Chem are
discussed in detail in Sections 5.3 and 5.4
respectively. The MP3 method is still occasionally used, while MP4 calculations
are quite commonly employed as part of the G2 and G3 thermochemical
methods [186,[187]. In the remainder of this section, the
theoretical basis of Møller-Plesset theory is reviewed.
5.2.2 Theoretical Background
The Hartree-Fock wavefunction Ψ0 and energy E0 are
approximate solutions (eigenfunction and eigenvalue) to the exact
Hamiltonian eigenvalue problem or Schrödinger's electronic wave equation,
Eq. (4.5). The HF wavefunction and energy are, however, exact solutions for the
Hartree-Fock Hamiltonian H0 eigenvalue problem. If we assume that the
Hartree-Fock wavefunction Ψ0 and energy E0 lie near the exact wave
function Ψ and energy E, we can now write the exact Hamiltonian
operator as
where V is the small perturbation and λ is a dimensionless parameter.
Expanding the exact wavefunction and energy in terms of the HF wavefunction
and energy yields
E=E(0)+λE(1)+λ2E(2)+λ3E(3)+… |
| (5.2) |
and
Ψ = Ψ0 +λΨ(1)+λ2Ψ(2)+λ3Ψ(3)+… |
| (5.3) |
Substituting these expansions into the Schrödinger equation and collecting terms according
to powers of λ yields
H0 Ψ(1)+VΨ0 = E(0)Ψ(1)+E(1)Ψ0 |
| (5.5) |
H0 Ψ(2)+VΨ(1)=E(0)Ψ(2)+E(1)Ψ(1)+E(2)Ψ0 |
| (5.6) |
and so forth. Multiplying each of the above equations by Ψ0 and
integrating over all space yields the following expression for the nth-order
(MPn) energy:
Thus, the Hartree-Fock energy
is simply the sum of the zeroth- and first- order energies
The correlation energy can then be written
Ecorr = E0(2) +E0(3) +E0(4) +… |
| (5.12) |
of which the first term is the MP2 energy.
It can be shown that the MP2 energy can be written (in terms of spin-orbitals) as
E0(2) = − |
1
4
|
|
virt ∑
ab
|
|
occ ∑
ij
|
|
| 〈 ab || ij 〉 |2
εa +εb −εi −εj
|
|
| (5.13) |
where
〈 ab|| ij 〉 = 〈 ab | ab ij ij 〉 −〈 ab | ab ji ji 〉 |
| (5.14) |
and
〈 ab | ab cd cd 〉 = | ⌠ ⌡
|
ψa (r1 )ψc (r1 ) | ⎡ ⎣
|
1
r12
| ⎤ ⎦
|
ψb (r2 )ψd (r2 )dr1 dr2 |
| (5.15) |
which can be written in terms of the two-electron repulsion integrals
〈 ab | ab cd cd 〉 = |
∑
μ
|
|
∑
ν
|
|
∑
λ
|
|
∑
σ
|
Cμa Cνc Cλb Cσd ( μν|λσ ) |
| (5.16) |
Expressions for higher order terms follow similarly, although with much greater
algebraic and computational complexity. MP3 and particularly MP4 (the third and
fourth order contributions to the correlation energy) are both occasionally
used, although they are increasingly supplanted by the coupled-cluster methods
described in the following sections. The disk and memory requirements for MP3
are similar to the self-consistent pair correlation methods discussed in
Section 5.7 while the computational cost of MP4 is similar to the
"(T)" corrections discussed in Section 5.8.
5.3 Exact MP2 Methods
5.3.1 Algorithm
Second order Møller-Plesset theory (MP2) [117] probably the
simplest useful wavefunction-based electron correlation method. Revived in
the mid-1970s, it remains highly popular today, because it offers systematic
improvement in optimized geometries and other molecular properties relative to
Hartree-Fock (HF) theory [6]. Indeed, in a recent comparative
study of small closed-shell molecules [214], MP2 outperformed
much more expensive singles and doubles coupled-cluster theory for such
properties! Relative to state-of-the-art Kohn-Sham density functional
theory (DFT) methods, which are the most economical methods to account for
electron correlation effects, MP2 has the advantage of properly incorporating
long-range dispersion forces. The principal weaknesses of MP2 theory are for
open shell systems, and other cases where the HF determinant is a poor starting
point.
Q-Chem contains an efficient conventional semi-direct method to evaluate the
MP2 energy and gradient [215]. These methods require OVN memory
(O, V, N are the numbers of occupied, virtual and total orbitals,
respectively), and disk space which is bounded from above by OVN2/2. The
latter can be reduced to IVN2/2 by treating the occupied orbitals in batches
of size I, and re-evaluating the two-electron integrals O/I times. This
approach is tractable on modern workstations for energy and gradient
calculations of at least 500 basis functions or so, or molecules of between 15
and 30 first row atoms, depending on the basis set size. The computational cost
increases between the 3rd and 5th power of the size of the molecule,
depending on which part of the calculation is time-dominant.
The algorithm and implementation in Q-Chem is improved over earlier
methods [216,[217], particularly in the following areas:
- Uses pure functions, as opposed to Cartesians, for all fifth-order steps.
This leads to large computational savings for basis sets containing pure
functions.
- Customized loop unrolling for improved efficiency.
- The sortless semi-direct method avoids a read and write operation
resulting in a large I/O savings.
- Reduction in disk and memory usage.
- No extra integral evaluation for gradient calculations.
- Full exploitation of frozen core approximation.
The implementation offers the user the following alternatives:
- Direct algorithm (energies only).
- Disk-based sortless semi-direct algorithm (energies and gradients).
- Local occupied orbital method (energies only).
The semi-direct algorithm is the only choice for gradient calculations. It is
also normally the most efficient choice for energy calculations. There are two
classes of exceptions:
- If the amount of disk space available is not significantly larger than
the amount of memory available, then the direct algorithm is preferred.
- If the calculation involves a very large basis set, then the local
orbital method may be faster, because it performs the transformation in a
different order. It does not have the large memory requirement (no
OVN array needed), and always evaluates the integrals four times. The
AO2MO_DISK option is also ignored in this algorithm, which
requires up to O2VN megabytes of disk space.
There are three important options that should be wisely chosen by the user in
order to exploit the full efficiency of Q-Chem's direct and semi-direct MP2
methods (as discussed above, the LOCAL_OCCUPIED method has different
requirements).
-
MEM_STATIC : The value specified for this $rem variable must be
sufficient to permit efficient integral evaluation (10-80Mb) and to hold
a large temporary array whose size is OVN, the product of the number of
occupied, virtual and total numbers of orbitals.
- AO2MO_DISK: The value specified for this $rem variable should be
as large as possible (i.e., perhaps 80% of the free space on your
$QCSCRATCH partition where temporary job files are held). The
value of this variable will determine how many times the two-electron
integrals in the atomic orbital basis must be re-evaluated, which is a
major computational step in MP2 calculations.
- N_FROZEN_CORE: The computational requirements for MP2 are
proportional to the number of occupied orbitals for some steps, and the
square of that number for other steps. Therefore the CPU time can be
significantly reduced if your job employs the frozen core approximation.
Additionally the memory and disk requirements are reduced when the frozen
core approximation is employed.
5.3.2 The Definition of Core Electron
The number of core electrons in an atom is relatively well defined, and
consists of certain atomic shells, (note that ECPs are available in
`small-core' and `large-core' varieties, see Chapter 8 for
further details). For example, in phosphorus the core consists of 1s, 2s,
and 2p shells, for a total of ten electrons. In molecular systems, the core
electrons are usually chosen as those occupying the n/2 lowest energy
orbitals, where n is the number of core electrons in the constituent atoms.
In some cases, particularly in the lower parts of the periodic table, this
definition is inappropriate and can lead to significant errors in the
correlation energy. Vitaly Rassolov has implemented an alternative definition
of core electrons within Q-Chem which is based on a Mulliken population
analysis, and which addresses this problem [218].
The current implementation is restricted to n-kl type basis sets such as
3-21 or 6-31, and related bases such as 6-31+G(d). There are essentially two
cases to consider, the outermost 6G functions (or 3G in the case of the 3-21G
basis set) for Na, Mg, K and Ca, and the 3d functions for the elements Ga-Kr.
Whether or not these are treated as core or valence is determined by the
CORE_CHARACTER $rem, as summarized in Table 5.3.2.
CORE_CHARACTER | Outermost 6G (3G) | 3d (Ga-Kr) |
| for Na, Mg, K, Ca | | |
1 | valence | valence |
2 | valence | core |
3 | core | core |
4 | core | valence |
Table 5.1: A summary of the effects of different core definitions
5.3.3 Algorithm Control and Customization
The direct and semi-direct integral transformation algorithms used by Q-Chem
(e.g., MP2, CIS(D)) are limited by available disk space, D, and memory, C,
the number of basis functions, N, the number of virtual orbitals, V and the
number of occupied orbitals, O, as discussed above. The generic description
of the key $rem variables are:
MEM_STATIC
Sets the memory for Fortran AO integral calculation and transformation modules. |
TYPE:
DEFAULT:
64 | corresponding to 64 Mb. |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
For direct and semi-direct MP2 calculations, this must exceed OVN +
requirements for AO integral evaluation (32-160 Mb), as discussed above. |
|
| MEM_TOTAL
Sets the total memory available to Q-Chem, in megabytes. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
Use default, or set to the physical memory of your machine. Note that if more
than 1GB is specified for a CCMAN job, the memory is allocated as
follows |
12% | MEM_STATIC |
50% | CC_MEMORY |
35% | Other memory requirements:
|
|
|
|
|
AO2MO_DISK
Sets the amount of disk space (in megabytes) available for MP2 calculations. |
TYPE:
DEFAULT:
2000 | Corresponding to 2000 Mb. |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
Should be set as large as possible, discussed in Section 5.3.1. |
|
| CD_ALGORITHM
Determines the algorithm for MP2 integral transformations. |
TYPE:
DEFAULT:
OPTIONS:
DIRECT | Uses fully direct algorithm (energies only). |
SEMI_DIRECT | Uses disk-based semi-direct algorithm. |
LOCAL_OCCUPIED | Alternative energy algorithm (see 5.3.1). |
RECOMMENDATION:
Semi-direct is usually most efficient, and will normally be chosen by default. |
|
|
|
N_FROZEN_CORE
Sets the number of frozen core orbitals in a post-Hartree-Fock calculation. |
TYPE:
DEFAULT:
OPTIONS:
FC | Frozen Core approximation (all core orbitals frozen). |
n | Freeze n core orbitals. |
RECOMMENDATION:
While the default is not to freeze orbitals, MP2 calculations are more
efficient with frozen core orbitals. Use FC if possible. |
|
| N_FROZEN_VIRTUAL
Sets the number of frozen virtual orbitals in a post-Hartree-Fock
calculation. |
TYPE:
DEFAULT:
OPTIONS:
n | Freeze n virtual orbitals. |
RECOMMENDATION:
|
|
|
CORE_CHARACTER
Selects how the core orbitals are determined in the frozen-core
approximation. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use energy-based definition. |
1-4 | Use Mulliken-based definition (see Table 5.3.2 for details). |
RECOMMENDATION:
Use default, unless performing calculations on molecules with heavy elements. |
|
| PRINT_CORE_CHARACTER
Determines the print level for the CORE_CHARACTER option. |
TYPE:
DEFAULT:
OPTIONS:
0 | No additional output is printed. |
1 | Prints core characters of occupied MOs. |
2 | Print level 1, plus prints the core character of AOs. |
RECOMMENDATION:
Use default, unless you are uncertain about what the core character is. |
|
|
|
5.3.4 Example
Example 5.0 Example of an MP2/6-31G* calculation employing the frozen core
approximation. Note that the EXCHANGE $rem variable will default to
HF
$molecule
0 1
O
H1 O oh
H2 O oh H1 hoh
oh = 1.01
hoh = 105
$end
$rem
CORRELATION mp2
BASIS 6-31g*
N_FROZEN_CORE fc
$end
5.4 Local MP2 Methods
5.4.1 Local Triatomics in Molecules (TRIM) Model
The development of what may be called "fast methods" for evaluating electron
correlation is a problem of both fundamental and practical importance, because
of the unphysical increases in computational complexity with molecular size
which afflict "exact" implementations of electron correlation methods.
Ideally, the development of fast methods for treating electron correlation
should not impact either model errors or numerical errors associated with the
original electron correlation models. Unfortunately this is not possible at
present, as may be appreciated from the following rough argument. Spatial
locality is what permits re-formulations of electronic structure methods that
yield the same answer as traditional methods, but faster. The one-particle
density matrix decays exponentially with a rate that relates to the HOMO-LUMO
gap in periodic systems. When length scales longer than this characteristic
decay length are examined, sparsity will emerge in both the one-particle
density matrix and also pair correlation amplitudes expressed in terms of
localized functions. Very roughly, such a length scale is about 5 to 10 atoms
in a line, for good insulators such as alkanes. Hence sparsity emerges beyond
this number of atoms in 1-D, beyond this number of atoms squared in 2-D, and
this number of atoms cubed in 3-D. Thus for three-dimensional systems,
locality only begins to emerge for systems of between hundreds and thousands of
atoms.
If we wish to accelerate calculations on systems below this size regime, we
must therefore introduce additional errors into the calculation, either as
numerical noise through looser tolerances, or by modifying the theoretical
model, or perhaps both. Q-Chem's approach to local electron correlation is
based on modifying the theoretical models describing correlation with an
additional well-defined local approximation. We do not attempt to accelerate
the calculations by introducing more numerical error because of the
difficulties of controlling the error as a function of molecule size, and the
difficulty of achieving reproducible significant results. From this
perspective, local correlation becomes an integral part of specifying the
electron correlation treatment. This means that the considerations necessary
for a correlation treatment to qualify as a well-defined theoretical model
chemistry apply equally to local correlation modeling. The local approximations
should be
- Size-consistent: meaning that the energy of a super-system of two
non-interacting molecules should be the sum of the energy obtained from
individual calculations on each molecule.
- Uniquely defined: Require no input beyond nuclei, electrons, and
an atomic orbital basis set. In other words, the model should be uniquely
specified without customization for each molecule.
- Yield continuous potential energy surfaces: The model
approximations should be smooth, and not yield energies that exhibit
jumps as nuclear geometries are varied.
To ensure that these model chemistry criteria are met, Q-Chem's local MP2
methods [219,[220] express the double substitutions (i.e., the
pair correlations) in a redundant basis of atom-labeled functions. The
advantage of doing this is that local models satisfying model chemistry
criteria can be defined by performing an atomic truncation of the double
substitutions. A general substitution in this representation will then involve
the replacement of occupied functions associated with two given atoms by empty
(or virtual) functions on two other atoms, coupling together four different
atoms. We can force one occupied to virtual substitution (of the two that
comprise a double substitution) to occur only between functions on the same
atom, so that only three different atoms are involved in the double
substitution. This defines the triatomics in molecules (TRIM) local
model for double substitutions. The TRIM model offers the potential for
reducing the computational requirements of exact MP2 theory by a factor
proportional to the number of atoms. We could also force each occupied to
virtual substitution to be on a given atom, thereby defining a more drastic
diatomics in molecules (DIM) local correlation model.
The simplest atom-centered basis that is capable of spanning the occupied
space is a minimal basis of core and valence atomic orbitals on each
atom. Such a basis is necessarily redundant because it also contains sufficient
flexibility to describe the empty valence anti-bonding orbitals necessary to
correctly account for non-dynamical electron correlation effects such as
bond-breaking. This redundancy is actually important for the success of the
atomic truncations because occupied functions on adjacent atoms to some extent
describe the same part of the occupied space. The minimal functions we use to
span the occupied space are obtained at the end of a large basis set
calculation, and are called extracted polarized atomic orbitals
(EPAOs) [221]. We discuss them briefly below. It is even possible to
explicitly perform an SCF calculation in terms of a molecule-optimized minimal
basis of polarized atomic orbitals (PAOs) (see Chapter 4).
To span the virtual space, we use the full set of atomic orbitals,
appropriately projected into the virtual space.
We summarize the situation. The number of functions spanning the occupied
subspace will be the minimal basis set dimension, M, which is greater than
the number of occupied orbitals, O, by a factor of up to about two. The
virtual space is spanned by the set of projected atomic orbitals whose number
is the atomic orbital basis set size N, which is fractionally greater than
the number of virtuals VNO. The number of double substitutions in such a
redundant representation will be typically three to five times larger than the
usual total. This will be more than compensated by reducing the number of
retained substitutions by a factor of the number of atoms, A, in the local
triatomics in molecules model, or a factor of A2 in the diatomics in
molecules model.
The local MP2 energy in the TRIM and DIM models are given by the following
expressions, which can be compared against the full MP2 expression given
earlier in Eq. (5.13). First, for the DIM model:
EDIM MP2 = − |
1
2
|
|
∑
―P,―Q
|
|
( |
-
P
|
| |
-
Q
|
) ( |
-
P
|
|| |
-
Q
|
) |
∆―P + ∆―Q
|
|
| (5.17) |
The sums run over the linear number of atomic single excitations after they
have been canonicalized. Each term in the denominator is thus an energy
difference between occupied and virtual levels in this local basis.
Similarly, the TRIM model corresponds to the following local MP2 energy:
ETRIM MP2 = − |
∑
―P,jb
|
|
∆―P + εb − εj
|
− EDIM MP2 |
| (5.18) |
where the sum is now mixed between atomic substitutions ―P, and nonlocal
occupied j to virtual b substitutions. See
Refs. for a full derivation and discussion.
The accuracy of the local TRIM and DIM models has been tested in a series of
calculations [219,[220]. In particular, the TRIM model
has been shown to be quite faithful to full MP2 theory via the following tests:
- The TRIM model recovers around 99.7% of the MP2 correlation energy for
covalent bonding. This is significantly higher than the roughly 98-99%
correlation energy recovery typically exhibited by the Saebo-Pulay local
correlation method [222]. The DIM model recovers around
95% of the correlation energy.
- The performance of the TRIM model for relative energies is very robust,
as shown in Ref. for the challenging case of
torsional barriers in conjugated molecules. The RMS error in these
relative energies is only 0.031 kcal/mol, as compared to around 1
kcal/mol when electron correlation effects are completely neglected.
- For the water dimer with the aug-cc-pVTZ basis, 96% of the MP2
contribution to the binding energy is recovered with the TRIM model, as
compared to 62% with the Saebo-Pulay local correlation method.
- For calculations of the MP2 contribution to the G3 and G3(MP2) energies
with the larger molecules in the G3-99 database [188],
introduction of the TRIM approximation results in an RMS error relative
to full MP2 theory of only 0.3 kcal/mol, even though the absolute
magnitude of these quantities is on the order of tens of kcal/mol.
5.4.2 EPAO Evaluation Options
When a local MP2 job (requested by the LOCAL_MP2 option for
CORRELATION) is performed, the first new step after the SCF
calculation is converged is to extract a minimal basis of polarized atomic
orbitals (EPAOs) that spans the occupied space. There are three valid choices
for this basis, controlled by the PAO_METHOD and
EPAO_ITERATE keywords described below.
- Uniterated EPAOs: The initial guess EPAOs are the default for
local MP2 calculations, and are defined as follows. For each atom, the
covariant density matrix (SPS) is diagonalized, giving eigenvalues which
are approximate natural orbital occupancies, and eigenvectors which are
corresponding atomic orbitals. The m eigenvectors with largest
populations are retained (where m is the minimal basis dimension for
the current atom). This nonorthogonal minimal basis is symmetrically
orthogonalized, and then modified as discussed in
Ref. to ensure that these functions rigorously span the
occupied space of the full SCF calculation that has just been performed.
These orbitals may be denoted as EPAO(0) to indicate that no iterations
have been performed after the guess. In general, the quality of the local
MP2 results obtained with this option is very similar to the EPAO option
below, but it is much faster and fully robust. For the example of the
torsional barrier calculations discussed above [219], the
TRIM RMS deviations of 0.03 kcal/mol from full MP2 calculations are
increased to only 0.04 kcal/mol when EPAO(0) orbitals are employed rather
than EPAOs.
- EPAOs: EPAOs are defined by minimizing a localization functional
as described in Ref. . These functions were designed
to be suitable for local MP2 calculations, and have yielded excellent
results in all tests performed so far. Unfortunately the functional is
difficult to converge for large molecules, at least with the algorithms
that have been developed to this stage. Therefore it is not the default,
but is switched on by specifying a (large) value for
EPAO_ITERATE, as discussed below.
- PAO: If the SCF calculation is performed in terms of a
molecule-optimized minimal basis, as described in Chapter 4, then the
resulting PAO-SCF calculation can be corrected with either conventional
or local MP2 for electron correlation. PAO-SCF calculations alter the
SCF energy, and are therefore not the default. This can be enabled by
specifying PAO_METHOD as PAO, in a job which also requests
CORRELATION as LOCAL_MP2.
PAO_METHOD
Controls the type of PAO calculations requested. |
TYPE:
DEFAULT:
EPAO | For local MP2, EPAOs are chosen by default. |
OPTIONS:
EPAO | Find EPAOs by minimizing delocalization function. |
PAO | Do SCF in a molecule-optimized minimal basis. |
RECOMMENDATION:
|
| EPAO_ITERATE
Controls iterations for EPAO calculations (see PAO_METHOD). |
TYPE:
DEFAULT:
0 | Use uniterated EPAOs based on atomic blocks of SPS. |
OPTIONS:
n | Optimize the EPAOs for up to n iterations. |
RECOMMENDATION:
Use default. For molecules that are not too large, one can test the
sensitivity of the results to the type of minimal functions by the use of
optimized EPAOs in which case a value of n=500 is reasonable. |
|
|
|
EPAO_WEIGHTS
Controls algorithm and weights for EPAO calculations (see PAO_METHOD). |
TYPE:
DEFAULT:
115 | Standard weights, use 1st and 2nd order optimization |
OPTIONS:
15 | Standard weights, with 1st order optimization only. |
RECOMMENDATION:
Use default, unless convergence failure is encountered. |
|
5.4.3 Algorithm Control and Customization
A local MP2 calculation (requested by the LOCAL_MP2 option for
CORRELATION) consists of the following steps:
- After the SCF is converged, a minimal basis of EPAOs are obtained.
- The TRIM (and DIM) local MP2 energies are then evaluated (gradients are
not yet available).
Details of the efficient implementation of the local MP2 method described above
are reported in the recent thesis of Dr. Michael Lee [223]. Here we
simply summarize the capabilities of the program. The computational advantage
associated with these local MP2 methods varies depending upon the size of
molecule and the basis set. As a rough general estimate, TRIM MP2 calculations
are feasible on molecule sizes about twice as large as those for which
conventional MP2 calculations are feasible on a given computer, and this is
their primary advantage. Our implementation is well suited for large basis set
calculations. The AO basis two-electron integrals are evaluated four times.
DIM MP2 calculations are performed as a by-product of TRIM MP2 but no
separately optimized DIM algorithm has been implemented.
The resource requirements for local MP2 calculations are as follows:
- Memory: The memory requirement for the integral transformation
does not exceed OON, and is thresholded so that it asymptotically grows
linearly with molecule size. Additional memory of approximately 32N2
is required to complete the local MP2 energy evaluation.
- Disk: The disk space requirement is only about 8OVN, but is
not governed by a threshold. This is a very large reduction from the case
of a full MP2 calculation, where, in the case of four integral
evaluations, OVN2/4 disk space is required. As the local MP2 disk
space requirement is not adjustable, the AO2MO_DISK keyword is
ignored for LOCAL_MP2 calculations.
The evaluation of the local MP2 energy does not require any further
customization. An adequate amount of MEM_STATIC (80 to 160 Mb) should
be specified to permit efficient AO basis two-electron integral evaluation,
but all large scratch arrays are allocated from MEM_TOTAL.
5.4.4 Examples
Example 5.0 A relative energy evaluation using the local TRIM model for MP2
with the 6-311G** basis set. The energy difference is the internal rotation
barrier in propenal, with the first geometry being planar trans, and the second
the transition structure.
$molecule
0 1
C
C 1 1.32095
C 2 1.47845 1 121.19
O 3 1.18974 2 123.83 1 180.00
H 1 1.07686 2 121.50 3 0.00
H 1 1.07450 2 122.09 3 180.00
H 2 1.07549 1 122.34 3 180.00
H 3 1.09486 2 115.27 4 180.00
$end
$rem
CORRELATION local_mp2
BASIS 6-311g**
$end
@@@
$molecule
0 1
C
C 1 1.31656
C 2 1.49838 1 123.44
O 3 1.18747 2 123.81 1 92.28
H 1 1.07631 2 122.03 3 -0.31
H 1 1.07484 2 121.43 3 180.28
H 2 1.07813 1 120.96 3 180.34
H 3 1.09387 2 115.87 4 179.07
$end
$rem
CORRELATION local_mp2
BASIS 6-311g**
$end
5.5 Auxiliary Basis Set (Resolution-of-Identity) MP2 Methods
For a molecule of fixed size, increasing the number of basis functions
per atom, n, leads to O(n4) growth in the number of significant
four-center two-electron integrals, since the number of non-negligible
product charge distributions, |μν〉, grows as O(n2). As a result,
the use of large (high-quality) basis expansions is computationally costly.
Perhaps the most practical way around this "basis set quality" bottleneck is
the use of auxiliary basis expansions [224,[225,[226].
The ability to use auxiliary
basis sets to accelerate a variety of electron correlation methods, including
both energies and analytical gradients, is a major feature of
Q-Chem.
The auxiliary basis {|K〉} is used to approximate products of Gaussian
basis functions:
|μν〉 ≈ | |
~
μν
|
〉 = |
∑
K
|
|K〉CμνK |
| (5.19) |
Auxiliary basis expansions were introduced long ago, and are now widely
recognized as an effective and powerful approach, which is sometimes
synonymously called resolution of the identity (RI) or density fitting (DF).
When using auxiliary basis expansions, the rate of growth of computational cost
of large-scale electronic structure calculations with n is reduced to
approximately n3.
If n is fixed and molecule size increases, auxiliary basis expansions reduce
the pre-factor associated with the computation, while not altering the
scaling. The important point is that the pre-factor can be reduced by 5 or 10
times or more. Such large speedups are possible because the number of
auxiliary functions required to obtain reasonable accuracy, X, has been shown
to be only about 3 or 4 times larger than N.
The auxiliary basis expansion coefficients, C, are determined by
minimizing the deviation between the fitted distribution and the actual
distribution, 〈μν−~μν | μν−~μν〉, which leads to the following set of linear equations:
|
∑
L
|
〈 K| L 〉 CμνL = 〈 K| μν 〉 |
| (5.20) |
Evidently solution of the fit equations requires only two- and three-center
integrals, and as a result the (four-center) two-electron integrals can be
approximated as the following optimal expression for a given choice of
auxiliary basis set:
〈μν|λσ〉 ≈ 〈 |
~
μν
|
| |
~
λσ
|
〉 = |
∑
| K,LCμL〈L|K 〉CλσK |
| (5.21) |
In the limit where the auxiliary basis is complete (i.e. all products of
AOs are included), the fitting procedure described above will be exact.
However, the auxiliary basis is invariably incomplete (as mentioned above,
X ≈ 3N) because this is essential for obtaining increased computational
efficiency.
Standardized auxiliary basis sets have been developed by the
Karlsruhe group for second order perturbation (MP2) calculations [227,[228]
of the correlation energy. With these basis
sets, small absolute errors (e.g., below 60 μHartree per atom in MP2) and
even smaller relative errors in computed energies are found, while the
speed-up can be 3-30 fold. This development has made the routine use of
auxiliary basis sets for electron correlation calculations possible.
Correlation calculations that can take advantage of auxiliary basis expansions
are described in the remainder of this section (MP2, and MP2-like methods) and
in Section 5.14 (simplified active space coupled cluster methods
such as PP, PP(2), IP, RP). These methods automatically employ auxiliary basis
expansions when a valid choice of auxiliary basis set is supplied using the
AUX_BASIS keyword which is used in the same way as the BASIS
keyword.
The PURECART $rem is no longer needed here, even if using
a auxiliary basis that does not have a predefined value. There
is a built-in automatic procedure that provides the effect
of the PURECART $rem in these cases by default.
5.5.1 RI-MP2 Energies and Gradients.
Following common convention, the MP2 energy evaluated approximately using an
auxiliary basis is referred to as "resolution of the identity" MP2, or RI-MP2
for short. RI-MP2 energy and gradient calculations are enabled simply by
specifying the AUX_BASIS keyword discussed above. As discussed above,
RI-MP2 energies [224] and gradients [229,[230]
are significantly faster than the best
conventional MP2 energies and gradients, and cause negligible loss of accuracy,
when an appropriate standardized auxiliary basis set is employed. Therefore
they are recommended for jobs where turnaround time is an issue. Disk
requirements are very modest; one merely needs to hold various 3-index
arrays. Memory requirements grow more slowly than our conventional MP2
algorithms-only quadratically with molecular size. The minimum memory
requirement is approximately 3X2, where X is the number of auxiliary
basis functions, for both energy and analytical gradient evaluations, with some
additional memory being necessary for integral evaluation and other small
arrays.
In fact, for molecules that are not too large (perhaps no more than 20 or 30
heavy atoms) the RI-MP2 treatment of electron correlation is so efficient that
the computation is dominated by the initial Hartree-Fock calculation. This is
despite the fact that as a function of molecule size, the cost of the RI-MP2
treatment still scales more steeply with molecule size (it is just that the
pre-factor is so much smaller with the RI approach). Its scaling remains
5th order with the size of the molecule, which only dominates the
initial SCF calculation for larger molecules. Thus, for RI-MP2 energy
evaluation on moderate size molecules (particularly in large basis sets), it is
desirable to use the dual basis HF method to further improve execution times
(see Section 4.7).
5.5.2 Example
Example 5.0 Q-Chem input for an RI-MP2 geometry optimization.
$molecule
0 1
O
H 1 0.9
F 1 1.4 2 100.
$end
$rem
JOBTYPE opt
CORRELATION rimp2
BASIS cc-pvtz
AUX_BASIS rimp2-cc-pvtz
SYMMETRY false
$end
For the size of required memory, the followings need to be considered.
| MEM_STATIC
Sets the memory for AO-integral evaluations and their transformations. |
TYPE:
DEFAULT:
64 | corresponding to 64 Mb. |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
For RI-MP2 calculations, 150(ON + V) of MEM_STATIC is required.
Because a number of matrices with N2 size also need to be
stored, 32-160 Mb of additional MEM_STATIC is needed. |
|
|
|
MEM_TOTAL
Sets the total memory available to Q-Chem, in megabytes. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
Use default, or set to the physical memory of your machine.
The minimum requirement is 3X2. |
|
5.5.3 OpenMP Implementation of RI-MP2
An OpenMP RI-MP2 energy algorithm is used by default in Q-Chem 4.1 onward. This can be invoked by using CORR=primp2 for older versions, but note that in 4.01 and below, only RHF / RI-MP2 was supported. Now UHF / RI-MP2 is supported, and the formation of the `B' matrices as well as three center integrals are parallelized. This algorithm uses the remaining memory from the MEM_TOTAL allocation for all computation, which can drastically reduce hard drive reads in the formation of t-amplitudes.
Example 5.0 Example of OpenMP-parallel RI-MP2 job.
$molecule
0 1
C1
H1 C1 1.0772600000
H2 C1 1.0772600000 H1 131.6082400000
$end
$rem
jobtype SP
exchange HF
correlation pRIMP2
basis cc-pVTZ
aux_basis rimp2-cc-pVTZ
purecart 11111
symmetry false
thresh 12
scf_convergence 8
max_sub_file_num 128
!time_mp2 true
$end
5.5.4 GPU Implementation of RI-MP2
5.5.4.1 Requirements
Q-Chem currently offers the possibility of accelerating RI-MP2 calculations
using graphics processing units (GPUs).
Currently, this is implemented for CUDA-enabled NVIDIA graphics
cards only, such as (in historical order from 2008) the GeForce,
Quadro, Tesla and Fermi cards. More information about
CUDA-enabled cards is available at
- ://www.nvidia.com/object/cuda_gpus.html@
-
://www.nvidia.com/object/cuda_gpus.html@
It should be noted that these GPUs have specific power and motherboard requirements.
Software requirements include the installation of the appropriate NVIDIA
CUDA driver (at least version 1.0, currently 3.2) and linear algebra library,
CUBLAS (at least version 1.0, currently 2.0). These can be downloaded jointly
in NVIDIA's developer website:
- ://developer.nvidia.com/object/cuda_3_2_downloads.html@
-
://developer.nvidia.com/object/cuda_3_2_downloads.html@
We have implemented a mixed-precision algorithm in order to
get better than single precision when users only have
single-precision GPUs. This is accomplished by noting that RI-MP2 matrices
have a large fraction of numerically "small" elements and a
small fraction of numerically "large" ones. The latter can
greatly affect the accuracy of the calculation in single-precision only
calculations, but calculation
involves a relatively small number of compute cycles. So, given a threshold
value δ, we perform a separation between "small" and "large"
elements and accelerate the former compute-intensive operations
using the GPU (in single-precision) and compute the latter
on the CPU (using double-precision). We are thus able to
determine how much "double-precision" we desire by tuning
the δ parameter, and tailoring the balance between computational speed and accuracy.
5.5.4.2 Options
CUDA_RI-MP2
Enables GPU implementation of RI-MP2 |
TYPE:
DEFAULT:
OPTIONS:
FALSE | GPU-enabled MGEMM off |
TRUE | GPU-enabled MGEMM on |
RECOMMENDATION:
Necessary to set to 1 in order to run GPU-enabled RI-MP2 |
|
| USECUBLAS_THRESH
Sets threshold of matrix size sent to GPU
(smaller size not worth sending to GPU). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default value. Anything less can
seriously hinder the GPU acceleration |
|
|
|
USE_MGEMM
Use the mixed-precision matrix scheme (MGEMM)
if you want to make calculations in your card in single-precision
(or if you have a single-precision-only GPU), but leave some parts
of the RI-MP2 calculation in double precision) |
TYPE:
DEFAULT:
OPTIONS:
0 | MGEMM disabled |
1 | MGEMM enabled |
RECOMMENDATION:
Use when having single-precision cards |
|
| MGEMM_THRESH
Sets MGEMM threshold to determine the separation
between "large" and "small" matrix elements.
A larger threshold value will result in a value closer
to the single-precision result. Note that the desired factor
should be multiplied by 10000 to ensure an integer value. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
For small molecules and basis sets up to triple-ζ, the
default value suffices to not deviate too much from the
double-precision values. Care should be taken to reduce
this number for larger molecules and also larger basis-sets. |
|
|
|
5.5.4.3 Input examples
Example 5.0 RI-MP2 double-precision calculation
$comment
RI-MP2 double-precision example
$end
$molecule
0 1
c
h1 c 1.089665
h2 c 1.089665 h1 109.47122063
h3 c 1.089665 h1 109.47122063 h2 120.
h4 c 1.089665 h1 109.47122063 h2 -120.
$end
$rem
jobtype sp
exchange hf
correlation rimp2
basis cc-pvdz
aux_basis rimp2-cc-pvdz
cuda_rimp2 1
$end
Example 5.0 RI-MP2 calculation with MGEMM
$comment
MGEMM example
$end
$molecule
0 1
c
h1 c 1.089665
h2 c 1.089665 h1 109.47122063
h3 c 1.089665 h1 109.47122063 h2 120.
h4 c 1.089665 h1 109.47122063 h2 -120.
$end
$rem
jobtype sp
exchange hf
correlation rimp2
basis cc-pvdz
aux_basis rimp2-cc-pvdz
cuda_rimp2 1
USE_MGEMM 1
mgemm_thresh 10000
$end
5.5.5 Opposite-Spin (SOS-MP2, MOS-MP2, and O2) Energies and Gradients
The accuracy of MP2 calculations can be significantly improved by
semi-empirically scaling the opposite-spin and same-spin correlation components
with separate scaling factors, as shown by Grimme [231]. Results
of similar quality can be obtained by just scaling the opposite spin
correlation (by 1.3), as was recently demonstrated [232].
Furthermore this SOS-MP2 energy can be evaluated using the RI approximation
together with a Laplace transform technique, in effort that scales only with
the 4th power of molecular size. Efficient algorithms for the energy [232]
and the analytical gradient [233] of this method
are available in Q-Chem 3.0, and offer advantages in speed over MP2 for
larger molecules, as well as statistically significant improvements in
accuracy.
However, we note that the SOS-MP2 method does systematically underestimate
long-range dispersion (for which the appropriate scaling factor is 2 rather
than 1.3) but this can be accounted for by making the scaling factor
distance-dependent, which is done in the modified opposite spin variant
(MOS-MP2) that has recently been proposed and tested [234]. The
MOS-MP2 energy and analytical gradient are also available in Q-Chem 3.0 at a
cost that is essentially identical with SOS-MP2. Timings show that the
4th-order implementation of SOS-MP2 and MOS-MP2 yields substantial
speedups over RI-MP2 for molecules in the 40 heavy atom regime and larger. It is
also possible to customize the scale factors for particular applications, such
as weak interactions, if required.
A fourth order scaling SOS-MP2 / MOS-MP2 energy calculation can be invoked by
setting the CORRELATION keyword to either SOSMP2 or
MOSMP2. MOS-MP2 further requires the specification of the $rem
variable OMEGA, which tunes the level of attenuation of the MOS
operator [234]:
gω(r12) = |
1
r12
|
+cMOS |
erf( ωr12 )
r12
|
|
| (5.22) |
The recommended OMEGA value is ω = 0.6 a.u. [234].
The fast algorithm makes use of auxiliary basis expansions
and therefore, the keyword AUX_BASIS should be
set consistently with the user's choice of BASIS. Fourth-order
scaling analytical gradient for both SOS-MP2 and MOS-MP2 are also available and
is automatically invoked when JOBTYPE is set to OPT or
FORCE. The minimum memory requirement is 3X2, where X = the
number of auxiliary basis functions, for both energy and analytical gradient
evaluations. Disk space requirement for closed shell calculations is ∼ 2OVX for energy evaluation and ∼ 4OVX for analytical gradient evaluation.
More recently, Brueckner orbitals (BO) are introduced into SOSMP2 and MOSMP2 methods to
resolve the problems of symmetry breaking and spin contamination
that are often associated with Hartree-Fock orbitals.
So the molecular orbitals are optimized with the mean-field energy
plus a correlation energy taken as the opposite-spin component of the second-order
many-body correlation energy, scaled by an empirically chosen parameter.
This "optimized second-order opposite-spin" abbreviated as O2 method [235]
requires fourth-order computation on each orbital iteration. O2 is shown to yield predictions of
structure and frequencies for closed-shell molecules that are very similar to scaled MP2
methods. However, it yields substantial improvements for open-shell molecules,
where problems with spin contamination and symmetry breaking are shown to be greatly
reduced.
Summary of key $rem variables to be specified:
CORRELATION | SOSMP2 |
| MOSMP2 |
JOBTYPE | sp (default) single point energy evaluation |
| opt geometry optimization with analytical gradient |
| force force evaluation with analytical gradient |
BASIS | user's choice (standard or user-defined: GENERAL or
MIXED) |
AUX_BASIS | corresponding auxiliary basis (standard or user-defined: |
| AUX_GENERAL or AUX_MIXED |
OMEGA | no default n; use ω = n/1000. The recommended value is |
| n=600 (ω = 0.6 a.u.) |
N_FROZEN_CORE | Optional |
N_FROZEN_VIRTUAL | Optional
|
5.5.6 Examples
Example 5.0 Example of SOS-MP2 geometry optimization
$molecule
0 3
C1
H1 C1 1.07726
H2 C1 1.07726 H1 131.60824
$end
$rem
JOBTYPE opt
CORRELATION sosmp2
BASIS cc-pvdz
AUX_BASIS rimp2-cc-pvdz
UNRESTRICTED true
SYMMETRY false
$end
Example 5.0 Example of MOS-MP2 energy evaluation with frozen core
approximation
$molecule
0 1
Cl
Cl 1 2.05
$end
$rem
JOBTYPE sp
CORRELATION mosmp2
OMEGA 600
BASIS cc-pVTZ
AUX_BASIS rimp2-cc-pVTZ
N_FROZEN_CORE fc
THRESH 12
SCF_CONVERGENCE 8
$end
Example 5.0 Example of O2 methodology applied to O(N4) SOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED TRUE
JOBTYPE FORCE Options are SP/FORCE/OPT
EXCHANGE HF
DO_O2 1 O2 with O(N^4) SOS-MP2 algorithm
SOS_FACTOR 100 Opposite Spin scaling factor = 100/100 = 1.0
SCF_ALGORITHM DIIS_GDM
SCF_GUESS GWH
BASIS sto-3g
AUX_BASIS rimp2-vdz
SCF_CONVERGENCE 8
THRESH 14
SYMMETRY FALSE
PURECART 1111
$end
Example 5.0 Example of O2 methodology applied to O(N4) MOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED TRUE
JOBTYPE FORCE Options are SP/FORCE/OPT
EXCHANGE HF
DO_O2 2 O2 with O(N^4) MOS-MP2 algorithm
OMEGA 600 Omega = 600/1000 = 0.6 a.u.
SCF_ALGORITHM DIIS_GDM
SCF_GUESS GWH
BASIS sto-3g
AUX_BASIS rimp2-vdz
SCF_CONVERGENCE 8
THRESH 14
SYMMETRY FALSE
PURECART 1111
$end
5.5.7 RI-TRIM MP2 Energies
The triatomics in molecules (TRIM) local correlation approximation to MP2
theory [219] was described in detail in Section 5.4.1 which
also discussed our implementation of this approach based on conventional
four-center two-electron integrals. Q-Chem 3.0 also includes an auxiliary
basis implementation of the TRIM model. The new RI-TRIM MP2 energy algorithm [236]
greatly accelerates these local correlation calculations
(often by an order of magnitude or more for the correlation part), which scale
with the 4th power of molecule size. The electron correlation part of
the calculation is speeded up over normal RI-MP2 by a factor proportional to
the number of atoms in the molecule. For a hexadecapeptide, for instance, the
speedup is approximately a factor of 4 [236]. The TRIM model
can also be applied to the scaled opposite spin models discussed above. As for
the other RI-based models discussed in this section, we recommend using
RI-TRIM MP2 instead of the conventional TRIM MP2 code whenever run-time of the
job is a significant issue. As for RI-MP2 itself, TRIM MP2 is invoked by
adding AUX_BASIS $rems to the input deck,
in addition to requesting CORRELATION = RILMP2.
Example 5.0 Example of RI-TRIM MP2 energy evaluation
$molecule
0 3
C1
H1 C1 1.07726
H2 C1 1.07726 H1 131.60824
$end
$rem
CORRELATION rilmp2
BASIS cc-pVDZ
AUX_BASIS rimp2-cc-pVDZ
PURECART 1111
UNRESTRICTED true
SYMMETRY false
$end
5.5.8 Dual-Basis MP2
The successful computational cost speedups of the previous sections often leave the
cost of the underlying SCF calculation dominant. The dual-basis method provides a means
of accelerating the SCF by roughly an order of magnitude, with minimal associated error
(see Section 4.7). This dual-basis reference energy may be combined
with RI-MP2 calculations for both energies [181,[185] and
analytic first derivatives [183]. In the latter case, further savings
(beyond the SCF alone) are demonstrated in the gradient due to the ability to solve
the response (Z-vector) equations in the smaller basis set. Refer to
Section 4.7 for details and job control options.
5.6 Short-Range Correlation Methods
5.6.1 Attenuated MP2
MP2(attenuator, basis) approximates MP2 by splitting the Coulomb operator in two pieces and preserving only short-range two-electron interactions, akin to the CASE approximation[205,[204], but without modification of the underlying SCF calculation. While MP2 is a comparatively efficient method for estimating the correlation energy, it converges slowly with basis set size - and, even in the complete basis limit, contains fundamentally inaccurate physics for long-range interactions. Basis set superposition error and the MP2-level treatment of long-range interactions both typically artificially increase correlation energies for noncovalent interactions. Attenuated MP2 improves upon MP2 for inter- and intramolecular interactions, with significantly better performance for relative and binding energies of noncovalent complexes, frequently outperforming complete basis set estimates of MP2[237,[238].
Attenuated MP2, denoted MP2(attenuator, basis) is implemented in Q-Chem based on the complementary terf function, below.
s(r)=terfc(r,r0)= |
1
2
|
{ erfc [ ω(r−r0) ] + erfc [ ω(r+r0) ] } |
| (5.23) |
By choosing the terfc short-range operator, we optimally preserve the short-range behavior of the Coulomb operator while smoothly and rapidly switching off around the distance r0. Since this directly addresses basis set superposition error, parametrization must be done for specific basis sets. This has been performed for the basis sets, aug-cc-pVDZ[237] and aug-cc-pVTZ[238]. Other basis sets are not recommended for general use until further testing has been done.
Energies and gradients are functional with and without the resolution of the identity approximation using correlation keywords ATTMP2 and ATTRIMP2.
5.6.2 Examples
Example 5.0 Example of RI-MP2(terfc, aug-cc-pVDZ) energy evaluation
$molecule
0 1
O -1.551007 -0.114520 0.000000
H -1.934259 0.762503 0.000000
H -0.599677 0.040712 0.000000
$end
$rem
jobtype sp
exchange hf
correlation attrimp2
basis aug-cc-pvdz
aux_basis rimp2-aug-cc-pvdz
n_frozen_core fc
$end
Example 5.0 Example of MP2(terfc, aug-cc-pVTZ) geometry optimization
$molecule
0 1
H 0.0 0.0 0.0
H 0.0 0.0 0.9
$end
$rem
jobtype opt
exchange hf
correlation attmp2
basis aug-cc-pvtz
n_frozen_core fc
$end
5.7 Coupled-Cluster Methods
The following sections give short summaries of the various coupled-cluster based
methods
available in Q-Chem, most of which are variants of coupled-cluster
theory. The basic object-oriented tools necessary to permit the implementation
of these methods in Q-Chem was accomplished by Profs. Anna Krylov and David
Sherrill, working at Berkeley with Martin Head-Gordon, and then continuing
independently at the University of Southern California and Georgia Tech,
respectively. While at Berkeley, Krylov and Sherrill also developed the
optimized orbital coupled-cluster method, with additional assistance from Ed
Byrd. The extension of this code to MP3, MP4, CCSD and QCISD is the work of Prof.
Steve Gwaltney at Berkeley, while the extensions to QCCD were implemented by Ed
Byrd at Berkeley.
The original tensor library and CC / EOM suite of methods are
handled by the CCMAN module of Q-Chem. Recently, a new code (termed CCMAN2)
has been developed in Krylov group by Evgeny Epifanovsky and others,
and a gradual transition from CCMAN to CCMAN2 has began.
During the transition time, both
codes will be available for users via the CCMAN2 keyword.
CORRELATION
Specifies the correlation level of theory handled by CCMAN/CCMAN2. |
TYPE:
DEFAULT:
OPTIONS:
CCMP2 | Regular MP2 handled by CCMAN/CCMAN2 |
MP3 | CCMAN |
MP4SDQ | CCMAN |
MP4 | CCMAN |
CCD | CCMAN |
CCD(2) | CCMAN |
CCSD | CCMAN and CCMAN2 |
CCSD(T) | CCMAN and CCMAN2 |
CCSD(2) | CCMAN |
CCSD(fT) | CCMAN |
CCSD(dT) | CCMAN |
QCISD | CCMAN |
QCISD(T) | CCMAN |
OD | CCMAN |
OD(T) | CCMAN |
OD(2) | CCMAN |
VOD | CCMAN |
VOD(2) | CCMAN |
QCCD | CCMAN |
QCCD(T) | CCMAN |
QCCD(2) | CCMAN |
VQCCD | CCMAN |
VQCCD(T) | CCMAN |
VQCCD(2) | CCMAN |
RECOMMENDATION:
Consult the literature for guidance. |
|
5.7.1 Coupled Cluster Singles and Doubles (CCSD)
The standard approach for treating pair correlations self-consistently are
coupled-cluster methods where the cluster operator contains all single and
double substitutions [239], abbreviated as CCSD. CCSD yields
results that are only slightly superior to MP2 for structures and frequencies
of stable closed-shell molecules. However, it is far superior for reactive
species, such as transition structures and radicals, for which the performance
of MP2 is quite erratic.
A full textbook presentation of CCSD is beyond the scope of this manual, and
several comprehensive references are available. However, it may be useful to
briefly summarize the main equations. The CCSD wavefunction is:
| ΨCCSD 〉 = exp | ⎛ ⎝
|
^
T
|
1
|
+ |
^
T
|
2
| ⎞ ⎠
|
| Φ0 〉 |
| (5.24) |
where the single and double excitation operators may be defined by their
actions on the reference single determinant (which is normally taken as the
Hartree-Fock determinant in CCSD):
|
^
T
|
1
|
| Φ0 〉 = |
occ ∑
i
|
|
virt ∑
a
|
tia | Φia 〉 |
| (5.25) |
|
^
T
|
2
|
| Φ0 〉 = |
1
4
|
|
occ ∑
ij
|
|
virt ∑
ab
|
tijab | Φijab 〉 |
| (5.26) |
It is not feasible to determine the CCSD energy by variational minimization of
〈E 〉CCSD with respect to the singles and doubles amplitudes
because the expressions terminate at the same level of complexity as full
configuration interaction (!). So, instead, the Schrödinger equation is
satisfied in the subspace spanned by the reference determinant, all single
substitutions, and all double substitutions. Projection with these functions
and integration over all space provides sufficient equations to determine the
energy, the singles and doubles amplitudes as the solutions of sets of
nonlinear equations. These equations may be symbolically written as follows:
| |
|
| |
| |
|
|
|
Φ0 | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
1
|
+ |
1
2
|
|
^
T
|
2 1
|
+ |
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| | (5.27) |
| |
|
|
|
Φia | ⎢ ⎢
|
^
H
|
−ECCSD | ⎢ ⎢
|
ΨCCSD |
|
|
| |
| |
|
|
|
Φia | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
1
|
+ |
1
2
|
|
^
T
|
2 1
|
+ |
^
T
|
2
|
+ |
^
T
|
1
|
|
^
T
|
2
|
+ |
1
3!
|
|
^
T
|
3 1
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| | (5.28) |
| |
|
|
|
Φijab | ⎢ ⎢
|
^
H
|
−ECCSD | ⎢ ⎢
|
ΨCCSD |
|
|
| |
| |
|
|
|
Φijab | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
1
|
+ |
1
2
|
|
^
T
|
2 1
|
+ |
^
T
|
2
|
+ |
^
T
|
1
|
|
^
T
|
2
|
+ |
1
3!
|
|
^
T
|
3 1
|
|
| |
| |
|
+ |
1
2
|
|
^
T
|
2 2
|
+ |
1
2
|
|
^
T
|
2 1
|
|
^
T
|
2
|
+ |
1
4!
|
|
^
T
|
4 1
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| | (5.29) |
|
The result is a set of equations which yield an energy that is not necessarily
variational (i.e., may not be above the true energy), although it is strictly
size-consistent. The equations are also exact for a pair of electrons, and, to
the extent that molecules are a collection of interacting electron pairs, this
is the basis for expecting that CCSD results will be of useful accuracy.
The computational effort necessary to solve the CCSD equations can be shown to
scale with the 6th power of the molecular size, for fixed choice of basis
set. Disk storage scales with the 4th power of molecular size, and
involves a number of sets of doubles amplitudes, as well as two-electron
integrals in the molecular orbital basis. Therefore the improved accuracy
relative to MP2 theory comes at a steep computational cost. Given these
scalings it is relatively straightforward to estimate the feasibility (or
non feasibility) of a CCSD calculation on a larger molecule (or with a larger
basis set) given that a smaller trial calculation is first performed.
Q-Chem supports both energies and analytic gradients for CCSD for
RHF and UHF references (including frozen-core). For ROHF, only energies and
unrelaxed properties are available.
5.7.2 Quadratic Configuration Interaction (QCISD)
Quadratic configuration interaction with singles and doubles (QCISD) [240]
is a widely used alternative to CCSD, that shares its main
desirable properties of being size-consistent, exact for pairs of electrons,
as well as being also non variational. Its computational cost also scales in the
same way with molecule size and basis set as CCSD, although with slightly
smaller constants. While originally proposed independently of CCSD based on
correcting configuration interaction equations to be size-consistent, QCISD is
probably best viewed as approximation to CCSD. The defining equations are given
below (under the assumption of Hartree-Fock orbitals, which should always be
used in QCISD). The QCISD equations can clearly be viewed as the CCSD equations
with a large number of terms omitted, which are evidently not very numerically
significant:
EQCISD = |
|
Φ0 | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| (5.30) |
0= |
|
Φia | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
^
T
|
1
|
+ |
^
T
|
2
|
+ |
^
T
|
1
|
|
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| (5.31) |
0= |
|
Φijab | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
1
|
+ |
^
T
|
2
|
+ |
1
2
|
|
^
T
|
2
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| (5.32) |
QCISD energies are available in Q-Chem, and are requested with the QCISD
keyword. As discussed in Section 5.8, the non iterative QCISD(T)
correction to the QCISD solution is also available to approximately incorporate
the effect of higher substitutions.
5.7.3 Optimized Orbital Coupled Cluster Doubles (OD)
It is possible to greatly simplify the CCSD equations by omitting the single
substitutions (i.e., setting the T1 operator to zero). If the same single
determinant reference is used (specifically the Hartree-Fock determinant),
then this defines the coupled-cluster doubles (CCD) method, by the following
equations:
| |
|
|
|
Φ0 | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| | (5.33) |
| |
|
|
|
Φijab | ⎢ ⎢
|
^
H
| ⎢ ⎢
|
| ⎛ ⎝
|
1+ |
^
T
|
2
|
+ |
1
2
|
|
^
T
|
2 2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| | (5.34) |
|
The CCD method cannot itself usually be recommended because while pair
correlations are all correctly included, the neglect of single substitutions
causes calculated energies and properties to be significantly less reliable
than for CCSD. Single substitutions play a role very similar to orbital
optimization, in that they effectively alter the reference determinant to be
more appropriate for the description of electron correlation (the Hartree-Fock
determinant is optimized in the absence of electron correlation).
This suggests an alternative to CCSD and QCISD that has some additional
advantages. This is the optimized orbital CCD method (OO-CCD), which we
normally refer to as simply optimized doubles (OD) [241]. The
OD method is defined by the CCD equations above, plus the additional set of
conditions that the cluster energy is minimized with respect to orbital
variations. This may be mathematically expressed by
where the rotation angle θia mixes the ith occupied orbital
with the ath virtual (empty) orbital. Thus the orbitals that define
the single determinant reference are optimized to minimize the coupled-cluster
energy, and are variationally best for this purpose. The resulting orbitals are
approximate Brueckner orbitals.
The OD method has the advantage of formal simplicity (orbital variations and
single substitutions are essentially redundant variables). In cases where
Hartree-Fock theory performs poorly (for example artificial symmetry
breaking, or non-convergence), it is also practically advantageous to use the OD
method, where the HF orbitals are not required, rather than CCSD or QCISD.
Q-Chem supports both energies and analytical gradients using the OD method.
The computational cost for the OD energy is more than twice that of the CCSD or
QCISD method, but the total cost of energy plus gradient is roughly similar,
although OD remains more expensive. An additional advantage of the OD method is
that it can be performed in an active space, as discussed later, in
Section 5.9.
5.7.4 Quadratic Coupled Cluster Doubles (QCCD)
The non variational determination of the energy in the CCSD, QCISD, and OD
methods discussed in the above subsections is not normally a practical problem.
However, there are some cases where these methods perform poorly. One such
example are potential curves for homolytic bond dissociation, using closed
shell orbitals, where the calculated energies near dissociation go
significantly below the true energies, giving potential curves with unphysical
barriers to formation of the molecule from the separated fragments [242].
The Quadratic Coupled Cluster Doubles (QCCD) method [243]
recently proposed by Troy Van Voorhis at Berkeley uses a
different energy functional to yield improved behavior in problem cases of this
type. Specifically, the QCCD energy functional is defined as
EQCCD = |
|
Φ0 | ⎛ ⎝
|
1+ |
^
Λ
|
2
|
+ |
1
2
|
|
^
Λ
|
2
2
| ⎞ ⎠
|
| ⎢ ⎢
|
^
H
| ⎢ ⎢
|
exp | ⎛ ⎝
|
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| (5.36) |
where the amplitudes of both the ∧T2 and ∧Λ 2
operators are determined by minimizing the QCCD energy functional.
Additionally, the optimal orbitals are determined by minimizing the QCCD
energy functional with respect to orbital rotations mixing occupied and
virtual orbitals.
To see why the QCCD energy should be an improvement on the OD energy, we first
write the latter in a different way than before. Namely, we can write a CCD
energy functional which when minimized with respect to the ∧T2 and
∧Λ2 operators, gives back the same CCD equations defined earlier.
This energy functional is
ECCD = |
|
Φ0 | ⎛ ⎝
|
1+ |
^
Λ
|
2
| ⎞ ⎠
|
| ⎢ ⎢
|
^
H
| ⎢ ⎢
|
exp | ⎛ ⎝
|
^
T
|
2
| ⎞ ⎠
|
Φ0 |
|
C
|
|
| (5.37) |
Minimization with respect to the ∧Λ2 operator gives the equations
for the ∧T2 operator presented previously, and, if those equations are
satisfied then it is clear that we do not require knowledge of the ∧Λ2 operator itself to evaluate the energy.
Comparing the two energy functionals, Eqs. (5.36) and (5.37), we see
that the QCCD functional includes up through quadratic terms of the Maclaurin
expansion of exp(∧Λ2) while the conventional CCD functional
includes only linear terms. Thus the bra wavefunction and the ket wavefunction
in the energy expression are treated more equivalently in QCCD than in CCD.
This makes QCCD closer to a true variational treatment [242]
where the bra and ket wavefunctions are treated precisely equivalently, but
without the exponential cost of the variational method.
In practice QCCD is a dramatic improvement relative to any of the conventional
pair correlation methods for processes involving more than two active electrons
(i.e., the breaking of at least a double bond, or, two spatially close single
bonds). For example calculations, we refer to the original paper [243],
and the follow-up paper describing the full implementation [244].
We note that these improvements carry a computational price.
While QCCD scales formally with the 6th power of molecule size like
CCSD, QCISD, and OD, the coefficient is substantially larger. For this reason,
QCCD calculations are by default performed as OD calculations until they are
partly converged. Q-Chem also contains some configuration interaction models
(CISD and CISDT). The CI methods are inferior to CC due to size-consistency
issues, however, these models may be useful for benchmarking and development
purposes.
5.7.5 Resolution-of-identity with CC (RI-CC)
The RI approximation (see Section 5.5)
can be used in coupled-cluster calculations, which substantially
reduces the cost of integral transformation and disk storage requirements.
The RI approximations may be used for integrals only such that integrals
are generated in conventional MO form and canonical CC/EOM
calculations are performed, or in a more complete version when
modified CC/EOM equations are used such that the integrals are used in their RI
representation. The latter version allows for more substantial
savings in storage and in computational speed-up.
The RI for integrals is invoked when AUX_BASIS is specified.
All two-electron integrals are used in RI decomposed form in
CC when AUX_BASIS is specified.
By default, the integrals will be stored in the RI form and special CC/EOM code
will be invoked. Keyword DIRECT_RI allows one to use RI generated integrals
in conventional form (by transforming RI integrals back to the standard format)
invoking conventional CC procedures.
Note:
RI for integrals is available for all CCMAN/CCMAN2 methods. CCMAN requires that the
unrestricted reference be used, CCMAN2 does not have this limitation.
In addition, while RI is available for jobs that need analytical gradients,
only energies and properties are computed using RI. Energy derivatives are calculated
using regular electron repulsion integral derivatives. Full RI implementation
(with integrals used in decomposed form) is only
available for CCMAN2. For maximum computational efficiency,
combine with FNO (see Sections
5.10 and 6.6.6)
when appropriate. |
5.7.6 Cholesky decomposition with CC (CD-CC)
Two-electron integrals can be decomposed using Cholesky
decomposition [245] giving rise to the same representation as
in RI and substantially reducing the cost of integral transformation,
disk storage requirements, and improving parallel performance:
(μν|λσ) ≈ |
M ∑
P=1
|
BμνP BλσP, |
| (5.38) |
The rank of Cholesky decomposition, M, is typically 3-10 times larger than
the number of basis functions N [246]; it depends on the
decomposition threshold δ and is considerably smaller than the full rank of the
matrix,
N(N+1)/2 [246,[247,[248].
Cholesky decomposition removes linear dependencies in product densities [246] (μν| allowing one to obtain compact approximation to the original matrix
with accuracy, in principle, up to machine precision.
Decomposition threshold δ is the only parameter that controls
accuracy and the rank of the decomposition. Cholesky decomposition is invoked by
specifying CHOLESKY_TOL that defines the accuracy with which decomposition
should be performed. For most calculations tolerance of δ = 10−3 gives a good
balance between accuracy and compactness of the rank. Tolerance of δ = 10−2 can
be used for exploratory calculations and δ = 10−4 for high-accuracy calculations. Similar to RI, Cholesky-decomposed integrals can be transformed back, into
the canonical MO form, using DIRECT_RI keyword.
Note:
Cholesky decomposition is available for all CCMAN2 methods.
Analytic gradients are not yet available;
only energies and properties are computed using CD.
For maximum computational efficiency, combine with FNO (see Sections
5.10 and 6.6.6)
when appropriate. |
5.7.7 Job Control Options
There are a large number of options for the coupled-cluster singles and
doubles methods. They are documented in Appendix C, and, as the reader will
find upon following this link, it is an extensive list indeed. Fortunately,
many of them are not necessary for routine jobs. Most of the options for
non-routine jobs concern altering the default iterative procedure, which is
most often necessary for optimized orbital calculations (OD, QCCD), as well as
the active space and EOM methods discussed later in Section
5.9. The more common options relating to convergence control
are discussed there, in Section 5.9.5. Below we list the options
that one should be aware of for routine calculations.
For memory options and parallel execution, see Section 5.13.
CC_CONVERGENCE
Overall convergence criterion for the coupled-cluster codes. This is designed
to ensure at least n significant digits in the calculated energy, and
automatically sets the other convergence-related variables
(CC_E_CONV, CC_T_CONV, CC_THETA_CONV,
CC_THETA_GRAD_CONV) [10−n]. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to 10−n convergence criterion. Amplitude convergence is set |
| automatically to match energy convergence. |
RECOMMENDATION:
|
Note:
For single point calculations, CC_E_CONV=6 and CC_T_CONV=4.
Tighter amplitude convergence (CC_T_CONV=5) is used for gradients and EOM
calculations. |
| CC_DOV_THRESH
Specifies minimum allowed values for the coupled-cluster energy denominators.
Smaller values are replaced by this constant during early iterations only, so
the final results are unaffected, but initial convergence is improved when the
HOMO-LUMO gap is small or when non-conventional references are used. |
TYPE:
DEFAULT:
OPTIONS:
abcde | Integer code is mapped to abc×10−de, e.g.,
2502 corresponds to 0.25 |
RECOMMENDATION:
Increase to 0.25, 0.5 or 0.75 for non convergent coupled-cluster calculations. |
|
|
|
CC_SCALE_AMP
If not 0, scales down the step for updating coupled-cluster amplitudes in cases of problematic convergence. |
TYPE:
DEFAULT:
OPTIONS:
abcd | Integer code is mapped to abcd×10−2, e.g.,
90 corresponds to 0.9 |
RECOMMENDATION:
Use 0.9 or 0.8 for non convergent coupled-cluster calculations. |
|
| CC_MAX_ITER
Maximum number of iterations to optimize the coupled-cluster energy. |
TYPE:
DEFAULT:
OPTIONS:
n | up to n iterations to achieve convergence. |
RECOMMENDATION:
|
|
|
CC_PRINT
Controls the output from post-MP2 coupled-cluster module of Q-Chem |
TYPE:
DEFAULT:
OPTIONS:
0→7 | higher values can lead to deforestation... |
RECOMMENDATION:
Increase if you need more output and don't like trees |
|
| CHOLESKY_TOL
Tolerance of Cholesky decomposition of two-electron integrals |
TYPE:
DEFAULT:
OPTIONS:
n to define tolerance of 10−n |
RECOMMENDATION:
2 - qualitative calculations, 3 - appropriate for most cases, 4 - quantitative
(error in total energy typically less than 1e-6 hartree) |
|
|
|
DIRECT_RI
Controls use of RI and Cholesky integrals in conventional (undecomposed) form |
TYPE:
DEFAULT:
OPTIONS:
FALSE - use all integrals in decomposed format |
TRUE - transform all RI or Cholesky integral back to conventional format |
RECOMMENDATION:
By default all integrals are used in decomposed format allowing significant reduction of
memory use. If all integrals are transformed back (TRUE option) no memory reduction is
achieved and decomposition error is introduced, however, the integral transformation
is performed significantly faster and conventional CC/EOM algorithms are used. |
|
5.7.8 Examples
Example 5.0 A series of jobs evaluating the correlation energy (with core
orbitals frozen) of the ground state of the NH2 radical with three methods
of coupled-cluster singles and doubles type: CCSD itself, OD, and QCCD.
$molecule
0 2
N
H1 N 1.02805
H2 N 1.02805 H1 103.34
$end
$rem
CORRELATION ccsd
BASIS 6-31g*
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION od
BASIS 6-31g*
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION qccd
BASIS 6-31g*
N_FROZEN_CORE fc
$end
Example 5.0 A job evaluating CCSD energy of water using RI-CCSD
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
CORRELATION ccsd
BASIS aug-cc-pvdz
max_sub_file_num 256
cc_memory 20000
mem_static 2000
AUX_BASIS rimp2-aug-cc-pvdz
$end
Example 5.0 A job evaluating CCSD energy of water using CD-CCSD (tolerance = 10−3)
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
CORRELATION ccsd
BASIS aug-cc-pvdz
max_sub_file_num 256
cc_memory 20000
mem_static 2000
cholesky_tol 3
$end
Example 5.0 A job evaluating CCSD energy of water using CD-CCSD (tolerance = 10−3) with FNO
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
CORRELATION ccsd
BASIS aug-cc-pvdz
max_sub_file_num 256
cc_memory 20000
mem_static 2000
cholesky_tol 3
CC_fno_thresh 9950
$end
5.8 Non-iterative Corrections to Coupled Cluster Energies
5.8.1 (T) Triples Corrections
To approach chemical accuracy in reaction energies and related properties, it
is necessary to account for electron correlation effects that involve three
electrons simultaneously, as represented by triple substitutions relative to
the mean field single determinant reference, which arise in MP4. The best
standard methods for including triple substitutions are the CCSD(T) [249]
and QCISD(T) methods [240] The accuracy of
these methods is well-documented for many cases [250], and in
general is a very significant improvement relative to the starting point
(either CCSD or QCISD). The cost of these corrections scales with the
7th power of molecule size (or the 4th power of the number of
basis functions, for a fixed molecule size), although no additional disk resources
are required relative to the starting coupled-cluster calculation. Q-Chem
supports the evaluation of CCSD(T) and QCISD(T) energies, as well as the
corresponding OD(T) correction to the optimized doubles method discussed in the
previous subsection. Gradients and properties are not yet available for
any of these (T) corrections.
5.8.2 (2) Triples and Quadruples Corrections
While the (T) corrections discussed above have been extraordinarily successful,
there is nonetheless still room for further improvements in accuracy, for at
least some important classes of problems. They contain judiciously chosen terms
from 4th- and 5th-order Møller-Plesset perturbation
theory, as well as higher order terms that result from the fact that the
converged cluster amplitudes are employed to evaluate the 4th- and 5th-order
order terms. The (T) correction therefore depends upon the bare
reference orbitals and orbital energies, and in this way its effectiveness
still depends on the quality of the reference determinant. Since we are
correcting a coupled-cluster solution rather than a single determinant, this is
an aspect of the (T) corrections that can be improved. Deficiencies of the (T)
corrections show up computationally in cases where there are near-degeneracies
between orbitals, such as stretched bonds, some transition states, open shell
radicals, and diradicals.
Prof. Steve Gwaltney, while working at Berkeley with Martin Head-Gordon, has
suggested a new class of non iterative correction that offers the prospect of
improved accuracy in problem cases of the types identified above [251].
Q-Chem contains Gwaltney's implementation of this new
method, for energies only. The new correction is a true second order correction
to a coupled-cluster starting point, and is therefore denoted as (2). It is
available for two of the cluster methods discussed above, as OD(2) and
CCSD(2) [251,[252]. Only energies are available at present.
The basis of the (2) method is to partition not the regular Hamiltonian into
perturbed and unperturbed parts, but rather to partition a
similarity-transformed Hamiltonian, defined as ~H=e−∧T∧He∧T. In the truncated space (call it the p-space)
within which the cluster problem is solved (e.g., singles and doubles for
CCSD), the coupled-cluster wavefunction is a true eigenvalue of ~H.
Therefore we take the zero order Hamiltonian, ~H(0),
to be the full ~H in the p-space, while in the space of
excluded substitutions (the q-space) we take only the one-body part of ~H
(which can be made diagonal). The fluctuation potential describing
electron correlations in the q-space is ~H−~H(0),
and the (2) correction then follows from second order perturbation
theory.
The new partitioning of terms between the perturbed and unperturbed
Hamiltonians inherent in the (2) correction leads to a correction that shows
both similarities and differences relative to the existing (T) corrections.
There are two types of higher correlations that enter at second order: not only
triple substitutions, but also quadruple substitutions. The quadruples are
treated with a factorization ansatz, that is exact in 5th order
Møller-Plesset theory [253], to reduce their computational
cost from N9 to N6. For large basis sets this can still be larger
than the cost of the triples terms, which scale as the 7th power of
molecule size, with a factor twice as large as the usual (T) corrections.
These corrections are feasible for molecules containing between four and ten
first row atoms, depending on computer resources, and the size of the basis set
chosen. There is early evidence that the (2) corrections are superior to the
(T) corrections for highly correlated systems [251]. This shows
up in improved potential curves, particularly at long range and may also extend
to improved energetic and structural properties at equilibrium in problematical
cases. It will be some time before sufficient testing on the new (2)
corrections has been done to permit a general assessment of the performance of
these methods. However, they are clearly very promising, and for this reason
they are available in Q-Chem.
5.8.3 (dT) and (fT) corrections
Alternative inclusion of non-iterative N7 triples corrections is described in
Section 6.6.17. These methods called (dT) and (fT) are of similar
accuracy to other triples corrections. CCSD(dT) and CCSD(fT) are equivalent to
the CR-CCSD(T)L and CR-CCSD(T)2 methods of Piecuch and co-workers.
5.8.4 Job Control Options
The evaluation of a non iterative (T) or (2) correction after a coupled-cluster
singles and doubles level calculation (either CCSD, QCISD or OD) is controlled
by the correlation keyword, and the specification of any frozen orbitals via
N_FROZEN_CORE (and possibly N_FROZEN_VIRTUAL).
There is only one additional job control option. For the (2) correction, it is
possible to apply the frozen core approximation in the reference coupled
cluster calculation, and then correlate all orbitals in the (2) correction.
This is controlled by CC_INCL_CORE_CORR, described below.
The default is to include core and core-valence correlation automatically in
the CCSD(2) or OD(2) correction, if the reference CCSD or OD calculation was
performed with frozen core orbitals. The reason for this choice is that core
correlation is economical to include via this method (the main cost increase is
only linear in the number of core orbitals), and such effects are important to
account for in accurate calculations. This option should be made false if a job
with explicitly frozen core orbitals is desired. One good reason for freezing
core orbitals in the correction is if the basis set is physically inappropriate
for describing core correlation (e.g., standard Pople basis sets, and Dunning
cc-pVxZ basis sets are designed to describe valence-only correlation effects).
Another good reason is if a direct comparison is desired against another method
such as CCSD(T) which is always used in the same orbital window as the CCSD
reference.
CC_INCL_CORE_CORR
Whether to include the correlation contribution from frozen core orbitals in
non iterative (2) corrections, such as OD(2) and CCSD(2). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless no core-valence or core correlation is desired (e.g., for
comparison with other methods or because the basis used cannot describe core
correlation). |
|
5.8.5 Example
Example 5.0 Two jobs that compare the correlation energy calculated via the
standard CCSD(T) method with the new CCSD(2) approximation, both using the
frozen core approximation. This requires that CC_INCL_CORE_CORR
must be specified as FALSE in the CCSD(2) input.
$molecule
0 2
O
H O 0.97907
$end
$rem
CORRELATION ccsd(t)
BASIS cc-pvtz
N_FROZEN_CORE fc
$end
@@@
$molecule
read
$end
$rem
CORRELATION ccsd(2)
BASIS cc-pvtz
N_FROZEN_CORE fc
CC_INCL_CORE_CORR false
$end
Example 5.0 Water: Ground state CCSD(dT) calculation using RI
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.957
HOH = 104.5
$end
$rem
JOBTYPE SP
BASIS cc-pvtz
AUX_BASIS rimp2-cc-pvtz
CORRELATION CCSD(dT)
$end
5.9 Coupled Cluster Active Space Methods
5.9.1 Introduction
Electron correlation effects can be qualitatively divided into two classes.
The first class is static or nondynamical correlation: long wavelength
low-energy correlations associated with other electron configurations that are
nearly as low in energy as the lowest energy configuration. These correlation
effects are important for problems such as homolytic bond breaking, and are the
hardest to describe because by definition the single configuration
Hartree-Fock description is not a good starting point. The second class is
dynamical correlation: short wavelength high-energy correlations associated
with atomic-like effects. Dynamical correlation is essential for
quantitative accuracy, but a reasonable description of static
correlation is a prerequisite for a calculation being qualitatively
correct.
In the methods discussed in the previous several subsections, the objective was
to approximate the total correlation energy. However, in some cases, it is
useful to model directly the nondynamical and dynamical correlation
energies separately. The reasons for this are pragmatic: with approximate
methods, such a separation can give a more balanced treatment of electron
correlation along bond-breaking coordinates, or reaction coordinates that
involve diradicaloid intermediates. The nondynamical correlation energy is
conveniently defined as the solution of the Schrödinger equation within a
small basis set composed of valence bonding, antibonding and lone pair
orbitals: the so-called full valence active space. Solved exactly, this is
the so-called full valence complete active space SCF (CASSCF) [254],
or equivalently, the fully optimized reaction space (FORS) method [255].
Full valence CASSCF and FORS involve computational complexity which increases
exponentially with the number of atoms, and is thus unfeasible beyond systems
of only a few atoms, unless the active space is further restricted on a
case-by-case basis. Q-Chem includes two relatively economical methods that
directly approximate these theories using a truncated coupled-cluster doubles
wavefunction with optimized orbitals [256]. They are active space
generalizations of the OD and QCCD methods discussed previously in
Sections 5.7.3 and 5.7.4, and are discussed in the following two
subsections. By contrast with the exponential growth of computational cost with
problem size associated with exact solution of the full valence CASSCF problem,
these cluster approximations have only 6th-order growth of computational
cost with problem size, while often providing useful accuracy.
The full valence space is a well-defined theoretical chemical model. For these
active space coupled-cluster doubles methods, it consists of the union of
valence levels that are occupied in the single determinant reference,
and those that are empty. The occupied levels that are to be replaced can only
be the occupied valence and lone pair orbitals, whose number is defined by the
sum of the valence electron counts for each atom (i.e., 1 for H, 2 for He, 1 for
Li, etc..). At the same time, the empty virtual orbitals to which the double
substitutions occur are restricted to be empty (usually antibonding) valence
orbitals. Their number is the difference between the number of valence atomic
orbitals, and the number of occupied valence orbitals given above. This
definition (the full valence space) is the default when either of the
"valence" active space methods are invoked (VOD or VQCCD)
There is also a second useful definition of a valence active space, which we
shall call the 1:1 or perfect pairing active space. In this definition, the
number of occupied valence orbitals remains the same as above. The number of
empty correlating orbitals in the active space is defined as being exactly the
same number, so that each occupied orbital may be regarded as being associated
1:1 with a correlating virtual orbital. In the water molecule,
for example, this means that the lone pair electrons as well as the
bond-orbitals are correlated. Generally the 1:1 active space recovers more
correlation for molecules dominated by elements on the right of the periodic
table, while the full valence active space recovers more correlation for
molecules dominated by atoms to the left of the periodic table.
If you wish to specify either the 1:1 active space as described above, or
some other choice of active space based on your particular chemical problem,
then you must specify the numbers of active occupied and virtual orbitals.
This is done via the standard "window options", documented earlier in the
Chapter.
Finally we note that the entire discussion of active spaces here leads only
to specific numbers of active occupied and virtual orbitals. The orbitals
that are contained within these spaces are optimized by minimizing the trial
energy with respect to all the degrees of freedom previously discussed: the
substitution amplitudes, and the orbital rotation angles mixing occupied and
virtual levels. In addition, there are new orbital degrees of freedom to be
optimized to obtain the best active space of the chosen size, in the sense
of yielding the lowest coupled-cluster energy. Thus rotation angles mixing
active and inactive occupied orbitals must be varied until the energy is
stationary. Denoting inactive orbitals by primes and active orbitals without
primes, this corresponds to satisfying
Likewise, the rotation angles mixing active and inactive virtual orbitals
must also be varied until the coupled-cluster energy is minimized with
respect to these degrees of freedom:
5.9.2 VOD and VOD(2) Methods
The VOD method is the active space version of the OD method described earlier
in Section 5.7.3. Both energies and gradients are available for VOD, so
structure optimization is possible. There are a few important comments to make
about the usefulness of VOD. First, it is a method that is capable of
accurately treating problems that fundamentally involve 2 active electrons in a
given local region of the molecule. It is therefore a good alternative for
describing single bond-breaking, or torsion around a double bond, or some
classes of diradicals. However it often performs poorly for problems where
there is more than one bond being broken in a local region, with the
non variational solutions being quite possible. For such problems the newer
VQCCD method is substantially more reliable.
Assuming that VOD is a valid zero order description for the electronic
structure, then a second order correction, VOD(2), is available for energies
only. VOD(2) is a version of OD(2) generalized to valence active spaces. It
permits more accurate calculations of relative energies by accounting for
dynamical correlation.
5.9.3 VQCCD
The VQCCD method is the active space version of the QCCD method described
earlier in Section 5.7.3. Both energies and gradients are available for
VQCCD, so that structure optimization is possible. VQCCD is applicable to a
substantially wider range of problems than the VOD method, because the modified
energy functional is not vulnerable to non variational collapse. Testing to
date suggests that it is capable of describing double bond breaking to similar
accuracy as full valence CASSCF, and that potential curves for triple
bond-breaking are qualitatively correct, although quantitatively in error by a
few tens of kcal/mol. The computational cost scales in the same manner with
system size as the VOD method, albeit with a significantly larger prefactor.
5.9.4 Local Pair Models for Valence Correlations Beyond Doubles
Working with Prof. Head-Gordon at Berkeley, John Parkhill has developed
implementations for pair models which couple 4 and 6 electrons together
quantitatively. Because these truncate the coupled cluster equations at
quadruples and hextuples respectively they have been termed the "Perfect Quadruples"
and "Perfect Hextuples" models. These can be viewed as local approximations
to CASSCF. The PQ and PH models are executed through an extension of Q-Chem's
coupled cluster code, and several options defined for those models will
have the same effects although the mechanism may be different
(CC_DIIS_START, CC_DIIS_SIZE, CC_DOV_THRESH, CC_CONV, etc..).
In the course of implementation, the non-local coupled cluster models
were also implemented up to ∧T6. Because the algorithms are
explicitly sparse their costs relative to the existing implementations
of CCSD are much higher (and should never be used in lieu
of an existing CCMAN code), but this capability may be useful for
development purposes, and when computable, models above CCSDTQ are
highly accurate.
To use PQ, PH, their dynamically correlated "+SD"
versions or this machine generated cluster code set: "CORRELATION MGC".
MGC_AMODEL
Choice of approximate cluster model. |
TYPE:
DEFAULT:
Determines | how the CC equations are approximated: |
OPTIONS:
0% | Local Active-Space Amplitude iterations. |
| (pre-calculate GVB orbitals with |
| your method of choice (RPP is good)). |
| |
7% | Optimize-Orbitals using the VOD 2-step solver. |
| (Experimental only use with MGC_AMPS = 2, 24 ,246) |
| |
8% | Traditional Coupled Cluster up to CCSDTQPH. |
9% | MR-CC version of the Pair-Models. (Experimental) |
RECOMMENDATION:
|
| MGC_NLPAIRS
Number of local pairs on an amplitude. |
TYPE:
DEFAULT:
OPTIONS:
Must be greater than 1, which corresponds to the PP model.
2 for PQ, and 3 for PH. |
RECOMMENDATION:
|
|
|
MGC_AMPS
Choice of Amplitude Truncation |
TYPE:
DEFAULT:
OPTIONS:
2 ≤ n ≤ 123456, a sorted list of integers for every amplitude |
which will be iterated. Choose 1234 for PQ and 123456 for PH |
RECOMMENDATION:
|
| MGC_LOCALINTS
Pair filter on an integrals. |
TYPE:
DEFAULT:
OPTIONS:
Enforces a pair filter on the 2-electron integrals, significantly |
reducing computational cost. Generally useful. for more than 1 pair locality. |
RECOMMENDATION:
|
|
|
MGC_LOCALINTER
Pair filter on an intermediate. |
TYPE:
DEFAULT:
OPTIONS:
Any nonzero value enforces the pair constraint on intermediates, |
significantly reducing computational cost. Not recommended for ≤ 2 pair locality |
RECOMMENDATION:
|
5.9.5 Convergence Strategies and More Advanced Options
These optimized orbital coupled-cluster active space methods enable the use
of the full valence space for larger systems than is possible with
conventional complete active space codes. However, we should note at the
outset that often there are substantial challenges in converging valence
active space calculations (and even sometimes optimized orbital coupled
cluster calculations without an active space). Active space calculations
cannot be regarded as "routine" calculations in the same way as SCF
calculations, and often require a considerable amount of computational trial
and error to persuade them to converge. These difficulties are largely
because of strong coupling between the orbital degrees of freedom and the
amplitude degrees of freedom, as well as the fact that the energy surface is
often quite flat with respect to the orbital variations defining the active
space.
Being aware of this at the outset, and realizing that the program has nothing
against you personally is useful information for the uninitiated user of these
methods. What the program does have, to assist in the struggle to achieve a
converged solution, are accordingly many convergence options, fully documented
in Appendix C. In this section, we describe the basic options and the ideas
behind using them as a starting point. Experience plays a critical role,
however, and so we encourage you to experiment with toy jobs that give rapid
feedback in order to become proficient at diagnosing problems.
If the default procedure fails to converge, the first useful option to employ
is CC_PRECONV_T2Z, with a value of between 10 and 50. This is useful
for jobs in which the MP2 amplitudes are very poor guesses for the converged
cluster amplitudes, and therefore initial iterations varying only the
amplitudes will be beneficial:
CC_PRECONV_T2Z
Whether to pre-converge the cluster amplitudes before beginning orbital
optimization in optimized orbital cluster methods. |
TYPE:
DEFAULT:
0 | (FALSE) |
10 | If CC_RESTART, CC_RESTART_NO_SCF or
CC_MP2NO_GUESS are TRUE |
OPTIONS:
0 | No pre-convergence before orbital optimization. |
n | Up to n iterations in this pre-convergence procedure. |
RECOMMENDATION:
Experiment with this option in cases of convergence failure. |
|
Other options that are useful include those that permit some damping of step
sizes, and modify or disable the standard DIIS procedure. The main choices are
as follows.
CC_DIIS
Specify the version of Pulay's Direct Inversion of the Iterative Subspace
(DIIS) convergence accelerator to be used in the coupled-cluster code. |
TYPE:
DEFAULT:
OPTIONS:
0 | Activates procedure 2 initially, and procedure 1 when gradients are smaller |
| than DIIS12_SWITCH. |
1 | Uses error vectors defined as differences between parameter vectors from |
| successive iterations. Most efficient near convergence. |
2 | Error vectors are defined as gradients scaled by square root of the |
| approximate diagonal Hessian. Most efficient far from convergence. |
RECOMMENDATION:
DIIS1 can be more stable. If DIIS problems are encountered in the early stages
of a calculation (when gradients are large) try DIIS1. |
|
| CC_DIIS_START
Iteration number when DIIS is turned on. Set to a large number to disable
DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Occasionally DIIS can cause optimized orbital coupled-cluster calculations to
diverge through large orbital changes. If this is seen, DIIS should be
disabled. |
|
|
|
CC_DOV_THRESH
Specifies minimum allowed values for the coupled-cluster energy denominators.
Smaller values are replaced by this constant during early iterations only, so
the final results are unaffected, but initial convergence is improved when the
guess is poor. |
TYPE:
DEFAULT:
2502 | Corresponding to 0.25 |
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
Increase to 0.5 or 0.75 for non convergent coupled-cluster calculations. |
|
| CC_THETA_STEPSIZE
Scale factor for the orbital rotation step size. The optimal rotation steps
should be approximately equal to the gradient vector. |
TYPE:
DEFAULT:
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
| If the initial step is smaller than 0.5, the program will increase step |
| when gradients are smaller than the value of THETA_GRAD_THRESH, |
| up to a limit of 0.5. |
RECOMMENDATION:
Try a smaller value in cases of poor convergence and very large orbital
gradients. For example, a value of 01001 translates to 0.1 |
|
|
|
An even stronger-and more-or-less last resort-option permits iteration of the
cluster amplitudes without changing the orbitals:
CC_PRECONV_T2Z_EACH
Whether to pre-converge the cluster amplitudes before each change of the
orbitals in optimized orbital coupled-cluster methods. The maximum number of
iterations in this pre-convergence procedure is given by the value of this
parameter. |
TYPE:
DEFAULT:
OPTIONS:
0 | No pre-convergence before orbital optimization. |
n | Up to n iterations in this pre-convergence procedure. |
RECOMMENDATION:
A very slow last resort option for jobs that do not converge. |
|
5.9.6 Examples
Example 5.0 Two jobs that compare the correlation energy of the water
molecule with partially stretched bonds, calculated via the two coupled-cluster
active space methods, VOD, and VQCCD. These are relatively "easy" jobs to
converge, and may be contrasted with the next example, which is not easy to
converge. The orbitals are restricted.
$molecule
0 1
O
H 1 r
H 1 r a
r = 1.5
a = 104.5
$end
$rem
CORRELATION vod
EXCHANGE hf
BASIS 6-31G
$end
@@@
$molecule
read
$end
$rem
CORRELATION vqccd
EXCHANGE hf
BASIS 6-31G
$end
Example 5.0 The water molecule with highly stretched bonds, calculated via
the two coupled-cluster active space methods, VOD, and VQCCD. These are
"difficult" jobs to converge. The convergence options shown permitted the job
to converge after some experimentation (thanks due to Ed Byrd for this!). The
difficulty of converging this job should be contrasted with the previous
example where the bonds were less stretched. In this case, the VQCCD method
yields far better results than VOD!.
$molecule
0 1
O
H 1 r
H 1 r a
r = 3.0
a = 104.5
$end
$rem
CORRELATION vod
EXCHANGE hf
BASIS 6-31G
SCF_CONVERGENCE 9
THRESH 12
CC_PRECONV_T2Z 50
CC_PRECONV_T2Z_EACH 50
CC_DOV_THRESH 7500
CC_THETA_STEPSIZE 3200
CC_DIIS_START 75
$end
@@@
$molecule
read
$end
$rem
CORRELATION vqccd
EXCHANGE hf
BASIS 6-31G
SCF_CONVERGENCE 9
THRESH 12
CC_PRECONV_T2Z 50
CC_PRECONV_T2Z_EACH 50
CC_DOV_THRESH 7500
CC_THETA_STEPSIZE 3200
CC_DIIS_START 75
$end
5.10 Frozen Natural Orbitals in CCD, CCSD, OD, QCCD, and QCISD Calculations
Large computational savings are possible if the virtual space is truncated
using the frozen natural orbital (FNO) approach.
For example, using a fraction f of the full virtual space results in a 1/(1−f)4-fold speed up for each CCSD iteration
(CCSD scales with the forth power of the virtual space size).
FNO-based truncation for ground-states CC methods was introduced by Bartlett
and co-workers [257,[258,[259].
Extension of the FNO approach to ionized states within
EOM-CC formalism was recently introduced and benchmarked [260]
(see Section 6.6.6).
The FNOs are computed as the eigenstates of the virtual-virtual block of the MP2 density matrix
[O(N5) scaling], and the eigenvalues are the occupation numbers associated with the respective FNOs.
By using a user-specified threshold, the FNOs with the smallest
occupations are frozen in CC calculations. This could be done in
CCSD, CCSD(T), CCSD(2), CCSD(dT), CCSD(fT) as well as CCD, OD,QCCD, VQCCD, and all
possible triples corrections for these wavefunctions.
The truncation can be performed using two different schemes.
The first approach is to simply specify the total number of virtual
orbitals to retain, e.g., as the percentage of total virtual orbitals,
as was done in Refs. .
The second approach is to specify the percentage of total
natural occupation (in the virtual space) that
needs to be recovered in the truncated space.
These two criteria are referred to as the
POVO (percentage of virtual orbitals) and OCCT (occupation threshold)
cutoffs, respectively [260].
Since the OCCT criterion is based on the correlation in a specific molecule,
it yields more consistent results than POVO.
For ionization energy calculations employing 99-99.5% natural occupation
threshold should yields errors (relative to the full virtual space values)
below 1 kcal/mol [260].
The errors decrease linearly as a function of
the total natural occupation recovered, which
can be exploited by extrapolating truncated calculations to the full virtual space values.
This extrapolation scheme is called the extrapolated FNO
(XFNO) procedure [260].
The linear behavior is exhibited by the total energies of the ground
and the ionized states as a function of OCCT.
Therefore, the XFNO scheme can be employed even when the two states are not
calculated on the same level, e.g., in adiabatic energy differences and EOM-IP-CC(2,3)
calculations (more on this in Ref. ).
The FNO truncation often causes slower convergence of the CCSD and EOM procedures.
Nevertheless, despite larger number of iterations, the FNO-based truncation of orbital space
reduces computational cost considerably, with a negligible decline in accuracy [260].
5.10.1 Job Control Options
| CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO) |
TYPE:
DEFAULT:
OPTIONS:
range | 0000-10000 |
abcd | Corresponding to ab.cd% |
RECOMMENDATION:
|
|
|
CC_FNO_USEPOP
Selection of the truncation scheme |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
5.10.2 Example
Example 5.0 CCSD(T) calculation using FNO with POVO=65%
$molecule
0 1
O
H 1 1.0
H 1 1.0 2 100.
$end
$rem
correlation = CCSD(T)
basis = 6-311+G(2df,2pd)
CC_fno_thresh 6500 65% of the virtual space
CC_fno_usepop 0
$end
5.11 Non-Hartree-Fock Orbitals in Correlated Calculations
In cases of problematic open-shell references, e.g., strongly spin-contaminated doublet radicals,
one may choose to use DFT orbitals, which can yield significantly improved results [261]. This can be achieved by first doing DFT calculation and then
reading the orbitals and turning the Hartree-Fock procedure off. A more convenient way is just to
specify EXCHANGE, e.g., EXCHANGE=B3LYP means that B3LYP orbitals will be
computed and used in the CCMAN / CCMAN2 module.
5.11.1 Example
Example 5.0 CCSD calculation of triplet methylene using B3LYP orbitals
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
jobtype SP single point
exchange b3lyp
LEVCOR ccsd
BASIS cc-pVDZ
N_FROZEN_CORE 1
$end
5.12 Analytic Gradients and Properties for Coupled-Cluster Methods
Analytic gradients are available for CCSD, OO-CCD/VOD, CCD, and QCCD/VQCCD methods for
both closed- and open-shell references (UHF and RHF only), including frozen
core and / or virtual functionality. In addition, gradients for selected GVB models are
available.
For the CCSD and OO-CCD wavefunctions, Q-Chem can also calculate
dipole moments, 〈R2〉 (as well as XX, YY and ZZ components separately, which is
useful for assigning different Rydberg states, e.g., 3px vs. 3s, etc.), and the
〈S2〉 values. Interface of the CCSD and (V)OO-CCD codes with the NBO 5.0
package is also available.
This code is closely related to EOM-CCSD properties / gradient calculations
(Section 6.6.10).
Solvent models available for CCSD are described in Chapter 10.2.
Limitations: Gradients and fully relaxed properties
for ROHF and non-HF (e.g., B3LYP) orbitals as well as RI approximation are not yet available.
Note:
If gradients or properties are computed with frozen core / virtual, the algorithm will
replace frozen orbitals to restricted. This will not affect the energies, but will change the
orbital numbering in the CCMAN printout. |
5.12.1 Job Control Options
CC_REF_PROP
Whether or not the non-relaxed (expectation value) or full response (including
orbital relaxation terms) one-particle CCSD
properties will be calculated. The properties currently include permanent
dipole moment, the second moments 〈X2〉, 〈Y2〉, and
〈Z2〉 of electron density, and the total
〈R2〉
= 〈X2〉+〈Y2〉+〈Z2〉 (in atomic units).
Incompatible with JOBTYPE=FORCE, OPT, FREQ. |
TYPE:
DEFAULT:
FALSE | (no one-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations need to be solved (lambda CCSD equations) for properties
with the cost approximately the same as CCSD equations. Use default if you do
not need properties. The cost of the properties calculation itself is low. The
CCSD one-particle density can be analyzed with NBO package by specifying
NBO=TRUE, CC_REF_PROP=TRUE and JOBTYPE=FORCE.
|
|
| CC_REF_PROP_TE
Request for calculation of non-relaxed two-particle CCSD properties. The
two-particle properties currently include 〈S2〉. The one-particle
properties also will be calculated, since the additional cost of the
one-particle properties calculation is inferior compared to the cost of
〈S2〉. The variable CC_REF_PROP must be also set to
TRUE. |
TYPE:
DEFAULT:
FALSE | (no two-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
The two-particle properties are computationally expensive, since
they require calculation and use of the two-particle density matrix (the cost
is approximately the same as the cost of an analytic gradient calculation). Do
not request the two-particle properties unless you really need them. |
|
|
|
CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed.
The variable CC_REF_PROP must be also set to TRUE. |
TYPE:
DEFAULT:
FALSE | (no orbital response will be calculated) |
OPTIONS:
RECOMMENDATION:
Not available for non UHF/RHF references and for the methods that do not have
analytic gradients (e.g., QCISD). |
|
5.12.2 Examples
Example 5.0 CCSD geometry optimization
of HHeF followed up by properties calculations
$molecule
0 1
H .000000 .000000 -1.886789
He .000000 .000000 -1.093834
F .000000 .000000 .333122
$end
$rem
JOBTYPE OPT
CORRELATION CCSD
BASIS aug-cc-pVDZ
GEOM_OPT_TOL_GRADIENT 1
GEOM_OPT_TOL_DISPLACEMENT 1
GEOM_OPT_TOL_ENERGY 1
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE SP
CORRELATION CCSD
BASIS aug-cc-pVDZ
SCF_GUESS READ
CC_REF_PROP 1
CC_FULLRESPONSE 1
$end
Example 5.0 CCSD on 1,2-dichloroethane gauche conformation using SCRF solvent model
$molecule
0 1
C 0.6541334418569877 -0.3817051480045552 0.8808840579322241
C -0.6541334418569877 0.3817051480045552 0.8808840579322241
Cl 1.7322599856434779 0.0877596094659600 -0.4630557359272908
H 1.1862455146007043 -0.1665749506296433 1.7960750032785453
H 0.4889356972641761 -1.4444403797631731 0.8058465784063975
Cl -1.7322599856434779 -0.0877596094659600 -0.4630557359272908
H -1.1862455146007043 0.1665749506296433 1.7960750032785453
H -0.4889356972641761 1.4444403797631731 0.8058465784063975
$end
$rem
JOBTYPE SP
EXCHANGE HF
CORRELATION CCSD
BASIS 6-31g**
N_FROZEN_CORE FC
CC_SAVEAMPL 1 Save CC amplitudes on disk
SOLVENT_METHOD SCRF
SOL_ORDER 15 L=15 Multipole moment order
SOLUTE_RADIUS 36500 3.65 Angstrom Solute Radius
SOLVENT_DIELECTRIC 89300 8.93 Dielectric (Methylene Chloride)
$end
5.13 Memory Options and Parallelization of Coupled-Cluster Calculations
The coupled-cluster suite of methods, which includes ground-state methods
mentioned earlier in this Chapter and excited-state methods in the next Chapter,
has been parallelized to take advantage of the multi-core architecture.
The code is parallelized at the level of the tensor library such that
the most time consuming operation, tensor contraction, is performed on
different processors (or different cores of the same processor) using
shared memory and shared scratch disk space[262].
Parallelization on multiple CPUs or CPU cores is achieved by breaking down
tensor operations into batches and running each batch in a separate thread.
Because each thread occupies one CPU core entirely, the maximum number of
threads must not exceed the total available number of CPU cores. If multiple
computations are performed simultaneously, they together should not run
more threads than available cores. For example, an eight-core node can
accommodate one eight-thread calculation, two four-thread calculations,
and so on.
The number of threads to be used in the calculation is specified as a command
line option ( -nt nthreads) Here nthreads should be given a
positive integer value. If this option is not specified, the job will run
in serial mode using single thread only.
Note:
The use of $QCTHREADS environment variable to specify the number of parallel
threads in coupled-cluster calculations is obsolete. For Q-Chem release 4.0.1 and above,
the number of threads to be used in coupled-cluster calculations must be explicitly
specified with command line option `-nt' or it defaults to single-thread execution. |
Setting the memory limit correctly is also very important for high performance
when running larger jobs. To estimate the amount of memory required for
coupled-clusters and related calculations, one can use the following formula:
Memory = |
(Number of basis set functions)4
131072
|
Mb |
| (5.41) |
If the new code (CCMAN2) is used and the calculation is based on a RHF reference,
the amount of memory needed is a half of that given by the formula.
In addition, if gradients are calculated, the amount should be multiplied
by two.
Because the size of data increases steeply with the size of the molecule
computed, both CCMAN and CCMAN2 are able to use disk space to supplement
physical RAM if so required. The strategies of memory management in older
CCMAN and newer CCMAN2 slightly differ, and that should be taken into account
when specifying memory related keywords in the input file.
The MEM_STATIC keyword specifies the amount of memory in megabytes
to be made available to routines that run prior to coupled-clusters
calculations: Hartree-Fock and electronic repulsion integrals evaluation.
A safe recommended value is 500 Mb.
The value of MEM_STATIC should rarely exceed 1000-2000 Mb even
for relatively large jobs.
The memory limit for coupled-clusters calculations is set by
CC_MEMORY. When running older CCMAN, its value is used as
the recommended amount of memory, and the calculation can in fact use less
or run over the limit. If the job is to run exclusively on a node,
CC_MEMORY should be given 50% of all RAM. If the calculation runs
out of memory, the amount of CC_MEMORY should be reduced
forcing CCMAN to use memory saving algorithms.
CCMAN2 uses a different strategy. It allocates the entire amount of RAM given
by CC_MEMORY before the calculation and treats that as a strict
memory limit. While that significantly improves the stability of larger jobs,
it also requires the user to set the correct value of CC_MEMORY
to ensure high performance.
The default value of approximately 1.5 Gb is not appropriate for large
calculations, especially if the node has more resources available. When running
CCMAN2 exclusively on a node, CC_MEMORY should be set to
75-80% of the total available RAM.
Note:
When running small jobs, using too large CC_MEMORY in CCMAN2
is not recommended because Q-Chem will allocate more resources than needed for
the calculation, which will affect
other jobs that you may wish to run on the same node. |
In addition, the user should verify
that the disk and RAM together have enough space by using the above formula.
In cases when CC_MEMORY set up is in conflict with the
available space on a particular platform, the CC job may segfault
at run time. In such cases readjusting the CC_MEMORY
value in the input is necessary so as to eliminate the segfaulting.
In addition to memory settings, the user may need to adjust
MAX_SUB_FILE_NUM which determines the maximum size
of tmp files.
MEM_STATIC
Sets the memory for individual Fortran program modules |
TYPE:
DEFAULT:
240 | corresponding to 240 Mb or 12% of MEM_TOTAL |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
For direct and semi-direct MP2 calculations, this must exceed OVN +
requirements for AO integral evaluation (32-160 Mb). Up to 2000 Mb for large
coupled-clusters calculations. |
|
| CC_MEMORY
Specifies the maximum size, in Mb, of the buffers for in-core storage of
block-tensors in CCMAN and CCMAN2. |
TYPE:
DEFAULT:
50% of MEM_TOTAL. If MEM_TOTAL is not set, use 1.5 Gb.
A minimum of |
192 Mb is hard-coded. |
OPTIONS:
RECOMMENDATION:
Larger values can give better I/O performance and are recommended for systems
with large memory (add to your .qchemrc file. When running
CCMAN2 exclusively on a node, CC_MEMORY should be set to
75-80% of the total available RAM. ) |
|
|
|
MAX_SUB_FILE_NUM
Sets the maximum number of sub files allowed. |
TYPE:
DEFAULT:
16 Corresponding to a total of 32Gb for a given file. |
OPTIONS:
n | User-defined number of gigabytes. |
RECOMMENDATION:
Leave as default, or adjust according to your system limits. |
|
5.14 Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active Space
The methods described below are related to valence bond theory and are handled by
the GVBMAN module. The following models are available:
CORRELATION
Specifies the correlation level in GVB models handled by GVBMAN. |
TYPE:
DEFAULT:
OPTIONS:
PP | |
CCVB | |
GVB_IP | |
GVB_SIP | |
GVB_DIP | |
OP | |
NP | |
2P | |
RECOMMENDATION:
As a rough guide, use PP for biradicaloids, and CCVB for polyradicaloids involving strong spin correlations. Consult the literature for further guidance. |
|
Molecules where electron correlation is strong are characterized by small
energy gaps between the nominally occupied orbitals (that would comprise the
Hartree-Fock wavefunction, for example) and nominally empty orbitals. Examples
include so-called diradicaloid molecules [263], or molecules with
partly broken chemical bonds (as in some transition-state structures). Because the
energy gap is small, electron configurations other than the reference
determinant contribute to the molecular wavefunction with considerable
amplitude, and omitting them leads to a significant error.
Including all possible configurations however, is a vast overkill. It is common to
restrict the configurations that one generates to be constructed not from all
molecular orbitals, but just from orbitals that are either "core" or
"active". In this section, we consider just one type of active space, which
is composed of two orbitals to represent each electron pair: one nominally
occupied (bonding or lone pair in character) and the other nominally empty, or
correlating (it is typically antibonding in character). This is usually called
the perfect pairing active space, and it clearly is well-suited to represent
the bonding-antibonding correlations that are associated with bond-breaking.
The quantum chemistry within this (or any other) active space is given by a
Complete Active Space SCF (CASSCF) calculation, whose exponential cost growth
with molecule size makes it prohibitive for systems with more than about 14
active orbitals. One well-defined coupled cluster (CC) approximation based
on CASSCF is to include only double substitutions in the valence space
whose orbitals are then optimized. In the framework of conventional
CC theory, this defines the valence optimized doubles (VOD) model [256],
which scales as O(N6) (see Section 5.9.2).
This is still too expensive to be readily applied to large molecules.
The methods described in this section bridge the gap between sophisticated but
expensive coupled cluster methods and inexpensive methods such as DFT, HF and
MP2 theory that may be (and indeed often are) inadequate for describing
molecules that exhibit strong electron correlations such as diradicals. The
coupled cluster perfect pairing (PP) [264,[265], imperfect
pairing (IP) [266] and restricted coupled cluster
(RCC) [267] models are local approximations to VOD that include only a
linear and quadratic number of double substitution amplitudes respectively.
They are close in spirit to generalized valence bond (GVB)-type
wavefunctions [268], because in fact they are all coupled cluster models for
GVB that share the same perfect pairing active space. The most powerful method in the family, the Coupled Cluster Valence Bond (CCVB) method [269,[270,[271], is a valence bond approach that goes well beyond the power of GVB-PP and related methods, as discussed below in Sec. 5.14.2.
5.14.1 Perfect pairing (PP)
To be more specific, the coupled cluster PP wavefunction is written as
|Ψ〉 = exp | ⎛ ⎝
|
nactive ∑
i=1
|
ti |
^
a
|
† i∗
|
|
^
a
|
† ―i∗
|
|
^
a
|
―i
|
|
^
a
|
i
| ⎞ ⎠
|
| Φ〉 |
| (5.42) |
where nactive is the number of active electrons, and the ti are the
linear number of unknown cluster amplitudes, corresponding to exciting the two
electrons in the ith electron pair from their bonding orbital pair to their
antibonding orbital pair. In addition to ti, the core and the active orbitals
are optimized as well to minimize the PP energy. The algorithm used for this is
a slight modification of the GDM method, described for SCF calculations in
Section 4.6.4. Despite the simplicity of the PP wavefunction, with
only a linear number of correlation amplitudes, it is still a useful
theoretical model chemistry for exploring strongly correlated systems.
This is because it is exact for a single electron pair in the PP active space,
and it is also exact for a collection of non-interacting electron pairs in
this active space. Molecules, after all, are in a sense a collection of
interacting electron pairs! In practice, PP on molecules recovers between 60%
and 80% of the correlation energy in its active space.
If the calculation is perfect pairing (CORRELATION = PP), it is
possible to look for unrestricted solutions in addition to restricted ones.
Unrestricted orbitals are the default for molecules with odd
numbers of electrons, but can also be specified for molecules with even numbers
of electrons. This is accomplished by setting GVB_UNRESTRICTED =
TRUE. Given a restricted guess, this will, however usually converge to a
restricted solution anyway, so additional REM variables should be specified to
ensure an initial guess that has broken spin symmetry. This can be accomplished
by using an unrestricted SCF solution as the initial guess, using the
techniques described in Chapter 4. Alternatively a restricted set
of guess orbitals can be explicitly symmetry broken just before the calculation
starts by using GVB_GUESS_MIX, which is described below.
There is also the implementation of Unrestricted-in-Active Pairs (UAP) [272]
which is the default unrestricted implementation
for GVB methods. This method simplifies the process of unrestriction by
optimizing only one set of ROHF MO coefficients and a single rotation
angle for each occupied-virtual pair. These angles are used to construct
a series of 2x2 Given's rotation matrices which are applied to the ROHF
coefficients to determine the α spin MO coefficients and their
transpose is applied to the ROHF coefficients to determine the β
spin MO coefficients. This algorithm is fast and eliminates many of the
pathologies of the unrestricted GVB methods near the dissociation limit.
To generate a full potential curve we find it is best to start at the
desired UHF dissociation solution as a guess for GVB and follow it
inwards to the equilibrium bond distance.
GVB_UNRESTRICTED
Controls restricted versus unrestricted PP jobs. Usually handled
automatically. |
TYPE:
DEFAULT:
same value as UNRESTRICTED |
OPTIONS:
RECOMMENDATION:
Set this variable explicitly only to do a UPP job from an RHF
or ROHF initial guess. Leave this variable alone and specify
UNRESTRICTED=TRUE to access the new Unrestricted-in-Active-Pairs
GVB code which can return an RHF or ROHF solution if used with GVB_DO_ROHF |
|
| GVB_DO_ROHF
Sets the number of Unrestricted-in-Active Pairs to be kept restricted. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If n is the same value as GVB_N_PAIRS returns the ROHF solution
for GVB, only works with the UNRESTRICTED=TRUE
implementation of GVB with GVB_OLD_UPP=0 (it's default value) |
|
|
|
GVB_OLD_UPP
Which unrestricted algorithm to use for GVB. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use Unrestricted-in-Active Pairs |
1 | Use Unrestricted Implementation described in Ref. |
RECOMMENDATION:
Only works for Unrestricted PP and no other GVB model. |
|
| GVB_GUESS_MIX
Similar to SCF_GUESS_MIX, it breaks alpha / beta symmetry for UPP by
mixing the alpha HOMO and LUMO orbitals according to the user-defined fraction
of LUMO to add the HOMO. 100 corresponds to a 1:1 ratio of HOMO and LUMO in
the mixed orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined, 0 ≤ n ≤ 100 |
RECOMMENDATION:
25 often works well to break symmetry without overly
impeding convergence. |
|
|
|
Whilst all of the description in this section refers to PP solved projectively, it is also possible, as described in Sec. 5.14.2 below, to solve variationally for the PP energy. This variational PP solution is the reference wavefunction for the CCVB method. In most cases use of spin-pure CCVB is preferable to attempting to improve restricted PP by permitting the orbitals to spin polarize.
5.14.2 Coupled Cluster Valence Bond (CCVB)
Cases where PP needs improvement include molecules with several strongly
correlated electron pairs that are all localized in the same region of space,
and therefore involve significant inter-pair, as well as intra-pair
correlations. For some systems of this type,
Coupled Cluster Valence Bond (CCVB) [269,[270] is an
appropriate method. CCVB is designed to
qualitatively treat
the breaking of covalent bonds. At the most basic theoretical level, as a molecular system dissociates into a collection
of open-shell fragments, the energy should approach the sum of the ROHF energies of the fragments. CCVB is able to
reproduce this for a wide class of problems, while maintaining proper spin symmetry. Along with this, CCVB's main
strength, come many of the spatial symmetry breaking issues common to the GVB-CC methods.
Like the other methods discussed in this section, the leading contribution to the CCVB wavefunction
is the perfect pairing wavefunction, which is shown in eqn. . One important difference is that CCVB uses the PP
wavefunction as a reference in the same way that other GVBMAN methods use a reference determinant.
The PP wavefunction is a product of simple, strongly orthogonal singlet geminals. Ignoring normalizations, two equivalent
ways of displaying these geminals are
(_i _i + t_i _i^* _i^*) ( - ) (Natural-orbital form)
_i _i^ ( - ) (Valence-bond form),
where on the left and right we have the spatial part (involving ϕ and χ orbitals) and the spin coupling,
respectively.
The VB-form orbitals are non-orthogonal within a pair and are generally AO-like.
The VB form is used in CCVB and the NO form is used in the other GVBMAN methods. It turns out that occupied UHF
orbitals can also be rotated (without affecting the energy) into the VB form (here the spin part would be just
αβ), and as such
we store the CCVB orbital coefficients in the same way as is done in UHF (even though no one spin is assigned to an orbital
in CCVB).
These geminals are uncorrelated
in the same way that molecular orbitals are uncorrelated in a HF calculation. Hence, they are able to describe
uncoupled, or independent, single-bond-breaking processes, like that found in C2H6 → 2 CH3,
but not coupled multiple-bond-breaking processes, such as the dissociation of N2. In the latter system the three bonds
may be described by three singlet geminals, but this picture must somehow translate into the coupling of two spin-quartet
N atoms into an overall singlet, as found at dissociation.
To achieve this sort of thing in a GVB context, it is necessary to correlate the geminals. The part of this correlation
that is essential to bond breaking is
obtained by replacing clusters of singlet geminals with triplet geminals, and recoupling the triplets to an overall
singlet. A triplet geminal is obtained from a singlet by simply modifying the spin component accordingly. We thus
obtain the CCVB wavefunction:
-
= - _0 + _k<l t_kl - _(kl)
+ _k<l<m<n + .
In this expansion, the summations go over the active singlet pairs, and the indices shown in the labellings of the kets
correspond to pairs that are being coupled as described just above. We see that this wavefunction couples clusters
composed of even numbers of geminals. In addition, we see that the amplitudes for clusters containing more than 2
geminals are parameterized by the amplitudes for the 2-pair clusters. This approximation is important for computational
tractability, but actually is just one in a family of CCVB methods: it is possible to include coupled clusters of odd
numbers of pairs, and also to introduce independent parameters for the higher-order amplitudes. At present, only the
simplest level is included in Q-Chem.
Older methods which attempt to describe substantially the same electron correlation effects as CCVB are the IP [266] and RCC [267] wavefunctions. In general CCVB should be used preferentially. It turns out that CCVB relates to the GVB-IP model. In fact, if we were to expand the CCVB wavefunction relative to a
set of determinants, we would see that for each pair of singlet pairs, CCVB contains only one of the two pertinent
GVB-IP doubles
amplitudes. Hence, for CCVB the various computational requirements and timings are very similar to those for GVB-IP.
The main difference between the two models lies in how the doubles amplitudes are used to parameterize the
quadruples, sextuples, etc., and this is what allows CCVB to give correct energies at full bond dissociation.
A CCVB calculation is invoked by setting CORRELATION = CCVB. The number of active singlet geminals must
be specified by GVB_N_PAIRS. After this, an initial guess is chosen. There are three main
options for this, specified by the following keyword
CCVB_GUESS
Specifies the initial guess for CCVB calculations |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard GVBMAN guess (orbital localization via GVB_LOCAL + Sano procedure). |
2 | Use orbitals from previous GVBMAN calculation, along with SCF_GUESS = read. |
3 | Convert UHF orbitals into pairing VB form. |
RECOMMENDATION:
Option 1 is the most useful overall. The success of GVBMAN methods is often dependent on localized orbitals, and this guess shoots for these. Option 2 is useful for comparing results to other GVBMAN methods, or if other GVBMAN methods are able to obtain a desired result more efficiently. Option 3 can be useful for bond-breaking situations when a pertinent UHF solution has been found. It works best for small systems, or if the unrestriction is a local phenomenon within a larger molecule. If the unrestriction is nonlocal and the system is large, this guess will often produce a solution that is not the global minimum. Any UHF solution has a certain number of pairs that are unrestricted, and this will be output by the program. If GVB_N_PAIRS exceeds this number, the standard GVBMAN initial-guess procedure will be used to obtain a guess for the excess pairs |
|
For potential energy surfaces, restarting from a previously computed CCVB solution is recommended. This is invoked by GVB_RESTART = TRUE. Whenever this is used, or any time orbitals are being read directly into CCVB from another calculation, it is important to also set:
-
SCF_GUESS = READ
- MP2_RESTART_NO_SCF = TRUE
- SCF_ALGORITHM = DIIS
This bypasses orthogonalization schemes used elsewhere within Q-Chem that are likely to jumble the CCVB guess.
In addition to the parent CCVB method as discussed up until now, we have included two related schemes for energy optimization, whose operation is controlled by
the following keyword:
CCVB_METHOD
Optionally modifies the basic CCVB method |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard CCVB model |
3 | Independent electron pair approximation (IEPA) to CCVB |
4 | Variational PP (the CCVB reference energy) |
RECOMMENDATION:
Option 1 is generally recommended. Option 4 is useful for preconditioning, and for obtaining localized-orbital solutions, which may be used in subsequent calculations. It is also useful for cases in which the regular GVBMAN PP code becomes variationally unstable. Option 3 is a simple independent-amplitude approximation to CCVB. It avoids the cubic-scaling amplitude equations of CCVB, and also is able to reach the correct dissociation energy for any molecular system (unlike regular CCVB which does so only for cases in which UHF can reach a correct dissociate limit). However the IEPA approximation to CCVB is sometimes variationally unstable, which we have yet to observe in regular CCVB. |
|
Example 5.0 N2 molecule in the intermediately dissociated region. In this case, SCF_ALGORITHM DIIS is necessary to obtain the symmetry unbroken RHF solution, which itself is necessary to obtain the proper CCVB solution. Note that many keywords general to GVBMAN are also used in CCVB.
$molecule
0 1
N 0 0 0
N 0 0 2.0
$end
$rem
jobtype = sp
unrestricted = false
basis = 6-31g*
exchange = hf
correlation = ccvb
gvb_n_pairs = 3
ccvb_method = 1
ccvb_guess = 1
gvb_local = 2
gvb_orb_max_iter = 100000
gvb_orb_conv = 7
gvb_restart = false
scf_convergence = 10
thresh = 14
scf_guess = sad
mp2_restart_no_scf = false
scf_algorithm = diis
max_scf_cycles = 2000
symmetry = false
sym_ignore = true
print_orbitals = true
$end
5.14.3 Second order correction to perfect pairing: PP(2)
The PP and CCVB models are potential replacements for HF theory as a zero order
description of electronic structure and can be used as a starting point for
perturbation theory. They neglect all correlations that involve electron
configurations with one or more orbitals that are outside the active space.
Physically this means that the so-called "dynamic correlations", which
correspond to atomic-like correlations involving high angular momentum virtual
levels are neglected. Therefore, the GVB models may not be very accurate
for describing energy differences that are sensitive to this neglected
correlation energy, e.g., atomization energies. It is desirable to
correct them for this neglected correlation in a way that is similar to how the
HF reference is corrected via MP2 perturbation theory.
For this purpose, the leading (second order) correction to the PP model, termed
PP(2) [273], has been formulated and efficiently implemented for
restricted and unrestricted orbitals (energy only). PP(2) improves upon many of
the worst failures of MP2 theory (to which it is analogous), such as for open
shell radicals. PP(2) also greatly improves relative energies relative to PP
itself. PP(2) is implemented using a resolution of the identity (RI) approach
to keep the computational cost manageable. This cost scales in the same 5th-order
way with molecular size as RI-MP2, but with a pre-factor
that is about 5 times larger. It is therefore vastly cheaper than CCSD or
CCSD(T) calculations which scale with the 6th and 7th powers
of system size respectively. PP(2) calculations are requested with
CORRELATION = PP(2). Since the only available algorithm uses auxiliary
basis sets, it is essential to also provide a valid value for
AUX_BASIS to have a complete input file.
The example below shows a PP(2) input file for the challenging case of the N2
molecule with a stretched bond. For this reason a number of the non-standard
options discussed in Sec. 5.14.1 and Sec. 5.14.4
for orbital convergence are enabled here. First, this
case is an unrestricted calculation on a molecule with an even number of
electrons, and so it is essential to break the alpha / beta spin symmetry in
order to find an unrestricted solution. Second, we have chosen to leave the
lone pairs uncorrelated, which is accomplished by specifying
GVB_N_PAIRS.
Example 5.0 A non-standard PP(2) calculation. UPP(2) for stretched N2 with
only 3 correlating pairs Try Boys localization scheme for initial guess.
$molecule
0 1
N
N 1 1.65
$end
$rem
UNRESTRICTED true
CORRELATION pp(2)
EXCHANGE hf
BASIS cc-pvdz
AUX_BASIS rimp2-cc-pvdz must use RI with PP(2)
% PURECART 11111
SCF_GUESS_MIX 10 mix SCF guess 100{\%}
GVB_GUESS_MIX 25 mix GVB guess 25{\%} also!
GVB_N_PAIRS 3 correlate only 3 pairs
GVB_ORB_CONV 6 tighter convergence
GVB_LOCAL 1 use Boys initial guess
$end
5.14.4 Other GVBMAN methods and options
In Q-Chem, the unrestricted and restricted GVB methods are
implemented with a resolution of the identity (RI) algorithm
that makes them computationally very efficient [274,[275].
They can be applied to systems with more than 100 active electrons, and both
energies and analytical gradients are available. These methods are requested
via the standard CORRELATION keyword. If
AUX_BASIS is not specified, the calculation uses four-center
two-electron integrals by default. Much faster auxiliary basis algorithms (see
5.5 for an introduction), which are used for the correlation energy
(not the reference SCF energy), can be enabled by specifying a valid string for
AUX_BASIS. The example below illustrates a simple IP calculation.
Example 5.0 Imperfect pairing with auxiliary basis set for geometry
optimization.
$molecule
0 1
H
F 1 1.0
$end
$rem
JOBTYPE opt
CORRELATION gvb_ip
BASIS cc-pVDZ
AUX_BASIS rimp2-cc-pVDZ
% PURECART 11111
$end
If further improvement in the orbitals are needed, the GVB_SIP, GVB_DIP, OP, NP
and 2P models are also included [272]. The GVB_SIP model
includes all the amplitudes of GVB_IP plus a set of quadratic
amplitudes the represent the single ionization of a pair. The GVB_DIP
model includes the GVB_SIP models amplitudes and the doubly ionized
pairing amplitudes which are analogous to the correlation of the occupied
electrons of the ith pair exciting into the virtual orbitals of
the jth pair. These two models have the implementation limit of no
analytic orbital gradient, meaning that a slow finite differences calculation
must be performed to optimize their orbitals, or they must be computed using
orbitals from a different method. The 2P model is the same as the GVB_DIP model,
except it only allows the amplitudes to couple via integrals that span only two
pairs. This allows for a fast implementation of it's analytic orbital gradient
and enables the optimization of it's own orbitals. The OP method is like the
2P method except it removes the "direct"-like IP amplitudes and all of the
same-spin amplitudes. The NP model is the GVB_IP model with the DIP amplitudes.
This model is the one that works best with the symmetry breaking corrections
that will be discussed later. All GVB methods except GVB_SIP and
GVB_DIP have an analytic nuclear gradient implemented for both regular
and RI four-center two-electron integrals.
There are often considerable challenges in converging the orbital optimization
associated with these GVB-type calculations. The situation is somewhat
analogous to SCF calculations but more severe because there are more orbital
degrees of freedom that affect the energy (for instance, mixing occupied active
orbitals amongst each other, mixing active virtual orbitals with each other, mixing
core and active occupied, mixing active virtual and inactive virtual).
Furthermore, the energy changes associated with many of these new orbital
degrees of freedom are rather small and delicate. As a consequence, in cases
where the correlations are strong, these GVB-type jobs often require many more
iterations than the corresponding GDM calculations at the SCF level. This is a
reflection of the correlation model itself. To deal with convergence issues, a
number of REM values are available to customize the calculations, as listed
below.
GVB_ORB_MAX_ITER
Controls the number of orbital iterations allowed in GVB-CC calculations.
Some jobs, particularly unrestricted PP jobs can require 500-1000 iterations. |
TYPE:
DEFAULT:
OPTIONS:
User-defined number of iterations. |
RECOMMENDATION:
Default is typically adequate, but some jobs, particularly UPP jobs, can
require 500-1000 iterations if converged tightly. |
|
| GVB_ORB_CONV
The GVB-CC wavefunction is considered converged when the root-mean-square
orbital gradient and orbital step sizes are less than
10−GVB_ORB_CONV. Adjust THRESH simultaneously. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use 6 for PP(2) jobs or geometry optimizations.
Tighter convergence (i.e. 7 or higher) cannot always be reliably achieved. |
|
|
|
GVB_ORB_SCALE
Scales the default orbital step size by n/1000. |
TYPE:
DEFAULT:
1000 | Corresponding to 100% |
OPTIONS:
RECOMMENDATION:
Default is usually fine, but for some stretched geometries it
can help with convergence to use smaller values. |
|
| GVB_AMP_SCALE
Scales the default orbital amplitude iteration step size by n/1000 for IP/RCC.
PP amplitude equations are solved analytically, so this parameter does not
affect PP. |
TYPE:
DEFAULT:
1000 | Corresponding to 100% |
OPTIONS:
RECOMMENDATION:
Default is usually fine, but in some highly-correlated
systems it can help with convergence to use smaller values. |
|
|
|
GVB_RESTART
Restart a job from previously-converged GVB-CC orbitals. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Useful when trying to converge to the same GVB
solution at slightly different geometries, for example. |
|
| GVB_REGULARIZE
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spin-unrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term -(c/10000)(etijp−1)/(e1−1) |
TYPE:
DEFAULT:
0 for restricted | 1 for unrestricted |
OPTIONS:
RECOMMENDATION:
Should be increased if unrestricted amplitudes do not converge or
converge slowly at dissociation. Set this to zero to remove
all dynamically-valued amplitude regularization. |
|
|
|
GVB_POWER
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spin-unrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term included in
the energy denominator: -(c/10000)(etijp−1)/(e1−1) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should be decreased if unrestricted amplitudes do not converge or
converge slowly at dissociation, and should be kept even valued. |
|
| GVB_SHIFT
Value for a statically valued energy shift in the energy
denominator used to solve the coupled cluster amplitude equations, n/10000. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Default is fine, can be used in lieu of the dynamically
valued amplitude regularization if it does not aid convergence. |
|
|
|
Another issue that a user of these methods should be aware of is the fact that
there is a multiple minimum challenge associated with GVB
calculations. In SCF calculations it is sometimes possible to converge to more
than one set of orbitals that satisfy the SCF equations at a given geometry.
The same problem can arise in GVB calculations, and based on our experience to
date, the problem in fact is more commonly encountered in GVB calculations than
in SCF calculations. A user may therefore want to (or have to!) tinker with the
initial guess used for the calculations. One way is to set GVB_RESTART
= TRUE (see above), to replace the default initial guess (the converged SCF
orbitals which are then localized). Another way is to change the localized
orbitals that are used in the initial guess, which is controlled by the
GVB_LOCAL variable, described below. Sometimes different localization
criteria, and thus different initial guesses, lead to different converged
solutions. Using the new amplitude regularization keywords enables some control
over the solution GVB optimizes [276].
A calculation can be performed with amplitude regularization
to find a desired solution, and then the calculation can be rerun with GVB_RESTART
= TRUE and the regularization turned off to remove the energy penalty of regularization.
GVB_LOCAL
Sets the localization scheme used in the initial guess wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
0 | No Localization |
1 | Boys localized orbitals |
2 | Pipek-Mezey orbitals |
RECOMMENDATION:
Different initial guesses can sometimes lead to different solutions.
It can be helpful to try both to ensure the global minimum has been found. |
|
| GVB_DO_SANO
Sets the scheme used in determining the active virtual orbitals
in a Unrestricted-in-Active Pairs GVB calculation. |
TYPE:
DEFAULT:
OPTIONS:
0 | No localization or Sano procedure |
1 | Only localizes the active virtual orbitals |
2 | Uses the Sano procedure |
RECOMMENDATION:
Different initial guesses can sometimes lead to different
solutions. Disabling sometimes can aid in finding more non-local solutions for the orbitals. |
|
|
|
Other $rem variables relevant to GVB calculations are given below. It is
possible to explicitly set the number of active electron pairs using the
GVB_N_PAIRS variable. The default is to make all valence electrons
active. Other reasonable choices are certainly possible. For instance all
electron pairs could be active (nactive = nβ). Or
alternatively one could make only formal bonding electron pairs active
(nactive = NSTO−3G − nα). Or in some cases, one might
want only the most reactive electron pair to be active (nactive = 1).
Clearly making physically appropriate choices for this variable is essential
for obtaining physically appropriate results!
GVB_N_PAIRS
Alternative to CC_REST_OCC and CC_REST_VIR for setting
active space size in GVB and valence coupled cluster methods. |
TYPE:
DEFAULT:
PP active space (1 occ and 1 virt for each valence electron pair) |
OPTIONS:
RECOMMENDATION:
Use the default unless one wants to study a special active space. When using
small active spaces, it is important to ensure that the proper orbitals are
incorporated in the active space. If not, use the $reorder_mo feature to adjust
the SCF orbitals appropriately. |
|
| GVB_PRINT
Controls the amount of information printed during a GVB-CC job. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should never need to go above 0 or 1. |
|
|
|
GVB_TRUNC_OCC
Controls how many pairs' occupied orbitals are truncated from the GVB active space |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing the n
lowest energy occupied orbitals from the GVB active space
while leaving their paired virtual orbitals in the active space.
Only the models including the SIP and DIP amplitudes (ie NP and 2P)
benefit from this all other models this equivalent to just
reducing the total number of pairs. |
|
| GVB_TRUNC_VIR
Controls how many pairs' virtual orbitals are truncated from the GVB active space |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing
the n highest energy occupied orbitals from the GVB
active space while leaving their paired virtual orbitals
in the active space. Only the models including the SIP
and DIP amplitudes (ie NP and 2P) benefit from this all
other models this equivalent to just reducing the total number of pairs. |
|
|
|
GVB_REORDER_PAIRS
Tells the code how many GVB pairs to switch around |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for the user to change the order the active
pairs are placed in after the orbitals are read in or are
guessed using localization and the Sano procedure. Up to 5
sequential pair swaps can be made, but it is best to leave this alone. |
|
| GVB_REORDER_1
Tells the code which two pairs to swap first |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified
in GVB_REORDER_PAIRS ≥ 1. |
|
|
|
GVB_REORDER_2
Tells the code which two pairs to swap second |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified in GVB_REORDER_PAIRS ≥ 2. |
|
| GVB_REORDER_3
Tells the code which two pairs to swap third |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002.
Must be specified in GVB_REORDER_PAIRS ≥ 3. |
|
|
|
GVB_REORDER_4
Tells the code which two pairs to swap fourth |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must
be specified in GVB_REORDER_PAIRS ≥ 4. |
|
| GVB_REORDER_5
Tells the code which two pairs to swap fifth |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must be
specified in GVB_REORDER_PAIRS ≥ 5. |
|
|
|
it is known that symmetry breaking of the orbitals to favor
localized solutions over non-local solutions is an issue with
GVB methods in general. A combined coupled-cluster perturbation
theory approach to solving symmetry breaking (SB) using perturbation
theory level double amplitudes that connect up to three pairs
has been examined in the literature [277,[278],
and it seems to alleviate the SB problem to a large extent.
It works in conjunction with the GVB_IP, NP, and 2P levels of
correlation for both restricted and unrestricted wavefunctions
(barring that there is no restricted implementation of the 2P model,
but setting GVB_DO_ROHF to the same number as the number of pairs in the system is equivalent).
GVB_SYMFIX
Should GVB use a symmetry breaking fix |
TYPE:
DEFAULT:
OPTIONS:
0 | no symmetry breaking fix |
1 | symmetry breaking fix with virtual orbitals spanning the active space |
2 | symmetry breaking fix with virtual orbitals spanning the whole virtual space |
RECOMMENDATION:
It is best to stick with type 1 to get a symmetry breaking correction
with the best results coming from CORRELATION=NP and GVB_SYMFIX=1. |
|
| GVB_SYMPEN
Sets the pre-factor for the amplitude regularization term for the SB amplitudes |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the pre-factor for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e(c*100)*t2−1). |
|
|
|
GVB_SYMSCA
Sets the weight for the amplitude regularization term for the SB amplitudes |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the weight for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e(c*100)*t2−1). |
|
We have already mentioned a few issues associated with the GVB calculations: the
neglect of dynamic correlation [which can be remedied with PP(2)], the
convergence challenges and the multiple minimum issues. Another weakness of
these GVB methods is the occasional symmetry-breaking artifacts that are a
consequence of the limited number of retained pair correlation amplitudes. For
example, benzene in the PP approximation prefers D3h symmetry over
D6h by 3 kcal/mol (with a 2 distortion), while in IP, this difference
is reduced to 0.5 kcal/mol and less than 1 [266]. Likewise
the allyl radical breaks symmetry in the unrestricted PP model [265],
although to a lesser extent than in restricted open shell
HF. Another occasional weakness is the limitation to the perfect pairing active
space, which is not necessarily appropriate for molecules with expanded valence
shells, such as in some transition metal compounds (e.g. expansion from 4s3d
into 4s4p3d) or possibly hypervalent molecules (expansion from 3s3p into
3s3p3d). The singlet strongly orthogonal geminal method (see the next
section) is capable of dealing with expanded valence shells and could be used
for such cases. The perfect pairing active space is satisfactory for most
organic and first row inorganic molecules.
To summarize, while these GVB methods are powerful and can yield much insight
when used properly, they do have enough pitfalls for not to be
considered true "black box" methods.
5.15 Geminal Models
5.15.1 Reference wavefunction
Computational models that use single reference wavefunction describe molecules
in terms of independent electrons interacting via mean Coulomb and exchange
fields. It is natural to improve this description by using correlated electron
pairs, or geminals, as building blocks for molecular wavefunctions.
Requirements of computational efficiency and size consistency constrain
geminals to have Sz=0 [279], with each geminal spanning its own
subspace of molecular orbitals [280]. Geminal wavefunctions were
introduced into computational chemistry by Hurley, Lennard-Jones, and Pople [281].
An excellent review of the history and properties of geminal
wavefunctions is given by Surjan [282].
We implemented a size consistent model chemistry based on Singlet type Strongly
orthogonal Geminals (SSG). In SSG, the number of molecular orbitals in each
singlet electron pair is an adjustable parameter chosen to minimize total
energy. Open shell orbitals remain uncorrelated. The SSG wavefunction is
computed by setting SSG $rem variable to 1. Both spin-restricted
(RSSG) and spin-unrestricted (USSG) versions are available, chosen by the
UNRESTRICTED $rem variable.
The wavefunction has the form
| |
|
|
^
A
|
[ψ1(r1,r2) … ψnβ(r2nβ−1,r2nβ) ϕi(r2nβ+1) … ϕj(rnβ+nα)] |
| |
| |
|
|
∑
k ∈ A
|
|
DAi
√2
|
[ϕk(r1) |
-
ϕk
|
(r2)−ϕk(r2) |
-
ϕk
|
(r1)] |
| | (5.43) |
| |
|
| |
| |
|
| |
|
with the coefficients C, D, and subspaces A chosen to minimize the energy
evaluated with the exact Hamiltonian ∧H. A constraint
―Ckλ=Ckλ for all MO coefficients yields a
spin-restricted version of SSG.
SSG model can use any orbital-based initial guess. It is often advantageous
to compute Hartree-Fock orbitals and then read them as initial guess for SSG.
The program distinguishes Hartree-Fock and SSG initial guess wavefunctions,
and in former case makes preliminary assignment of individual orbital pairs
into geminals. The verification of orbital assignments is performed every ten
wavefunction optimization steps, and the orbital pair is reassigned if total
energy is lowered.
The convergence algorithm consists of combination of three types of
minimization steps. The direct minimization steps [283] seeks a
minimum along the gradient direction, rescaled by the quantity analogous to the
orbital energy differences in SCF theory [279]. If the orbitals
are nearly degenerate or inverted, a perturbative re-optimization of single
geminal is performed. Finally, new set of the coefficients C and D is
formed from a linear combination of previous iterations, in a manner similar to
DIIS algorithm [173,[174]. The size of iterative subspace
is controlled by the DIIS_SUBSPACE_SIZE keyword.
After convergence is achieved, SSG reorders geminals based on geminal energy.
The energy, along with geminal expansion coefficients, is printed for each
geminal. Presence of any but the leading coefficient with large absolute value
(value of 0.1 is often used for the definition of "large") indicates the
importance of electron correlation in the system. The Mulliken population
analysis is also performed for each geminal, which enables easy assignment of
geminals into such chemical objects as core electron pairs, chemical bonds, and
lone electron pairs.
As an example, consider the sample calculation of ScH molecule with 6-31G basis
set at the experimental bond distance of 1.776 Å. In its singlet ground
state the molecule has 11 geminals. Nine of them form core electrons on Sc.
Two remaining geminals are:
Geminal 10 E = -1.342609
0.99128 -0.12578 -0.03563 -0.01149 -0.01133 -0.00398
Geminal 11 E = -0.757086
0.96142 -0.17446 -0.16872 -0.12414 -0.03187 -0.01227 -0.01204 -0.00435 -0.00416 -0.00098
Mulliken population analysis shows that geminal 10 is delocalized between Sc
and H, indicating a bond. It is moderately correlated, with second expansion
coefficient of a magnitude 0.126. The geminal of highest energy is localized
on Sc. It represents 4s2 electrons and describes their excitation into 3d
orbitals. Presence of three large expansion coefficients show that this effect
cannot be described within GVB framework [284].
5.15.2 Perturbative corrections
The SSG description of molecular electronic structure can be improved
by perturbative description of missing inter-geminal correlation
effects. We have implemented Epstein-Nesbet form of perturbation
theory [285,[286] that permits a balanced description of one- and
two-electron contributions to excited states' energies in SSG model.
This form of perturbation theory is especially accurate for
calculation of weak intermolecular forces. Also, two-electron
[i―j,j―i] integrals are included in the reference
Hamiltonian in addition to intra-geminal [i―j,i―j]
integrals that are needed for reference wavefunction to be an
eigenfunction of the reference Hamiltonian [287].
All perturbative contributions to the SSG(EN2) energy (second-order
Epstein-Nesbet perturbation theory of SSG wavefunction) are analyzed in
terms of largest numerators, smallest denominators, and total energy
contributions by the type of excitation. All excited states are
subdivided into dispersion-like with correlated excitation within one
geminal coupled to the excitation within another geminal, single, and
double electron charge transfer. This analysis permits careful
assessment of the quality of SSG reference wavefunction. Formally,
the SSG(EN2) correction can be applied both to RSSG and USSG
wavefunctions. Experience shows that molecules with broken or nearly
broken bonds may have divergent RSSG(EN2) corrections. USSG(EN2)
theory is balanced, with largest perturbative corrections to the
wavefunction rarely exceeding 0.1 in magnitude.
SSG
Controls the calculation of the SSG wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute the SSG wavefunction |
1 | Do compute the SSG wavefunction |
RECOMMENDATION:
See also the UNRESTRICTED and DIIS_SUBSPACE_SIZE $rem
variables. |
|
Chapter 6 Open-Shell and Excited-State Methods
6.1 General Excited-State Features
As for ground state calculations, performing an adequate excited-state
calculation involves making an appropriate choice of method and basis set. The
development of effective approaches to modelling electronic excited states has
historically lagged behind advances in treating the ground state. In part this
is because of the much greater diversity in the character of the wavefunctions
for excited states, making it more difficult to develop broadly applicable
methods without molecule-specific or even state-specific specification of the
form of the wavefunction. Recently, however, a hierarchy of single-reference
ab initio methods has begun to emerge for the treatment of excited
states. Broadly speaking, Q-Chem contains methods that are capable of giving
qualitative agreement, and in many cases quantitative agreement with experiment
for lower optically allowed states. The situation is less satisfactory for
states that involve two-electron excitations, although even here reasonable
results can sometimes be obtained. Moreover, some of the excited state methods
can treat open-shell wavefunctions, e.g. diradicals, ionized
and electron attachment states and more.
In excited-state calculations, as for ground state calculations, the user must
strike a compromise between cost and accuracy. The few sections of this Chapter
summarize Q-Chem's capabilities in four general classes of excited state
methods:
- Single-electron wavefunction-based methods (Section 6.2).
These are excited state treatments of roughly the same level of
sophistication as the Hartree-Fock ground state method, in the sense
that electron correlation is essentially ignored. Single excitation
configuration interaction (CIS) is the workhorse method of this type.
The spin-flip variant of CIS extends it to diradicals.
- Time-dependent density functional theory (TDDFT)
(Section 6.3). TDDFT is the most useful extension of density
functional theory to excited states that has been developed so far. For a
cost that is little greater than the simple wavefunction methods such as
CIS, a significantly more accurate method results. TDDFT can be extended
to treat di- and tri-radicals and bond-breaking by adopting the spin-flip
approach (see Section 6.3.1 for details).
- The Maximum Overlap Method (MOM) for excited SCF states
(Section 6.5). This method overcomes some of the deficiencies of TDDFT
and, in particular, can be used for modeling charge-transfer and Rydberg
transitions.
- Wavefunction-based electron correlation treatments
(Sections 6.4, 6.7, 6.8 and 6.6).
Roughly speaking, these are excited state analogs of the ground state
wavefunction-based electron correlation methods discussed in
Chapter 5. They are more accurate than the methods
of Section 6.2, but also significantly more computationally
expensive. These methods can also describe certain multi-configurational
wavefunctions, for example, problematic doublet radicals, diradicals,
triradicals, and more.
In general, a basis set appropriate for a ground state density functional
theory or a Hartree-Fock calculation will be appropriate for describing
valence excited states. However, many excited states involve significant
contributions from diffuse Rydberg orbitals, and, therefore, it is often
advisable to use basis sets that include additional diffuse functions. The
6-31+G* basis set is a reasonable compromise for the low-lying valence excited
states of many organic molecules. To describe true Rydberg excited states,
Q-Chem allows the user to add two or more sets of diffuse functions (see
Chapter 7). For example the 6-311(2+)G* basis includes two sets
of diffuse functions on heavy atoms and is generally adequate for description
of both valence and Rydberg excited states.
Q-Chem supports four main types of excited state calculation:
- Vertical absorption spectrum
This is the calculation of the excited states of the molecule at the
ground state geometry, as appropriate for absorption spectroscopy. The
methods supported for performing a vertical absorption calculation are:
CIS, RPA, XCIS, SF-XCIS, CIS(D), ADC(2)-s, ADC(2)-x,
RAS-SF, EOM-CCSD and EOM-OD, each of which will be discussed in turn.
- Visualization
It is possible to visualize the excited
states either by attachment / detachment density analysis (available for CIS, RPA, and TDDFT only)
or by plotting
the transition density (see $plots descriptions in
Chapters 3 and 10). Transition densities can be calculated
for CIS and EOM-CCSD methods.
The theoretical basis of the attachment / detachment density analysis
is discussed in Section 6.4 of this Chapter. In addition
Dyson orbitals can be calculated and plotted for the ionization from the ground and
electronically excited states for the CCSD and EOM-CCSD wavefunctions.
For the RAS-SF method (Section 6.8), one can plot the natural orbitals
of a computed electronic state.
- Excited-state optimization
Optimization of the geometry of stationary points on excited state
potential energy surfaces is valuable for understanding the geometric
relaxation that occurs between the ground and excited state. Analytic
first derivatives are available for UCIS, RCIS, TDDFT and EOM-CCSD, EOM-OD
excited state optimizations may also be performed using finite difference
methods, however, these can be very time-consuming to compute.
- Optimization of the crossings between potential energy surfaces
Seams between potential energy surfaces can be located and optimized by using analytic
gradients within CCSD and EOM-CCSD formalisms.
- Properties
Properties such as transition dipoles, dipole moments, spatial extent
of electron densities and 〈S2〉 values can be computed for
EOM-CCSD, EOM-OD, RAS-SF and CIS wavefunctions.
- Excited-state vibrational analysis
Given an optimized excited state geometry, Q-Chem can calculate the
force constants at the stationary point to predict excited state
vibrational frequencies. Stationary points can also be characterized as
minima, transition structures or nth-order saddle points.
Analytic excited state vibrational analysis can only be performed using
the UCIS, RCIS and TDDFT methods, for which efficient analytical second
derivatives are available. EOM-CCSD frequencies are also available using
analytic first derivatives and second derivatives obtained from finite
difference methods. EOM-OD frequencies are only available through finite
difference calculations.
EOM-CC, and most of the CI codes are part of CCMAN and CCMAN2.
6.2 Non-Correlated Wavefunction Methods
Q-Chem includes several excited state methods which do not incorporate
correlation: CIS, XCIS and RPA. These methods are sufficiently inexpensive
that calculations on large molecules are possible, and are roughly
comparable to the HF treatment of the ground state in terms of performance.
They tend to yield qualitative rather than quantitative insight. Excitation
energies tend to exhibit errors on the order of an electron volt, consistent
with the neglect of electron correlation effects, which are generally
different in the ground state and the excited state.
6.2.1 Single Excitation Configuration Interaction (CIS)
The derivation of the CI-singles [289,[290] energy and wave
function begins by selecting the HF single-determinant wavefunction as
reference for the ground state of the system:
ΨHF = |
1
|
|
det
| { χ1 χ 2…χi χ j…χn } |
| (6.1) |
where n is the number of electrons, and the spin orbitals
are expanded in a finite basis of N atomic orbital basis functions. Molecular
orbital coefficients {cμi} are usually found by SCF procedures which
solve the Hartree-Fock equations
where S is the overlap matrix, C is the matrix of molecular
orbital coefficients, ε is a diagonal matrix of orbital
eigenvalues and F is the Fock matrix with elements
Fμυ = Hμυ + |
∑
λσ
|
|
∑
i
|
cμi cυi ( μλ || υσ ) |
| (6.4) |
involving the core Hamiltonian and the anti-symmetrized two-electron
integrals
( μν||λσ )= | ⌠ ⌡
|
| ⌠ ⌡
|
ϕμ (r1 )ϕν (r2 )( 1 | / |
1 r12 r12 )[ ϕλ (r1 )ϕσ (r2 )−ϕλ (r2 )ϕσ (r1 ) ]dr1 dr2 |
| (6.5) |
On solving Eq. (6.3), the total energy of the ground state single determinant
can be expressed as
EHF = |
∑
μυ
|
PμυHF Hμυ + |
1
2
|
|
∑
μυλσ
|
PμυHF PλσHF ( μλ || υσ ) +Vnuc |
| (6.6) |
where PHF is the HF density matrix and Vnuc is the nuclear repulsion
energy.
Equation (6.1) represents only one of many possible determinants made
from orbitals of the system; there are in fact n(N−n) possible singly
substituted determinants constructed by replacing an orbital occupied in the
ground state (i, j, k,…) with an orbital unoccupied in the ground
state (a, b, c, …). Such wavefunctions and energies can be written
Ψia = |
1
|
|
det
| { χ1 χ 2…χa χ j…χn } |
| (6.7) |
Eia = EHF +εa −εi −( ia || ia ) |
| (6.8) |
where we have introduced the anti-symmetrized two-electron integrals in the
molecular orbital basis
( pq || rs )= |
∑
μυλσ
|
cμp cυq cλr cσs ( μλ || υσ ) |
| (6.9) |
These singly excited wavefunctions and energies could be considered crude
approximations to the excited states of the system. However, determinants of
the form Eq. (6.7) are deficient in that they:
- do not yield pure spin states
- resemble more closely ionization rather than excitation
- are not appropriate for excitation into degenerate states
These deficiencies can be partially overcome by representing the excited
state wavefunction as a linear combination of all possible singly excited
determinants,
where the coefficients {aia} can be obtained by diagonalizing the
many-electron Hamiltonian, A, in the space of all single
substitutions. The appropriate matrix elements are:
Aia,jb = 〈 Ψia |H| Ψjb 〉 = [ EHF +εa −εj ]δij δab −( ja || ib ) |
| (6.11) |
According to Brillouin's, theorem single substitutions do not interact directly with a
reference HF determinant, so the resulting eigenvectors from the CIS excited
state represent a treatment roughly comparable to that of the HF ground state.
The excitation energy is simply the difference between HF ground state energy
and CIS excited state energies, and the eigenvectors of A correspond
to the amplitudes of the single-electron promotions.
CIS calculations can be performed in Q-Chem using restricted
(RCIS) [289,[290], unrestricted (UCIS), or restricted open shell
(ROCIS) [291] spin orbitals.
6.2.2 Random Phase Approximation (RPA)
The Random Phase Approximation (RPA) [292,[293] also known
as time-dependent Hartree-Fock (TD-HF) is an alternative to CIS for
uncorrelated calculations of excited states. It offers some advantages for
computing oscillator strengths, and is roughly comparable
in accuracy to CIS for excitation energies to singlet states, but is inferior
for triplet states. RPA energies are non-variational.
6.2.3 Extended CIS (XCIS)
The motivation for the extended CIS procedure (XCIS) [294] stems
from the fact that ROCIS and UCIS are less effective for radicals that CIS is
for closed shell molecules. Using the attachment / detachment density analysis
procedure [295], the failing of ROCIS and UCIS methodologies for the
nitromethyl radical was traced to the neglect of a particular class of double
substitution which involves the simultaneous promotion of an α spin
electron from the singly occupied orbital and the promotion of a β spin
electron into the singly occupied orbital. The spin-adapted configurations
| ⎢ ⎢
|
~
Ψ
|
a i
|
(1) |
|
= |
1
√6
|
( |Ψ―i―a〉−|Ψia〉)+ |
2
√6
|
|Ψp―ia―p〉 |
| (6.12) |
are of crucial importance. (Here, a, b, c, … are virtual orbitals;
i, j, k, … are occupied orbitals; and p, q, r, … are singly-occupied orbitals.)
It is quite likely that similar excitations are
also very significant in other radicals of interest.
The XCIS proposal, a more satisfactory generalization of CIS to open shell
molecules, is to simultaneously include a restricted class of double
substitutions similar to those in Eq. (6.12). To illustrate this, consider
the resulting orbital spaces of an ROHF calculation: doubly occupied (d),
singly occupied (s) and virtual (v). From this starting point we can
distinguish three types of single excitations of the same multiplicity as the
ground state: d → s, s → v and d → v. Thus, the spin-adapted
ROCIS wavefunction is
| ΨROCIS 〉 = |
1
√2
|
|
dv ∑
ia
|
aia ( |Ψia〉+|Ψ―i―a 〉 ) + |
sv ∑
pa
|
apa |Ψpa〉+ |
ds ∑
ip
|
a―i―p |Ψ―i―p〉 |
| (6.13) |
The extension of CIS theory to incorporate higher excitations maintains the
ROHF as the ground state reference and adds terms to the ROCIS wavefunction
similar to that of Eq. (6.13), as well as those where the double
excitation occurs through different orbitals in the α and β space:
|
| ΨXCIS 〉 = |
1
√2
|
|
dv ∑
ia
|
aia ( |Ψia〉+|Ψ―i―a〉 ) + |
sv ∑
pa
|
apa |Ψpa〉+ |
ds ∑
ip
|
a―i―p |Ψ―i―p〉 |
|
+ |
dvs ∑
iap
|
|
~
a
|
a i
|
(p)| |
~
Ψ
|
a i
|
(p)〉+ |
dv,ss ∑
ia,p ≠ q
|
ap―ia―q |Ψp―ia―q〉 |
|
|
|
| (6.14) |
XCIS is defined only from a restricted open shell Hartree-Fock ground state
reference, as it would be difficult to uniquely define singly occupied orbitals
in a UHF wavefunction. In addition, β unoccupied orbitals,
through which the spin-flip double excitation proceeds, may not match the
half-occupied α orbitals in either character or even symmetry.
For molecules with closed shell ground states, both the HF ground and CIS
excited states emerge from diagonalization of the Hamiltonian in the space
of the HF reference and singly excited substituted configuration state
functions. The XCIS case is different because the restricted class of double
excitations included could mix with the ground state and lower its energy.
This mixing is avoided to maintain the size consistency of the ground state
energy.
With the inclusion of the restricted set of doubles excitations in the
excited states, but not in the ground state, it could be expected that some
fraction of the correlation energy be recovered, resulting in anomalously
low excited state energies. However, the fraction of the total number of
doubles excitations included in the XCIS wavefunction is very small and
those introduced cannot account for the pair correlation of any pair of
electrons. Thus, the XCIS procedure can be considered one that neglects
electron correlation.
The computational cost of XCIS is approximately four times greater than CIS
and ROCIS, and its accuracy for open shell molecules is generally comparable
to that of the CIS method for closed shell molecules. In general, it
achieves qualitative agreement with experiment. XCIS is available for
doublet and quartet excited states beginning from a doublet ROHF treatment
of the ground state, for excitation energies only.
6.2.4 Spin-Flip Extended CIS (SF-XCIS)
Spin-flip extended CIS (SF-XCIS) [296] is a
spin-complete extension of the spin-flip single excitation configuration
interaction (SF-CIS) method [297].
The method includes all configurations in which no more
than one virtual level of the high spin triplet reference
becomes occupied and no more than one doubly occupied level becomes vacant.
SF-XCIS is defined only from a restricted open shell Hartree-Fock triplet ground state reference.
The final SF-XCIS wavefunctions correspond to spin-pure Ms=0 (singlet or triplet) states.
The fully balanced treatment of the half-occupied reference orbitals makes it very
suitable for applications with two strongly correlated electrons, such as
single bond dissociation, systems with important diradical character or the study of excited states
with significant double excitation character.
The computational cost of SF-XCIS scales in the same way with molecule size as CIS itself,
with a pre-factor 13 times larger.
6.2.5 Basic Job Control Options
See also JOBTYPE, BASIS, EXCHANGE and CORRELATION.
EXCHANGE must be HF and CORRELATION must be
NONE. The minimum input required above a ground state HF calculation
is to specify a nonzero value for CIS_N_ROOTS.
CIS_N_ROOTS
Sets the number of CI-Singles (CIS) excited state roots to find |
TYPE:
DEFAULT:
0 | Do not look for any excited states |
OPTIONS:
n | n > 0 Looks for n CIS excited states |
RECOMMENDATION:
|
| CIS_SINGLETS
Solve for singlet excited states in RCIS calculations (ignored for UCIS) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for singlet states |
FALSE | Do not solve for singlet states. |
RECOMMENDATION:
|
|
|
CIS_TRIPLETS
Solve for triplet excited states in RCIS calculations (ignored for UCIS) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for triplet states |
FALSE | Do not solve for triplet states. |
RECOMMENDATION:
|
| RPA
Do an RPA calculation in addition to a CIS calculation |
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an RPA calculation |
True | Do an RPA calculation. |
RECOMMENDATION:
|
|
|
SPIN_FLIP
Selects whether to perform a standard excited state calculation, or a
spin-flip calculation. Spin multiplicity should be set to 3 for systems with
an even number of electrons, and 4 for systems with an odd number of electrons. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| SPIN_FLIP_XCIS
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an SF-XCIS calculation |
True | Do an SF-XCIS calculation (requires ROHF triplet ground state). |
RECOMMENDATION:
|
|
|
SFX_AMP_OCC_A
Defines a customer amplitude guess vector in SF-XCIS method |
TYPE:
DEFAULT:
OPTIONS:
n | builds a guess amplitude with an α-hole in the nth orbital
(requires SFX_AMP_VIR_B). |
RECOMMENDATION:
Only use when default guess is not satisfactory |
|
| SFX_AMP_VIR_B
Defines a customer amplitude guess vector in SF-XCIS method |
TYPE:
DEFAULT:
OPTIONS:
n | builds a guess amplitude with a β-particle in the nth orbital
(requires SFX_AMP_OCC_A). |
RECOMMENDATION:
Only use when default guess is not satisfactory |
|
|
|
XCIS
Do an XCIS calculation in addition to a CIS calculation |
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an XCIS calculation |
True | Do an XCIS calculation (requires ROHF ground state). |
RECOMMENDATION:
|
6.2.6 Customization
N_FROZEN_CORE
Controls the number of frozen core orbitals |
TYPE:
DEFAULT:
0 | No frozen core orbitals |
OPTIONS:
FC | Frozen core approximation |
n | Freeze n core orbitals |
RECOMMENDATION:
There is no computational advantage to using frozen core for CIS, and
analytical derivatives are only available when no orbitals are frozen. It is
helpful when calculating CIS(D) corrections (see Sec. 6.4). |
|
| N_FROZEN_VIRTUAL
Controls the number of frozen virtual orbitals. |
TYPE:
DEFAULT:
0 | No frozen virtual orbitals |
OPTIONS:
n | Freeze n virtual orbitals |
RECOMMENDATION:
There is no computational advantage to using frozen virtuals for CIS, and
analytical derivatives are only available when no orbitals are frozen. |
|
|
|
MAX_CIS_CYCLES
Maximum number of CIS iterative cycles allowed |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of cycles |
RECOMMENDATION:
Default is usually sufficient. |
|
| MAX_CIS_SUBSPACE
Maximum number of subspace vectors allowed in the CIS iterations |
TYPE:
DEFAULT:
As many as required to converge all roots |
OPTIONS:
n | User-defined number of subspace vectors |
RECOMMENDATION:
The default is usually appropriate, unless a large number of states are
requested for a large molecule.
The total memory required to store the subspace vectors is bounded
above by 2nOV, where O and V represent the number of occupied
and virtual orbitals,
respectively. n can be reduced to save memory, at the cost of a
larger number of CIS iterations. Convergence may be impaired if n
is not much larger than CIS_N_ROOTS. |
|
|
|
CIS_CONVERGENCE
CIS is considered converged when error is less than 10−CIS_CONVERGENCE |
TYPE:
DEFAULT:
6 | CIS convergence threshold 10−6 |
OPTIONS:
RECOMMENDATION:
|
| CIS_RELAXED_DENSITY
Use the relaxed CIS density for attachment/detachment density analysis |
TYPE:
DEFAULT:
OPTIONS:
False | Do not use the relaxed CIS density in analysis |
True | Use the relaxed CIS density in analysis. |
RECOMMENDATION:
|
|
|
CIS_GUESS_DISK
Read the CIS guess from disk (previous calculation) |
TYPE:
DEFAULT:
OPTIONS:
False | Create a new guess |
True | Read the guess from disk |
RECOMMENDATION:
Requires a guess from previous calculation. |
|
| CIS_GUESS_DISK_TYPE
Determines the type of guesses to be read from disk |
TYPE:
DEFAULT:
OPTIONS:
0 | Read triplets only |
1 | Read triplets and singlets |
2 | Read singlets only |
RECOMMENDATION:
Must be specified if CIS_GUESS_DISK is TRUE. |
|
|
|
STS_MOM
Control calculation of the transition moments between excited states in the
CIS and TDDFT calculations (including SF-CIS and SF-DFT). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate state-to-state transition moments. |
TRUE | Do calculate state-to-state transition moments. |
RECOMMENDATION:
When set to true requests the state-to-state dipole transition moments for
all pairs of excited states and for each excited state with the ground
state. |
|
Note:
This option is not available for SF-XCIS. |
| CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not calculate excited-state moments. |
TRUE | (or 1) Calculate moments for each excited state.
|
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial
additional calculation.) The MULTIPOLE_ORDER controls how many
multipole moments are printed. |
|
|
|
6.2.7 CIS Analytical Derivatives
While CIS excitation energies are relatively inaccurate, with errors of the
order of 1 eV, CIS excited state properties, such as structures and
frequencies, are much more useful. This is very similar to the manner in
which ground state Hartree-Fock (HF) structures and frequencies are much
more accurate than HF relative energies. Generally speaking, for low-lying
excited states, it is expected that CIS vibrational frequencies will be
systematically 10% higher or so relative to
experiment [298,[299,[300]. If the excited states are of
pure valence character, then basis set requirements are generally similar to
the ground state. Excited states with partial Rydberg character require the
addition of one or preferably two sets of diffuse functions.
Q-Chem includes efficient analytical first and second derivatives of the CIS
energy [301,[302], to yield analytical gradients, excited
state vibrational frequencies, force constants, polarizabilities, and infrared
intensities. Their evaluation is controlled by two $rem variables, listed
below. Analytical gradients can be evaluated for any job where the CIS
excitation energy calculation itself is feasible.
JOBTYPE
Specifies the type of calculation |
TYPE:
DEFAULT:
OPTIONS:
SP | Single point energy |
FORCE | Analytical Force calculation |
OPT | Geometry Minimization |
TS | Transition Structure Search |
FREQ | Frequency Calculation |
RECOMMENDATION:
|
| CIS_STATE_DERIV
Sets CIS state for excited state optimizations and vibrational analysis |
TYPE:
DEFAULT:
0 | Does not select any of the excited states |
OPTIONS:
RECOMMENDATION:
Check to see that the states do no change order during an optimization |
|
|
|
The semi-direct method [294] used to evaluate the frequencies is
generally similar to the semi-direct method used to evaluate Hartree-Fock
frequencies for the ground state. Memory and disk requirements (see below) are
similar, and the computer time scales approximately as the cube of the system
size for large molecules.
The main complication associated with running analytical CIS second derivatives
is ensuring Q-Chem has sufficient memory to perform the calculations. For
most purposes, the defaults will be adequate, but if a large calculation fails
due to a memory error, then the following additional information may be useful
in fine tuning the input, and understanding why the job failed. Note that the
analytical CIS second derivative code does not currently support frozen core or
virtual orbitals (unlike Q-Chem's MP2 code). Unlike MP2 calculations,
applying frozen core/virtual orbital approximations does not lead to large
computational savings in CIS calculations as all computationally expensive
steps are performed in the atomic basis.
The memory requirements for CIS (and HF) analytical frequencies are primarily
extracted from "C" memory, which is defined as
"C" memory = MEM_TOTAL - MEM_STATIC
"C" memory must be large enough to contain a number of arrays whose size is
3×NAtoms×NBasis2 (NAtoms is the number of atoms and
NBasis refers to the number of basis functions). The value of the $rem
variable MEM_STATIC should be set sufficiently large to permit
efficient integral evaluation. If too large, it reduces the amount of "C"
memory available. If too small, the job may fail due to insufficient scratch
space. For most purposes, a value of about 80 Mb is sufficient, and by default
MEM_TOTAL is set to a very large number (larger than physical memory
on most computers) and thus malloc (memory allocation) errors may occur
on jobs where the memory demands exceeds physical memory.
6.2.8 Examples
Example 6.0 A basic CIS excitation energy calculation on formaldehyde at the
HF/6-31G* optimized ground state geometry, which is obtained in the first part
of the job. Above the first singlet excited state, the states have Rydberg
character, and therefore a basis with two sets of diffuse functions is used.
$molecule
0 1
C
O 1 CO
H 1 CH 2 A
H 1 CH 2 A 3 D
CO = 1.2
CH = 1.0
A = 120.0
D = 180.0
$end
$rem
jobtype = opt
exchange = hf
basis = 6-31G*
$end
@@@
$molecule
read
$end
$rem
exchange = hf
basis = 6-311(2+)G*
cis_n_roots = 15 Do 15 states
cis_singlets = true Do do singlets
cis_triplets = false Don't do Triplets
$end
Example 6.0 An XCIS calculation of excited states of an unsaturated radical,
the phenyl radical, for which double substitutions make considerable
contributions to low-lying excited states.
$comment
C6H5 phenyl radical C2v symmetry MP2(full)/6-31G* = -230.7777459
$end
$molecule
0 2
c1
x1 c1 1.0
c2 c1 rc2 x1 90.0
x2 c2 1.0 c1 90.0 x1 0.0
c3 c1 rc3 x1 90.0 c2 tc3
c4 c1 rc3 x1 90.0 c2 -tc3
c5 c3 rc5 c1 ac5 x1 -90.0
c6 c4 rc5 c1 ac5 x1 90.0
h1 c2 rh1 x2 90.0 c1 180.0
h2 c3 rh2 c1 ah2 x1 90.0
h3 c4 rh2 c1 ah2 x1 -90.0
h4 c5 rh4 c3 ah4 c1 180.0
h5 c6 rh4 c4 ah4 c1 180.0
rh1 = 1.08574
rh2 = 1.08534
rc2 = 2.67299
rc3 = 1.35450
rh4 = 1.08722
rc5 = 1.37290
tc3 = 62.85
ah2 = 122.16
ah4 = 119.52
ac5 = 116.45
$end
$rem
basis = 6-31+G*
exchange = hf
mem_static = 80
intsbuffersize = 15000000
scf_convergence = 8
cis_n_roots = 5
xcis = true
$end
Example 6.0 A SF-XCIS calculation of ground and excited states of
trimethylenemethane (TMM) diradical,
for which double substitutions make considerable contributions to low-lying exc
ited states.
$comment
TMM ground and excited states
$end
$molecule
0 3
C
C 1 CC1
C 1 CC2 2 A2
C 1 CC2 2 A2 3 180.0
H 2 C2H 1 C2CH 3 0.0
H 2 C2H 1 C2CH 4 0.0
H 3 C3Hu 1 C3CHu 2 0.0
H 3 C3Hd 1 C3CHd 4 0.0
H 4 C3Hu 1 C3CHu 2 0.0
H 4 C3Hd 1 C3CHd 3 0.0
CC1 = 1.35
CC2 = 1.47
C2H = 1.083
C3Hu = 1.08
C3Hd = 1.08
C2CH = 121.2
C3CHu = 120.3
C3CHd = 121.3
A2 = 121.0
$end
$rem
unrestricted = false SF-XCIS runs from ROHF triplet reference
exchange = HF
basis = 6-31G*
scf_convergence = 10
scf_algorithm = DM
max_scf_cycles = 100
spin_flip_xcis = true Do SF-XCIS
cis_n_roots = 3
cis_singlets = true Do singlets
cis_triplets = true Do triplets
$end
Example 6.0 This example illustrates a CIS geometry optimization followed by
a vibrational frequency analysis on the lowest singlet excited state of
formaldehyde. This n→π∗ excited state is non-planar, unlike the ground
state. The optimization converges to a non-planar structure with zero forces,
and all frequencies real.
$comment
singlet n -> pi* state optimization and frequencies for formaldehyde
$end
$molecule
0 1
C
O 1 CO
H 1 CH 2 A
H 1 CH 2 A 3 D
CO = 1.2
CH = 1.0
A = 120.0
D = 150.0
$end
$rem
jobtype = opt
exchange = hf
basis = 6-31+G*
cis_state_deriv = 1 Optimize state 1
cis_n_roots = 3 Do 3 states
cis_singlets = true Do do singlets
cis_triplets = false Don't do Triplets
$end
@@@
$molecule
read
$end
$rem
jobtype = freq
exchange = hf
basis = 6-31+G*
cis_state_deriv = 1 Focus on state 1
cis_n_roots = 3 Do 3 states
cis_singlets = true Do do singlets
cis_triplets = false Don't do Triplets
$end
6.2.9 Non-Orthogonal Configuration Interaction
In some systems such as transition metals, some open shell systems and dissociating molecules where there
are low-lying excited states, which manifest themselves as different solutions to the SCF equations.
By using SCF Metadynamics (see Chapter 4), these can be successfully located, but
often there is little physical reason to choose one SCF
solution as a reference over another, and it is appropriate to have a method which treats
these on an equal footing.
In particular these SCF solutions are not subject to non-crossing rules, and do in fact often cross each
other as geometry is changed, so the lowest energy state may switch abruptly with consequent discontinuities
in the energy gradients. To achieve a smoother, more qualitatively correct surface, these SCF solutions
can be used as a basis for a Configuration Interaction calculation, where the resultant wavefunction will either
smoothly interpolate between these states. As the SCF states are not orthogonal to each other
(one cannot be constructed as a single determinant
made out of the orbitals of another), and so the CI is a little more complicated and denoted Non-Orthogonal Configuration Interaction (NOCI) [303].
This can be viewed as an alternative to CASSCF within an "active space" of SCF states of interest, and has the advantage that the SCF states, and thus the NOCI wavefunctions are size-consistent.
In common with CASSCF, it is able to describe complicated phenomena such as avoided crossings (where states mix instead of passing through each other), and conical intersections (where through symmetry or accidental reasons, there is no coupling between the states, and they pass cleanly through each other at a degeneracy).
Another use for a NOCI calculation is that of symmetry purification. At some geometries, the SCF states break spatial or spin symmetry to achieve a lower energy single determinant than if these symmetries were conserved. As these symmetries still exist within the Hamiltonian, its true eigenfunctions will preserve these symmetries. In the case of spin, this manifests itself as spin-contamination, and for spatial symmetries, the orbitals will usually adopt a more localized structure.
To recover a (yet lower energy) wavefunction retaining the correct symmetries, one can include these symmetry broken states (with all relevant symmetry permutations) in a NOCI calculation, and the resultant eigenfunction will have the true symmetries restored as a linear combination of these broken symmetry states.
A common example would be for a UHF state which has an indefinite spin (value of S not Ms). By including a UHF solution along with its spin-flipped version (where all alpha and beta orbitals have been switched) in NOCI, the resultant wavefunction will be a more pure spin state (though there is still no guarantee of finding an eigenfunction of ∧S2), reducing spin contamination in the same way as the Half-Projected Hartree-Fock method [304].
As an example using an Ms=0 UHF wavefunction and its spin-flipped version will produce two new NOCI eigenfunctions, one with even S (a mixture of S=0, S=2, ...), and one with odd (mixing S=1, S=3,...), which may be use as approximations to singlet and triplet wavefunctions.
NOCI can be enabled by specifying CORRELATION NOCI, and will automatically use all of the states located with SCF metadynamics. To merely include the two spin-flipped versions of a UHF wavefunction, this can be specified without turning metadynamics on.
For more customization, a $noci section can be included in the input file to specify the states to include:
Example 6.0 $noci section example
This section specifies (first line) that states 1,2, and 4 are to be included as well as the spin-flipped version of state 2 (the -2 indicates this). The second line (optional) indicates which (zero-based) eigenvalue is to be returned to Q-Chem (the third in this case).
Analytic gradients are not available for NOCI, but finite difference geometry optimizers are available.
NOCI_PRINT
Specify the debug print level of NOCI |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase this for more debug information |
|
6.3 Time-Dependent Density Functional Theory (TDDFT)
6.3.1 Brief Introduction to TDDFT
Excited states may be obtained from density functional theory by
time-dependent density functional theory [305,[306], which
calculates poles in the response of the ground state density to a time-varying
applied electric field. These poles are Bohr frequencies or excitation
energies, and are available in Q-Chem [307], together with the
CIS-like Tamm-Dancoff approximation [137]. TDDFT is becoming
very popular as a method for studying excited states because the computational
cost is roughly similar to the simple CIS method (scaling as roughly the square
of molecular size), but a description of differential electron correlation
effects is implicit in the method. The excitation energies for low-lying
valence excited states of molecules (below the ionization threshold, or more
conservatively, below the first Rydberg threshold) are often remarkably
improved relative to CIS, with an accuracy of roughly 0.1-0.3 eV being observed
with either gradient corrected or local density functionals.
However, standard density functionals do not yield a potential with the correct
long-range Coulomb tail, owing to the so-called self-interaction problem, and
therefore excitation energies corresponding to states that sample this tail (e.g., diffuse Rydberg
states and some charge transfer excited states) are not given
accurately [136,[308,[100]. The extent to which a particular excited state
is characterized by charge transfer can be assessed using an a spatial overlap metric
proposed by Peach, Benfield, Helgaker, and Tozer (PBHT) [309]. (However, see
Ref. for a cautionary note regarding this metric.)
It is advisable to only employ TDDFT for
low-lying valence excited states that are below the first ionization potential
of the molecule. This makes radical cations a particularly favorable choice of
system, as exploited in Ref. . TDDFT for low-lying valence
excited states of radicals is in general a remarkable improvement relative to
CIS, including some states, that, when treated by wavefunction-based methods
can involve a significant fraction of double excitation character [307].
The calculation of the nuclear gradients of full
TDDFT and within the Tamm-Dancoff approximation is also implemented [312].
Standard TDDFT also does not yield a good description of static correlation
effects (see Section 5.9), because it is based on a single reference
configuration of Kohn-Sham orbitals. Recently, a new variation of TDDFT called
spin-flip density functional theory (SFDFT) was developed by Yihan Shao,
Martin Head-Gordon and Anna Krylov to address this issue [61].
SFDFT is different from standard TDDFT in two ways:
- The reference is a high-spin triplet (quartet) for a system with an even
(odd) number of electrons;
- One electron is spin-flipped from an alpha Kohn-Sham orbital to a beta
orbital during the excitation.
SF-DFT can describe the ground state as well as a few low-lying excited states,
and has been applied to bond-breaking processes, and di- and tri-radicals
with degenerate or near-degenerate frontier orbitals.
Recently, we also implemented[313] a SFDFT method with a non-collinear exchange-correlation potential
from Tom Ziegler et al. [314,[315], which is in many case an improvement
over collinear SFDFT [61]. Recommended functionals for SF-DFT calculations
are 5050 and PBE50 (see Ref. [313] for extensive benchmarks).
See also Section 6.6.3 for details on wavefunction-based spin-flip models.
6.3.2 TDDFT within a Reduced Single-Excitation Space
Much of chemistry and biology occurs in solution or on surfaces. The molecular
environment can have a large effect on electronic structure and may change
chemical behavior. Q-Chem is able to compute excited states within a local
region of a system through performing the TDDFT (or CIS) calculation with a
reduced single excitation subspace [316]. This allows the excited
states of a solute molecule to be studied with a large number of solvent
molecules reducing the rapid rise in computational cost. The success of this
approach relies on there being only weak mixing between the electronic
excitations of interest and those omitted from the single excitation space. For
systems in which there are strong hydrogen bonds between solute and solvent, it
is advisable to include excitations associated with the neighboring solvent
molecule(s) within the reduced excitation space.
The reduced single excitation space is constructed from excitations between a
subset of occupied and virtual orbitals. These can be selected from an analysis
based on Mulliken populations and molecular orbital coefficients. For this
approach the atoms that constitute the solvent needs to be defined.
Alternatively, the orbitals can be defined directly. The atoms or orbitals are
specified within a $solute block. These approach is implemented within
the TDA and has been used to study the excited states of formamide in
solution [317], CO on the Pt(111) surface [318], and the
tryptophan chromophore within proteins [319].
6.3.3 Job Control for TDDFT
Input for time-dependent density functional theory calculations follows very
closely the input already described for the uncorrelated excited state methods
described in the previous section (in particular, see
Section 6.2.5). There are several points to be aware of:
- The exchange and correlation functionals are specified exactly as for a
ground state DFT calculation, through EXCHANGE and
CORRELATION.
- If RPA is set to TRUE, a full TDDFT calculation will be
performed. This is not the default. The default is RPA =
FALSE, which leads to a calculation employing the Tamm-Dancoff
approximation (TDA), which is usually a good approximation to full TDDFT.
- If SPIN_FLIP is set to TRUE when performing a TDDFT
calculation, a SFDFT calculation will also be performed. At present,
SFDFT is only implemented within TDDFT/TDA so RPA must be set to
FALSE. Remember to set the spin multiplicity to
3 for systems
with an even-number of electrons (e.g., diradicals), and
4 for odd-number electron systems (e.g., triradicals).
TRNSS
Controls whether reduced single excitation space is used |
TYPE:
DEFAULT:
FALSE | Use full excitation space |
OPTIONS:
TRUE | Use reduced excitation space |
RECOMMENDATION:
|
| TRTYPE
Controls how reduced subspace is specified |
TYPE:
DEFAULT:
OPTIONS:
1 | Select orbitals localized on a set of atoms |
2 | Specify a set of orbitals |
3 | Specify a set of occupied orbitals, include excitations to all virtual orbitals |
RECOMMENDATION:
|
|
|
N_SOL
Specifies number of atoms or orbitals in $solute |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CISTR_PRINT
TYPE:
DEFAULT:
OPTIONS:
TRUE | Increase output level |
RECOMMENDATION:
|
|
|
CUTOCC
Specifies occupied orbital cutoff |
TYPE:
INTEGER: CUTOFF=CUTOCC/100 |
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CUTVIR
Specifies virtual orbital cutoff |
TYPE:
INTEGER: CUTOFF=CUTVIR/100 |
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
PBHT_ANALYSIS
Controls whether overlap analysis of electronic excitations is performed. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform overlap analysis |
TRUE | Perform overlap analysis |
RECOMMENDATION:
|
| PBHT_FINE
Increases accuracy of overlap analysis |
TYPE:
DEFAULT:
OPTIONS:
FALSE | |
TRUE | Increase accuracy of overlap analysis |
RECOMMENDATION:
|
|
|
SRC_DFT
Selects form of the short-range corrected functional |
TYPE:
DEFAULT:
OPTIONS:
1 | SRC1 functional |
2 | SRC2 functional |
RECOMMENDATION:
|
| OMEGA
Sets the Coulomb attenuation parameter for the short-range component. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
|
|
OMEGA2
Sets the Coulomb attenuation parameter for the long-range component. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω2 = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
| HF_SR
Sets the fraction of Hartree-Fock exchange at r12=0. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to HF_SR = n/1000 |
RECOMMENDATION:
|
|
|
HF_LR
Sets the fraction of Hartree-Fock exchange at r12=∞. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to HF_LR = n/1000 |
RECOMMENDATION:
|
| WANG_ZIEGLER_KERNEL
Controls whether to use the Wang-Ziegler
non-collinear exchange-correlation kernel in a SFDFT calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use non-collinear kernel |
TRUE | Use non-collinear kernel |
RECOMMENDATION:
|
|
|
6.3.4 TDDFT coupled with C-PCM for excitation energies and
properties calculations
As described in Section 10.2 (and especially
Section 10.2.2), continuum solvent models such as
C-PCM allow one to include solvent effect in the calculations.
TDDFT / C-PCM allows excited-state modeling in solution.
Q-Chem also features TDDFT coupled with C-PCM which
extends TDDFT to calculations of properties of
electronically-excited molecules in solution.
In particular, TDDFT / C-PCM allows one
to perform geometry optimization and
vibrational analysis [320].
When TDDFT / C-PCM is applied to calculate vertical excitation energies,
the solvent around vertically excited solute is out of equilibrium.
While the solvent electron density equilibrates fast
to the density of the solute (electronic response),
the relaxation of nuclear degrees of freedom (e.g., orientational polarization)
takes place on a slower timescale. To describe this situation, an optical
dielectric constant is employed. To distinguish between equilibrium and non-equilibrium calculations, two dielectric constants
are used in these calculations: a static constant (ε0), equal
to the equilibrium bulk value, and a fast constant (εfast)
related to the response of the medium to high frequency perturbations.
For vertical excitation energy calculations (corresponding
to the unrelaxed solvent nuclear degrees of freedom),
an optical dielectric constant (DIELECTRIC_INFI) is recommended.
For the optimization and vibrational analysis, the static dielectric
constant (DIELECTRIC) should be used [320].
For more details on the PCM calculations and PCM job control options
are given in Section 10.2.
DIELECTRIC_INFI
The optical dielectric constant of the PCM solvent. |
TYPE:
DEFAULT:
OPTIONS:
ε | Use an optical dielectric constant of ε > 0. |
RECOMMENDATION:
The default corresponds to water at T=298 K. |
|
The example below illustrates TDDFT / C-PCM calculations of vertical excitation energies.
Example 6.0 TDDFT/C-PCM low-lying vertical excitation energy
$molecule
0 1
C 0 0 0.0
O 0 0 1.21
$end
$rem
EXCHANGE B3lyp
CIS_N_ROOTS 10
cis_singlets true
cis_triplets true
RPA TRUE
BASIS 6-31+G*
XC_GRID 1
solvent_method pcm
$end
$pcm
Theory CPCM
Method SWIG
Solver Inversion
Radii Bondi
$end
$pcm_solvent
Dielectric 78.39
Dielectric_infi 1.777849 non-equilibrium solvent
$end
$pcm_solvent
Dielectric 78.39
Dielectric_infi 1.777849 non-equilibrium solvent
$end
6.3.5 Analytical Excited-State Hessian in TDDFT
To carry out vibrational frequency analysis of an excited
state with TDDFT [321,[322], an optimization of
the excited-state geometry is always
necessary. Like the vibrational frequency analysis of the ground state, the
frequency analysis of the excited state should be also performed at a
stationary point on the excited state potential surface. The $rem
variable CIS_STATE_DERIV should be set to the
excited state for which an optimization and frequency
analysis is needed, in addition to the $rem keywords used for an excitation energy
calculation.
Compared to the numerical differentiation method, the analytical calculation
of geometrical second derivatives of the excitation energy needs much less
time but much more memory. The computational cost is mainly consumed by the
steps to solve both the CPSCF equations for the derivatives of molecular
orbital coefficients Cx and the CP-TDDFT equations for the
derivatives of the transition vectors, as well as to build the Hessian matrix.
The memory usages for these steps scale as O(3mN2), where N is
the number of basis functions and m is the number of atoms. For large
systems, it is thus essential to solve all the coupled-perturbed equations
in segments. In this case, the $rem variable CPSCF_NSEG is
always needed.
In the calculation of the analytical TDDFT excited-state Hessian, one has to
evaluate a large number of energy-functional derivatives: the first-order to
fourth-order functional derivatives with respect to the density variables as
well as their derivatives with respect to the nuclear coordinates.
Therefore, a very fine integration grid for DFT calculation should be
adapted to guarantee the accuracy of the results.
Analytical TDDFT / C-PCM Hessian has been implemented in Q-Chem.
Normal mode analysis
for a system in solution can be performed with the frequency calculation by TDDFT / C-PCM method.
the $rem and $pcm variables for the excited state calculation with TDDFT / C-PCM included in the vertical excitation energy example above
are needed. When the properties of large systems are calculated, you must
pay attention to the memory limit. At present,
only a few exchange correlation functionals, including Slater+VWN, BLYP, B3LYP, are available for the analytical Hessian calculation.
Example:
Example 6.0 A B3LYP/6-31G* optimization in gas phase, followed by a frequency analysis for the first
excited state of the peroxy
$molecule
0 2
C 1.004123 -0.180454 0.000000
O -0.246002 0.596152 0.000000
O -1.312366 -0.230256 0.000000
H 1.810765 0.567203 0.000000
H 1.036648 -0.805445 -0.904798
H 1.036648 -0.805445 0.904798
$end
$rem
jobtype opt
exchange b3lyp
cis_state_deriv 1
basis 6-31G*
cis_n_roots 10
cis_singlets true
cis_triplets false
xc_grid 000075000302
RPA 0
$end
@@@
$molecule
Read
$end
$rem
jobtype freq
exchange b3lyp
cis_state_deriv 1
basis 6-31G*
cis_n_roots 10
cis_singlets true
cis_triplets false
RPA 0
xc_grid 000075000302
$end
Example 6.0 The optimization and Hessian calculation for low-lying excited state with TDDFT/C-PCM
$comment
9-Fluorenone + 2 methonal in methonal solution
$end
$molecule
0 1
6 -1.987249 0.699711 0.080583
6 -1.987187 -0.699537 -0.080519
6 -0.598049 -1.148932 -0.131299
6 0.282546 0.000160 0.000137
6 -0.598139 1.149219 0.131479
6 -0.319285 -2.505397 -0.285378
6 -1.386049 -3.395376 -0.388447
6 -2.743097 -2.962480 -0.339290
6 -3.049918 -1.628487 -0.186285
6 -3.050098 1.628566 0.186246
6 -2.743409 2.962563 0.339341
6 -1.386397 3.395575 0.388596
6 -0.319531 2.505713 0.285633
8 1.560568 0.000159 0.000209
1 0.703016 -2.862338 -0.324093
1 -1.184909 -4.453877 -0.510447
1 -3.533126 -3.698795 -0.423022
1 -4.079363 -1.292006 -0.147755
1 0.702729 2.862769 0.324437
1 -1.185378 4.454097 0.510608
1 -3.533492 3.698831 0.422983
1 -4.079503 1.291985 0.147594
8 3.323150 2.119222 0.125454
1 2.669309 1.389642 0.084386
6 3.666902 2.489396 -1.208239
1 4.397551 3.298444 -1.151310
1 4.116282 1.654650 -1.759486
1 2.795088 2.849337 -1.768206
1 2.669205 -1.389382 -0.084343
8 3.322989 -2.119006 -0.125620
6 3.666412 -2.489898 1.207974
1 4.396966 -3.299023 1.150789
1 4.115800 -1.655485 1.759730
1 2.794432 -2.850001 1.767593
$end
$rem
jobtype OPT
EXCHANGE B3lyp
CIS_N_ROOTS 10
cis_singlets true
cis_triplets true
cis_state_deriv 1 Lowest TDDFT state
RPA TRUE
BASIS 6-311G**
XC_GRID 000075000302
solvent_method pcm
$end
$pcm
Theory CPCM
Method SWIG
Solver Inversion
Radii Bondi
$end
$pcm_solvent
Dielectric 32.613
$end
@@@
$molecule
read
$end
$rem
jobtype freq
EXCHANGE B3lyp
CIS_N_ROOTS 10
cis_singlets true
cis_triplets true
RPA TRUE
cis_state_deriv 1 Lowest TDDFT state
BASIS 6-311G**
XC_GRID 000075000302
solvent_method pcm
mem_static 4000
mem_total 24000
cpscf_nseg 3
$end
$pcm
Theory CPCM
Method SWIG
Solver Inversion
Radii Bondi
$end
$pcm_solvent
Dielectric 32.613
$end
6.3.6 Various TDDFT-Based Examples
Example 6.0 This example shows two jobs which request variants of
time-dependent density functional theory calculations. The first job, using
the default value of RPA = FALSE, performs TDDFT in the
Tamm-Dancoff approximation (TDA). The second job, with RPA =
TRUE performs a both TDA and full TDDFT calculations.
$comment
methyl peroxy radical
TDDFT/TDA and full TDDFT with 6-31+G*
$end
$molecule
0 2
C 1.00412 -0.18045 0.00000
O -0.24600 0.59615 0.00000
O -1.31237 -0.23026 0.00000
H 1.81077 0.56720 0.00000
H 1.03665 -0.80545 -0.90480
H 1.03665 -0.80545 0.90480
$end
$rem
EXCHANGE b
CORRELATION lyp
CIS_N_ROOTS 5
BASIS 6-31+G*
SCF_CONVERGENCE 7
$end
@@@
$molecule
read
$end
$rem
EXCHANGE b
CORRELATION lyp
CIS_N_ROOTS 5
RPA true
BASIS 6-31+G*
SCF_CONVERGENCE 7
$end
Example 6.0 This example shows a calculation of the
excited states of a formamide-water complex within a
reduced excitation space of the orbitals located on formamide
$comment
formamide-water
TDDFT/TDA in reduced excitation space
$end
$molecule
0 1
H 1.13 0.49 -0.75
C 0.31 0.50 -0.03
N -0.28 -0.71 0.08
H -1.09 -0.75 0.67
H 0.23 -1.62 -0.22
O -0.21 1.51 0.47
O -2.69 1.94 -0.59
H -2.59 2.08 -1.53
H -1.83 1.63 -0.30
$end
$rem
EXCHANGE b3lyp
CIS_N_ROOTS 10
BASIS 6-31++G**
TRNSS TRUE
TRTYPE 1
CUTOCC 60
CUTVIR 40
CISTR_PRINT TRUE
$end
$solute
1
2
3
4
5
6
$end
Example 6.0 This example shows a calculation of the
core-excited states at the oxygen K-edge of CO with a short-range corrected functional.
$comment
TDDFT with short-range corrected (SRC1) functional for the
oxygen K-edge of CO
$end
$molecule
0 1
C 0.000000 0.000000 -0.648906
O 0.000000 0.000000 0.486357
$end
$rem
exchange gen
basis 6-311(2+,2+)G**
cis_n_roots 6
cis_triplets false
trnss true
trtype 3
n_sol 1
src_dft 1
omega 560
omega2 2450
HF_SR 500
HF_LR 170
$end
$solute
1
$end
$XC_Functional
X HF 1.00
X B 1.00
C LYP 0.81
C VWN 0.19
$end
Example 6.0 This example shows a calculation of the
core-excited states at the phosphorus K-edge with a short-range corrected functional.
$comment
TDDFT with short-range corrected (SRC2) functional for the
phosphorus K-edge of PH3
$end
$molecule
0 1
H 1.196206 0.000000 -0.469131
P 0.000000 0.000000 0.303157
H -0.598103 -1.035945 -0.469131
H -0.598103 1.035945 -0.469131
$end
$rem
exchange gen
basis 6-311(2+,2+)G**
cis_n_roots 6
cis_triplets false
trnss true
trtype 3
n_sol 1
src_dft 2
omega 2200
omega2 1800
HF_SR 910
HF_LR 280
$end
$solute
1
$end
$XC_Functional
X HF 1.00
X B 1.00
C LYP 0.81
C VWN 0.19
$end
Example 6.0 SF-TDDFT SP calculation of the 6 lowest states of the TMM diradical using recommended
50-50 functional
$molecule
0 3
C
C 1 CC1
C 1 CC2 2 A2
C 1 CC2 2 A2 3 180.0
H 2 C2H 1 C2CH 3 0.0
H 2 C2H 1 C2CH 4 0.0
H 3 C3Hu 1 C3CHu 2 0.0
H 3 C3Hd 1 C3CHd 4 0.0
H 4 C3Hu 1 C3CHu 2 0.0
H 4 C3Hd 1 C3CHd 3 0.0
CC1 = 1.35
CC2 = 1.47
C2H = 1.083
C3Hu = 1.08
C3Hd = 1.08
C2CH = 121.2
C3CHu = 120.3
C3CHd = 121.3
A2 = 121.0
$end
$rem
jobtype SP
EXCHANGE GENERAL Exact exchange
BASIS 6-31G*
SCF_GUESS CORE
SCF_CONVERGENCE 10
MAX_SCF_CYCLES 100
SPIN_FLIP 1
CIS_N_ROOTS 6
CIS_CONVERGENCE 10
MAX_CIS_CYCLES = 100
$end
$xc_functional
X HF 0.5
X S 0.08
X B 0.42
C VWN 0.19
C LYP 0.81
$end
Example 6.0 SFDFT with non-collinear exchange-correlation functional for low-lying states
of CH2
$comment
non-collinear SFDFT calculation for CH2
at 3B1 state geometry from EOM-CCSD(fT) calculation
$end
$molecule
0 3
C
H 1 rCH
H 1 rCH 2 HCH
rCH = 1.0775
HCH = 133.29
$end
$rem
JOBTYPE SP
UNRESTRICTED TRUE
EXCHANGE PBE0
BASIS cc-pVTZ
SPIN_FLIP 1
WANG_ZIEGLER_KERNEL TRUE
SCF_CONVERGENCE 10
CIS_N_ROOTS 6
CIS_CONVERGENCE 10
$end
6.4 Correlated Excited State Methods: the CIS(D) Family
CIS(D) [323,[324] is a simple size-consistent doubles
correction to CIS which has a computational cost scaling as the fifth
power of the basis set for each excited state. In this sense, CIS(D) can be
considered as an excited state analog of the ground state MP2 method. CIS(D)
yields useful improvements in the accuracy of excitation energies relative to
CIS, and yet can be applied to relatively large molecules using
Q-Chem's efficient integrals transformation package. In addition, as in the
case of MP2 method, the efficiency can be significantly improved through the
use of the auxiliary basis expansions (Section 5.5) [325].
6.4.1 CIS(D) Theory
The CIS(D) excited state procedure is a second-order perturbative
approximation to the computationally expensive CCSD, based on a single
excitation configuration interaction (CIS) reference. The coupled-cluster
wavefunction, truncated at single and double excitations, is the exponential of
the single and double substitution operators acting on the Hartree-Fock
determinant:
| Ψ〉 = exp( T1 +T2 )| Ψ0 〉 |
| (6.15) |
Determination of the singles and doubles amplitudes requires solving the two
equations
〈 Ψia |H−E | ⎢ ⎢
| ⎛ ⎝
|
1+T1 +T2 + |
1
2
|
T12 +T1 T2 + |
1
3!
|
T13 | ⎞ ⎠
|
Ψ0 |
|
=0 |
| (6.16) |
and
〈 Ψijab |H−E | ⎢ ⎢
| ⎛ ⎝
|
1+T1 +T2 + |
1
2
|
T12 +T1 T2 + |
1
3!
|
T13 + |
1
2
|
T22 + |
1
2
|
T12 T2 + |
1
4!
|
T14 | ⎞ ⎠
|
Ψ0 |
|
=0 |
| (6.17) |
which lead to the CCSD excited state equations. These can be written
〈 Ψia |H−E | ⎢ ⎢
| ⎛ ⎝
|
U1 +U2 +T1 U1 +T1 U2 +U1 T2 + |
1
2
|
T12 U1 | ⎞ ⎠
|
Ψ0 |
|
=ωbia |
| (6.18) |
and
|
〈 Ψia |H−E | ⎢ ⎢
| ⎛ ⎝
|
U1 +U2 +T1 U1 +T1 U2 +U1 T2 + |
1
2
|
T12 U1 +T2 U2 |
|
+ |
1
2
|
T12 U2 +T1 T2 U1 + |
1
3!
|
T13 U1 | ⎢ ⎢
|
Ψ0 〉 = ωbijab |
|
|
|
| (6.19) |
This is an eigenvalue equation Ab = ωb for the transition
amplitudes (b vectors), which are also contained in the U operators.
The second-order approximation to the CCSD eigenvalue equation yields a
second-order contribution to the excitation energy which can be written in the
form
ω(2) = b(0)t A(1) b(1)+ b(0)t A(2) b(0) |
| (6.20) |
or in the alternative form
ω(2)=ωCIS(D) = ECIS(D)−EMP2 |
| (6.21) |
where
ECIS(D) = 〈 ΨCIS |V| U2 ΨHF 〉 +〈 ΨCIS |V| T2 U1 ΨHF 〉 |
| (6.22) |
and
EMP2 = 〈 ΨHF |V| T2 ΨHF 〉 |
| (6.23) |
The output of a CIS(D) calculation contains useful information beyond the
CIS(D) corrected excitation energies themselves. The stability of the CIS(D)
energies is tested by evaluating a diagnostic, termed the "theta diagnostic" [326].
The theta diagnostic calculates a mixing angle that measures
the extent to which electron correlation causes each pair of calculated CIS
states to couple. Clearly the most extreme case would be a mixing angle of
45°, which would indicate breakdown of the validity of the initial CIS
states and any subsequent corrections. On the other hand, small mixing angles on
the order of only a degree or so are an indication that the calculated results
are reliable. The code can report the largest mixing angle for each state to all
others that have been calculated.
6.4.2 Resolution of the Identity CIS(D) Methods
Because of algorithmic similarity with MP2 calculation,
the "resolution of the identity" approximation can also be used in CIS(D).
In fact, RI-CIS(D) is orders of magnitudes more efficient than previously
explained CIS(D) algorithms for effectively all molecules with more than
a few atoms. Like in MP2,
this is achieved by reducing the prefactor of the computational load.
In fact, the overall cost still scales with the fifth power of the system size.
Presently in Q-Chem, RI approximation is supported for
closed-shell restricted CIS(D) and open-shell unrestricted
UCIS(D) energy calculations. The theta diagnostic is not implemented for RI-CIS(D).
6.4.3 SOS-CIS(D) Model
As in MP2 case, the accuracy of CIS(D) calculations can be improved by semi-empirically
scaling the opposite-spin components of CIS(D) expression:
ESOS−CIS(D) = cU 〈 ΨCIS |V| U2OS ΨHF 〉 + cT 〈 ΨCIS |V| T2OS U1 ΨHF 〉 |
| (6.24) |
with the corresponding ground state energy
ESOS−MP2 = cT 〈 ΨHF |V| T2OS ΨHF 〉 |
| (6.25) |
More importantly, this SOS-CIS(D) energy can be evaluated with the 4th power
of the molecular size by adopting Laplace transform technique [325].
Accordingly, SOS-CIS(D) can be applied to the calculations of
excitation energies for relatively large molecules.
6.4.4 SOS-CIS(D0) Model
CIS(D) and its cousins explained in the above are all based on a second-order non-degenerate
perturbative correction scheme on the CIS energy ("diagonalize-and-then-perturb" scheme).
Therefore, they may fail when multiple excited states come close in terms of their energies.
In this case, the system can be handled by applying quasi-degenerate perturbative correction scheme
("perturb-and-then-diagonalize" scheme). The working expression can be obtained by
slightly modifying CIS(D) expression shown in
Section 6.4.1 [327].
First, starting from Eq. (6.20), one can be explicitly write the CIS(D) energy
as [328,[327]
ωCIS + ω(2) = b(0)t ASS(0) b(0) + b(0)t ASS(2) b(0) − b(0)t ASD(1)( DDD(0) − ωCIS )−1 ADS(1) b(0) |
| (6.26) |
To avoid the failures of the perturbation theory near degeneracies, the entire single and double
blocks of the response matrix should be diagonalized. Because such a diagonalization
is a non-trivial non-linear problem, an additional approximation from the binomial expansion of the
( DDD(0) − ωCIS )−1 is further
applied [327]:
( DDD(0) − ωCIS )−1 = ( DDD(0) )−1 ( 1 + ω( DDD(0) )−1 + ω2 ( DDD(0) )−2 + ... ) |
| (6.27) |
The CIS(D0) energy ω is defined as the eigensolution of the response matrix with
the zero-th order expansion of this equation. Namely,
( ASS(0) + ASS(2) − ASD(1) ( DDD(0) )−1 ADS(1)) b = ωb |
| (6.28) |
Similar to SOS-CIS(D), SOS-CIS(D0) theory is defined by taking the opposite-spin
portions of this equation and then scaling them with two semi-empirical
parameters [328]:
( ASS(0) + cT ASSOS(2) − cU ASDOS(1) ( DDD(0) )−1 ADSOS(1)) b = ωb |
| (6.29) |
Using the Laplace transform and the auxiliary basis expansion techniques,
this can also be handled with a 4th-order scaling computational effort.
In Q-Chem, an efficient 4th-order scaling analytical gradient of SOS-CIS(D0)
is also available. This can be used to perform excited state geometry optimizations on
the electronically excited state surfaces.
6.4.5 CIS(D) Job Control and Examples
The legacy CIS(D) algorithm in Q-Chem is handled by the CCMAN/CCMAN2
modules of Q-Chem's and shares many of the $rem options.
RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0) do not depend on the coupled-cluster
routines. Users who will not use this legacy CIS(D) method may skip to
Section 6.4.6.
As with all post-HF calculations, it is important to ensure there are sufficient
resources available for the necessary integral calculations and
transformations. For CIS(D), these resources are controlled using the $rem
variables CC_MEMORY,
MEM_STATIC and MEM_TOTAL (see Section 5.7.7).
To request a CIS(D) calculation
the CORRELATION $rem should be set to CIS(D) and
the number of excited states to calculate should be specified
by EOM_EE_STATES (or EOM_EE_SINGLETS
and EOM_EE_TRIPLETS when appropriate).
Alternatively,
CIS(D) will be performed when CORRELATION=CI and
EOM_CORR=CIS(D).
The SF-CIS(D) is invoked by using EOM_SF_STATES.
EOM_EE_STATES
Sets the number of excited state roots to find. For
closed-shell reference, defaults into EOM_EE_SINGLETS. For open-shell references,
specifies all low-lying states. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EE_SINGLETS
Sets the number of singlet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EE_TRIPLETS
Sets the number of triplet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_SF_STATES
Sets the number of spin-flip target states roots to find. |
TYPE:
DEFAULT:
0 | Do not look for any spin-flip states. |
OPTIONS:
[i,j,k…] | Find i SF states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
Note:
It is a symmetry of a transition rather than that of a target state which
is specified in excited state calculations. The symmetry of the target state
is a product of the symmetry of the reference state and the transition.
For closed-shell molecules, the former is fully symmetric and the symmetry of the
target state is the same as that of transition, however, for open-shell references
this is not so. |
| CC_STATE_TO_OPT
Specifies which state to optimize. |
TYPE:
DEFAULT:
OPTIONS:
[i,j] | optimize the jth state of the ith irrep. |
RECOMMENDATION:
|
|
|
Note:
Since there are no analytic gradients for CIS(D), the symmetry should be turned off for geometry
optimization and frequency calculations,
and CC_STATE_TO_OPT should be specified assuming C1 symmetry, i.e.,
as [1,N] where N is the number of state to optimize (the states are numbered from 1). |
Example 6.0
CIS(D) excitation energy calculation for ozone at the experimental ground state geometry C2v
$molecule
0 1
O
O 1 RE
O 2 RE 1 A
RE=1.272
A=116.8
$end
$rem
jobtype SP
BASIS 6-31G*
N_FROZEN_CORE 3 use frozen core
correlation CIS(D)
EOM_EE_SINGLETS [2,2,2,2] find 2 lowest singlets in each irrep.
EOM_EE_TRIPLETS [2,2,2,2] find two lowest triplets in each irrep.
$end
Example 6.0
CIS(D) geometry optimization for the lowest triplet state of water. The symmetry is automatically
turned off for finite difference calculations
$molecule
0 1
o
h 1 r
h 1 r 2 a
r 0.95
a 104.0
$end
$rem
jobtype opt
basis 3-21g
correlation cis(d)
eom_ee_triplets 1 calculate one lowest triplet
cc_state_to_opt [1,1] optimize the lowest state (first state in first irrep)
$end
Example 6.0
CIS(D) excitation energy and transition property calculation (between all states)
for ozone at the experimental ground state geometry C2v
$molecule
0 1
O
O 1 RE
O 2 RE 1 A
RE=1.272
A=116.8
$end
$rem
jobtype SP
BASIS 6-31G*
purcar 2 Non-spherical (6D)
correlation CIS(D)
eom_ee_singlets [2,2,2,2]
eom_ee_triplets [2,2,2,2]
cc_trans_prop = 1
$end
6.4.6 RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0): Job Control
These methods are activated by setting
the $rem keyword CORRELATION to RICIS(D), SOSCIS(D),
and SOSCIS(D0), respectively. Other keywords are the same as in
CIS method explained in Section 6.2.1. As these methods rely on the RI approximation,
AUX_BASIS needs
to be set by following the same guide as in RI-MP2
(Section 5.5).
CORRELATION
Excited state method of choice |
TYPE:
DEFAULT:
OPTIONS:
RICIS(D) | Activate RI-CIS(D) |
SOSCIS(D) | Activate SOS-CIS(D) |
SOSCIS(D0) | Activate SOS-CIS(D0) |
RECOMMENDATION:
|
| CIS_N_ROOTS
Sets the number of excited state roots to find |
TYPE:
DEFAULT:
0 | Do not look for any excited states |
OPTIONS:
n | n > 0 Looks for n excited states |
RECOMMENDATION:
|
|
|
CIS_SINGLETS
Solve for singlet excited states (ignored for spin unrestricted systems) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for singlet states |
FALSE | Do not solve for singlet states. |
RECOMMENDATION:
|
| CIS_TRIPLETS
Solve for triplet excited states (ignored for spin unrestricted systems) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for triplet states |
FALSE | Do not solve for triplet states. |
RECOMMENDATION:
|
|
|
SET_STATE_DERIV
Sets the excited state index for analytical gradient calculation
for geometry optimizations and vibrational analysis with SOS-CIS(D0) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Check to see that the states do no change order during an optimization.
For closed-shell systems, either CIS_SINGLETS or CIS_TRIPLETS
must be set to false. |
|
| MEM_STATIC
Sets the memory for individual program modules |
TYPE:
DEFAULT:
64 | corresponding to 64 Mb |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
At least 150(N2 + N)D of MEM_STATIC is required
(N: number of basis functions, D: size of a double precision storage, usually 8).
Because a number of matrices with N2 size also need to be
stored, 32-160 Mb of additional MEM_STATIC is needed. |
|
|
|
MEM_TOTAL
Sets the total memory available to Q-Chem |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes |
RECOMMENDATION:
The minimum memory requirement of RI-CIS(D) is approximately
MEM_STATIC + max(3SVXD, 3X2D)
(S: number of excited states, X: number of auxiliary basis
functions, D: size of a double precision storage, usually 8). However, because
RI-CIS(D) uses a batching scheme for efficient evaluations of electron
repulsion integrals, specifying more memory will significantly speed up the
calculation. Put as much memory as possible if you are not sure
what to use, but never put any more than what is available.
The minimum memory requirement of SOS-CIS(D) and SOS-CIS(D0) is approximately
MEM_STATIC + 20 X2 D. SOS-CIS(D0) gradient calculation
becomes more efficient when 30 X2 D more memory space is given.
Like in RI-CIS(D), put as much memory as possible if you are not sure what to use.
The actual memory size used in these calculations will be printed out in the output file
to give a guide about the required memory. |
|
| AO2MO_DISK
Sets the scratch space size for individual program modules |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
The minimum disk requirement of RI-CIS(D) is approximately
3SOVXD. Again, the batching scheme will become more efficient with
more available disk space. There is no simple formula for SOS-CIS(D) and SOS-CIS(D0)
disk requirement. However, because the disk space is abundant in modern computers,
this should not pose any problem. Just put the available disk space size in this case.
The actual disk usage information will also be printed in the output file. |
|
|
|
SOS_FACTOR
Sets the scaling parameter cT |
TYPE:
DEFAULT:
130 | corresponding to 1.30 |
OPTIONS:
RECOMMENDATION:
|
| SOS_UFACTOR
Sets the scaling parameter cU |
TYPE:
DEFAULT:
151 | For SOS-CIS(D), corresponding to 1.51 |
140 | For SOS-CIS(D0), corresponding to 1.40
|
OPTIONS:
RECOMMENDATION:
|
|
|
6.4.7 Examples
Example 6.0 Q-Chem input for an RI-CIS(D) calculation.
$molecule
0 1
C 0.667472 0.000000 0.000000
C -0.667472 0.000000 0.000000
H 1.237553 0.922911 0.000000
H 1.237553 -0.922911 0.000000
H -1.237553 0.922911 0.000000
H -1.237553 -0.922911 0.000000
$end
$rem
jobtype sp
exchange hf
basis aug-cc-pVDZ
mem_total 1000
mem_static 100
ao2mo_disk 1000
aux_basis rimp2-aug-cc-pVDZ
purecart 1111
correlation ricis(d)
cis_n_roots 10
cis_singlets true
cis_triplets false
$end
Example 6.0 Q-Chem input for an SOS-CIS(D) calculation.
$molecule
0 1
C -0.627782 0.141553 0.000000
O 0.730618 -0.073475 0.000000
H -1.133677 -0.033018 -0.942848
H -1.133677 -0.033018 0.942848
$end
$rem
jobtype sp
exchange hf
basis aug-cc-pVDZ
mem_total 1000
mem_static 100
ao2mo_disk 500000 ! 0.5 Terabyte of disk space available
aux_basis rimp2-aug-cc-pVDZ
purecart 1111
correlation soscis(d)
cis_n_roots 5
cis_singlets true
cis_triplets true
$end
Example 6.0 Q-Chem input for an SOS-CIS(D0) geometry optimization
on S2 surface.
$molecule
0 1
o
h 1 r
h 1 r 2 a
r 0.95
a 104.0
$end
$rem
jobtype = opt
exchange = hf
correlation = soscis(d0)
basis = 6-31G**
aux_basis = rimp2-VDZ
purecart = 1112
set_state_deriv = 2
cis_n_roots = 5
cis_singlets = true
cis_triplets = false
$end
6.5 Maximum Overlap Method (MOM) for SCF Excited States
The Maximum Overlap Method (MOM) [176] is a useful alternative to CIS and
TDDFT for obtaining low-cost excited states. It works by modifying the orbital
selection step in the SCF procedure. By choosing orbitals that most resemble
those from the previous cycle, rather than those with the lowest
eigenvalues, excited SCF determinants are able to be obtained.
The MOM has several advantages over existing low-cost excited state methods.
Current implementations of TDDFT usually struggle to accurately model
charge-transfer and Rydberg transitions, both of which can be well-modeled
using the MOM. The MOM also allows the user to target very high energy states,
such as those involving excitation of core electrons [329], which are
hard to capture using other excited state methods.
In order to calculate an excited state using MOM, the user must correctly
identify the orbitals involved in the transition. For example, in a
π→π∗ transition, the π and π∗ orbitals must be
identified and this usually requires a preliminary calculation. The user then
manipulates the orbital occupancies using the $occupied section, removing an
electron from the π and placing it in the π∗. The MOM is then
invoked to preserve this orbital occupancy.
The success of the MOM relies on the quality of the initial guess for the
calculation. If the virtual orbitals are of poor quality then the calculation
may `fall down' to a lower energy state of the same symmetry. Often the
virtual orbitals of the corresponding cation are more appropriate for using as
initial guess orbitals for the excited state.
Because the MOM states are single determinants, all of Q-Chem's existing
single determinant properties and derivatives are available. This allows, for
example, analytic harmonic frequencies to be computed on excited states. The
orbitals from a Hartree-Fock MOM calculation can also be used in an MP2
calculation. For all excited state calculations, it is important to add
diffuse functions to the basis set. This is particularly true if Rydberg
transitions are being sought. For DFT based methods, it is also advisable to
increase the size of the quadrature grid so that the more diffuse densities are
accurately integrated.
Example 6.0 Calculation of the lowest singlet state of CO.
$comment
CO spin-purified calculation
$end
$molecule
0 1
C
O C 1.05
$end
$rem
JOBTYPE SP
EXCHANGE B3LYP
BASIS 6-31G*
$end
@@@
$molecule
read
$end
$rem
JOBTYPE SP
EXCHANGE B3LYP
BASIS 6-31G*
SCF_GUESS READ
MOM_START 1
UNRESTRICTED TRUE
OPSING TRUE
$end
$occupied
1 2 3 4 5 6 7
1 2 3 4 5 6 8
$end
The following $rem is used to invoke the MOM:
MOM_START
Determines when MOM is switched on to preserve orbital occupancies. |
TYPE:
DEFAULT:
OPTIONS:
0 (FALSE) | MOM is not used |
n | MOM begins on cycle n. |
RECOMMENDATION:
For calculations on excited states, an initial calculation without MOM is
usually required to get satisfactory starting orbitals. These orbitals
should be read in using SCF_GUESS=true and
MOM_START set to 1.
|
|
Example 6.0 Input for obtaining the 2A′ excited state of
formamide corresponding to the π→π∗ transition. The
1A′ ground state is obtained if MOM is not used in the second
calculation. Note the use of diffuse functions and a larger quadrature grid to
accurately model the larger excited state.
$molecule
1 2
C
H 1 1.091480
O 1 1.214713 2 123.107874
N 1 1.359042 2 111.982794 3 -180.000000 0
H 4 0.996369 1 121.060099 2 -0.000000 0
H 4 0.998965 1 119.252752 2 -180.000000 0
$end
$rem
exchange B3LYP
basis 6-311(2+,2+)G(d,p)
xc_grid 000100000194
$end
@@@
$molecule
0 1
C
H 1 1.091480
O 1 1.214713 2 123.107874
N 1 1.359042 2 111.982794 3 -180.000000 0
H 4 0.996369 1 121.060099 2 -0.000000 0
H 4 0.998965 1 119.252752 2 -180.000000 0
$end
$rem
exchange B3LYP
basis 6-311(2+,2+)G(d,p)
xc_grid 000100000194
mom_start 1
scf_guess read
unrestricted true
$end
$occupied
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 13
$end
6.6 Coupled-Cluster Excited-State and Open-Shell Methods
6.6.1 Excited States via EOM-EE-CCSD and EOM-EE-OD
One can describe electronically excited states at a level
of theory similar to that associated with coupled-cluster theory for the ground
state by applying either linear response theory [330] or equation-of-motion
methods [331]. A number of groups have demonstrated
that excitation energies based on a coupled-cluster singles and doubles ground
state are generally very accurate for states that are primarily single electron
promotions. The error observed in calculated excitation energies to such states
is typically 0.1-0.2 eV, with 0.3 eV as a conservative estimate,
including both valence and Rydberg excited states.
This, of course, assumes that a basis set large and flexible enough to describe
the valence and Rydberg states is employed. The accuracy of excited state
coupled-cluster methods is much lower for excited states that involve a
substantial double excitation character, where errors may be 1 eV
or even more. Such errors arise because the description of electron correlation
of an excited state with substantial double excitation character requires
higher truncation of the excitation operator.
The description of these states can be improved by
including triple excitations, as in the EOM(2,3) or EOM-CCSD(dT)/(dT) methods.
Q-Chem includes coupled-cluster methods for excited states based on the
optimized orbital coupled-cluster doubles (OD), and the coupled cluster singles
and doubles (CCSD) methods, described earlier. OD excitation energies have been
shown to be essentially identical in numerical performance to CCSD excited
states [332].
These methods, while far more computationally
expensive than TDDFT, are nevertheless useful as proven high accuracy methods
for the study of excited states of small molecules. Moreover, they are capable of
describing both valence and Rydberg excited states, as well as states of a
charge-transfer character.
Also, when studying a
series of related molecules it can be very useful to compare the performance of
TDDFT and coupled-cluster theory for at least a small example to understand its
performance. Along similar lines, the CIS(D) method described earlier as an
economical correlation energy correction to CIS excitation energies is
in fact
an approximation to EOM-CCSD. It is useful to assess
the performance of CIS(D) for a class of problems by benchmarking against the
full coupled-cluster treatment. Finally, Q-Chem includes extensions of EOM
methods to treat ionized or electron attachment systems, as well as di- and
tri-radicals.
EOM-EE Ψ(Ms=0)=R(Ms=0) Ψ0(Ms=0)
Picture Omitted
EOM-IP Ψ(N)=R(−1) Ψ0(N+1)
Picture Omitted
EOM-EA Ψ(N)=R(+1) Ψ0(N−1)
Picture Omitted
EOM-SF Ψ(Ms=0)=R(Ms=−1) Ψ0(Ms=1)
Picture Omitted
Figure 6.1: In the EOM formalism, target states Ψ are described as
excitations from a reference state Ψ 0: Ψ = R Ψ 0, where R is a
general excitation operator. Different EOM models are defined by choosing the
reference and the form of the operator R. In the EOM models for
electronically excited states (EOM-EE, upper panel), the reference is the
closed-shell ground state Hartree-Fock determinant, and the operator R
conserves the number of α and β electrons. Note that
two-configurational open-shell singlets can be correctly described by EOM-EE
since both leading determinants appear as single electron excitations. The
second and third panels present the EOM-IP/EA models. The reference states
for EOM-IP/EA are determinants for N+1/N−1 electron states, and the
excitation operator R is ionizing or electron-attaching, respectively. Note
that both the EOM-IP and EOM-EA sets of determinants are spin-complete and
balanced with respect to the target multi-configurational ground and excited
states of doublet radicals. Finally, the EOM-SF method (the lowest panel)
employs the hight-spin triplet state as a reference, and the operator R
includes spin-flip, i.e., does not conserve the number of α and
β electrons. All the determinants present in the target low-spin states
appear as single excitations, which ensures their balanced treatment both in the
limit of large and small HOMO-LUMO gaps.
6.6.2 EOM-XX-CCSD and CI Suite of Methods
Q-Chem features the most complete set of EOM-CCSD models [333]
that enables
accurate, robust, and efficient calculations of electronically excited states
(EOM-EE-CCSD or EOM-EE-OD) [334,[335,[331,[332,[336];
ground and excited states of diradicals and triradicals (EOM-SF-CCSD and
EOM-SF-OD [337,[336]);
ionization potentials and electron
attachment energies as well as problematic doublet radicals, cation or anion
radicals, (EOM-IP/EA-CCSD) [338,[339,[340],
as well as EOM-DIP-CCSD and EOM-2SF-CCSD.
Conceptually, EOM is very similar to configuration
interaction (CI): target EOM states are found by diagonalizing the similarity
transformed Hamiltonian ―H = e−TH eT,
where T and R are general excitation operators with respect to the
reference determinant |Φ0〉. In the EOM-CCSD models, T and R are
truncated at single and double excitations, and the amplitudes T satisfy the
CC equations for the reference state |Φ0〉:
The computational scaling of EOM-CCSD and CISD methods is identical, i.e.,
O(N6), however EOM-CCSD is numerically superior to CISD because
correlation effects are "folded in" in the transformed Hamiltonian, and
because EOM-CCSD is rigorously size-intensive.
By combining different types of excitation operators and references
|Φ0〉, different groups of target states can be accessed as explained
in Fig. 6.1. For example, electronically excited states can be
described when the reference |Φ0〉 corresponds to the ground state wave
function, and operators R conserve the number of electrons and a total
spin [331]. In the ionized/electron attached EOM
models [339,[340], operators R are not electron
conserving (i.e., include different number of creation and annihilation
operators)-these models can accurately treat ground and excited states of
doublet radicals and some other open-shell systems. For example, singly
ionized EOM methods, i.e., EOM-IP-CCSD and EOM-EA-CCSD, have proven very useful
for doublet radicals whose theoretical treatment is often plagued by symmetry
breaking. Finally, the EOM-SF method [337,[336] in which the
excitation operators include spin-flip allows one to access diradicals,
triradicals, and bond-breaking.
Q-Chem features EOM-EE / SF / IP / EA-CCSD methods for both closed and
open-shell references (RHF / UHF / ROHF), including frozen core / virtual options.
All EOM models take full advantage of molecular point group symmetry.
Analytic gradients are available for RHF and UHF references, for the full
orbital space, and with frozen core / virtual orbitals [341].
Properties calculations (permanent and transition dipole moments, 〈
S2〉, 〈R2 〉, etc.) are also available. The current implementation
of the EOM-XX-CCSD methods enables calculations of medium-size molecules,
e.g., up to 15-20 heavy atoms. Using RI approximation 5.7.5 or Cholesky decomposition 5.7.6 helps to reduce integral
transformation time and disk usage enabling calculations on much
larger systems.
Q-Chem includes two implementations of EOM-IP-CCSD. The proper
implementation [342] is used by default is more efficient and robust.
The EOM_FAKE_IPEA keyword invokes is a pilot implementation in which
EOM-IP-CCSD calculation is set up by adding a very diffuse orbital to a requested basis set, and
by solving EOM-EE-CCSD equations for the target states that include excitations of an electron to
this diffuse orbital. Our current implementation of EOM-EA-CCSD also uses this trick.
Fake IP/EA calculations are only recommended for Dyson orbital
calculations and debug purposes.
The computational cost of EOM-IP calculations can be
considerably reduced (with negligible decline in accuracy) by
truncating virtual orbital space using FNO scheme (see Section 6.6.6).
Finally, a more economical CI variant of EOM-IP-CCSD, IP-CISD is also available.
This is an N5 approximation of IP-CCSD, and is recommended for geometry optimizations
of problematic doublet states [343].
EOM and CI methods are handled by the CCMAN / CCMAN2 modules.
6.6.3 Spin-Flip Methods for Di- and Triradicals
The spin-flip method [337,[297,[344]
addresses the bond-breaking problem associated with a
single-determinant description of the wavefunction. Both closed and open shell
singlet states are described within a single reference as spin-flipping,
(e.g., α→ β excitations from the triplet reference state, for
which both dynamical and non-dynamical correlation effects are smaller than for
the corresponding singlet state. This is because the exchange hole, which
arises from the Pauli exclusion between same-spin electrons, partially
compensates for the poor description of the coulomb hole by the mean-field
Hartree-Fock model. Furthermore, because two α electrons cannot form a
bond, no bond breaking occurs as the internuclear distance is stretched, and
the triplet wavefunction remains essentially single-reference in character.
The
spin-flip approach has also proved useful in the description of di- and
tri-radicals as well as some problematic doublet states.
The spin-flip method is available for the CIS, CIS(D), CISD, CISDT, OD, CCSD,
and EOM-(2,3) levels of theory and the spin complete SF-XCIS (see Section 6.2.4).
An N7 non-iterative triples corrections
are also available.
For the OD and CCSD models, the following non-relaxed
properties are also available: dipoles, transition dipoles, eigenvalues of the
spin-squared operator (〈S2〉), and densities. Analytic gradients are also for
SF-CIS and EOM-SF-CCSD methods. To invoke a spin-flip calculation the
EOM_SF_STATES $rem should be used, along with the
associated $rem settings for the chosen level of correlation (CORRELATION, and, optionally,
EOM_CORR). Note that the
high multiplicity triplet or quartet reference states should be used.
Several double SF methods have also been implemented [345].
To invoke these methods, use EOM_DSF_STATES.
6.6.4 EOM-DIP-CCSD
Double-ionization potential (DIP) is another non-electron-conserving variant of
EOM-CCSD [346,[347,[348].
In DIP, target states are reached by
detaching two electrons from the reference state:
Ψk = RN−2 Ψ0 (N+2), \protect |
| (6.33) |
and the excitation operator R has the following form:
As a reference state in the EOM-DIP calculations one usually takes a
well-behaved closed-shell state. EOM-DIP is a useful tool for
describing molecules with electronic degeneracies of the type "2n−2
electrons on n degenerate orbitals". The simplest examples of such systems
are diradicals with two-electrons-on-two-orbitals pattern. Moreover,
DIP is a preferred method for four-electrons-on-three-orbitals wavefunctions.
Accuracy of the EOM-DIP-CCSD method
is similar to accuracy of other EOM-CCSD models, i.e., 0.1-0.3 eV.
The scaling of EOM-DIP-CCSD is O(N6), analogous to that of other
EOM-CCSD methods. However, its computational cost is less compared to, e.g.,
EOM-EE-CCSD, and it increases more slowly with the basis set size.
An EOM-DIP calculation is invoked by using EOM_DIP_STATES,
or EOM_DIP_SINGLETS and EOM_DIP_TRIPLETS.
6.6.5 Charge Stabilization for EOM-DIP and Other Methods
Unfortunately, the performance of EOM-DIP deteriorates when the reference
state is unstable with respect to
electron-detachment [347,[348], which is usually the case
for dianion references employed to describe neutral diradicals by EOM-DIP. Similar
problems are encountered by all excited-state methods when dealing with excited states
lying above ionization or electron-detachment thresholds.
To remedy this problem,
one can employ charge stabilization methods, as described in
Refs. [347,[348]. In this approach (which can also be used
with any other electronic structure method implemented in Q-Chem), an additional
Coulomb potential is introduced to stabilize unstable wave functions. The following
keywords invoke stabilization potentials:
SCALE_NUCLEAR_CHARGE and ADD_CHARGED_CAGE. In the former case, the potential
is generated by increasing nuclear charges by a specified amount.
In the latter, the potential is generated by a cage built out of point charges comprising
the molecule. There are two cages available: dodecahedral and spherical.
The shape, radius, number of points, and the total charge of the cage are set
by the user.
Note:
A perturbative correction estimating the effect of the external Coulomb
potential on EOM energy will be computed when target state densities are calculated, e.g.,
when CC_EOM_PROP is set to TRUE. |
Note:
Charge stabilization techniques can be used with other methods such as
EOM-EE, CIS, and TDDFT to improve the description of resonances. It can also be
employed to describe metastable ground states. |
6.6.6 Frozen Natural Orbitals in CC and IP-CC Calculations
Large computational savings are possible if the virtual space is truncated
using the frozen natural orbital (FNO) approach (see Section 5.10).
Extension of the FNO approach to ionized states within
EOM-CC formalism was recently introduced and benchmarked [260].
In addition to ground-state coupled-cluster calculations, FNOs can also be used in EOM-IP-CCSD,
EOM-IP-CCSD(dT/fT) and EOM-IP-CC(2,3). In IP-CC the FNOs are computed
for the reference (neutral) state and then are used to describe several
target (ionized) states of interest. Different truncation scheme are described in
Section 5.10.
6.6.7 Equation-of-Motion Coupled-Cluster Job Control
It is important to ensure there are sufficient
resources available for the necessary integral calculations and
transformations. For CCMAN/CCMAN2 algorithms, these resources are controlled
using the $rem variables CC_MEMORY, MEM_STATIC and
MEM_TOTAL (see Section 5.13).
There is a rich range of input control options for coupled-cluster excited
state or other EOM calculations. The minimal requirement is the input for the
reference state
CCSD or OD calculation (see Chapter 5), plus
specification of the number of target states requested through
EOM_XX_STATES (XX specifies the type of the target states, e.g.,
EE, SF, IP, EA, DIP, DSF, etc.).
Users must be aware of the point group
symmetry of the system being studied and also the symmetry of the initial and target states
of interest, as well as symmetry of transition. It is possible to turn off the use of symmetry by
CC_SYMMETRY. If set to FALSE the molecule will
be treated as having C1 symmetry and all states will be of A symmetry.
Note:
In finite-difference calculations, the symmetry is turned off automatically,
and the user must ensure that EOM_XX_STATES is adjusted accordingly. |
Note:
Mixing different EOM models in a single calculation is only allowed in Dyson
orbitals calculations. |
By default, the level of correlation of the EOM part of the wavefunction
(i.e., maximum excitation level in the EOM operators R) is set
to match CORRELATION, however, one can mix different correlation levels
for the reference and EOM states by using EOM_CORR. To request a CI
calculation, set CORRELATION=CI and select type of CI expansion
by EOM_CORR. The table below shows default and allowed
CORRELATION and EOM_CORR combinations.
CORRELATION | Default | Allowed | Target states | CCMAN/CCMAN2 |
| EOM_CORR | EOM_CORR | | |
CI | none | CIS, CIS(D) | EE,SF | y/n |
| | CISD | EE,SF,IP | y/n |
| | SDT, DT | EE,SF,DSF | y/n |
CIS(D) | CIS(D) | N/A | EE,SF | y/n |
CCSD, OD | CISD | | EE,SF,IP,EA,DIP | y/y |
| | SD(dT),SD(fT) | EE,SF, fake IP/EA | y/n |
| | SD(dT),SD(fT), SD(sT) | IP | y/n |
| | SDT, DT | EE,SF,IP,EA,DIP,DSF | y/n |
Table 6.1: Default and allowed CORRELATION and EOM_CORR combinations
as well as valid target state types. The last column shows if a method is available in CCMAN or CCMAN2.
The table below shows the correct combinations of CORRELATION and EOM_CORR
for standard EOM and CI models.
Method | CORRELATION | EOM_CORR | Target states selection |
CIS | CI | CIS | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
SF-CIS | CI | CIS | EOM_SF_STATES |
CIS(D) | CI | CIS(D) | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
SF-CIS(D) | CI | CIS(D) | EOM_SF_STATES |
CISD | CI | CISD | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
SF-CISD | CI | CISD | EOM_SF_STATES |
IP-CISD | CI | CISD | EOM_IP_STATES |
CISDT | CI | SDT | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
SF-CISDT | CI | SDT or DT | EOM_SF_STATES |
EOM-EE-CCSD | CCSD | | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
EOM-SF-CCSD | CCSD | | EOM_SF_STATES |
EOM-IP-CCSD | CCSD | | EOM_IP_STATES |
EOM-EA-CCSD | CCSD | | EOM_EA_STATES |
EOM-DIP-CCSD | CCSD | | EOM_DIP_STATES |
| | | EOM_DIP_SNGLETS,EOM_DIP_TRIPLETS |
EOM-2SF-CCSD | CCSD | SDT or DT | EOM_DSF_STATES |
EOM-EE-(2,3) | CCSD | SDT | EOM_EE_STATES |
| | | EOM_EE_SNGLETS,EOM_EE_TRIPLETS |
EOM-SF-(2,3) | CCSD | SDT | EOM_SF_STATES |
EOM-IP-(2,3) | CCSD | SDT | EOM_IP_STATES |
EOM-SF-CCSD(dT) | CCSD | SD(dT) | EOM_SF_STATES |
EOM-SF-CCSD(fT) | CCSD | SD(fT) | EOM_SF_STATES |
EOM-IP-CCSD(dT) | CCSD | SD(dT) | EOM_IP_STATES |
EOM-IP-CCSD(fT) | CCSD | SD(fT) | EOM_IP_STATES |
EOM-IP-CCSD(sT) | CCSD | SD(sT) | EOM_IP_STATES |
Table 6.2: Commonly used EOM and CI models. 'SINGLETS' and 'TRIPLETS' are only
available for closed-shell references.
The most relevant EOM-CC input options follow.
EOM_EE_STATES
Sets the number of excited state roots to find. For
closed-shell reference, defaults into EOM_EE_SINGLETS. For open-shell references,
specifies all low-lying states. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EE_SINGLETS
Sets the number of singlet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EE_TRIPLETS
Sets the number of triplet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_SF_STATES
Sets the number of spin-flip target states roots to find. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i SF states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_DSF_STATES
Sets the number of doubly spin-flipped target states roots to find. |
TYPE:
DEFAULT:
0 | Do not look for any DSF states. |
OPTIONS:
[i,j,k…] | Find i doubly spin-flipped states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_IP_STATES
Sets the number of ionized target states roots to find. By default, β electron will be removed
(see EOM_IP_BETA). |
TYPE:
DEFAULT:
0 | Do not look for any IP states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_IP_ALPHA
Sets the number of ionized target states derived by removing α electron (Ms=−[1/2]). |
TYPE:
DEFAULT:
0 | Do not look for any IP/α states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_IP_BETA
Sets the number of ionized target states derived by removing β electron (Ms=[1/2],
default for EOM-IP). |
TYPE:
DEFAULT:
0 | Do not look for any IP/β states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EA_STATES
Sets the number of attached target states roots to find. By default, α electron will be attached
(see EOM_EA_ALPHA). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EA_ALPHA
Sets the number of attached target states derived by attaching α electron (Ms=[1/2],
default in EOM-EA). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EA_BETA
Sets the number of attached target states derived by attaching β electron (Ms=−[1/2],
EA-SF). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_DIP_STATES
Sets the number of DIP roots to find. For
closed-shell reference, defaults into EOM_DIP_SINGLETS. For open-shell references,
specifies all low-lying states. |
TYPE:
DEFAULT:
0 | Do not look for any DIP states. |
OPTIONS:
[i,j,k…] | Find i DIP states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_DIP_SINGLETS
Sets the number of singlet DIP roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any singlet DIP states. |
OPTIONS:
[i,j,k…] | Find i DIP singlet states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_DIP_TRIPLETS
Sets the number of triplet DIP roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any DIP triplet states. |
OPTIONS:
[i,j,k…] | Find i DIP triplet states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
Note:
It is a symmetry of a transition rather than that of a target state which
is specified in excited state calculations. The symmetry of the target state
is a product of the symmetry of the reference state and the transition.
For closed-shell molecules, the former is fully symmetric and the symmetry of the
target state is the same as that of transition, however, for open-shell references
this is not so. |
Note:
For the EOM_XX_STATES options, Q-Chem will increase the number of roots if it suspects degeneracy,
or change it to a smaller value, if it cannot generate enough guess vectors to
start the calculations. |
| EOM_FAKE_IPEA
If TRUE, calculates fake EOM-IP or EOM-EA energies and properties using the diffuse orbital trick.
Default for EOM-EA and Dyson orbital calculations. |
TYPE:
DEFAULT:
FALSE (use proper EOM-IP code) |
OPTIONS:
RECOMMENDATION:
|
|
|
Note:
When EOM_FAKE_IPEA is set to TRUE, it can change the convergence of
Hartree-Fock iterations compared to the same job without EOM_FAKE_IPEA,
because a very diffuse basis function is added to a center of symmetry
before the Hartree-Fock iterations start. For the same reason,
BASIS2 keyword is incompatible with EOM_FAKE_IPEA. In order to
read Hartree-Fock guess from a previous job, you must specify
EOM_FAKE_IPEA (even if you do not request for any correlation or excited
states) in that previous job. Currently, the second moments of electron density and Mulliken
charges and spin densities are incorrect for the EOM-IP/EA-CCSD target states. |
| EOM_DAVIDSON_CONVERGENCE
Convergence criterion for the RMS residuals of excited state vectors |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to 10−n convergence criterion |
RECOMMENDATION:
Use default. Should normally be set to the same value as EOM_DAVIDSON_THRESHOLD. |
|
|
|
EOM_DAVIDSON_THRESHOLD
Specifies threshold for including a new expansion vector in the iterative
Davidson diagonalization. Their norm must be above this threshold. |
TYPE:
DEFAULT:
00105 | Corresponding to 0.00001 |
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
Use default unless converge problems are encountered. Should normally be set
to the same values as EOM_DAVIDSON_CONVERGENCE, if convergence problems arise
try setting to a value less than EOM_DAVIDSON_CONVERGENCE. |
|
| EOM_DAVIDSON_MAXVECTORS
Specifies maximum number of vectors in the subspace for the Davidson
diagonalization. |
TYPE:
DEFAULT:
OPTIONS:
n | Up to n vectors per root before the subspace is reset |
RECOMMENDATION:
Larger values increase disk storage but accelerate and stabilize convergence. |
|
|
|
EOM_DAVIDSON_MAX_ITER
Maximum number of iteration allowed for Davidson diagonalization procedure. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of iterations |
RECOMMENDATION:
Default is usually sufficient |
|
| EOM_NGUESS_DOUBLES
Specifies number of excited state guess vectors which are double excitations. |
TYPE:
DEFAULT:
OPTIONS:
n | Include n guess vectors that are double excitations |
RECOMMENDATION:
This should be set to the expected number of doubly excited states (see also
EOM_PRECONV_DOUBLES), otherwise they may not be found. |
|
|
|
EOM_NGUESS_SINGLES
Specifies number of excited state guess vectors that are single excitations. |
TYPE:
DEFAULT:
Equal to the number of excited states requested |
OPTIONS:
n | Include n guess vectors that are single excitations |
RECOMMENDATION:
Should be greater or equal than the number of excited states requested. |
|
| EOM_PRECONV_SINGLES
When not zero, singly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging singles. |
RECOMMENDATION:
Sometimes helps with problematic convergence. |
|
|
|
EOM_PRECONV_DOUBLES
When not zero, doubly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging doubles. |
RECOMMENDATION:
Occasionally necessary to ensure a doubly excited state is found. Also used in DSF calculations
instead of EOM_PRECONV_SINGLES |
|
| EOM_PRECONV_SD
When not zero, singly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging singles and doubles. |
RECOMMENDATION:
Occasionally necessary to ensure a doubly excited state is found. Also, very useful in EOM(2,3)
calculations. |
|
|
|
None
EOM_IPEA_FILTER
If TRUE, filters the EOM-IP/EA amplitudes obtained using the diffuse orbital implementation
(see EOM_FAKE_IPEA). Helps with convergence. |
TYPE:
DEFAULT:
FALSE (EOM-IP or EOM-EA amplitudes will not be filtered) |
OPTIONS:
RECOMMENDATION:
|
| CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO) |
TYPE:
DEFAULT:
OPTIONS:
range | 0000-10000 |
abcd | Corresponding to ab.cd% |
RECOMMENDATION:
|
|
|
CC_FNO_USEPOP
Selection of the truncation scheme |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| SCALE_NUCLEAR_CHARGE
Scales charge of each nuclei by a certain value. The nuclear repulsion energy
is calculated for the unscaled nuclear charges. |
TYPE:
DEFAULT:
OPTIONS:
n a total positive charge of (1+n / 100)e is added to the molecule. |
RECOMMENDATION:
|
|
|
ADD_CHARGED_CAGE
Add a point charge cage of a given radius and total charge. |
TYPE:
DEFAULT:
OPTIONS:
0 no cage. |
1 dodecahedral cage. |
2 spherical cage. |
RECOMMENDATION:
Spherical cage is expected to yield more accurate results, especially for small radii. |
|
| CAGE_RADIUS
Defines radius of the charged cage. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CAGE_POINTS
Defines number of point charges for the spherical cage. |
TYPE:
DEFAULT:
OPTIONS:
n n point charges are used. |
RECOMMENDATION:
|
| CAGE_CHARGE
Defines the total charge of the cage. |
TYPE:
DEFAULT:
400 Add a cage charged +4e. |
OPTIONS:
n total charge of the cage is n / 100 a.u. |
RECOMMENDATION:
|
|
|
6.6.8 Examples
Example 6.0 EOM-EE-OD and EOM-EE-CCSD calculations of the singlet excited states of formaldehyde
$molecule
0 1
O
C,1,R1
H,2,R2,1,A
H,2,R2,1,A,3,180.
R1=1.4
R2=1.0
A=120.
$end
$rem
correlation od
basis 6-31+g
eom_ee_states [2,2,2,2]
$end
@@@
$molecule
read
$end
$rem
correlation ccsd
basis 6-31+g
eom_ee_singlets [2,2,2,2]
eom_ee_triplets [2,2,2,2]
$end
Example 6.0 EOM-EE-CCSD calculations of the singlet excited states of PYP using
Cholesky decomposition
$molecule
0 1
...too long to enter...
$end
$rem
correlation ccsd
n_frozen_core fc
BASIS aug-cc-pVDZ
PURECART 1112
max_sub_file_num 64
cc_memory 8000
mem_static 500
CC_T_CONV 4
CC_E_CONV 6
cholesky_tol 2 using CD/1e-2 threshold
eom_ee_singlets = [2,2]
$end
Example 6.0 EOM-SF-CCSD calculations for methylene from high-spin 3B2 reference
$molecule
0 3
C
H 1 rCH
H 1 rCH 2 aHCH
rCH = 1.1167
aHCH = 102.07
$end
$rem
jobtype SP
CORRELATION CCSD
BASIS 6-31G*
SCF_GUESS CORE
EOM_NGUESS_SINGLES 4
EOM_SF_STATES [2,0,0,2] Two singlet A1 states and singlet and triplet B2 states
$end
Example 6.0 EOM-IP-CCSD calculations for NO3 using closed-shell anion reference
$molecule
-1 1
N
O 1 r1
O 1 r2 2 A2
O 1 r2 2 A2 3 180.0
r1 = 1.237
r2 = 1.237
A2 = 120.00
$end
$rem
jobtype SP single point
LEVCOR CCSD
BASIS 6-31G*
EOM_IP_STATES [1,1,2,1] ground and excited states of the radical
$end
Example 6.0 EOM-IP-CCSD calculation using FNO with OCCT=99%.
$molecule
0 1
O
H 1 1.0
H 1 1.0 2 100.
$end
$rem
correlation = CCSD
eom_ip_states [1,0,1,1]
basis = 6-311+G(2df,2pd)
CC_fno_thresh 9900 99% of the total natural population recovered
$end
Example 6.0 DSF-CIDT calculation of methylene starting with quintet reference
$molecule
0 5
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
correlation ci
eom_corr sdt
basis 6-31G
eom_dsf_states [0,2,2,0]
eom_nguess_singles 0
eom_nguess_doubles 2
$end
Example 6.0 EOM-EA-CCSD job for cyano radical. We first do Hartree-Fock calculation for the cation in the basis set with one extremely diffuse orbital
(EOM_FAKE_IPEA) and use these orbitals in the second job.
We need make sure that the diffuse orbital is occupied using the OCCUPIED keyword.
No SCF iterations are performed as the diffuse electron and the molecular core are uncoupled.
The attached states show up as "excited" states in which electron is promoted from the diffuse orbital to the molecular ones.
$molecule
+1 1
C
N 1 bond
bond 1.1718
$end
$rem
jobtype sp
exchange hf
basis 6-311+G*
purecart 111
scf_convergence 8
correlation none
eom_fake_ipea true
$end
@@@
$molecule
0 2
C
N 1 bond
bond 1.1718
$end
$rem
jobtype sp
basis 6-311+G*
purecart 111
scf_guess read
max_scf_cycles 0
correlation ccsd
cc_dov_thresh 2501 use threshold for CC iterations with problematic convergence
eom_ea_states [2,0,0,0]
eom_fake_ipea true
$end
$occupied
1 2 3 4 5 6 14
1 2 3 4 5 6
$end
Example 6.0 EOM-DIP-CCSD calculation of electronic states in methylene
using charged cage stabilization method.
$molecule
-2 1
C 0.000000 0.000000 0.106788
H -0.989216 0.000000 -0.320363
H 0.989216 0.000000 -0.320363
$end
$rem
jobtype = sp
basis = 6-311g(d,p)
scf_algorithm = diis_gdm
symmetry = false
correlation = ccsd
cc_symmetry = false
eom_dip_singlets = [1] ! Compute one EOM-DIP singlet state
eom_dip_triplets = [1] ! Compute one EOM-DIP triplet state
eom_davidson_convergence = 5
cc_eom_prop = true ! Compute excited state properties
add_charged_cage = 2 ! Install a charged sphere around the molecule
cage_radius = 225 ! Radius = 2.25 A
cage_charge = 500 ! Charge = +5 a.u.
cage_points = 100 ! Place 100 point charges
cc_memory = 256 ! Use 256Mb of memory, increase for larger jobs
$end
Example 6.0 EOM-EE-CCSD calculation of excited states in NO−
using scaled nuclear charge stabilization method.
$molecule
-1 1
N -1.08735 0.0000 0.0000
O 1.08735 0.0000 0.0000
$end
$rem
jobtype = sp
input_bohr = true
basis = 6-31g
symmetry = false
cc_symmetry = false
correlation = ccsd
eom_ee_singlets = [2] ! Compute two EOM-EE singlet excited states
eom_ee_triplets = [2] ! Compute two EOM-EE triplet excited states
cc_ref_prop = true ! Compute ground state properties
cc_eom_prop = true ! Compute excited state properties
cc_memory = 256 ! Use 256Mb of memory, increase for larger jobs
scale_nuclear_charge = 180 ! Adds +1.80e charge to the molecule
$end
6.6.9 Non-Hartree-Fock Orbitals in EOM Calculations
In cases of problematic open-shell references, e.g., strongly spin-contaminated
doublet, triplet or quartet states, one may choose to use DFT orbitals. This
can be achieved by first doing DFT calculation and then
reading the orbitals and turning Hartree-Fock off. A more convenient way is just to
specify EXCHANGE, e.g., if EXCHANGE=B3LYP, B3LYP orbitals will be computed and used in
the CCMAN/CCMAN2 module.
6.6.10 Analytic Gradients and Properties for the CCSD and EOM-XX-CCSD Methods
Analytic gradients are available for the CCSD and all EOM-CCSD methods for
both closed- and open-shell references (UHF and RHF only), including frozen
core / virtual functionality [341] (see also Section 5.12).
Application limit: same as for the single-point CCSD or EOM-CCSD
calculations.
Limitations: Gradients for ROHF and non-HF (e.g., B3LYP) orbitals are not yet available.
For the CCSD and EOM-CCSD wavefunctions, Q-Chem currently can calculate
permanent and transition dipole moments, oscillator strengths, 〈R2〉
(as well as XX, YY and ZZ components separately, which is
useful for assigning different Rydberg states, e.g., 3px vs. 3s, etc.), and the
〈S2〉 values. Interface of the CCSD and EOM-CCSD codes with the NBO 5.0
package is also available. Similar functionality is available for some EOM-OD and
CI models.
The coupled-cluster package in Q-Chem can calculate properties of target
EOM states including transition dipoles and geometry optimizations. The target
state of interest is selected by CC_STATE_TO_OPT $rem, which
specifies the symmetry and the number of the EOM state.
Users must be aware of the point group
symmetry of the system being studied and also the symmetry of the excited (target) state
of interest. It is possible to turn off the use of symmetry using the
CC_SYMMETRY. If set to FALSE the molecule will
be treated as having C1 symmetry and all states will be of A symmetry.
6.6.11 Equation-of-Motion Coupled-Cluster Optimization and Properties Job Control
CC_STATE_TO_OPT
Specifies which state to optimize. |
TYPE:
DEFAULT:
OPTIONS:
[i,j] | optimize the jth state of the ith irrep. |
RECOMMENDATION:
|
Note:
The state number should be smaller or equal to the number of excited states calculated in the corresponding irrep. |
Note:
If analytic gradients are not available, the finite difference
calculations will be performed and the symmetry will be turned off. In this case,
CC_STATE_TO_OPT should be specified assuming C1 symmetry, i.e.,
as [1,N] where N is the number of state to optimize (the states are numbered from 1). |
| CC_EOM_PROP
Whether or not the non-relaxed (expectation value) one-particle EOM-CCSD
target state properties will be calculated. The properties currently include
permanent dipole moment, the second moments 〈X2〉, 〈Y2〉, and
〈Z2〉 of electron density, and the total 〈R2〉
= 〈X2〉
+〈Y2〉+〈Z2〉 (in atomic units). Incompatible with
JOBTYPE=FORCE, OPT, FREQ. |
TYPE:
DEFAULT:
FALSE (no one-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations (EOM-CCSD equations for the left eigenvectors) need to be
solved for properties, approximately doubling the cost of calculation for each
irrep. Sometimes the equations for left and right eigenvectors converge to
different sets of target states. In this case, the simultaneous iterations of
left and right vectors will diverge, and the properties for several or all the
target states may be incorrect! The problem can be solved by varying the number
of requested states, specified with EOM_XX_STATES, or the number of guess vectors
(EOM_NGUESS_SINGLES). The cost of the one-particle properties
calculation itself is low. The one-particle density of an EOM-CCSD target
state can be analyzed with NBO package by specifying the state with
CC_STATE_TO_OPT and requesting
NBO=TRUE and CC_EOM_PROP=TRUE. |
|
|
|
CC_TRANS_PROP
Whether or not the transition dipole moment (in atomic units) and oscillator
strength for the EOM-CCSD target states will be calculated. By default, the
transition dipole moment is calculated between the CCSD reference and the
EOM-CCSD target states. In order to calculate transition dipole moment between
a set of EOM-CCSD states and another EOM-CCSD state, the
CC_STATE_TO_OPT
must be specified for this state. |
TYPE:
DEFAULT:
FALSE (no transition dipole and oscillator strength will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations (for the left EOM-CCSD eigenvectors plus lambda CCSD
equations in case if transition properties between the CCSD reference and
EOM-CCSD target states are requested) need to be solved for transition
properties, approximately doubling the computational cost. The cost of the
transition properties calculation itself is low. |
|
| EOM_REF_PROP_TE
Request for calculation of non-relaxed two-particle EOM-CC properties. The
two-particle properties currently include 〈S2〉. The one-particle
properties also will be calculated, since the additional cost of the
one-particle properties calculation is inferior compared to the cost of
〈S2〉. The variable CC_EOM_PROP must be also set to
TRUE. Alternatively, CC_CALC_SSQ can be used to
request 〈S2〉 calculation. |
TYPE:
DEFAULT:
FALSE | (no two-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
The two-particle properties are computationally expensive since
they require calculation and use of the two-particle density matrix (the cost
is approximately the same as the cost of an analytic gradient calculation). Do
not request the two-particle properties unless you really need them. |
|
|
|
CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed.
The variable CC_EOM_PROP must be also set to TRUE. |
TYPE:
DEFAULT:
FALSE | (no orbital response will be calculated) |
OPTIONS:
RECOMMENDATION:
Not available for non-UHF/RHF references. Only available for EOM/CI methods for which analytic
gradients are available. |
|
| CC_SYMMETRY
Controls the use of symmetry in coupled-cluster calculations |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Use the point group symmetry of the molecule |
FALSE | Do not use point group symmetry (all states will be of A symmetry). |
RECOMMENDATION:
It is automatically turned off for any finite difference calculations, e.g. second derivatives. |
|
|
|
6.6.12 Examples
Example 6.0 Geometry optimization for the excited open-shell singlet state, 1B2, of
methylene followed by the calculations of the fully relaxed one-electron properties using EOM-EE-CCSD
$molecule
0 1
C
H 1 rCH
H 1 rCH 2 aHCH
rCH = 1.083
aHCH = 145.
$end
$rem
jobtype OPT
CORRELATION CCSD
BASIS cc-pVTZ
SCF_GUESS CORE
SCF_CONVERGENCE 9
EOM_EE_SINGLETS [0,0,0,1]
EOM_NGUESS_SINGLES 2
cc_state_to_opt [4,1]
EOM_DAVIDSON_CONVERGENCE 9 use tighter convergence for EOM amplitudes
$end
@@@
$molecule
READ
$end
$rem
jobtype SP
CORRELATION CCSD
BASIS cc-pVTZ
SCF_GUESS READ
EOM_EE_SINGLETS [0,0,0,1]
EOM_NGUESS_SINGLES 2
CC_EOM_PROP 1 calculate properties for EOM states
CC_FULLRESPONSE 1 use fully relaxed properties
$end
Example 6.0 Property and transition property calculation on the lowest singlet
state of CH2 using EOM-SF-CCSD
$molecule
0 3
C
H 1 rch
H 1 rch 2 ahch
rch = 1.1167
ahch = 102.07
$end
$rem
CORRELATION ccsd
EXCHANGE hf
BASIS cc-pvtz
SCF_GUESS core
SCF_CONVERGENCE 9
EOM_SF_STATES [2,0,0,3] Get three 1^B2 and two 1^A1 SF states
CC_EOM_PROP 1
CC_TRANS_PROP 1
CC_STATE_TO_OPT [4,1] First EOM state in the 4th irrep
$end
Example 6.0 Geometry optimization with tight convergence for the 2A1 excited state of CH2Cl,
followed by calculation of non-relaxed and fully relaxed
permanent dipole moment and 〈S2〉.
$molecule
0 2
H
C 1 CH
CL 2 CCL 1 CCLH
H 2 CH 3 CCLH 1 DIH
CH=1.096247
CCL=2.158212
CCLH=122.0
DIH=180.0
$end
$rem
JOBTYPE OPT
CORRELATION CCSD
BASIS 6-31G* Basis Set
SCF_GUESS SAD
EOM_DAVIDSON_CONVERGENCE 9 EOM amplitude convergence
CC_T_CONV 9 CCSD amplitudes convergence
EOM_EE_STATES [0,0,0,1]
cc_state_to_opt [4,1]
EOM_NGUESS_SINGLES 2
GEOM_OPT_TOL_GRADIENT 2
GEOM_OPT_TOL_DISPLACEMENT 2
GEOM_OPT_TOL_ENERGY 2
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE SP
CORRELATION CCSD
BASIS 6-31G* Basis Set
SCF_GUESS READ
EOM_EE_STATES [0,0,0,1]
CC_NGUESS_SINGLES 2
CC_EOM_PROP 1 calculate one-electron properties
CC_EOM_PROP_TE 1 and two-electron properties (S^2)
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE SP
CORRELATION CCSD
BASIS 6-31G* Basis Set
SCF_GUESS READ
EOM_EE_STATES [0,0,0,1]
EOM_NGUESS_SINGLES 2
CC_EOM_PROP 1 calculate one-electron properties
CC_EOM_PROP_TE 1 and two-electron properties (S^2)CC_EXSTATES_PROP 1
CC_FULL_RESPONSE 1 same as above, but do fully relaxed properties
$end
Example 6.0 CCSD calculation on three A2 and one B2 state of
formaldehyde. Transition properties will be calculated between the third A2
state and all other EOM states
$molecule
0 1
O
C 1 1.4
H 2 1.0 1 120
H 3 1.0 1 120
$end
$rem
BASIS 6-31+G
CORRELATION CCSD
EOM_EE_STATES [0,3,0,1]
CC_STATE_TO_OPT [2,3]
CC_TRANS_PROP true
$end
Example 6.0 EOM-IP-CCSD geometry optimization of X 2B2 state of H2O+.
$molecule
0 1
H 0.774767 0.000000 0.458565
O 0.000000 0.000000 -0.114641
H -0.774767 0.000000 0.458565
$end
$rem
jobtype opt
exchange hf
correlation ccsd
basis 6-311G
eom_ip_states [0,0,0,1]
cc_state_to_opt [4,1]
$end
6.6.13 EOM(2,3) Methods for Higher-Accuracy and Problematic Situations
In the EOM-CC(2,3) approach [349], the transformed
Hamiltonian ―H is diagonalized in the basis of the reference, singly,
doubly, and triply excited determinants, i.e., the excitation operator R is
truncated at triple excitations. The excitation operator T, however, is
truncated at double excitation level, and its amplitudes are found from the
CCSD equations, just like for EOM-CCSD [or EOM-CC(2,2)] method.
The accuracy of the EOM-CC(2,3) method closely follows that of full EOM-CCSDT
[which can be also called EOM-CC(3,3)], whereas computational cost of the
former model is less.
The inclusion of triple excitations is necessary for achieving chemical
accuracy (1 kcal/mol) for ground state properties. It is even more so for
excited states. In particular, triple excitations are crucial for doubly
excited states [349], excited states of some radicals and SF
calculations (diradicals, triradicals, bond-breaking) when a reference
open-shell state is heavily spin-contaminated. Accuracy of EOM-CCSD and
EOM-CC(2,3) is compared in Table 6.6.13.
System | EOM-CCSD | EOM-CC(2,3) | |
Singly-excited electronic states | 0.1-0.2 eV | 0.01 eV |
Doubly-excited electronic states | ≥ 1 eV | 0.1-0.2 eV |
Severe spin-contamination of the reference | ∼ 0.5 eV | ≤ 0.1 eV |
Breaking single bond (EOM-SF) | 0.1-0.2 eV | 0.01 eV |
Breaking double bond (EOM-2SF) | ∼ 1 eV | 0.1-0.2 eV |
Table 6.3: Performance of the EOM-CCSD and EOM-CC(2,3) methods
The applicability of the EOM-EE / SF-CC(2,3) models to larger
systems can be extended by using their
active-space variants, in which triple
excitations are restricted to semi-internal ones.
Since the computational scaling of EOM-CC(2,3) method is O(N8), these
calculations can be performed only for relatively small systems. Moderate size
molecules (10 heavy atoms) can be tackled by either using the active space
implementation or tiny basis sets. To achieve high accuracy for these systems,
energy additivity schemes can be used. For example,
one can extrapolate EOM-CCSDT / large basis set values by
combining large basis set EOM-CCSD calculations with small basis set
EOM-CCSDT ones.
Running the full EOM-CC(2,3) calculations is straightforward, however, the
calculations are expensive with the bottlenecks being storage of the data on a
hard drive and the CPU time. Calculations with around 80 basis functions are
possible for a molecule consisting of four first row atoms (NO dimer). The
number of basis functions can be larger for smaller systems.
Note:
In EE calculations, one needs to always solve for at least one low-spin
root in the first symmetry irrep in order to obtain the correlated EOM energy
of the reference. The triples correction to the total reference energy must be
used to evaluate EOM-(2,3) excitation energies. |
Note:
EOM-CC(2,3) works for EOM-EE, EOM-SF, and EOM-IP/EA. In EOM-IP, "triples" correspond to
3h2p excitations, and the computational scaling of EOM-IP-CC(2,3) is less. |
6.6.14 Active-Space EOM-CC(2,3): Tricks of the Trade
Active space calculations are less demanding with respect to the size of a hard
drive. The main bottlenecks here are the memory usage and the CPU time. Both
arise due to the increased number of orbital blocks in the active space
calculations. In the current implementation, each block can contain from 0 up
to 16 orbitals of the same symmetry irrep, occupancy, and spin-symmetry. For
example, for a typical molecule of C2v symmetry, in a small/moderate
basis set (e.g., TMM in 6-31G*), the number of blocks for each index is:
occupied: (α+ β)×( a1 + a2 + b1 + b2) = 2×4 = 8
virtuals: (α+ β)×( 2 a1 + a2 + b1 + 2 b2) = 2×6 = 12
(usually there are more than 16 a1 and b2 virtual orbitals).
In EOM-CCSD, the total number of blocks is O2V2 = 82 ×122 = 9216 .
In EOM-CC(2,3) the number of blocks in the EOM part is O3V3 = 83 ×123 = 884736 . In active space EOM-CC(2,3), additional fragmentation of blocks
occurs to distinguish between the restricted and active orbitals. For example,
if the active space includes occupied and virtual orbitals of all symmetry
irreps (this will be a very large active space), the number of occupied and
virtual blocks for each index is 16 and 20, respectively, and the total number
of blocks increases to 3.3×107. Not all of the blocks contain real
information, some blocks are zero because of the spatial or spin-symmetry
requirements. For the C2v symmetry group, the number of non-zero
blocks is about 10-12 times less than the total number of blocks, i.e.,
3×106. This is the number of non-zero blocks in one vector.
Davidson diagonalization procedure requires (2*MAX_VECTORS + 2*NROOTS)
vectors, where MAX_VECTORS is the maximum number of vectors in the subspace,
and NROOTS is the number of the roots to solve for. Taking NROOTS=2 and
MAX_VECTORS=20, we obtain 44 vectors with the total number of non-zero blocks
being 1.3×108.
In CCMAN implementation, each block is a logical unit of information. Along with
real data, which are kept on a hard drive at all the times except of their
direct usage, each non-zero block contains an auxiliary information about its
size, structure, relative position with respect to other blocks, location on a
hard drive, and so on. The auxiliary information about blocks is always
kept in memory. Currently, the approximate size of this auxiliary information
is about 400 bytes per block. It means, that in order to keep information about
one vector (3×106 blocks), 1.2 GB of memory is required! The information
about 44 vectors amounts 53 GB. Moreover, the huge number of blocks
significantly slows down the code.
To make the calculations of active space EOM-CC(2,3) feasible, we need to
reduce the total number of blocks. One way to do this is to reduce the symmetry
of the molecule to lower or C1 symmetry group (of course, this will result
in more expensive calculation). For example, lowering the symmetry group from
C2v to Cs would results in reducing the total number of
blocks in active space EOM-CC(2,3) calculations in about 26 = 64 times, and
the number of non-zero blocks in about 30 times (the relative portion of
non-zero blocks in Cs symmetry group is smaller compared to that in
C2v).
Alternatively, one may keep the MAX_VECTORS and NROOTS parameters of
Davidson's diagonalization procedure as small as possible (this mainly concerns
the MAX_VECTORS parameter). For example, specifying MAX_VECTORS = 12 instead
of 20 would require 30% less memory.
One more trick concerns specifying the active space. In a desperate situation
of a severe lack of memory, should the two previous options fail, one can try
to modify (increase) the active space in such a way that the fragmentation of
active and restricted orbitals would be less. For example, if there is one
restricted occupied b1 orbital and one active occupied B1 orbital, adding
the restricted b1 to the active space will reduce the number of blocks, by
the price of increasing the number of FLOPS. In principle, adding extra orbital
to the active space should increase the accuracy of calculations, however, a
special care should be taken about the (near) degenerate pairs of orbitals,
which should be handled in the same way, i.e., both active or both restricted.
6.6.15 Job Control for EOM-CC(2,3)
EOM-CC(2,3) is invoked by CORRELATION=CCSD and EOM_CORR=SDT.
The following options are available:
EOM_PRECONV_SD
Solves the EOM-CCSD equations, prints energies, then uses EOM-CCSD vectors
as initial vectors in EOM-CC(2,3). Very convenient for calculations using energy additivity schemes. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Turning this option on is recommended |
|
| CC_REST_AMPL
Forces the integrals, T, and R amplitudes to be determined in the full
space even though the CC_REST_OCC and CC_REST_VIR
keywords are used. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do apply restrictions |
1 | Do not apply restrictions |
RECOMMENDATION:
|
|
|
CC_REST_TRIPLES
Restricts R3 amplitudes to the active space, i.e., one electron should be
removed from the active occupied orbital and one electron should be added to
the active virtual orbital. |
TYPE:
DEFAULT:
OPTIONS:
1 | Applies the restrictions |
RECOMMENDATION:
|
| CC_REST_OCC
Sets the number of restricted occupied orbitals including frozen occupied
orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | Restrict n occupied orbitals. |
RECOMMENDATION:
|
|
|
CC_REST_VIR
Sets the number of restricted virtual orbitals including frozen virtual
orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | Restrict n virtual orbitals. |
RECOMMENDATION:
|
To select the active space, orbitals can be reordered by specifying the new
order in the $reorder_mosection. The section consists of two rows of
numbers (α and β sets), starting from 1, and ending with
n, where n is the number of the last orbital specified.
Example 6.0 Example $reorder_mosection with orbitals 16 and 17 swapped for both α and β electrons
$reorder_mo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
$end
6.6.16 Examples
Example 6.0 EOM-SF(2,3) calculations of methylene.
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
correlation ccsd
eom_corr sdt do EOM-(2,3)
basis 6-31G
eom_sf_states [2,0,0,2]
n_frozen_core 1
n_frozen_virtual 1
eom_preconv_sd 20 Get EOM-CCSD energies first (max_iter=20).
$end
Example 6.0 This is active-space EOM-SF(2,3) calculations for methane with
an elongated CC bond. HF MOs should be reordered as specified in
the $reorder_mosection such that active space for triples
consists of sigma and sigma* orbitals.
$molecule
0 3
C
H 1 CH
H 1 CHX 2 HCH
H 1 CH 2 HCH 3 A120
H 1 CH 2 HCH 4 A120
CH=1.086
HCH=109.4712206
A120=120.
CHX=1.8
$end
$rem
jobtype sp
correlation ccsd
eom_corr sdt
basis 6-31G*
eom_sf_states [1,0]
n_frozen_core 1
eom_preconv_sd 20 does eom-ccsd first, max_iter=20
cc_rest_triples 1 triples are restricted to the active space only
cc_rest_ampl 0 ccsd and eom singles and doubles are full-space
cc_rest_occ 4 specifies active space
cc_rest_vir 17 specifies active space
print_orbitals 10 (number of virtuals to print)
$end
$reorder_mo
1 2 5 4 3
1 2 3 4 5
$end
Example 6.0 EOM-IP-CC(2,3) calculation of three lowest electronic states of water cation.
$molecule
0 1
H 0.774767 0.000000 0.458565
O 0.000000 0.000000 -0.114641
H -0.774767 0.000000 0.458565
$end
$rem
jobtype sp
correlation ccsd
eom_corr sdt
basis 6-311G
eom_ip_states [1,0,1,1]
$end
6.6.17 Non-Iterative Triples Corrections to EOM-CCSD and CCSD
The effect of triple excitations to EOM-CCSD energies can be included via
perturbation theory in an economical N7 computational scheme.
Using EOM-CCSD wavefunctions as zero-order wavefunctions,
the second order triples correction to the μth EOM-EE or SF state is:
∆E(2)μ = − |
1
36
|
|
∑
i,j,k
|
|
∑
a,b,c
|
|
|
~
σ
|
abc ijk
|
(μ) σijkabc(μ) |
Dijkabc − ωμ
|
|
| (6.37) |
where i,j and k denote occupied orbitals, and
a,b and c are virtual orbital indices. ωμ is the
EOM-CCSD excitation energy of the μth state.
The quantities ~σ and σ are:
| |
|
〈Φ0| (L1μ + L2μ) (H e(T1+T2))c | Φijkabc〉 |
| | (6.38) |
| |
|
〈Φijkabc|[H e(T1+T2)(R0μ + R1μ + R2μ)]c | Φ0〉 |
| |
|
where, the L and R are left and right eigen-vectors for μth state.
Two different choices of the denominator,
Dijkabc, define the (dT) and (fT) variants of the correction.
In (fT), Dijkabc is just Hartree-Fock orbital energy differences.
A more accurate (but not fully orbital invariant) (dT) correction employs
the complete three body diagonal of ―H,
〈Φijkabc| (He(T1+T2))C|Φijkabc〉,
Dijkabcas a denominator.
For the reference (e.g., a ground-state CCSD wavefunction),
the (fT) and (dT) corrections are identical to the
CCSD(2)T and CR-CCSD(T)L corrections
of Piecuch and co-workers [350].
The EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT) methods
yield a systematic improvement over EOM-SF-CCSD bringing the errors
below 1 kcal/mol. For theoretical background and detailed benchmarks,
see Ref. .
Similar corrections are available for EOM-IP-CCSD [352], where
triples correspond to 3h2p excitations.
6.6.18 Job Control for Non-Iterative Triples Corrections
Triples corrections are requested by using EOM_CORR:
| EOM_CORR
Specifies the correlation level. |
TYPE:
DEFAULT:
None | No correction will be computed |
OPTIONS:
SD(DT) | EOM-CCSD(dT), available for EE, SF, and IP |
SD(FT) | EOM-CCSD(dT), available for EE, SF, and IP |
SD(ST) | EOM-CCSD(sT), available for IP |
RECOMMENDATION:
|
|
|
6.6.19 Examples
Example 6.0 EOM-EE-CCSD(fT) calculation of CH+
$molecule
1 1
C
H C CH
CH = 2.137130
$end
$rem
input_bohr true
jobtype sp
correlation ccsd
eom_corr sd(ft)
basis general
eom_ee_states [1,0,1,1]
eom_davidson_max_iter 60 increase number of Davidson iterations
$end
$basis
H 0
S 3 1.00
19.24060000 0.3282800000E-01
2.899200000 0.2312080000
0.6534000000 0.8172380000
S 1 1.00
0.1776000000 1.000000000
S 1 1.00
0.0250000000 1.000000000
P 1 1.00
1.00000000 1.00000000
****
C 0
S 6 1.00
4232.610000 0.2029000000E-02
634.8820000 0.1553500000E-01
146.0970000 0.7541100000E-01
42.49740000 0.2571210000
14.18920000 0.5965550000
1.966600000 0.2425170000
S 1 1.00
5.147700000 1.000000000
S 1 1.00
0.4962000000 1.000000000
S 1 1.00
0.1533000000 1.000000000
S 1 1.00
0.0150000000 1.000000000
P 4 1.00
18.15570000 0.1853400000E-01
3.986400000 0.1154420000
1.142900000 0.3862060000
0.3594000000 0.6400890000
P 1 1.00
0.1146000000 1.000000000
P 1 1.00
0.0110000000 1.000000000
D 1 1.00
0.750000000 1.00000000
****
$end
Example 6.0 EOM-SF-CCSD(dT) calculations of methylene
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
correlation ccsd
eom_corr sd(dt)
basis 6-31G
eom_sf_states [2,0,0,2]
n_frozen_core 1
n_frozen_virtual 1
$end
Example 6.0 EOM-IP-CCSD(dT) calculations of Mg
$molecule
0 1
Mg 0.000000 0.000000 0.000000
$end
$rem
jobtype sp
correlation ccsd
eom_corr sd(dt)
basis 6-31g
eom_ip_states [1,0,0,0,0,1,1,1]
$end
6.6.20 Potential Energy Surface Crossing Minimization
Potential energy surface crossing optimization procedure finds energy minima
on crossing seams. On the seam, the potential surfaces are degenerated in the
subspace perpendicular to the plane defined by two vectors: the gradient
difference
and the derivative coupling
At this time Q-Chem is unable to locate crossing minima for states which
have non-zero derivative coupling. Fortunately, often this is not the case.
Minima on the seams of conical intersections of states of different
multiplicity can be found as their derivative coupling is zero.
Minima on the seams of intersections of states of different
point group symmetry can be located as well.
To run a PES crossing minimization, CCSD and EOM-CCSD methods must
be employed for the ground and excited state calculations respectively.
6.6.20.1 Job Control Options
XOPT_STATE_1, XOPT_STATE_2
Specify two electronic states the intersection of which will be searched. |
TYPE:
[INTEGER, INTEGER, INTEGER] |
DEFAULT:
No default value (the option must be specified to run this calculation) |
OPTIONS:
[spin, irrep, state] | |
spin = 0 | Addresses states with low spin, |
| see also EOM_EE_SINGLETS. |
spin = 1 | Addresses states with high spin, |
| see also EOM_EE_TRIPLETS. |
irrep | Specifies the irreducible representation to which |
| the state belongs, for C2v point group symmetry |
| irrep = 1 for A1, irrep = 2 for A2, |
| irrep = 3 for B1, irrep = 4 for B2. |
state | Specifies the state number within the irreducible |
| representation, state = 1 means the lowest excited |
| state, state = 2 is the second excited state, etc.. |
0, 0, -1 | Ground state. |
RECOMMENDATION:
Only intersections of states with different spin
or symmetry can be calculated at this time. |
|
Note:
The spin can only be specified when using closed-shell RHF references. In the case of
open-shell references all states are treated together, see also
EOM_EE_STATES. |
| XOPT_SEAM_ONLY
Orders an intersection seam search only, no minimization is to perform. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Find a point on the intersection seam and stop. |
FALSE | Perform a minimization of the intersection seam. |
RECOMMENDATION:
In systems with a large number of degrees of freedom it might be useful
to locate the seam first setting this option to TRUE and use that geometry
as a starting point for the minimization. |
|
|
|
6.6.20.2 Examples
Example 6.0
Minimize the intersection of Ã2A1 and
~B2B1 states of the NO2 molecule
using EOM-IP-CCSD method
$molecule
-1 1
N1
O2 N1 rno
O3 N1 rno O2 aono
rno = 1.3040
aono = 106.7
$end
$rem
JOBTYPE opt Optimize the intersection seam
UNRESTRICTED true
CORRELATION ccsd
BASIS 6-31g
EOM_IP_STATES [1,0,1,0] C2v point group symmetry
EOM_FAKE_IPEA 1
XOPT_STATE_1 [0,1,1] 1A1 low spin state
XOPT_STATE_2 [0,3,1] 1B1 low spin state
GEOM_OPT_TOL_GRADIENT 30 Tighten gradient tolerance
$END
Example 6.0
Minimize the intersection of Ã1B2 and
~B1A2 states of the N3+ ion
using EOM-CCSD method
$molecule
1 1
N1
N2 N1 rnn
N3 N2 rnn N1 annn
rnn=1.46
annn=70.0
$end
$rem
JOBTYPE opt
CORRELATION ccsd
BASIS 6-31g
EOM_EE_SINGLES [0,2,0,2] C2v point group symmetry
XOPT_STATE_1 [0,4,1] 1B2 low spin state
XOPT_STATE_2 [0,2,2] 2A2 low spin state
XOPT_SEAM_ONLY true Find the seam only
GEOM_OPT_TOL_GRADIENT 100
$end
$opt
CONSTRAINT Set constraints on the N-N bond lengths
stre 1 2 1.46
stre 2 3 1.46
ENDCONSTRAINT
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE opt Optimize the intersection seam
CORRELATION ccsd
BASIS 6-31g
EOM_EE_SINGLETS [0,2,0,2]
XOPT_STATE_1 [0,4,1]
XOPT_STATE_2 [0,2,2]
GEOM_OPT_TOL_GRADIENT 30
$end
6.6.21 Dyson Orbitals for Ionization from Ground and Excited
States within EOM-CCSD Formalism
Dyson orbitals can be used to compute total photodetachment / photoionization cross sections, as well
as angular distribution of photoelectrons.
A Dyson orbital is the overlap between the N-electron molecular
wavefunction and the N-1/N+1 electron wavefunction of the corresponding cation / anion:
| |
|
|
1
N−1
|
| ⌠ ⌡
|
ΨN(1, …, n) ΨN−1(2, …, n) d2 …dn |
| | (6.41) |
| |
|
|
1
N+1
|
| ⌠ ⌡
|
ΨN(2, …, n+1) ΨN+1(1, …, n+1) d2 …d(n+1) |
| | (6.42) |
|
For the Hartree-Fock wavefunctions and within Koopmans' approximation, these are just the
canonical HF orbitals. For correlated wavefunctions, Dyson orbitals are linear
combinations of
the reference molecular orbitals:
The calculation of Dyson orbitals is straightforward within the EOM-IP / EA-CCSD methods,
where cation / anion and initial molecule states are defined with respect to the same MO basis. Since the
left and right CC vectors are not the same, one can define correspondingly two Dyson orbitals
(left-right and right-left):
| |
|
〈Φ0 eT1+T2 LEE |p+| RIP eT1+T2 Φ0〉 |
| | (6.46) |
| |
|
〈Φ0 eT1+T2 LIP |p| REE eT1+T2 Φ0〉 |
| | (6.47) |
|
The norm of these orbitals is proportional to the one-electron character of the transition.
Dyson orbitals also offer qualitative insight visualizing
the difference between molecular and ionized/attached states.
In ionization / photodetachment processes,
these orbitals can be also interpreted as the wavefunction of the leaving
electron. For additional details, see Refs. .
6.6.21.1 Dyson Orbitals Job Control
The calculation of Dyson orbitals is implemented for the ground (reference) and excited states
ionization / electron attachment. To obtain the ground state Dyson orbitals one needs to run an
EOM-IP / EA-CCSD calculation, request transition properties calculation by
setting CC_TRANS_PROP=TRUE and CC_DO_DYSON = TRUE. The Dyson orbitals
decomposition in the MO basis is printed in the output, for all transitions between the reference and all
IP / EA states. At the end of the file, also the coefficients of the Dyson orbitals in the AO basis are
available. In CCMAN2, EOM_FAKE_IPEA will be automatically set to TRUE.
CC_DO_DYSON
Whether the reference-state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations. |
TYPE:
DEFAULT:
FALSE (the option must be specified to run this calculation) |
OPTIONS:
RECOMMENDATION:
|
For calculating Dyson orbitals between excited states from the reference configuration and
IP / EA states,
CC_TRANS_PROP=TRUE and CC_DO_DYSON_EE = TRUE have to be added to the usual EOM-IP/EA-CCSD
calculation.
The EOM_IP_STATES keyword is used to specify the target ionized states.
The attached states are specified by EOM_EA_STATES. The EA-SF states
are specified by EOM_EA_BETA.
The excited (or spin-flipped) states
are specified by EOM_EE_STATES and EOM_SF_STATES
The Dyson orbital decomposition in MO and AO bases is printed for each
EE-IP/EA pair of states in the order: EE1 - IP/EA1, EE1 - IP/EA2,… , EE2 - IP/EA1, EE2 - IP/EA2,
…, and so on.
CC_DO_DYSON_EE
Whether excited-state or spin-flip state Dyson orbitals will be calculated for
EOM-IP/EA-CCSD calculations. |
TYPE:
DEFAULT:
FALSE (the option must be specified to run this calculation) |
OPTIONS:
RECOMMENDATION:
|
Dyson orbitals can be also plotted using IANLTY = 200 and the $plots utility. Only the sizes of
the box need to be specified, followed by a line of zeros:
$plots
comment
10 -2 2
10 -2 2
10 -2 2
0 0 0 0
$plots
All Dyson orbitals on the xyz Cartesian grid will be written in the resulting plot.mo file.
For RHF(UHF) reference, the columns order in plot.mo is:
ϕlr1α (ϕlr1β) ϕrl1α (ϕrl1β) ϕlr2α (ϕlr2β) …
In addition, setting the MAKE_CUBE_FILES keyword to TRUE will
create cube files for Dyson orbitals which can be viewed with VMD or other
programs (see Section 10.9.4 for details). Other means of
visualization (e.g., with MOLDEN_FORMAT=TRUE or GUI=2)
are currently not available.
6.6.21.2 Examples
Example 6.0 Plotting grd-ex and ex-grd state Dyson orbitals for
ionization of the oxygen molecule. The target states of the cation are
2Ag and 2B2u.
$molecule
0 3
O 0.000 0.000 0.000
O 1.222 0.000 0.000
$end
$rem
jobtype sp
basis 6-31G*
correlation ccsd
eom_ip_states [1,0,0,0,0,0,1,0] Target EOM-IP states
cc_trans_prop true request transition OPDMs to be calculated
cc_do_dyson true calculate Dyson orbitals
IANLTY 200
$end
$plots
plots excited states densities and trans densities
10 -2 2
10 -2 2
10 -2 2
0 0 0 0
$plots
Example 6.0 Plotting ex-ex state Dyson orbitals between the
1st 2A1 excited state of the HO radical and the
the 1st A1 and A2 excited states of HO−
$molecule
-1 1
H 0.000 0.000 0.000
O 1.000 0.000 0.000
$end
$rem
jobtype SP
correlation CCSD
BASIS 6-31G*
eom_ip_states [1,0,0,0] states of HO radical
eom_ee_states [1,1,0,0] excited states of HO-
CC_TRANS_PROP true calculate transition properties
CC_DO_DYSON_EE true calculate Dyson orbitals for ionization from ex. states
IANLTY 200
$end
$plots
plot excited states densities and trans densities
10 -2 2
10 -2 2
10 -2 2
0 0 0 0
$plots
6.6.22 Interpretation of EOM / CI Wavefunction and Orbital Numbering
Analysis of the leading wavefunction amplitudes is always necessary for
determining the character of the state (e.g., HOMO-LUMO excitation,
open-shell diradical, etc.).
The CCMAN module print out leading EOM / CI amplitudes using its internal orbital
numbering scheme, which is printed in the beginning. The typical CCMAN
EOM-CCSD output
looks like:
Root 1 Conv-d yes Tot Ene= -113.722767530 hartree (Ex Ene 7.9548 eV),
U1^2=0.858795, U2^2=0.141205 ||Res||=4.4E-07
Right U1:
Value i -> a
0.5358 7( B2 ) B -> 17( B2 ) B
0.5358 7( B2 ) A -> 17( B2 ) A
-0.2278 7( B2 ) B -> 18( B2 ) B
-0.2278 7( B2 ) A -> 18( B2 ) A
This means that this state is derived by excitation from occupied orbital #7
(which has b2 symmetry) to virtual orbital #17 (which is also of b2 symmetry).
The two leading amplitudes correspond to β→β and α→α
excitation (the spin part is denoted by A or B). The orbital numbering for this job is defined
by the following map:
The orbitals are ordered and numbered as follows:
Alpha orbitals:
Number Energy Type Symmetry ANLMAN number Total number:
0 -20.613 AOCC A1 1A1 1
1 -11.367 AOCC A1 2A1 2
2 -1.324 AOCC A1 3A1 3
3 -0.944 AOCC A1 4A1 4
4 -0.600 AOCC A1 5A1 5
5 -0.720 AOCC B1 1B1 6
6 -0.473 AOCC B1 2B1 7
7 -0.473 AOCC B2 1B2 8
0 0.071 AVIRT A1 6A1 9
1 0.100 AVIRT A1 7A1 10
2 0.290 AVIRT A1 8A1 11
3 0.327 AVIRT A1 9A1 12
4 0.367 AVIRT A1 10A1 13
5 0.454 AVIRT A1 11A1 14
6 0.808 AVIRT A1 12A1 15
7 1.196 AVIRT A1 13A1 16
8 1.295 AVIRT A1 14A1 17
9 1.562 AVIRT A1 15A1 18
10 2.003 AVIRT A1 16A1 19
11 0.100 AVIRT B1 3B1 20
12 0.319 AVIRT B1 4B1 21
13 0.395 AVIRT B1 5B1 22
14 0.881 AVIRT B1 6B1 23
15 1.291 AVIRT B1 7B1 24
16 1.550 AVIRT B1 8B1 25
17 0.040 AVIRT B2 2B2 26
18 0.137 AVIRT B2 3B2 27
19 0.330 AVIRT B2 4B2 28
20 0.853 AVIRT B2 5B2 29
21 1.491 AVIRT B2 6B2 30
The first column is CCMAN's internal numbering (e.g., 7 and 17 from the example above).
This is followed by the orbital energy, orbital type (frozen, restricted, active,
occupied, virtual), and orbital symmetry. Note that the orbitals are blocked by
symmetries and then ordered by energy within each symmetry block,
(i.e., first all occupied a1, then all a2, etc.), and numbered starting from 0.
The occupied and virtual orbitals are numbered separately, and frozen orbitals
are excluded from CCMAN numbering.
The two last columns give numbering in terms of the final ANLMAN printout (starting from 1),
e.g., our occupied orbital #7 will be numbered as 1B2 in the
final printout.
The last column gives the absolute orbital number (all occupied and all virtuals
together, starting from 1), which is often used by external visualization routines.
CCMAN2 numbers orbitals by their energy within each irrep keeping the
same numbering for occupied and virtual orbitals. This numbering is exactly the same
as in the final printout of the SCF wavefunction analysis.
Orbital energies are printed next to the respective amplitudes.
For example, a typical CCMAN2 EOM-CCSD output
will look like that:
EOMEE-CCSD transition 2/A1
Total energy = -75.87450159 a.u. Excitation energy = 11.2971 eV.
R1^2 = 0.9396 R2^2 = 0.0604 Res^2 = 9.51e-08
Amplitude Orbitals with energies
0.6486 1 (B2) A -> 2 (B2) A
-0.5101 0.1729
0.6486 1 (B2) B -> 2 (B2) B
-0.5101 0.1729
-0.1268 3 (A1) A -> 4 (A1) A
-0.5863 0.0404
-0.1268 3 (A1) B -> 4 (A1) B
-0.5863 0.0404
which means that for this state, the leading EOM
amplitude corresponds to the transition from the first b2
orbital (orbital energy −0.5101) to the second b2 orbital
(orbital energy 0.1729).
6.7 Correlated Excited State Methods: ADC(n) Family
The ADC(n) family of correlated excited state methods is a series of
size-extensive excited state methods based on perturbation theory. Each
order n of ADC presents the excited state equivalent to the well-known
nth order Møller-Plesset perturbation theory for the ground state.
Currently, the ADC variants ADC(0), ADC(1), ADC(2)-s and ADC(2)-x are
implemented into Q-Chem.
6.7.1 The Algebraic Diagrammatic Construction (ADC) Scheme
The Algebraic Diagrammatic Construction (ADC) scheme of the polarization
propagator is an excited state method originating from Green's function
theory. It has first been derived employing the diagrammatic perturbation
expansion of the polarization propagator using the Møller-Plesset partition
of the Hamiltonian [355]. An alternative derivation is available
in terms of the intermediate state representation (ISR) [356]
which will be presented in the following.
As starting point for the derivation of ADC equations via ISR serves the
exact N electron ground state |Ψ0N > . From
|Ψ0N > a complete set of correlated excited states is
obtained by applying physical excitation operators ∧CJ.
| ⎢ ⎢
|
-
Ψ
|
N J
|
> = |
^
C
|
J
|
|Ψ0N > |
| (6.48) |
with
| ⎧ ⎨
⎩
|
^
C
|
J
| ⎫ ⎬
⎭
|
= { c†a ci; c†a c†b ci cj, i < j, a < b; …} |
| (6.49) |
Yet, the resulting excited states do not form an orthonormal basis. To
construct an orthonormal basis out of the |―ΨJN〉 the
Gram-Schmidt orthogonalization scheme is employed successively on the
excited states in the various excitation classes starting from the exact ground
state, the singly excited states, the doubly excited states etc.. This procedure
eventually yields the basis of intermediate states
{|~ΨJN〉} in which the Hamiltonian of the
system can be represented forming the hermitian ADC matrix
MIJ = < |
~
Ψ
|
N I
| ⎢ ⎢
|
|
^
H
|
− E0N | ⎢ ⎢
|
~
Ψ
|
N J
|
> |
| (6.50) |
Here, the Hamiltonian of the system is shifted by the exact ground state
energy E0N. The solution of the secular ISR equation
yields the exact excitation energies Ωn as eigenvalues. From the
eigenvectors the exact excited states in terms of the intermediate states can
be constructed as
|ΨnN > = |
∑
J
|
XnJ | ⎢ ⎢
|
~
Ψ
|
N J
|
> |
| (6.52) |
This also allows for the calculation of dipole transition moments via
Tn = < ΨnN| |
^
μ
|
|Ψ0N > = |
∑
J
|
XnJ† < |
~
Ψ
|
N J
| ⎢ ⎢
|
|
^
μ
|
|Ψ0N > , |
| (6.53) |
as well as excited state properties via
On = < ΨnN| |
^
o
|
|ΨnN > = |
∑
I,J
|
XnI† XnJ < |
~
Ψ
|
N I
| ⎢ ⎢
|
|
^
o
|
|ΨJN > , |
| (6.54) |
where On is the property associated with operator ∧o.
Up to now, the exact N-electron ground state has been employed in the
derivation of the ADC scheme, thereby resulting in exact excitation energies
and exact excited state wavefunctions. Since the exact ground state is usually
not known, a suitable approximation must be used in the derivation of the ISR
equations. An obvious choice is the nth order Møller-Plesset ground state
yielding the nth order approximation of the ADC scheme. The appropriate
ADC equations have been derived in detail up to third order in
Refs. . Due to the dependency on
the Møller-Plesset ground state the nth order ADC scheme should only
be applied to molecular systems whose ground state is well described by the
respective MP(n) method.
As in Møller-Plesset perturbation theory, the first ADC scheme which goes beyond
the non-correlated wavefunction methods in Section 6.2 is ADC(2).
ADC(2) is available in a strict and an extended variant which are
usually referred to as ADC(2)-s and ADC(2)-x, respectively.
The strict variant ADC(2)-s
scales with the 5th power of the basis set. The quality of ADC(2)-s excitation
energies and corresponding excited states is comparable to the quality of those
obtained with CIS(D) (Section 6.4) or CC2. More precisely, excited states with
mostly single excitation character are well-described by ADC(2)-s, while excited states
with double excitation character are usually found to be too high in energy. An
improved treatment of doubly excited states can be obtained by the ADC(2)-x variant
which scales as the sixth power of the basis set. The excitation energies of doubly
excited states are substantially decreased in ADC(2)-x relative to the states with
mostly single excitation character. Thus, the comparison of results from ADC(2)-s
and ADC(2)-x calculations can be used as a test for the importance of doubly
excited states in the low-energy region of the spectrum.
6.7.2 ADC Job Control
For an ADC calculation it is important to ensure that there are sufficient resources
available for the necessary integral calculations and transformations. These resources
are controlled using the $rem variables MEM_STATIC and
MEM_TOTAL. The memory used by ADC is currently 80% of the difference
MEM_TOTAL − MEM_STATIC.
To request an ADC calculation the $rem variable ADC_ORDER should be
set to 0, 1, or 2, and the number of excited states to calculate should be specified using
ADC_STATES, ADC_SINGLETS, or ADC_TRIPLETS.
In calculations on molecular systems with point-group symmetry the latter $rem
variables normally refer to the number of calculated excited states per irreducible
representation. In general, the irreducible representation determines the symmetry
of the corresponding electronic transition and not the symmetry of the excited state wavefunction.
Users can switch off this behavior by setting CC_SYMMETRY to FALSE, thus disabling any
symmetry. Alternatively, users can select the irreducible representations of the electronic transitions
for which excited states are to be calculated by defining ADC_STATE_SYM.
ADC_ORDER
Controls the order in perturbation theory of ADC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Activate ADC(0). |
1 | Activate ADC(1). |
2 | Activate ADC(2)-s or ADC(2)-x. |
RECOMMENDATION:
|
| ADC_EXTENDED
Activates the ADC(2)-x variant. This option is ignored unless
ADC_ORDER is set to 2. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Activate ADC(2)-x. |
FALSE | Do an ADC(2)-s calculation. |
RECOMMENDATION:
|
|
|
ADC_PROP_ES
Controls the calculation of excited state properties (currently only dipole moments). |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculate excited state properties. |
FALSE | Only calculate transition properties from the ground state. |
RECOMMENDATION:
|
| ADC_STATES
Controls the number of excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable to define the number of excited states in case of unrestricted or
open-shell calculations. In restricted calculations it can also be used, if the same
number of singlet and triplet states is to be requested. |
|
|
|
ADC_SINGLETS
Controls the number of singlet excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable in case of
restricted calculation. |
|
| ADC_TRIPLETS
Controls the number of triplet excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable in case of restricted calculation. |
|
|
|
CC_SYMMETRY
Activates point-group symmetry in the ADC calculation. |
TYPE:
DEFAULT:
TRUE | If the system possesses any point-group symmetry. |
OPTIONS:
TRUE | Employ point-group symmetry |
FALSE | Do not use point-group symmetry |
RECOMMENDATION:
|
| ADC_STATE_SYM
Controls the irreducible representations of the electronic transitions for
which excited states should be calculated. This option is ignored, unless point-group
symmetry is present in the system and CC_SYMMETRY is set to TRUE. |
TYPE:
DEFAULT:
0 | States of all irreducible representations are calculated |
| (equivalent to setting the $rem variable to 111...). |
OPTIONS:
i1 i2 ... iN
| A sequence of 0 and 1 in which each digit represents one |
| irreducible representation. |
| 1 activates the calculation of the respective electronic transitions. |
RECOMMENDATION:
The irreducible representations are ordered according to the standard ordering in Q-Chem.
For example, in a system with D2 symmetry ADC_STATE_SYM = 0101 would activate the
calculation of B1 and B3 excited states. |
|
|
|
ADC_NGUESS_SINGLES
Controls the number of excited state guess vectors which are single excitations. If the
number of requested excited states exceeds the total number of guess vectors (singles and
doubles), this parameter is automatically adjusted, so that the number of guess vectors
matches the number of requested excited states. |
TYPE:
DEFAULT:
Equals to the number of excited states requested. |
OPTIONS:
RECOMMENDATION:
|
| ADC_NGUESS_DOUBLES
Controls the number of excited state guess vectors which are double excitations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
ADC_DO_DIIS
Activates the use of the DIIS algorithm for the calculation of ADC(2) excited states. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Use DIIS algorithm. |
FALSE | Do diagonalization using Davidson algorithm. |
RECOMMENDATION:
|
| ADC_DIIS_START
Controls the iteration step at which DIIS is turned on. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to a large number to switch off DIIS steps. |
|
|
|
ADC_DIIS_SIZE
Controls the size of the DIIS subspace. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| ADC_DIIS_MAXITER
Controls the maximum number of DIIS iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase in case of slow convergence. |
|
|
|
ADC_DIIS_ECONV
Controls the convergence criterion for the excited state energy during DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| ADC_DIIS_RCONV
Convergence criterion for the residual vector norm of the excited state during DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
ADC_DAVIDSON_MAXSUBSPACE
Controls the maximum subspace size for the Davidson procedure. |
TYPE:
DEFAULT:
5 × the number of excited states to be calculated. |
OPTIONS:
RECOMMENDATION:
Should be at least 2−4 × the number of excited states to calculate. The larger the value the more disk space is required. |
|
| ADC_DAVIDSON_MAXITER
Controls the maximum number of iterations of the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. |
|
|
|
ADC_DAVIDSON_CONV
Controls the convergence criterion of the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. |
|
| ADC_DAVIDSON_THRESH
Controls the threshold for the norm of expansion vectors to be added
during the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. The threshold
value should always be smaller or equal to the convergence criterion
ADC_DAVIDSON_CONV. |
|
|
|
ADC_PRINT
Controls the amount of printing during an ADC calculation. |
TYPE:
DEFAULT:
1 | Basic status information and results are printed. |
OPTIONS:
0 | Quiet: almost only results are printed. |
1 | Normal: basic status information and results are printed. |
2 | Debug1: more status information, extended timing information. |
... |
RECOMMENDATION:
|
6.7.3 Examples
Example 6.0 Q-Chem input for an ADC(2)-s calculation of singlet
exited states of methane with D2 symmetry. Only excited states
having irreducible representations B1 or B2 are to be calculated.
$molecule
0 1
C
H 1 r0
H 1 r0 2 d0
H 1 r0 2 d0 3 d1
H 1 r0 2 d0 4 d1
r0 = 1.085
d0 = 109.4712206
d1 = 120.0
$end
$rem
jobtype sp
exchange hf
basis 6-31g(d,p)
mem_total 4000
mem_static 100
cc_symmetry true
cc_orbs_per_block 32
adc_order 2
adc_singlets 4
adc_nguess_singles 3
adc_nguess_doubles 3
adc_state_sym 0110
adc_davidson_conv 6
adc_davidson_thresh 7
$end
Example 6.0 Q-Chem input for an unrestricted ADC(2)-s calculation
using DIIS.
$molecule
0 2
C 0.0 0.0 -0.630969
N 0.0 0.0 0.540831
$end
$rem
jobtype sp
exchange hf
basis aug-cc-pVDZ
mem_total 4000
mem_static 100
cc_symmetry false
cc_orbs_per_block 32
adc_order 2
adc_states 6
adc_nguess_singles 6
adc_do_diis true
$end
Example 6.0 Q-Chem input for a restricted ADC(2)-x calculation of
4 singlet and 2 triplet excited states.
$molecule
0 1
O 0.000 0.000 0.000
H 0.000 0.000 0.950
H 0.896 0.000 -0.317
$end
$rem
jobtype sp
exchange hf
basis 6-31g(d,p)
threads 2
mem_total 3000
mem_static 100
cc_symmetry false
cc_orbs_per_block 32
adc_order 2
adc_extended true
adc_print 1
adc_singlets 4
adc_triplets 2
adc_nguess_singles 4
adc_nguess_doubles 6
adc_davidson_thresh 9
adc_davidson_conv 8
$end
6.8 Restricted active space spin-flip (RAS-SF) and configuration interaction (RAS-CI) methods
The restricted active space spin-flip (RAS-SF) is a special form of configuration interaction
that is capable of describing the ground and low-lying excited states
with moderate computational cost in a single-reference
formulation [360,[361,[362,[363], including
strongly correlated systems. The RAS-SF approach is essentially a
much lower computational cost alternative to Complete Active Space
SCF (CAS-SCF) methods. RAS-SF typically works by performing a full CI calculation within
an active space that is defined by the half-occupied orbitals of a
restricted open shell HF (ROHF) reference determinant. In this way the difficulties
of state-specific orbital optimization in CAS-SCF are bypassed. Single excitations
into (hole) and out of (particle) the active space provide state-specific relaxation instead.
Unlike most CI-based methods, RAS-SF is size-consistent, as well as variational, and, the
increase in computational cost with system size is modest for a fixed number of spin flips.
Beware, however, for the increase in cost as a function of the number of spin-flips is
exponential! RAS-SF has been shown to be capable of tackling multiple low-lying electronic
states in polyradicals and reliably predicting ground state
multiplicities [360,[364,[365,[366,[361,[362].
RAS-SF can also be viewed as one particular case of a more general RAS-CI family of methods.
For instance, instead of defining the actice space via spin-flipping as above,
initial orbitals of other types can be read in, and electronic
excitations calculated this way may be viewed as a RAS-EE-CI method (though size-consistency
will generally be lost).
Similar to EOM-CC approaches (see Section 6.6),
other target RAS-CI wavefunctions can be constructed starting from any
electronic configuration as the reference and using a general excitation-type
operator. For instance, one can
construct an ionizing variant that removes
an arbitrary number of particles that is RAS-nIP-CI.
An electron attaching variant is RAS-nEA-CI [363].
Q-Chem features two versions of RAS-CI code with different, complementary,
functionalities.
One code (invoked by specifying CORR=RASCI) has been
written by David Casanova [363]; below we will refer to this code as RASCI1.
The second implementation (invoked by specifying CORR=RASCI2)
is primarily by Paul Zimmerman [361]; we will refer to it as RASCI2 below.
The RASCI1 code uses an integral-driven implementation (exact integrals) and spin-adaptation
of the CI configurations which results in a smaller diagonalization dimension.
The current Q-Chem implementation of RASCI1 only allows for the calculation of systems with an even
number of electrons, with the multiplicity of each state being printed alongside the state energy.
Shared memory parallel execution decreases compute time as all the underlying integrals routines are parallelized.
The RASCI2 code includes the ability to simulate closed and open shell
systems (i.e., even and odd numbers of electrons), fast integral evaluation
using the resolution of the identity (RI) approximation, shared memory parallel operation, and analysis
of the 〈S2〉 values and natural orbitals.
The natural orbitals are stored in the QCSCRATCH directory in a folder called
"NOs" in Molden-readable format. Shared memory parallel is invoked
as described in Section 2.7.1. A RASCI2 input
requires the specification of an auxiliary basis set analogous to RI-MP2
computations (see Section 5.5.1). Otherwise, the active space as
well as hole and particle excitations are specified in the same way
as in RASCI1.
Note:
Because RASCI2 uses the RI approximation, the total energies computed with the two codes will be slightly different; however, the energy differences
between different states should closely match each other. |
6.8.1 The Restricted Active Space (RAS) Scheme
In the RAS formalism, we divide the orbital space into three subspaces
called RAS1, RAS2 and RAS3 (Fig. 6.2).
The RAS-CI states are defined by the number of orbitals and the
restrictions in each subspace.
Picture Omitted
Figure 6.2: Orbital subspaces in RAS-CI employing a ROHF triplet reference.
The single reference RAS-CI electronic wavefunctions
are obtained by applying a spin-flipping or excitation operator ∧R
on the reference determinant ϕ(0).
The ∧R operator must obey the restrictions imposed in the subspaces
RAS1, RAS2 and RAS3, and can be decomposed as:
|
^
R
|
= |
^
r
|
RAS2
|
+ |
^
r
|
h
|
+ |
^
r
|
p
|
+ |
^
r
|
hp
|
+ |
^
r
|
2h
|
+ |
^
r
|
2p
|
+... |
| (6.56) |
where ∧rRAS2 contains all possible electronic promotions within the RAS2 space,
that is, a reduced full CI, and the rest of the terms generate configurations with different number
of holes (h superindex) in RAS1 and electrons in RAS3 (p superindex).
The current implementation truncates this series up to the inclusion of hole and particle
contributions, i.e. the first three terms on the right hand side of Eq. (6.56).
6.8.2 Job Control for the RASCI1 implementation
At present the RASCI1 and RASCI2 implementations employ different keywords (which will be reconciled
in a future version). This subsection applies to RASCI1 (even electron systems, spin adapted algorithm
using exact integrals).
The use of the RAS-CI1 methodology is controlled by setting the
$rem variable CORRELATION to RASCI and EXCHANGE should be set to HF.
The RASCI1 implementation is only compatible with even numbers of electrons and restricted orbitals,
i.e., UNRESTRICTED = FALSE.
The minimum input also requires specifying the desired (non-zero) value for
RAS_ROOTS,
the number of electrons in the "active" RAS2 space
and the number of orbitals in RAS1 and RAS2 subspaces.
RAS_ROOTS
Sets the number of RAS-CI roots to be computed. |
TYPE:
DEFAULT:
OPTIONS:
n | n > 0 Compute n RAS-CI states |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_ELEC
Sets the number of electrons in RAS2 (active electrons). |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_ACT
Sets the number of orbitals in RAS2 (active orbitals). |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_OCC
Sets the number of orbitals in RAS1 |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
These are the initial doubly occupied orbitals (RAS1) before
including hole type of excitations.
The RAS1 space starts from the lowest orbital up to RAS_OCC,
i.e. no frozen orbitals option available yet. Only works with RASCI. |
|
|
|
RAS_DO_HOLE
Controls the presence of hole excitations in the RAS-CI wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Include hole configurations (RAS1 to RAS2 excitations) |
FALSE | Do not include hole configurations |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_DO_PART
Controls the presence of particle excitations in the RAS-CI wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Include particle configurations (RAS2 to RAS3 excitations) |
FALSE | Do not include particle configurations |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_AMPL_PRINT
Defines the absolute threshold (×102) for the CI amplitudes to be printed. |
TYPE:
DEFAULT:
10 | 0.1 minimum absolute amplitude |
OPTIONS:
n | User-defined integer, n ≥ 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
RAS_ACT_ORB
Sets the user-selected active orbitals (RAS2 orbitals). |
TYPE:
DEFAULT:
From RAS_OCC+1 to RAS_OCC+RAS_ACT |
OPTIONS:
[i,j,k...] | The number of orbitals must be equal to the RAS_ACT variable |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_NATORB
Controls the computation of the natural orbital occupancies. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Compute natural orbital occupancies for all states |
FALSE | Do not compute natural orbital occupancies |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_NATORB_STATE
Allows to save the natural orbitals of a RAS-CI computed state. |
TYPE:
DEFAULT:
OPTIONS:
i | Saves the natural orbitals for the i-th state |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_GUESS_CS
Controls the number of closed shell guess configurations in RAS-CI. |
TYPE:
DEFAULT:
OPTIONS:
n | Imposes to start with n closed shell guesses |
RECOMMENDATION:
Only relevant for the computation of singlet states. Only works with RASCI. |
|
|
|
RAS_SPIN_MULT
Specifies the spin multiplicity of the roots to be computed |
TYPE:
DEFAULT:
OPTIONS:
0 | Compute any spin multiplicity |
2n+1 | User-defined integer, n ≥ 0 |
RECOMMENDATION:
Only for RASCI, which at present only allows for the computation of systems with an even
number of electrons. Thus, RAS_SPIN_MULT only can take odd values. |
|
6.8.3 Job control options for the RASCI2 implementation
At present the RASCI1 and RASCI2 implementations employ different keywords (which will be reconciled
in a future version). This subsection applies to RASCI2 (even and odd electron systems, determinant-driven algorithm
using the resolution of the identity approximation).
The use of the RAS-CI2 methodology is controlled by setting the
CORRELATION = RASCI2 and EXCHANGE = HF.
The minimum input also requires specifying the desired (non-zero) value for
RAS_N_ROOTS, and
the number of active occupied and virtual orbital comprising the "active" RAS2 space.
RASCI2 calculations also require
specification of an auxiliary basis via AUX_BASIS.
RAS_N_ROOTS
Sets the number of RAS-CI roots to be computed. |
TYPE:
DEFAULT:
OPTIONS:
n | n > 0 Compute n RAS-CI states |
RECOMMENDATION:
None. Only works with RASCI2 |
|
| RAS_ACT_OCC
Sets the number of occupied orbitals to enter the RAS active space. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
None. Only works with RASCI2 |
|
|
|
RAS_ACT_VIR
Sets the number of virtual orbitals to enter the RAS active space. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
None. Only works with RASCI2. |
|
| RAS_ACT_DIFF
Sets the number of alpha vs. beta electrons |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to 1 for an odd number of electrons or a cation, -1 for an anion.
Only works with RASCI2. |
|
|
|
Other $rem variables that can be used to control the evaluation of RASCI2 calculations
are SET_ITER for the maximum number of Davidson iterations, and
N_FROZEN_CORE and N_FROZEN_VIRTUAL to exclude core and/or virtual
orbitals from the RASCI2 calculation.
6.8.4 Examples
Example 6.0 Input for a RAS-2SF-CI calculation of
three states of the DDMX tetraradical using RASCI1.
The active space (RAS2) contains
4 electrons in the 4 singly occupied orbitals in the ROHF quintet reference.
Natural orbital occupancies are requested.
$molecule
0 5
C 0.000000 0.000000 1.092150
C -1.222482 0.000000 0.303960
C -2.390248 0.000000 1.015958
H -2.344570 0.000000 2.095067
H -3.363161 0.000000 0.537932
C -1.215393 0.000000 -1.155471
H -2.150471 0.000000 -1.702536
C 0.000000 0.000000 -1.769131
C 1.215393 0.000000 -1.155471
H 2.150471 0.000000 -1.702536
C 1.222482 0.000000 0.303960
C 2.390248 0.000000 1.015958
H 2.344570 0.000000 2.095067
H 3.363161 0.000000 0.537932
$end
$rem
jobtype sp
exchange hf
correlation rasci
basis 6-31g
unrestricted false
mem_total 4000
mem_static 100
ras_roots 3
ras_act 4
ras_elec 4
ras_occ 25
ras_spin_mult 0
ras_natorb true
$end
Example 6.0 Input for a RAS-2IP-CI calculation of
triplet states of F2 molecule using the dianion
closed shell F22− as the reference determinant. RASCI1 code is used
$molecule
-2 1
F
F 1 rFF
rFF = 1.4136
$end
$rem
jobtype sp
exchange hf
correlation rasci
basis cc-pVTZ
mem_total 4000
mem_static 100
ras_roots 2
ras_act 6
ras_elec 10
ras_occ 4
ras_spin_mult 3
$end
Example 6.0 Input for a FCI/6-31G calculation of water molecule
expanding the RAS2 space to the entire molecular orbital set. RASCI code is used.
$molecule
0 1
O 0.000 0.000 0.120
H -0.762 0.000 -0.479
H 0.762 0.000 -0.479
$end
$rem
jobtype sp
exchange hf
correlation rasci
basis 6-31G
mem_total 4000
mem_static 100
ras_roots 1
ras_act 13
ras_elec 10
ras_occ 0
ras_spin_mult 1
ras_do_hole false
ras_do_part false
$end
Example 6.0 Methylene single spin-flip calculation using RASCI2
$rem
jobtype sp
exchange HF
correlation RASCI2
basis cc-pVDZ
aux_basis rimp2-cc-pVDZ
unrestricted false
RAS_ACT_OCC 1 ! # alpha electrons
RAS_ACT_VIR 1 ! # virtuals in active space
RAS_ACT_DIFF 0 ! # set to 1 for odd # of e-s
ras_n_roots 4
set_iter 25 ! number of iterations in RASCI
$end
$molecule
0 3
C 0.0000000 0.0000000 0.0000000
H -0.8611113 0.0000000 0.6986839
H 0.8611113 0.0000000 0.6986839
$end
Example 6.0 Two methylene separated by 10 Å; double spin-flip calculation
using RASCI2. Note that the 〈S2〉 values for this case will not be uniquely
defined at the triply-degenerate ground state.
$rem
exchange HF
correlation RASCI2
basis cc-pVDZ
aux_basis rimp2-cc-pVDZ
RAS_ACT_OCC 2 ! # alpha electrons
RAS_ACT_VIR 2 ! # virtuals in active space
RAS_ACT_DIFF 0 ! # set to 1 for odd # of e-s
unrestricted false
jobtype sp
ras_n_roots 8
set_iter 25
$end
$molecule
0 5
C 0.0000000 0.0000000 0.0000000
H -0.8611113 0.0000000 0.6986839
H 0.8611113 0.0000000 0.6986839
C 0.0000000 10.0000000 0.0000000
H -0.8611113 10.0000000 0.6986839
H 0.8611113 10.0000000 0.6986839
$end
Example 6.0 RASCI2 calculation of the nitrogen cation using
double spin-flip.
$rem
exchange HF
correlation RASCI2
jobtype sp
basis 6-31G*
aux_basis rimp2-VDZ
RAS_ACT_OCC 3 ! # alpha electrons
RAS_ACT_VIR 3 ! # virtuals in active space
RAS_ACT_DIFF 1 ! # for odd # e-s, cation
unrestricted false
n_frozen_core 2
n_frozen_virtual 2
ras_n_roots 8
set_iter 25
$end
$molecule
1 6
N 0.0000000 0.0000000 0.0000000
N 0.0000000 0.0000000 4.5
$end
6.9 How to Compute Ionization Energies of Core Electrons and Excited
States Involving Excitations of Core Electrons
In experiments using high-energy radiation (such as X-ray spectroscopy)
core electrons can be ionized or excited to low-lying virtual orbitals.
There are two ways to compute ionization energies of core electrons in Q-Chem.
The first approach is a simple energy difference calculation in which core
ionization is computed from energy differences computed for the neutral and
core-ionized state. It is illustrated by this example:
Example 6.0 Q-Chem input for
calculating chemical shift for 1s-level of methane (CH4).
The first job is just an SCF calculation to obtain the orbitals and CCSD energy of the
neutral. The second job
solves the HF and CCSD equations for the core-ionized state.
$molecule
0,1
C 0.000000 0.000000 0.000000
H 0.631339 0.631339 0.631339
H -0.631339 -0.631339 0.631339
H -0.631339 0.631339 -0.631339
H 0.631339 -0.631339 -0.631339
$end
$rem
JOB_TYPE SP
exchange HF
CORRELATION CCSD
basis 6-31G*
MAX_CIS_CYCLES = 100
$end
@@@
$molecule
+1,2
C 0.000000 0.000000 0.000000
H 0.631339 0.631339 0.631339
H -0.631339 -0.631339 0.631339
H -0.631339 0.631339 -0.631339
H 0.631339 -0.631339 -0.631339
$end
$rem
JOB_TYPE SP
UNRESTRICTED TRUE
exchange HF
basis 6-31G*
MAX_CIS_CYCLES = 100
SCF_GUESS read Read MOs from previous job and use occupied as specified below
CORRELATION CCSD
MOM_START 1 Do not reorder orbitals in SCF procedure!
$end
$occupied
1 2 3 4 5
2 3 4 5
$end
In this job, we first compute the HF and CCSD energies of neutral CH4:
ESCF=−40.1949062375 and ECCSD=−40.35748087 (HF orbital energy
of the neutral gives Koopmans IE, which is
11.210 hartree = 305.03 eV). In the second job, we do the same for core-ionized CH4.
To obtain the desired SCF solution, MOM_START option and $occupied
keyword are
used. The resulting energies are ESCF=−29.4656758483 (〈S2〉 = 0.7730) and
ECCSD=−29.64793957. Thus, ∆ECCSD=(40.357481−29.647940)=10.709
hartree = 291.42 eV.
This approach can be further extended to obtain multiple excited states
involving core electrons by performing CIS, TDDFT, or EOM-EE calculations.
The second approach is illustrated by the following input:
Example 6.0 Q-Chem input for
calculating chemical shift for 1s-level of methane (CH4) using EOM-IP.
Here we solve SCF as usual, then reorder the MOs such that the core orbital
becomes the "HOMO", then solve the CCSD and EOM-IP equations with all valence orbitals
frozen and the core orbital being active.
$molecule
0,1
C 0.000000 0.000000 0.000000
H 0.631339 0.631339 0.631339
H -0.631339 -0.631339 0.631339
H -0.631339 0.631339 -0.631339
H 0.631339 -0.631339 -0.631339
$end
$rem
JOB_TYPE SP
exchange HF
basis 6-31G*
MAX_CIS_CYCLES = 100
CORRELATION CCSD
CCMAN2 = false
N_FROZEN_CORE 4 Freeze all valence orbitals
EOM_IP_STATES [1,0,0,0] Find one EOM_IP state
$end
$reorder_mo
5 2 3 4 1
5 2 3 4 1
$end
Here we use EOM-IP to compute core-ionized states. Since core states are very
high in energy, we use "frozen
core" trick to eliminate valence ionized states from the calculation. That is,
we reorder MOs such that our core is the last occupied orbital and then
freeze all the rest. The so computed EOM-IP energy is 245.57 eV. From the
EOM-IP amplitude, we note that this state of a Koopmans character (dominated
by single core ionization); thus, canonical HF MOs provide good representation
of the correlated Dyson orbital. The same strategy can be used to compute
core-excited states.
The accuracy of both calculations can be improved using triples corrections,
e.g., CCSD(T) and EOM-IP-CCSD(dT). It is also recommend using a better basis that
has more core functions.
6.9.1 Calculations of States Involving Core Electron Excitation/Ionization
with DFT and TDDFT
TDDFT is not suited to describe the
extended X-ray absorption fine structure (EXAFS) region, wherein
the core electron is ejected and scattered by the neighboring atoms.
Core-excitation energies computed with TDDFT with standard hybrid
functionals are many electron volts too low compared with experiment.
Exchange-correlation functionals specifically designed to treat core
excitations are available in Q-Chem. These short-range corrected (SRC) functionals
are a modification of the more familiar long-range corrected functionals
(discussed in Section 4.3.4). However, in SRC-DFT
the short-range component of the Coulomb operator utilizes predominantly Hartree-Fock exchange,
while the mid to long-range component is primarily treated with standard DFT exchange.
These functionals can be invoked by using the SRC_DFT rem. In addition, a number
of parameters (OMEGA, OMEGA2, HF_LR, HF_SR)
that control the shape of the short and long-range Hartree-Fock components
need to be specified.
Full details of these functionals and appropriate values for the
parameters can be found in Refs. .
An example of how to use these functionals is given below.
For the K-shell of heavy elements (2nd row of the periodic table) relativistic effects
become increasing important and a further correction for these effects is required.
Also calculations for L-shell excitations are
complicated by core-hole spin orbit coupling.
6.10 Visualization of Excited States
As methods for ab initio calculations of excited states are becoming
increasingly more routine, questions arise concerning how
best to extract chemical meaning
from such calculations. Recently, several new methods of analyzing molecular
excited states have been proposed, including attachment / detachment density
analysis [295] and natural transition orbitals [369].
This section describes the theoretical background behind these methods,
while details of the input for creating data suitable for plotting
these quantities is described separately in Chapter 10.
6.10.1 Attachment / Detachment Density Analysis
Consider the one-particle density matrices of the initial and final states of
interest, P1 and P2 respectively. Assuming that each state is
represented in a finite basis of spin-orbitals, such as the molecular orbital
basis, and each state is at the same geometry. Subtracting these matrices
yields the difference density
Now, the eigenvectors of the one-particle density matrix P describing a
single state are termed the natural orbitals, and provide the best orbital
description that is possible for the state, in that a CI expansion using the
natural orbitals as the single-particle basis is the most compact. The basis
of the attachment / detachment analysis is to consider what could be termed
natural orbitals of the electronic transition and their occupation numbers
(associated eigenvalues). These are defined as the eigenvectors U
defined by
The sum of the occupation numbers δp of these orbitals is then
where n is the net gain or loss of electrons in the transition. The net gain
in an electronic transition which does not involve ionization or electron
attachment will obviously be zero.
The detachment density
is defined as the sum of all natural orbitals of the difference density with
negative occupation numbers, weighted by the absolute value of their
occupations where d is a diagonal matrix with elements
The detachment density corresponds to the electron density associated with
single particle levels vacated in an electronic transition or hole density.
The attachment density
is defined as the sum of all natural orbitals of the difference density with
positive occupation numbers where a is a diagonal matrix with elements
The attachment density corresponds to the electron density associated with
the single particle levels occupied in the transition or particle density.
The difference between the attachment and detachment densities yields the
original difference density matrix
6.10.2 Natural Transition Orbitals
In certain situations, even the attachment/detachment densities may be
difficult to analyze. An important class of examples are systems with
multiple chromophores, which may support exciton states consisting of linear
combinations of localized excitations. For such states, both the
attachment and the detachment density are highly delocalized and
occupy basically the same region of space [101]. Lack of
phase information makes the
attachment / detachment densities difficult to analyze, while strong mixing
of the canonical MOs means that excitonic states are also difficult to
characterize in terms of MOs.
Analysis of these and other excited states is greatly simplified by
constructing Natural Transition Orbitals (NTOs) for the excited states.
(The basic idea behind NTOs is rather
old [370], and has been rediscovered several times [369,[371];
the term "natural transition orbitals" was coined in Ref. .) Let
T denote the transition density matrix from a CIS, RPA, or TDDFT
calculation. The dimension of this matrix is O×V, where O and V denote the
number of occupied and virtual MOs, respectively. The NTOs are defined by
transformations U and V obtained by singular
value decomposition (SVD) of the matrix T, i.e. [371]
The matrices U and V are unitary and
Λ is diagonal, with the latter containing at most O non-zero elements.
The matrix U is a unitary transformation from the canonical occupied MOs
to a set of NTOs that together represent the "hole"
orbital that is left by the excited electron, while V transforms the
canonical virtual MOs into a set of NTOs representing the excited electron.
(Equivalently, the "holes" are the eigenvectors
of the O×O matrix TT† and the particles are
eigenvectors of the V×V matrix T†T [369].)
These "hole" and "particle" NTOs come in pairs, and their relative importance in
describing the excitation is governed by the diagonal elements of Λ, which
are excitation amplitudes in the NTO basis. By virtue of the SVD in
Eq. , any excited state may be represented using at most O
excitation amplitudes and corresponding hole / particle NTO pairs.
[The discussion here assumes that V ≥ O, which is typically the case except possibly
in minimal basis sets. Although it is possible to use the transpose of
Eq. to obtain NTOs when V < O, this has not been implemented in
Q-Chem due to its limited domain of applicability.]
The SVD generalizes the concept of matrix
diagonalization to the case of rectangular matrices,
and therefore reduces as much as possible
the number of non-zero outer products needed for an exact representation of
T. In this sense, the NTOs represent the best possible particle / hole
picture of an excited state. The detachment density
is recovered as the sum of the squares of the "hole" NTOs, while the attachment
density is precisely the sum of the squares of the "particle" NTOs. Unlike the
attachment/detachment densities, however, NTOs
preserve phase information, which can be very helpful in characterizing the
diabatic character (e.g., ππ∗ or nπ∗) of excited
states in complex systems. Even when there is more than one significant NTO
amplitude, as in systems of electronically-coupled chromophores [101],
the NTOs still represent a significant compression of information,
as compared to the canonical MO basis.
NTOs are available within Q-Chem for CIS, RPA, and TDDFT excited states.
The simplest way to
visualize the NTOs is to generate them in a format suitable for viewing
with the freely-available MolDen or MacMolPlt programs, as
described in Chapter 10.
Chapter 7 Basis Sets
7.1 Introduction
A basis set is a set of functions combined linearly to model molecular
orbitals. Basis functions can be considered as representing the atomic orbitals
of the atoms and are introduced in quantum chemical calculations because the
equations defining the molecular orbitals are otherwise very difficult to
solve.
Many standard basis sets have been carefully optimized and tested over the
years. In principle, a user would employ the largest basis set available in
order to model molecular orbitals as accurately as possible. In practice, the
computational cost grows rapidly with the size of the basis set so a compromise
must be sought between accuracy and cost. If this is systematically pursued, it
leads to a "theoretical model chemistry" [6], that is, a
well-defined energy procedure (e.g., Hartree-Fock) in combination with a
well-defined basis set.
Basis sets have been constructed from Slater, Gaussian, plane wave and delta
functions. Slater functions were initially employed because they are considered
"natural" and have the correct behavior at the origin and in the asymptotic
regions. However, the two-electron repulsion integrals (ERIs) encountered when
using Slater basis functions are expensive and difficult to evaluate. Delta
functions are used in several quantum chemistry programs. However, while codes
incorporating delta functions are simple, thousands of functions are required
to achieve accurate results, even for small molecules. Plane waves are widely
used and highly efficient for calculations on periodic systems, but are not so
convenient or natural for molecular calculations.
The most important basis sets are contracted sets of atom-centered Gaussian
functions. The number of basis functions used depends on the number of core and
valence atomic orbitals, and whether the atom is light (H or He) or heavy
(everything else). Contracted basis sets have been shown to be computationally
efficient and to have the ability to yield chemical accuracy (see Appendix B).
The Q-Chem program has been optimized to exploit basis sets of
the contracted Gaussian function type and has a large number of built-in
standard basis sets (developed by Dunning and Pople, among others) which the
user can access quickly and easily.
The selection of a basis set for quantum chemical calculations is very
important. It is sometimes possible to use small basis sets to obtain good
chemical accuracy, but calculations can often be significantly improved by the
addition of diffuse and polarization functions. Consult the literature and
review articles [6,[373,[374,[375,[8]
to aid your selection and see the section "Further Reading" at the end of
this chapter.
7.2 Built-In Basis Sets
Q-Chem is equipped with many standard basis sets [376], and allows
the user to specify the required basis set by its standard symbolic
representation. The available built-in basis sets are of four types:
- Pople basis sets
- Dunning basis sets
- Correlation consistent Dunning basis sets
- Ahlrichs basis sets
In addition, Q-Chem supports the following features:
- Extra diffuse functions available for high quality excited state
calculations.
- Standard polarization functions.
- Basis sets are requested by symbolic representation.
- s, p, sp, d, f and g angular momentum types of basis functions.
- Maximum number of shells per atom is 100.
- Pure and Cartesian basis functions.
- Mixed basis sets (see section 7.5).
- Basis set superposition error (BSSE) corrections.
The following $rem keyword controls the basis set:
BASIS
Sets the basis set to be used |
TYPE:
DEFAULT:
OPTIONS:
General, Gen | User-defined. See section below |
Symbol | Use standard basis sets as in the table below |
Mixed | Use a combination of different basis sets |
RECOMMENDATION:
Consult literature and reviews to aid your selection. |
|
7.3 Basis Set Symbolic Representation
Examples are given in the tables below and follow the standard format generally
adopted for specifying basis sets. The single exception applies to additional
diffuse functions. These are best inserted in a similar manner to the
polarization functions; in parentheses with the light atom designation
following heavy atom designation. (i.e., heavy, light). Use
a period (.) as a place-holder (see examples).
| j | k | l | m | n | |
STO−j(k+,l+)G(m,n) | 2,3,6 | a | b | d | p |
j−21(k+,l+)G(m,n) | 3 | a | b | 2d | 2p |
j−31(k+,l+)G(m,n) | 4,6 | a | b | 3d | 3p |
j−311(k+,l+)G(m,n) | 6 | a | b | df,2df,3df | pd,2pd,3pd |
|
Table 7.1: Summary of Pople type basis sets available in the Q-Chem program.
m and n refer to the polarization functions on heavy and light atoms
respectively. ak is the number of sets of diffuse functions on heavy
bl is the number of sets of diffuse functions on light atoms
Symbolic Name | Atoms Supported | |
STO-2G | H, He, Li→Ne, Na→Ar, K, Ca, Sr |
STO-3G | H, He, Li→Ne, Na→Ar, K→Kr, Rb→Sb |
STO-6G | H, He, Li→Ne, Na→Ar, K→Kr |
3-21G | H, He, Li→Ne, Na→Ar, K→Kr, Rb→Xe, Cs |
4-31G | H, He, Li→Ne, P→Cl |
6-31G | H, He, Li→Ne, Na→Ar, K→Zn |
6-311G | H, He, Li→Ne, Na→Ar, Ga→Kr |
G3LARGE | H, He, Li→Ne, Na→Ar, K→Kr |
G3MP2LARGE | H, He, Li→Ne, Na→Ar, Ga→Kr |
|
Table 7.2: Atoms supported for Pople basis sets available in Q-Chem (see the
Table below for specific examples).
Symbolic Name | Atoms Supported | |
3-21G | H, He, Li→Ne, Na→Ar, K→Kr, Rb→Xe, Cs |
3-21+G | H, He, Na→Cl, Na→Ar, K, Ca, Ga→Kr |
3-21G* | H, He, Na→Cl |
6-31G | H, He, Li→Ne, Na→Ar, K→Zn, Ga→Kr |
6-31+G | H, He, Li→Ne, Na→Ar, Ga→Kr |
6-31G* | H, He, Li→Ne, Na→Ar, K→Zn, Ga→Kr |
6-31G(d,p) | H, He, Li→Ne, Na→Ar, K→Zn, Ga→Kr |
6-31G(.,+)G | H, He, Li→Ne, Na→Ar, Ga→Kr |
6-31+G* | H, He, Li→Ne, Na→Ar, Ga→Kr |
6-311G | H, He, Li→Ne, Na→Ar, Ga→Kr |
6-311+G | H, He, Li→Ne, Na→Ar |
6-311G* | H, He, Li→Ne, Na→Ar, Ga→Kr |
6-311G(d,p) | H, He, Li→Ne, Na→Ar, Ga→Kr |
G3LARGE | H, He, Li→Ne, Na→Ar, K→Kr |
G3MP2LARGE | H, He, Li→Ne, Na→Ar, Ga→Kr |
|
Table 7.3: Examples of extended Pople basis sets.
| SV(k+,l+)(md,np), DZ(k+,l+)(md,np), TZ(k+,l+)(md,np) | |
k | # sets of heavy atom diffuse functions |
l | # sets of light atom diffuse functions |
m | # sets of d functions on heavy atoms |
n | # sets of p functions on light atoms |
Table 7.4: Summary of Dunning-type basis sets available in the Q-Chem program.
Symbolic Name | Atoms Supported | |
SV | H, Li→Ne |
DZ | H, Li→Ne, Al→Cl |
TZ | H, Li→Ne |
|
Table 7.5: Atoms supported for old Dunning basis sets available in Q-Chem.
Symbolic Name | Atoms Supported | |
SV | H, Li→Ne |
SV* | H, B→Ne |
SV(d,p) | H, B→Ne |
DZ | H, Li→Ne, Al→ Cl |
DZ+ | H, B→Ne |
DZ++ | H, B→Ne |
DZ* | H, Li→Ne |
DZ** | H, Li→Ne |
DZ(d,p) | H, Li→Ne |
TZ | H, Li→ Ne |
TZ+ | H, Li→ Ne |
TZ++ | H, Li→ Ne |
TZ* | H, Li→ Ne |
TZ** | H, Li→ Ne |
TZ(d,p) | H, Li→ Ne |
|
Table 7.6: Examples of extended Dunning basis sets.
Symbolic Name | Atoms Supported | |
cc-pVDZ | H, He, B→Ne, Al→Ar, Ga→Kr |
cc-pVTZ | H, He, B→Ne, Al→Ar, Ga→Kr |
cc-pVQZ | H, He, B→Ne, Al→Ar, Ga→Kr |
cc-pCVDZ | B→Ne |
cc-pCVTZ | B→Ne |
cc-pCVQZ | B→Ne |
aug-cc-pVDZ | H, He, B→Ne, Al→Ar, Ga→Kr |
aug-cc-pVTZ | H, He, B→Ne, Al→Ar, Ga→Kr |
aug-cc-pVQZ | H, He, B→Ne, Al→Ar, Ga→Kr |
aug-cc-pCVDZ | B→F |
aug-cc-pCVTZ | B→Ne |
aug-cc-pCVQZ | B→Ne |
|
Table 7.7: Atoms supported Dunning correlation-consistent basis sets available in
Q-Chem.
Symbolic Name | Atoms Supported | |
TZV | Li→Kr |
VDZ | H→Kr |
VTZ | H→Kr |
Table 7.8: Atoms supported for Ahlrichs basis sets available in Q-Chem
7.3.1 Customization
Q-Chem offers a number of standard and special customization features. One
of the most important is that of supplying additional diffuse functions.
Diffuse functions are often important for studying anions and excited states of
molecules, and for the latter several sets of additional diffuse functions may
be required. These extra diffuse functions can be generated from the standard
diffuse functions by applying a scaling factor to the exponent of the original
diffuse function. This yields a geometric series of exponents for the diffuse
functions which includes the original standard functions along with more
diffuse functions.
When using very large basis sets, especially those that include many diffuse
functions, or if the system being studied is very large, linear dependence in
the basis set may arise. This results in an over-complete description of the
space spanned by the basis functions, and can cause a loss of uniqueness in the
molecular orbital coefficients. Consequently, the SCF may be slow to converge
or behave erratically. Q-Chem will automatically check for linear dependence
in the basis set, and will project out the near-degeneracies, if they exist.
This will result in there being slightly fewer molecular orbitals than there are
basis functions. Q-Chem checks for linear-dependence by considering the
eigenvalues of the overlap matrix. Very small eigenvalues are an indication
that the basis set is close to being linearly dependent. The size at which the
eigenvalues are considered to be too small is governed by the $rem variable
BASIS_LIN_DEP_THRESH. By default this is set to 6, corresponding to
a threshold of 10−6. This has been found to give reliable results,
however, if you have a poorly behaved SCF, and you suspect there maybe linear
dependence in you basis, the threshold should be increased.
PRINT_GENERAL_BASIS
Controls print out of built in basis sets in input format |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Print out standard basis set information |
FALSE | Do not print out standard basis set information |
RECOMMENDATION:
Useful for modification of standard basis sets. |
|
| BASIS_LIN_DEP_THRESH
Sets the threshold for determining linear dependence in the basis set |
TYPE:
DEFAULT:
6 | Corresponding to a threshold of 10−6 |
OPTIONS:
n | Sets the threshold to 10−n |
RECOMMENDATION:
Set to 5 or smaller if you have a poorly behaved SCF and you suspect linear
dependence in you basis set. Lower values (larger thresholds) may affect the
accuracy of the calculation. |
|
|
|
7.4 User-Defined Basis Sets ($basis)
7.4.1 Introduction
Users may, on occasion, prefer to use non-standard basis, and it is possible
to declare user-defined basis sets in Q-Chem input (see Chapter
3 on Q-Chem inputs). The format for inserting a non-standard
user-defined basis set is both logical and flexible, and is described in
detail in the job control section below.
Note that the SAD guess is not currently supported with non-standard or
user-defined basis sets. The simplest alternative is to specify the
GWH or CORE options for SCF_GUESS, but these are
relatively ineffective other than for small basis sets. The recommended
alternative is to employ basis set projection by specifying a standard basis
set for the BASIS2 keyword. See the section in Chapter 4
on initial guesses for more information.
7.4.2 Job Control
In order to use a user-defined basis set the BASIS $rem must be
set to GENERAL or GEN.
When using a non-standard basis set which incorporates d or higher angular
momentum basis functions, the $rem variable PURECART needs to be
initiated. This $rem variable indicates to the Q-Chem program how to handle
the angular form of the basis functions. As indicated above, each integer
represents an angular momentum type which can be defined as either pure (1) or
Cartesian (2). For example, 111 would specify all g, f and d basis
functions as being in the pure form. 121 would indicate g- and d-
functions are pure and f-functions Cartesian.
PURECART
TYPE:
Controls the use of pure (spherical harmonic) or Cartesian angular forms |
DEFAULT:
2111 | Cartesian h-functions and pure g,f,d functions |
OPTIONS:
hgfd | Use 1 for pure and 2 for Cartesian. |
RECOMMENDATION:
This is pre-defined for all standard basis sets |
|
In standard basis sets all functions are pure, except for the d functions
in n-21G-type bases (e.g., 3-21G) and n-31G bases (e.g., 6-31G, 6-31G*,6-31+G*, …).
In particular, the 6-311G series uses pure functions for both d and f.
7.4.3 Format for User-Defined Basis Sets
The format for the user-defined basis section is as follows:
$basis | | | | | |
| X | 0 | | | |
| L | K | scale | | |
| α1 | C1Lmin | C1Lmin+1 | … | C1Lmax |
| α2 | C2Lmin | C2Lmin+1 | … | C2Lmax |
| : | : | : | ··· | : |
| αK | CKLmin | CKLmin+1 | … | CKLmax |
**** | | | | | |
$end | | | | | |
where
X | Atomic symbol of the atom (atomic number not accepted) |
L | Angular momentum symbol (S, P, SP, D, F, G) |
K | Degree of contraction of the shell (integer) |
scale | Scaling to be applied to exponents (default is 1.00) |
ai | Gaussian primitive exponent (positive real number) |
CiL | Contraction coefficient for each angular momentum (non-zero real numbers). |
Atoms are terminated with **** and the complete basis set is
terminated with the $end keyword terminator. No blank lines can be
incorporated within the general basis set input. Note that more than one
contraction coefficient per line is one required for compound shells like SP.
As with all Q-Chem input deck information, all input is case-insensitive.
7.4.4 Example
Example 7.0 Example of adding a user-defined non-standard basis set. Note
that since d, f and g functions are incorporated, the $rem variable
PURECART must be set. Note the use of BASIS2 for the initial
guess.
$molecule
0 1
O
H O oh
H O oh 2 hoh
oh = 1.2
hoh = 110.0
$end
$rem
EXCHANGE hf
BASIS gen user-defined general basis
BASIS2 sto-3g sto-3g orbitals as initial guess
PURECART 112 Cartesian d functions, pure f and g
$end
$basis
H 0
S 2 1.00
1.30976 0.430129
0.233136 0.678914
****
O 0
S 2 1.00
49.9810 0.430129
8.89659 0.678914
SP 2 1.00
1.94524 0.0494720 0.511541
0.493363 0.963782 0.612820
D 1 1.00
0.39000 1.000000
F 1 1.00
4.10000 1.000000
G 1 1.00
3.35000 1.000000
****
$end
7.5 Mixed Basis Sets
In addition to defining a custom basis set, it is also possible to specify
different standard basis sets for different atoms. For example, in a large
alkene molecule the hydrogen atoms could be modeled by the STO-3G basis, while
the carbon atoms have the larger 6-31G(d) basis. This can be specified within
the $basis block using the more familiar basis set labels.
Note:
(1) It is not possible to augment a standard basis set in this way; the whole
basis needs to be inserted as for a user-defined basis (angular
momentum, exponents, contraction coefficients) and additional functions
added. Standard basis set exponents and coefficients can be easily
obtained by setting the PRINT_GENERAL_BASIS $rem variable to
TRUE.
(2) The PURECART flag must be set for all general basis
input containing d angular momentum or higher functions, regardless of
whether standard basis sets are entered in this non-standard manner. |
The user can also specify different basis sets for atoms of the same type, but
in different parts of the molecule. This allows a larger basis set to be used
for the active region of a system, and a smaller basis set to be used in the
less important regions. To enable this the BASIS keyword must be set
to MIXED and a $basis section included in the input deck that gives a
complete specification of the basis sets to be used. The format is exactly the
same as for the user-defined basis, except that the atom number (as ordered in
the $molecule section) must be specified in the field after the atomic
symbol. A basis set must be specified for every atom in the input, even if the
same basis set is to be used for all atoms of a particular element. Custom
basis sets can be entered, and the shorthand labeling of basis sets is also
supported.
The use of different basis sets for a particular element means the global
potential energy surface is no longer unique. The user should exercise caution
when using this feature of mixed basis sets, especially during geometry
optimizations and transition state searches.
7.5.1 Examples
Example 7.0 Example of adding a user defined non-standard basis set. The
user is able to specify different standard basis sets for different atoms.
$molecule
0 1
O
H O oh
H O oh 2 hoh
oh = 1.2
hoh = 110.0
$end
$rem
EXCHANGE hf
BASIS General user-defined general basis
PURECART 2 Cartesian D functions
BASIS2 sto-3g use STO-3G as initial guess
$end
$basis
H 0
6-31G
****
O 0
6-311G(d)
****
$end
Example 7.0 Example of using a mixed basis set for methanol. The user is
able to specify different standard basis sets for some atoms and supply
user-defined exponents and contraction coefficients for others. This might be
particularly useful in cases where the user has constructed exponents and
contraction coefficients for atoms not defined in a standard basis set so that
only the non-defined atoms need have the exponents and contraction
coefficients entered. Note that a basis set has to be specified for every atom
in the molecule, even if the same basis is to be used on an atom type. Note
also that the dummy atom is not counted.
$molecule
0 1
C
O C rco
H1 C rch1 O h1co
x C 1.0 O xcol h1 180.0
H2 C rch2 x h2cx h1 90.0
H3 C rch2 x h2cx h1 -90.0
H4 O roh C hoc h1 180.0
rco = 1.421
rch1 = 1.094
rch2 = 1.094
roh = 0.963
h1co = 107.2
xco = 129.9
h2cx = 54.25
hoc = 108.0
$end
$rem
exchange hf
basis mixed user-defined mixed basis
$end
$basis
C 1
3-21G
****
O 2
S 3 1.00
3.22037000E+02 5.92394000E-02
4.84308000E+01 3.51500000E-01
1.04206000E+01 7.07658000E-01
SP 2 1.00
7.40294000E+00 -4.04453000E-01 2.44586000E-01
1.57620000E+00 1.22156000E+00 8.53955000E-01
SP 1 1.00
3.73684000E-01 1.00000000E+00 1.00000000E+00
SP 1 1.00
8.45000000E-02 1.00000000E+00 1.00000000E+00
****
H 3
6-31(+,+)G(d,p)
****
H 4
sto-3g
****
H 5
sto-3g
****
H 6
sto-3g
****
$end
7.6 Dual basis sets
There are several types of calculation that can be performed within Q-Chem using two
atomic orbital basis sets instead of just one as we have been assuming in this chapter so far.
Such calculations are said to involve dual basis sets. Typically iterations are performed
in a smaller, primary, basis, which is specified by the $rem keyword BASIS2. Examples
of calculations that can be performed using dual basis sets include:
- An improved initial guess for an SCF calculation in the large basis. See Section 4.5.5.
- Dual basis self-consistent field calculations (Hartree-Fock and density functional theory). See discussion
in Section 4.7.
- Density functional perturbative corrections by "triple jumping". See Section 4.8.2.
- Dual basis MP2 calculations. See discussion in Section 5.5.1.
BASIS2
Defines the (small) second basis set. |
TYPE:
DEFAULT:
No default for the second basis set. |
OPTIONS:
Symbol | Use standard basis sets as for BASIS. |
BASIS2_GEN | General BASIS2 |
BASIS2_MIXED | Mixed BASIS2 |
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using
a basis larger than a minimal basis when BASIS2 is used for initial guess purposes.
Larger, standardized BASIS2 options are available for dual-basis calculations
as discussed in Section 4.7 and summarized in Table 4.7.2. |
|
In addition to built-in basis sets for BASIS2, it is also possible to enter user-defined
second basis sets using an additional $basis2 input section, whose syntax generally follows
the $basis input section documented above in Section 7.4.
7.7 Auxiliary basis sets for RI / density fitting
Whilst atomic orbital standard basis sets are used to expand one-electron functions such
as molecular orbitals, auxiliary basis sets are also used in many Q-Chem jobs to
efficiently approximate products of one-electron functions, such as arise in electron
correlation methods.
For a molecule of fixed size, increasing the number of basis functions
per atom, n, leads to O(n4) growth in the number of significant
four-center two-electron integrals, since the number of non-negligible
product charge distributions, |μν〉, grows as O(n2). As a result,
the use of large (high-quality) basis expansions is computationally costly.
Perhaps the most practical way around this "basis set quality" bottleneck is
the use of auxiliary basis expansions [224,[225,[226].
The ability to use auxiliary
basis sets to accelerate a variety of electron correlation methods, including
both energies and analytical gradients, is a major feature of
Q-Chem.
The auxiliary basis {|K〉} is used to approximate products of Gaussian
basis functions:
|μν〉 ≈ | |
~
μν
|
〉 = |
∑
K
|
|K〉CμνK |
| (7.1) |
Auxiliary basis expansions were introduced long ago, and are now widely
recognized as an effective and powerful approach, which is sometimes
synonymously called resolution of the identity (RI) or density fitting (DF).
When using auxiliary basis expansions, the rate of growth of computational cost
of large-scale electronic structure calculations with n is reduced to
approximately n3.
If n is fixed and molecule size increases, auxiliary basis expansions reduce
the pre-factor associated with the computation, while not altering the
scaling. The important point is that the pre-factor can be reduced by 5 or 10
times or more. Such large speedups are possible because the number of
auxiliary functions required to obtain reasonable accuracy, X, has been shown
to be only about 3 or 4 times larger than N.
The auxiliary basis expansion coefficients, C, are determined by
minimizing the deviation between the fitted distribution and the actual
distribution, 〈μν−~μν | μν−~μν〉, which leads to the following set of linear equations:
|
∑
L
|
〈 K| L 〉 CμνL = 〈 K| μν 〉 |
| (7.2) |
Evidently solution of the fit equations requires only two- and three-center
integrals, and as a result the (four-center) two-electron integrals can be
approximated as the following optimal expression for a given choice of
auxiliary basis set:
〈μν|λσ〉 ≈ 〈 |
~
μν
|
| |
~
λσ
|
〉 = |
∑
| K,LCμL〈L|K 〉CλσK |
| (7.3) |
In the limit where the auxiliary basis is complete (i.e. all products of
AOs are included), the fitting procedure described above will be exact.
However, the auxiliary basis is invariably incomplete (as mentioned above,
X ≈ 3N) because this is essential for obtaining increased computational
efficiency.
More details on Q-Chem's use of RI methods is given in Section 5.5 on
RI-MP2 and related methods, Section 5.14 on pairing methods,
Section 5.7.5 on coupled cluster methods,
Section 4.4.8 on DFT methods,
and Section 6.8 on restricted active space methods.
In the remainder of this section we focus on documenting the input associated with
the auxiliary basis itself.
Q-Chem contains a variety of built-in auxiliary basis sets, that can be specified
by the $rem keyword aux_basis.
AUX_BASIS
Sets the auxiliary basis set to be used |
TYPE:
DEFAULT:
No default auxiliary basis set |
OPTIONS:
General, Gen | User-defined. As for BASIS |
Symbol | Use standard auxiliary basis sets as in the table below |
Mixed | Use a combination of different basis sets |
RECOMMENDATION:
Consult literature and EMSL Basis Set Exchange to aid your selection. |
|
Symbolic Name | Atoms Supported | |
RIMP2-VDZ | H, He, Li→Ne, Na→Ar, K→Br |
RIMP2-TZVPP | H, He, Li→Ne, Na→Ar, Ga→Kr |
RIMP2-cc-pVDZ | H, He, Li→Ne, Na→Ar, Ga→Kr |
RIMP2-cc-pVTZ | H, He, Li→Ne, Na→Ar, Ga→Kr |
RIMP2-cc-pVQZ | H, He, Li→Ne, Na→Ar, Ga→Kr |
RIMP2-aug-cc-pVDZ | H, He, B→Ne, Al→Ar, Ga→Kr |
RIMP2-aug-cc-pVTZ | H, He, B→Ne, Al→Ar, Ga→Kr |
RIMP2-aug-cc-pVQZ | H, He, B→Ne, Al→Ar, Ga→Kr |
Table 7.9: Built-in auxiliary basis sets available in Q-Chem for electron correlation.
In addition to built-in auxiliary basis sets, it is also possible to enter user-defined
auxiliary basis sets using an $aux_basis input section, whose syntax generally follows
the $basis input section documented above in Section 7.4.
7.8 Basis Set Superposition Error (BSSE)
When calculating binding energies, the energies of the fragments are usually
higher than they should be due to the smaller effective basis set used for the
individual species. This leads to an overestimate of the binding energy called
the basis set superposition error. The effects of this can be corrected for by
performing the calculations on the individual species in the presence of the
basis set associated with the other species. This requires basis functions to
be placed at arbitrary points in space, not just those defined by the nuclear
centers. This can be done within Q-Chem by using ghost atoms. These atoms
have zero nuclear charge, but can support a user defined basis set. Ghost atom
locations are specified in the $molecule section, as for any other atom, and
the basis must be specified in a $basis section in the same manner as for a
mixed basis.
Example 7.0 A calculation on a water monomer in the presence of the full
dimer basis set. The energy will be slightly lower than that without the ghost
atom functions due to the greater flexibility of the basis set.
$molecule
0 1
O 1.68668 -0.00318 0.000000
H 1.09686 0.01288 -0.741096
H 1.09686 0.01288 0.741096
Gh -1.45451 0.01190 0.000000
Gh -2.02544 -0.04298 -0.754494
Gh -2.02544 -0.04298 0.754494
$end
$rem
EXCHANGE hf
CORRELATION mp2
BASIS mixed
$end
$basis
O 1
6-31G*
****
H 2
6-31G*
****
H 3
6-31G*
****
O 4
6-31G*
****
H 5
6-31G*
****
H 6
6-31G*
****
$end
Ghosts atoms can also be specified by placing @ in front of the corresponding atomic symbol in the $molecule section of the input file. If @ is used to designate the ghost atoms in the system then it is not necessary to use MIXED basis set and include the $basis section in the input.
Example 7.0
A calculation on ammonia in the presence of the basis set of ammonia borane.
$molecule
0 1
N 0.0000 0.0000 0.7288
H 0.9507 0.0001 1.0947
H -0.4752 -0.8234 1.0947
H -0.4755 0.8233 1.0947
@B 0.0000 0.0000 -0.9379
@H 0.5859 1.0146 -1.2474
@H 0.5857 -1.0147 -1.2474
@H -1.1716 0.0001 -1.2474
$end
$rem
JOBTYPE SP
EXCHANGE B3LYP
CORRELATION NONE
BASIS 6-31G(d,p)
PURECART 1112
$end
Finally, there is an alternative approach to counterpoise corrections that is also available in Q-Chem. The powerful Absolutely Localized Molecular Orbital (ALMO) methods can be very conveniently used for the fully automated evaluation of BSSE corrections with associated computational advantages also. This is described in detail, including examples, in Section 12.4.3.
Chapter 8 Effective Core Potentials
8.1 Introduction
The application of quantum chemical methods to elements in the lower half of
the Periodic Table is more difficult than for the lighter atoms. There are
two key reasons for this:
- the number of electrons in heavy atoms is large
- relativistic effects in heavy atoms are often non-negligible
Both of these problems stem from the presence of large numbers of core
electrons and, given that such electrons do not play a significant
direct role in chemical behavior, it is natural to ask whether it is
possible to model their effects in some simpler way. Such enquiries led to the
invention of Effective Core Potentials (ECPs) or pseudopotentials. For reviews
of relativistic effects in chemistry, see for example
Refs. .
If we seek to replace the core electrons around a given nucleus by a
pseudopotential, while affecting the chemistry as little as possible, the
pseudopotential should have the same effect on nearby valence electrons as the
core electrons. The most obvious effect is the simple electrostatic repulsion
between the core and valence regions but the requirement that valence orbitals
must be orthogonal to core orbitals introduces additional subtler effects that
cannot be neglected.
The most widely used ECPs today are of the form first proposed by Kahn
et al. [384] in the 1970s. These model the effects of the core by a
one-electron operator U(r) whose matrix elements are simply added to the
one-electron Hamiltonian matrix. The ECP operator is given by
U(r) = UL (r) + |
L−1 ∑
l=0
|
|
+l ∑
m=−l
|
| Ylm 〉 [ Ul (r)−UL (r) ] 〈 Ylm | |
| (8.1) |
where the | Ylm 〉 are spherical harmonic projectors and the
Ul(r) are linear combinations of Gaussians, multiplied by r−2, r−1
or r0. In addition, UL(r) contains a Coulombic term Nc/r, where
Nc is the number of core electrons.
One of the key issues in the development of pseudopotentials is the definition
of the "core". So-called "large-core" ECPs include all shells except the
outermost one, but "small-core" ECPs include all except the outermost
two shells. Although the small-core ECPs are more expensive to use
(because more electrons are treated explicitly), it is often found that their
enhanced accuracy justifies their use.
When an ECP is constructed, it is usually based either on non-relativistic, or
quasi-relativistic all-electron calculations. As one might expect, the
quasi-relativistic ECPs tend to yield better results than their
non-relativistic brethren, especially for atoms beyond the 3d block.
8.2 Built-In Pseudopotentials
8.2.1 Overview
Q-Chem is equipped with several standard ECP sets which are specified using
the ECP keyword within the $rem block. The built-in ECPs, which are
described in some detail at the end of this Chapter, fall into four families:
- The Hay-Wadt (or Los Alamos) sets (HWMB and LANL2DZ)
- The Stevens-Basch-Krauss-Jansien-Cundari set (SBKJC)
- The Christiansen-Ross-Ermler-Nash-Bursten sets (CRENBS and CRENBL)
- The Stuttgart-Bonn sets (SRLC and SRSC)
References and information about the definition and characteristics of most of
these sets can be found at the EMSL site of the Pacific Northwest National
Laboratory [376]:
http://www.emsl.pnl.gov/forms/basisform.html
Each of the built-in ECPs comes with a matching orbital basis set for the
valence electrons. In general, it is advisable to use these together and, if
you select a basis set other than the matching one, Q-Chem will print a
warning message in the output file. If you omit the BASIS $rem
keyword entirely, Q-Chem will automatically provide the matching one.
The following $rem variable controls which ECP is used:
ECP
Defines the effective core potential and associated basis set to be used |
TYPE:
DEFAULT:
OPTIONS:
General, Gen | User defined. ($ecp keyword required) |
Symbol | Use standard pseudopotentials discussed above. |
RECOMMENDATION:
Pseudopotentials are recommended for first row transition metals and heavier
elements. Consul the reviews for more details. |
|
8.2.2 Combining Pseudopotentials
If you wish, you can use different ECP sets for different elements in the
system. This is especially useful if you would like to use a particular ECP but
find that it is not available for all of the elements in your molecule. To
combine different ECP sets, you set the ECP and BASIS
keywords to "Gen" or "General" and then add a $ecp block and a $basis
block to your input file. In each of these blocks, you must name the ECP and
the orbital basis set that you wish to use, separating each element by a
sequence of four asterisks. There is also a built-in combination that can be
invoked specifying "ECP=LACVP". It assigns automatically
6-31G* or other suitable type basis
sets for atoms H-Ar, while uses LANL2DZ for heavier atoms.
8.2.3 Examples
Example 8.0 Computing the HF/LANL2DZ energy of AgCl at a bond length of 2.4
Å .
$molecule
0 1
Ag
Cl Ag r
r = 2.4
$end
$rem
EXCHANGE hf Hartree-Fock calculation
ECP lanl2dz Using the Hay-Wadt ECP
BASIS lanl2dz And the matching basis set
$end
Example 8.0 Computing the HF geometry of CdBr2 using the Stuttgart
relativistic ECPs. The small-core ECP and basis are employed on the Cd atom
and the large-core ECP and basis on the Br atoms.
$molecule
0 1
Cd
Br1 Cd r
Br2 Cd r Br1 180
r = 2.4
$end
$rem
JOBTYPE opt Geometry optimization
EXCHANGE hf Hartree-Fock theory
ECP gen Combine ECPs
BASIS gen Combine basis sets
PURECART 1 Use pure d functions
$end
$ecp
Cd
srsc
****
Br
srlc
****
$end
$basis
Cd
srsc
****
Br
srlc
****
$end
8.3 User-Defined Pseudopotentials
Many users will find that the library of built-in pseudopotentials is adequate
for their needs. However, if you need to use an ECP that is not built into
Q-Chem, you can enter it in much the same way as you can enter a
user-defined orbital basis set (see Chapter 7).
8.3.1 Job Control for User-Defined ECPs
To apply a user-defined pseudopotential, you must set the ECP and
BASIS keywords in $rem to "Gen". You then add a $ecp block that
defines your ECP, element by element, and a $basis block that defines your
orbital basis set, separating elements by asterisks.
The syntax within the $basis block is described in Chapter 7.
The syntax for each record within the $ecp block is as follows:.
$ecp
For each atom that will bear an ECP
Chemical symbol for the atom
ECP name ; the L value for the ECP ; number of core electrons removed
For each ECP component (in the order unprojected, ∧P0 , ∧P1 , , ∧PL−1
The component name
The number of Gaussians in the component
For each Gaussian in the component
The power of r ; the exponent ; the contraction coefficient
A sequence of four asterisks (i.e., ****)
$end
Note:
(1) All of the information in the $ecp block is case-insensitive.
(2) The L value may not exceed 4. That is, nothing beyond G projectors is allowed.
(3) The power of r (which includes the Jacobian r2 factor) must be 0, 1 or 2. |
8.3.2 Example
Example 8.0 Optimizing the HF geometry of AlH3 using a user-defined
ECP and basis set on Al and the 3-21G basis on H.
$molecule
0 1
Al
H1 Al r
H2 Al r H1 120
H3 Al r H1 120 H2 180
r = 1.6
$end
$rem
JOBTYPE opt Geometry optimization
EXCHANGE hf Hartree-Fock theory
ECP gen User-defined ECP
BASIS gen User-defined basis
$end
$ecp
Al
Stevens_ECP 2 10
d potential
1
1 1.95559 -3.03055
s-d potential
2
0 7.78858 6.04650
2 1.99025 18.87509
p-d potential
2
0 2.83146 3.29465
2 1.38479 6.87029
****
$end
$basis
Al
SP 3 1.00
0.90110 -0.30377 -0.07929
0.44950 0.13382 0.16540
0.14050 0.76037 0.53015
SP 1 1.00
0.04874 0.32232 0.47724
****
H
3-21G
****
$end
8.4 Pseudopotentials and Density Functional Theory
Q-Chem's pseudopotential package and DFT package are tightly integrated and
facilitate the application of advanced density functionals to molecules
containing heavy elements. Any of the local, gradient-corrected and hybrid
functionals discussed in Chapter 4 may be used and you may also
perform ECP calculations with user-defined hybrid functionals.
In a DFT calculation with pseudopotentials, the exchange-correlation energy is
obtained entirely from the non-core electrons. This will be satisfactory if
there are no chemically important core-valence effects but may introduce
significant errors, particularly if you are using a "large-core" ECP.
Q-Chem's default quadrature grid is SG-1 (see section
4.3.11) which was originally defined only for the elements up to
argon. In Q-Chem 2.0 and above, the SG-1 grid has been extended and it is
now defined for all atoms up to, and including, the actinides.
8.4.1 Example
Example 8.0 Optimization of the structure of XeF5+ using B3LYP
theory and the ECPs of Stevens and collaborators. Note that the BASIS
keyword has been omitted and, therefore, the matching SBKJC orbital basis set
will be used.
$molecule
1 1
Xe
F1 Xe r1
F2 Xe r2 F1 a
F3 Xe r2 F1 a F2 90
F4 Xe r2 F1 a F3 90
F5 Xe r2 F1 a F4 90
r1 = 2.07
r2 = 2.05
a = 80.0
$end
$rem
JOBTYP opt
EXCHANGE b3lyp
ECP sbkjc
$end
8.5 Pseudopotentials and Electron Correlation
The pseudopotential package is integrated with the electron correlation package
and it is therefore possible to apply any of Q-Chem's post-Hartree-Fock
methods to systems in which some of the atoms may bear pseudopotentials. Of
course, the correlation energy contribution arising from core electrons that
have been replaced by an ECP is not included. In this sense, correlation
energies with ECPs are comparable to correlation energies from frozen core
calculations. However, the use of ECPs effectively removes both core electrons
and the corresponding virtual (unoccupied) orbitals.
8.5.1 Example
Example 8.0 Optimization of the structure of Se8 using HF/LANL2DZ,
followed by a single-point energy calculation at the MP2/LANL2DZ level.
$molecule
0 1
x1
x2 x1 xx
Se1 x1 sx x2 90.
Se2 x1 sx x2 90. Se1 90.
Se3 x1 sx x2 90. Se2 90.
Se4 x1 sx x2 90. se3 90.
Se5 x2 sx x1 90. Se1 45.
Se6 x2 sx x1 90. Se5 90.
Se7 x2 sx x1 90. Se6 90.
Se8 x2 sx x1 90. Se7 90.
xx = 1.2
sx = 2.8
$end
$rem
JOBTYP opt
EXCHANGE hf
ECP lanl2dz
$end
@@@
$molecule
read
$end
$rem
JOBTYP sp Single-point energy
CORRELATION mp2 MP2 correlation energy
ECP lanl2dz Hay-Wadt ECP and basis
SCF_GUESS read Read in the MOs
$end
8.6 Pseudopotentials, Forces and Vibrational Frequencies
It is important to be able to optimize geometries using pseudopotentials
and for this purpose Q-Chem contains analytical first derivatives of
the nuclear potential energy term for pseudopotentials. However, as
documented in Section 8.6.2, these capabilities are more limited
than those for undifferentiated matrix elements. To avoid this limitation,
Q-Chem will switch seamlessly to numerical derivatives of the ECP matrix
elements when needed, which are combined with analytical evaluation of the
remainder of the force contributions (where available, as documented in
Table 9.1.
The pseudopotential package is also integrated with the vibrational analysis
package and it is therefore possible to compute the vibrational frequencies
(and hence the infrared and Raman spectra) of systems in which some of the
atoms may bear pseudopotentials.
Q-Chem cannot calculate analytic second derivatives of the nuclear
potential-energy term when ECP's are used, and must therefore resort to finite
difference methods. However, for HF and DFT calculations, it can compute
analytic second derivatives for all other terms in the Hamiltonian. The
program takes full advantage of this by only computing the potential-energy
derivatives numerically, and adding these to the analytically calculated second
derivatives of the remaining energy terms.
There is a significant speed advantage associated with this approach as, at
each finite-difference step, only the potential-energy term needs to be
calculated. This term requires only three-center integrals, which are far
fewer in number and much cheaper to evaluate than the four-center,
two-electron integrals associated with the electron-electron interaction
terms. Readers are referred to Table 9.1 for a full list of the analytic
derivative capabilities of Q-Chem.
8.6.1 Example
Example 8.0 Structure and vibrational frequencies of TeO2 using
Hartree-Fock theory and the Stuttgart relativistic large-core ECPs. Note
that the vibrational frequency job reads both the optimized structure and the
molecular orbitals from the geometry optimization job that precedes it. Note
also that only the second derivatives of the potential energy term will be
calculated by finite difference, all other terms will be calculated
analytically.
$molecule
0 1
Te
O1 Te r
O2 Te r O1 a
r = 1.8
a = 108
$end
$rem
JOBTYP opt
EXCHANGE hf
ECP srlc
$end
@@@
$molecule
read
$end
$rem
JOBTYP freq
EXCHANGE hf
ECP srlc
SCF_GUESS read
$end
8.6.2 A Brief Guide to Q-Chem's Built-In ECPs
The remainder of this Chapter consists of a brief reference guide to Q-Chem's
built-in ECPs. The ECPs vary in their complexity and their accuracy and the
purpose of the guide is to enable the user quickly and easily to decide which
ECP to use in a planned calculation.
The following information is provided for each ECP:
- The elements for which the ECP is available in Q-Chem. This is shown on
a schematic Periodic Table by shading all the elements that are
not supported.
- The literature reference for each element for which the ECP is available
in Q-Chem.
- The matching orbital basis set that Q-Chem will use for light (i.e..
non-ECP atoms). For example, if the user requests SRSC pseudopotentials-which
are defined only for atoms beyond argon- Q-Chem will use the
6-311G* basis set for all atoms up to Ar.
- The core electrons that are replaced by the ECP. For example, in the
LANL2DZ pseudopotential for the Fe atom, the core is [Ne], indicating
that the 1s, 2s and 2p electrons are removed.
- The maximum spherical harmonic projection operator that is used for each
element. This often, but not always, corresponds to the maximum orbital
angular momentum of the core electrons that have been replaced by the
ECP. For example, in the LANL2DZ pseudopotential for the Fe atom, the
maximum projector is of P-type.
- The number of valence basis functions of each angular momentum type that
are present in the matching orbital basis set. For example, in the
matching basis for the LANL2DZ pseudopotential for the Fe atom, there the
three s shells, three p shells and two d shells. This basis is
therefore almost of triple-split valence quality.
Finally, we note the limitations of the current ECP implementation within
Q-Chem:
- Energies can be calculated only for s, p, d and f basis functions
with G projectors. Consequently, Q-Chem cannot perform energy
calculations on actinides using SRLC.
- Analytical ECP gradients can be calculated only for s, p and d basis functions
with F projectors and only for s and p basis functions with G
projectors. This limitation does not affect evaluation of forces and frequencies
as discussed in Section 8.6.
8.6.3 The HWMB Pseudopotential at a Glance
HWMB is not available for shaded elements
(a) | No pseudopotential; Pople STO-3G basis used |
(b) | Wadt & Hay, J. Chem. Phys. 82 (1985) 285 |
(c) | Hay & Wadt, J. Chem. Phys. 82 (1985) 299 |
(d) | Hay & Wadt, J. Chem. Phys. 82 (1985) 270 |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (1s) |
Li-Ne | none | none | (2s,1p) |
Na-Ar | [Ne] | P | (1s,1p) |
K-Ca | [Ne] | P | (2s,1p) |
Sc-Cu | [Ne] | P | (2s,1p,1d) |
Zn | [Ar] | D | (1s,1p,1d) |
Ga-Kr | [Ar]+3d | D | (1s,1p) |
Rb-Sr | [Ar]+3d | D | (2s,1p) |
Y-Ag | [Ar]+3d | D | (2s,1p,1d) |
Cd | [Kr] | D | (1s,1p,1d) |
In-Xe | [Kr]+4d | D | (1s,1p) |
Cs-Ba | [Kr]+4d | D | (2s,1p) |
La | [Kr]+4d | D | (2s,1p,1d) |
Hf-Au | [Kr]+4d+4f | F | (2s,1p,1d) |
Hg | [Xe]+4f | F | (1s,1p,1d) |
Tl-Bi | [Xe]+4f+5d | F | (1s,1p) |
|
8.6.4 The LANL2DZ Pseudopotential at a Glance
LANL2DZ is not available for shaded elements
(a) | No pseudopotential; Pople 6-31G basis used |
(b) | Wadt & Hay, J. Chem. Phys. 82 (1985) 285 |
(c) | Hay & Wadt, J. Chem. Phys. 82 (1985) 299 |
(d) | Hay & Wadt, J. Chem. Phys. 82 (1985) 270 |
(e) | Hay, J. Chem. Phys. 79 (1983) 5469 |
(f) | Wadt, to be published |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (2s) |
Li-Ne | none | none | (3s,2p) |
Na-Ar | [Ne] | P | (2s,2p) |
K-Ca | [Ne] | P | (3s,3p) |
Sc-Cu | [Ne] | P | (3s,3p,2d) |
Zn | [Ar] | D | (2s,2p,2d) |
Ga-Kr | [Ar]+3d | D | (2s,2p) |
Rb-Sr | [Ar]+3d | D | (3s,3p) |
Y-Ag | [Ar]+3d | D | (3s,3p,2d) |
Cd | [Kr] | D | (2s,2p,2d) |
In-Xe | [Kr]+4d | D | (2s,2p) |
Cs-Ba | [Kr]+4d | D | (3s,3p) |
La | [Kr]+4d | D | (3s,3p,2d) |
Hf-Au | [Kr]+4d+4f | F | (3s,3p,2d) |
Hg | [Xe]+4f | F | (2s,2p,2d) |
Tl | [Xe]+4f+5d | F | (2s,2p,2d) |
Pb-Bi | [Xe]+4f+5d | F | (2s,2p) |
U-Pu | [Xe]+4f+5d | F | (3s,3p,2d,2f) |
|
8.6.5 The SBKJC Pseudopotential at a Glance
SBKJC is not available for shaded elements
(a) | No pseudopotential; Pople 3-21G basis used |
(b) | W.J. Stevens, H. Basch & M. Krauss, J. Chem. Phys. 81 (1984) 6026 |
(c) | W.J. Stevens, M. Krauss, H. Basch & P.G. Jasien, Can. J. Chem 70 (1992) 612 |
(d) | T.R. Cundari & W.J. Stevens, J. Chem. Phys. 98 (1993) 5555 |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (2s) |
Li-Ne | [He] | S | (2s,2p) |
Na-Ar | [Ne] | P | (2s,2p) |
K-Ca | [Ar] | P | (2s,2p) |
Sc-Ga | [Ne] | P | (4s,4p,3d) |
Ge-Kr | [Ar]+3d | D | (2s,2p) |
Rb-Sr | [Kr] | D | (2s,2p) |
Y-In | [Ar]+3d | D | (4s,4p,3d) |
Sn-Xe | [Kr]+4d | D | (2s,2p) |
Cs-Ba | [Xe] | D | (2s,2p) |
La | [Kr]+4d | F | (4s,4p,3d) |
Ce-Lu | [Kr]+4d | D | (4s,4p,1d,1f) |
Hf-Tl | [Kr]+4d+4f | F | (4s,4p,3d) |
Pb-Rn | [Xe]+4f+5d | F | (2s,2p) |
|
8.6.6 The CRENBS Pseudopotential at a Glance
CRENBS is not available for shaded elements
(a) | No pseudopotential; Pople STO-3G basis used |
(b) | Hurley, Pacios, Christiansen, Ross & Ermler, J. Chem. Phys. 84 (1986) 6840 |
(c) | LaJohn, Christiansen, Ross, Atashroo & Ermler, J. Chem. Phys. 87 (1987) 2812 |
(d) | Ross, Powers, Atashroo, Ermler, LaJohn & Christiansen, J. Chem. Phys. 93 (1990) 6654 |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (1s) |
Li-Ne | none | none | (2s,1p) |
Na-Ar | none | none | (3s,2p) |
K-Ca | none | none | (4s,3p) |
Sc-Zn | [Ar] | P | (1s,0p,1d) |
Ga-Kr | [Ar]+3d | D | (1s,1p) |
Y-Cd | [Kr] | D | (1s,1p,1d) |
In-Xe | [Kr]+4d | D | (1s,1p) |
La | [Xe] | D | (1s,1p,1d) |
Hf-Hg | [Xe]+4f | F | (1s,1p,1d) |
Tl-Rn | [Xe]+4f+5d | F | (1s,1p) |
|
8.6.7 The CRENBL Pseudopotential at a Glance
(a) | No pseudopotential; Pople 6-311G* basis used |
(b) | Pacios & Christiansen, J. Chem. Phys. 82 (1985) 2664 |
(c) | Hurley, Pacios, Christiansen, Ross & Ermler, J. Chem. Phys. 84 (1986) 6840 |
(d) | LaJohn, Christiansen, Ross, Atashroo & Ermler, J. Chem. Phys. 87 (1987) 2812 |
(e) | Ross, Powers, Atashroo, Ermler, LaJohn & Christiansen, J. Chem. Phys. 93 (1990) 6654 |
(f) | Ermler, Ross & Christiansen, Int. J. Quantum Chem. 40 (1991) 829 |
(g) | Ross, Gayen & Ermler, J. Chem. Phys. 100 (1994) 8145 |
(h) | Nash, Bursten & Ermler, J. Chem. Phys. 106 (1997) 5133 |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (3s) |
Li-Ne | [He] | S | (4s,4p) |
Na-Mg | [He] | S | (6s,4p) |
Al-Ar | [Ne] | P | (4s,4p) |
K-Ca | [Ne] | P | (5s,4p) |
Sc-Zn | [Ne] | P | (7s,6p,6d) |
Ga-Kr | [Ar] | P | (3s,3p,4d) |
Rb-Sr | [Ar]+3d | D | (5s,5p) |
Y-Cd | [Ar]+3d | D | (5s,5p,4d) |
In-Xe | [Kr] | D | (3s,3p,4d) |
Cs-La | [Kr]+4d | D | (5s,5p,4d) |
Ce-Lu | [Xe] | D | (6s,6p,6d,6f) |
Hf-Hg | [Kr]+4d+4f | F | (5s,5p,4d) |
Tl-Rn | [Xe]+4f | F | (3s,3p,4d) |
Fr-Ra | [Xe]+4f+5d | F | (5s,5p,4d) |
Ac-Pu | [Xe]+4f+5d | F | (5s,5p,4d,4f) |
Am-Lr | [Xe]+4f+5d | F | (0s,2p,6d,5f) |
|
8.6.8 The SRLC Pseudopotential at a Glance
SRLC is not available for shaded elements
(a) | No pseudopotential; Pople 6-31G basis used |
(b) | Fuentealba, Preuss, Stoll & Szentpaly, Chem. Phys. Lett. 89 (1982) 418 |
(c) | Fuentealba, Szentpály, Preuss & Stoll, J. Phys. B 18 (1985) 1287 |
(d) | Bergner, Dolg, Küchle, Stoll & Preuss, Mol. Phys. 80 (1993) 1431 |
(e) | Nicklass, Dolg, Stoll & Preuss, J. Chem. Phys. 102 (1995) 8942 |
(f) | Schautz, Flad & Dolg, Theor. Chem. Acc. 99 (1998) 231 |
(g) | Fuentealba, Stoll, Szentpaly, Schwerdtfeger & Preuss, J. Phys. B 16 (1983) L323 |
(h) | Szentpaly, Fuentealba, Preuss & Stoll, Chem. Phys. Lett. 93 (1982) 555 |
(i) | Küchle, Dolg, Stoll & Preuss, Mol. Phys. 74 (1991) 1245 |
(j) | Küchle, to be published |
|
Element | Core | Max Projector | Valence | |
H-He | none | none | (2s) |
Li-Be | [He] | P | (2s,2p) |
B-N | [He] | D | (2s,2p) |
O-F | [He] | D | (2s,3p) |
Ne | [He] | D | (4s,4p,3d,1f) |
Na-P | [Ne] | D | (2s,2p) |
S-Cl | [Ne] | D | (2s,3p) |
Ar | [Ne] | F | (4s,4p,3d,1f) |
K-Ca | [Ar] | D | (2s,2p) |
Zn | [Ar]+3d | D | (3s,2p) |
Ga-As | [Ar]+3d | F | (2s,2p) |
Se-Br | [Ar]+3d | F | (2s,3p) |
Kr | [Ar]+3d | G | (4s,4p,3d,1f) |
Rb-Sr | [Kr] | D | (2s,2p) |
In-Sb | [Kr]+4d | F | (2s,2p) |
Te-I | [Kr]+4d | F | (2s,3p) |
Xe | [Kr]+4d | G | (4s,4p,3d,1f) |
Cs-Ba | [Xe] | D | (2s,2p) |
Hg-Bi | [Xe]+4f+5d | G | (2s,2p,1d) |
Po-At | [Xe]+4f+5d | G | (2s,3p,1d) |
Rn | [Xe]+4f+5d | G | (2s,2p,1d) |
Ac-Lr | [Xe]+4f+5d | G | (5s,5p,4d,3f,2g) |
|
8.6.9 The SRSC Pseudopotential at a Glance
SRSC is not available for shaded elements
(a) | No pseudopotential; Pople 6-311G* basis used |
(b) | Leininger, Nicklass, Küchle, Stoll, Dolg & Bergner, Chem. Phys. Lett. 255 (1996) 274 |
(c) | Kaupp, Schleyer, Stoll & Preuss, J. Chem. Phys. 94 (1991) 1360 |
(d) | Dolg, Wedig, Stoll & Preuss, J. Chem. Phys. 86 (1987) 866 |
(e) | Andrae, Haeussermann, Dolg, Stoll & Preuss, Theor. Chim. Acta 77 (1990) 123 |
(f) | Dolg, Stoll & Preuss, J. Chem. Phys. 90 (1989) 1730 |
(g) | Küchle, Dolg, Stoll & Preuss, J. Chem. Phys. 100 (1994) 7535 |
|
Element | Core | Max Projector | Valence | |
H-Ar | none | none | (3s) |
Li-Ne | none | none | (4s,3p,1d) |
Na-Ar | none | none | (6s,5p,1d) |
K | [Ne] | F | (5s,4p) |
Ca | [Ne] | F | (4s,4p,2d) |
Sc-Zn | [Ne] | D | (6s,5p,3d) |
Rb | [Ar]+3d | F | (5s,4p) |
Sr | [Ar]+3d | F | (4s,4p,2d) |
Y-Cd | [Ar]+3d | F | (6s,5p,3d) |
Cs | [Kr]+4d | F | (5s,4p) |
Ba | [Kr]+4d | F | (3s,3p,2d,1f) |
Ce-Yb | [Ar]+3d | G | (5s,5p,4d,3f) |
Hf-Pt | [Kr]+4d+4f | G | (6s,5p,3d) |
Au | [Kr]+4d+4f | F | (7s,3p,4d) |
Hg | [Kr]+4d+4f | G | (6s,6p,4d) |
Ac-Lr | [Kr]+4d+4f | G | (8s,7p,6d,4f) |
|
Chapter 9 Molecular Geometry Critical Points, ab Initio Molecular Dynamics, and QM/MM Features
9.1 Equilibrium Geometries and Transition Structures
Molecular potential energy surfaces rely on the Born-Oppenheimer separation of
nuclear and electronic motion. Minima on such energy surfaces correspond to the
classical picture of equilibrium geometries and first-order saddle points for
transition structures. Both equilibrium and transition structures are
stationary points and therefore the energy gradients will vanish.
Characterization of the critical point requires consideration of the
eigenvalues of the Hessian (second derivative matrix). Equilibrium geometries
have Hessians whose eigenvalues are all positive. Transition structures, on the
other hand, have Hessians with exactly one negative eigenvalue. That is, a
transition structure is a maximum along a reaction path between two local
minima, but a minimum in all directions perpendicular to the path.
The quality of a geometry optimization algorithm is of major importance; even
the fastest integral code in the world will be useless if combined with an
inefficient optimization algorithm that requires excessive numbers of steps to
converge. Thus, Q-Chem incorporates the most advanced geometry optimization
features currently available through Jon Baker's Optimize package (see
Appendix A), a product of over ten years of research and development.
The key to optimizing a molecular geometry successfully is to proceed from the
starting geometry to the final geometry in as few steps as possible. Four
factors influence the path and number of steps:
- starting geometry
- optimization algorithm
- quality of the Hessian (and gradient)
- coordinate system
Q-Chem controls the last three of these, but the starting geometry is solely
determined by the user, and the closer it is to the converged geometry, the
fewer optimization steps will be required. Decisions regarding the optimizing
algorithm and the coordinate system are generally made by the Optimize
package to maximize the rate of convergence. Users are able to override these
decisions, but in general, this is not recommended.
Another consideration when trying to minimize the optimization time concerns
the quality of the gradient and Hessian. A higher quality Hessian (i.e.,
analytical vs. approximate) will in many cases lead to faster
convergence and hence, fewer optimization steps. However, the construction of
an analytical Hessian requires significant computational effort and may
outweigh the advantage of fewer optimization cycles. Currently available
analytical gradients and Hessians are summarized in Table 9.1.
Level of Theory | Analytical | Maximum Angular
| Analytical | Maximum Angular |
(Algorithm) | Gradients | Momentum Type
| Hessian | Momentum Type | |
DFT | | h | | f |
HF | | h | | f |
ROHF | | h | | |
MP2 | | h | | |
(V)OD | | h | | |
(V)QCCD | | h | | |
CIS (except RO) | | h | | f |
CFMM | | h | | |
|
Table 9.1: Gradients and Hessians currently available for geometry optimizations
with maximum angular momentum types for analytical derivative calculations (for
higher angular momentum, derivatives are computed numerically).
Analytical Hessian is not yet available to tau-dependent functionals, such as BMK, M05 and M06 series.
9.2 User-Controllable Parameters
9.2.1 Features
- Cartesian, Z-matrix or internal coordinate systems
- Eigenvector Following (EF) or GDIIS algorithms
- Constrained optimizations
- Equilibrium structure searches
- Transition structure searches
- Initial Hessian and Hessian update options
- Reaction pathways using intrinsic reaction coordinates (IRC)
9.2.2 Job Control
Note:
Users input starting geometry through the $molecule keyword. |
Users must first define what level of theory is required. Refer back to
previous sections regarding enhancements and customization of these features.
EXCHANGE, CORRELATION (if required) and BASIS $rem
variables must be set.
The remaining $rem variables are those specifically relating to the
Optimize package.
JOBTYPE
Specifies the calculation. |
TYPE:
DEFAULT:
Default is single-point, which should be changed to one of the following options. |
OPTIONS:
OPT | Equilibrium structure optimization. |
TS | Transition structure optimization. |
RPATH | Intrinsic reaction path following. |
RECOMMENDATION:
|
| GEOM_OPT_HESSIAN
Determines the initial Hessian status. |
TYPE:
DEFAULT:
OPTIONS:
DIAGONAL | Set up diagonal Hessian. |
READ | Have exact or initial Hessian. Use as is if Cartesian, or transform |
| if internals. |
RECOMMENDATION:
An accurate initial Hessian will improve the performance of the optimizer, but is
expensive to compute. |
|
|
|
GEOM_OPT_COORDS
Controls the type of optimization coordinates. |
TYPE:
DEFAULT:
OPTIONS:
0 | Optimize in Cartesian coordinates. |
1 | Generate and optimize in internal coordinates, if this fails abort. |
-1 | Generate and optimize in internal coordinates, if this fails at any stage of the |
| optimization, switch to Cartesian and continue. |
2 | Optimize in Z-matrix coordinates, if this fails abort. |
-2 | Optimize in Z-matrix coordinates, if this fails during any stage of the |
| optimization switch to Cartesians and continue. |
RECOMMENDATION:
Use the default; delocalized internals are more efficient. |
|
| GEOM_OPT_TOL_GRADIENT
Convergence on maximum gradient component. |
TYPE:
DEFAULT:
300 | ≡ 300×10−6 tolerance on maximum gradient component. |
OPTIONS:
n | Integer value (tolerance = n ×10−6). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
|
|
GEOM_OPT_TOL_DISPLACEMENT
Convergence on maximum atomic displacement. |
TYPE:
DEFAULT:
1200 ≡ 1200 ×10−6 tolerance on maximum atomic displacement. |
OPTIONS:
n | Integer value (tolerance = n ×10−6). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
| GEOM_OPT_TOL_ENERGY
Convergence on energy change of successive optimization cycles. |
TYPE:
DEFAULT:
100 ≡ 100 ×10−8 tolerance on maximum gradient component. |
OPTIONS:
n Integer value (tolerance = value n ×10−8). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
|
|
GEOM_OPT_MAX_CYCLES
Maximum number of optimization cycles. |
TYPE:
DEFAULT:
OPTIONS:
n | User defined positive integer. |
RECOMMENDATION:
The default should be sufficient for most cases. Increase if the initial
guess geometry is poor, or for systems with shallow potential wells. |
|
| GEOM_OPT_PRINT
Controls the amount of Optimize print output. |
TYPE:
DEFAULT:
3 | Error messages, summary, warning, standard information and gradient print
out. |
OPTIONS:
0 | Error messages only. |
1 | Level 0 plus summary and warning print out. |
2 | Level 1 plus standard information. |
3 | Level 2 plus gradient print out. |
4 | Level 3 plus Hessian print out. |
5 | Level 4 plus iterative print out. |
6 | Level 5 plus internal generation print out. |
7 | Debug print out. |
RECOMMENDATION:
|
|
|
9.2.3 Customization
GEOM_OPT_SYMFLAG
Controls the use of symmetry in Optimize. |
TYPE:
DEFAULT:
OPTIONS:
1 | Make use of point group symmetry. |
0 | Do not make use of point group symmetry. |
RECOMMENDATION:
|
| GEOM_OPT_MODE
Determines Hessian mode followed during a transition state search. |
TYPE:
DEFAULT:
OPTIONS:
0 | Mode following off. |
n | Maximize along mode n. |
RECOMMENDATION:
Use default, for geometry optimizations. |
|
|
|
GEOM_OPT_MAX_DIIS
Controls maximum size of subspace for GDIIS. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use GDIIS. |
-1 | Default size = min(NDEG, NATOMS, 4) NDEG = number of molecular |
| degrees of freedom. |
n | Size specified by user. |
RECOMMENDATION:
Use default or do not set n too large. |
|
| GEOM_OPT_DMAX
Maximum allowed step size. Value supplied is multiplied by 10−3. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
GEOM_OPT_UPDATE
Controls the Hessian update algorithm. |
TYPE:
DEFAULT:
OPTIONS:
-1 | Use the default update algorithm. |
0 | Do not update the Hessian (not recommended). |
1 | Murtagh-Sargent update. |
2 | Powell update. |
3 | Powell/Murtagh-Sargent update (TS default). |
4 | BFGS update (OPT default). |
5 | BFGS with safeguards to ensure retention of positive definiteness |
| (GDISS default). |
RECOMMENDATION:
|
| GEOM_OPT_LINEAR_ANGLE
Threshold for near linear bond angles (degrees). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference. |
TYPE:
DEFAULT:
100 | Corresponding to 0.001 Å. For calculating second derivatives. |
OPTIONS:
n | Use a step size of n×10−5. |
RECOMMENDATION:
Use default, unless on a very flat potential, in which case a larger
value should be used. See FDIFF_STEPSIZE_QFF for third and fourth derivatives. |
|
9.2.4 Example
Example 9.0
As outlined, the rate of convergence of the iterative optimization process is
dependent on a number of factors, one of which is the use of an initial
analytic Hessian. This is easily achieved by instructing Q-Chem to calculate
an analytic Hessian and proceed then to determine the required critical point
$molecule
0 1
O
H 1 oh
H 1 oh 2 hoh
oh = 1.1
hoh = 104
$end
$rem
JOBTYPE freq Calculate an analytic Hessian
EXCHANGE hf
BASIS 6-31g(d)
$end
$comment
Now proceed with the Optimization making sure to read in the analytic
Hessian (use other available information too).
$end
@@@
$molecule
read
$end
$rem
JOBTYPE opt
EXCHANGE hf
BASIS 6-31g(d)
SCF_GUESS read
GEOM_OPT_HESSIAN read Have the initial Hessian
$end
9.3 Constrained Optimization
9.3.1 Introduction
Constrained optimization refers to the optimization of molecular structures
(transition or equilibrium) in which certain parameters (e.g., bond lengths,
bond angles or dihedral angles) are fixed. Jon Baker's Optimize package
implemented in the Q-Chem program has been modified to handle constraints
directly in delocalized internal coordinates using the method of Lagrange
multipliers (see Appendix A). Constraints are imposed in an $opt
keyword section of the input file.
Features of constrained optimizations in Q-Chem are:
- Starting geometries do not have to satisfy imposed constraints.
- Delocalized internal coordinates are the most efficient system for large
molecules.
- Q-Chem's free format $opt section allows the user to apply
constraints with ease.
Note:
The $opt input section is case-insensitive and free-format, except that
there should be no space at the start of each line. |
9.3.2 Geometry Optimization with General Constraints
CONSTRAINT and ENDCONSTRAINT define the beginning and end,
respectively, of the constraint section of $opt within which users may
specify up to six different types of constraints:
interatomic distances
Values in angstroms; value > 0:
stre atom1 atom2 value
|
angles
Values in degrees, 0 ≤ value ≤ 180; atom2 is the middle atom
of the bend:
bend atom1 atom2 atom3 value
|
out-of-plane-bends
Values in degrees, −180 ≤ value ≤ 180 atom2; angle between
atom4 and the atom1-atom2-atom3 plane:
outp atom1 atom2 atom3 atom4 value
|
dihedral angles
Values in degrees, −180 ≤ value ≤ 180; angle the plane
atom1-atom2-atom3 makes with the plane atom2-atom3-atom4:
tors atom1 atom2 atom3 atom4 value
|
coplanar bends
Values in degrees, −180 ≤ value ≤ 180; bending of
atom1-atom2-atom3 in the plane atom2-atom3-atom4:
linc atom1 atom2 atom3 atom4 value
|
perpendicular bends
Values in degrees, −180 ≤ value ≤ 180; bending of
atom1-atom2-atom3 perpendicular to the plane
atom2-atom3-atom4:
linp atom1 atom2 atom3 atom4 value
|
9.3.3 Frozen Atoms
Absolute atom positions can be frozen with the FIXED section. The
section starts with the FIXED keyword as the first line and ends with
the ENDFIXED keyword on the last. The format to fix a coordinate or
coordinates of an atom is:
atom coordinate_reference
coordinate_reference can be any combination of up to three characters
X, Y and Z to specify the coordinate(s) to be fixed: X, Y, Z,
XY, XZ, YZ, XYZ. The fixing characters must be next
to each other. e.g.,
means the x-coordinate and y-coordinate of atom 2 are fixed, whereas
will yield erroneous results.
Note:
When the FIXED section is specified within $opt, the
optimization coordinates will be Cartesian. |
9.3.4 Dummy Atoms
DUMMY defines the beginning of the dummy atom section and
ENDDUMMY its conclusion. Dummy atoms are used to help define constraints
during constrained optimizations in Cartesian coordinates. They cannot be used
with delocalized internals.
All dummy atoms are defined with reference to a list of real atoms, that is,
dummy atom coordinates are generated from the coordinates of the real atoms
from the dummy atoms defining list (see below). There are three types of dummy
atom:
- Positioned at the arithmetic mean of up to seven real atoms in the defining
list.
- Positioned a unit distance along the normal to a plane defined by three
atoms, centered on the middle atom of the three.
- Positioned a unit distance along the bisector of a given angle.
The format for declaring dummy atoms is:
DUMMY
idum type list_length defining_list
ENDDUMMY
idum | Center number of defining atom (must be one greater
than the total number of real atoms for the first
dummy atom, two greater for second etc.). |
type | Type of dummy atom (either 1, 2 or 3; see above). |
list_length | Number of atoms in the defining list. |
defining_list | List of up to seven atoms defining the position of the
dummy atom. |
Once defined, dummy atoms can be used to define standard internal (distance,
angle) constraints as per the constraints section, above.
Note:
The use of dummy atoms of type 1 has never progressed beyond the
experimental stage. |
9.3.5 Dummy Atom Placement in Dihedral Constraints
Bond and dihedral angles cannot be constrained in Cartesian optimizations to
exactly 0° or ±180°. This is because the corresponding
constraint normals are zero vectors. Also, dihedral constraints near these two
limiting values (within, say 20°) tend to oscillate and are difficult
to converge.
These difficulties can be overcome by defining dummy atoms and redefining the
constraints with respect to the dummy atoms. For example, a dihedral constraint
of 180° can be redefined to two constraints of 90° with
respect to a suitably positioned dummy atom. The same thing can be done with a
180° bond angle (long a familiar use in Z-matrix construction).
Typical usage is as follows:
Internal Coordinates | Cartesian Coordinates |
| |
$opt
CONSTRAINT
tors I J K L 180.0
ENDCONSTRAINT
$end | $opt
DUMMY
M 2 I J K
ENDDUMMY
CONSTRAINT
tors I J K M 90
tors M J K L 90
ENDCONSTRAINT
$end |
Table 9.2: Comparison of dihedral angle constraint method for adopted
coordinates.
The order of atoms is important to obtain the correct signature on the dihedral
angles. For a 0° dihedral constraint, J and K should be switched in
the definition of the second torsion constraint in Cartesian coordinates.
Note:
In almost all cases the above discussion is somewhat academic, as
internal coordinates are now best imposed using delocalized internal
coordinates and there is no restriction on the constraint values. |
9.3.6 Additional Atom Connectivity
Normally delocalized internal coordinates are generated automatically from the
input Cartesian coordinates. This is accomplished by first determining the
atomic connectivity list (i.e., which atoms are formally bonded) and then
constructing a set of individual primitive internal coordinates comprising all
bond stretches, all planar bends and all proper torsions that can be generated
based on the atomic connectivity. The delocalized internal are in turn
constructed from this set of primitives.
The atomic connectivity depends simply on distance and there are default bond
lengths between all pairs of atoms in the code. In order for delocalized
internals to be generated successfully, all atoms in the molecule must be
formally bonded so as to form a closed system. In molecular complexes with
long, weak bonds or in certain transition states where parts of the molecule
are rearranging or dissociating, distances between atoms may be too great for
the atoms to be regarded as formally bonded, and the standard atomic
connectivity will separate the system into two or more distinct parts. In this
event, the generation of delocalized internal coordinates will fail. Additional
atomic connectivity can be included for the system to overcome this difficulty.
CONNECT defines the beginning of the additional connectivity section
and ENDCONNECT the end. The format of the CONNECT section is:
CONNECT
atom list_length list
ENDCONNECT
atom | Atom for which additional connectivity is being defined. |
list_length | Number of atoms in the list of bonded atoms. |
list | List of up to 8 atoms considered as being bonded to the
given atom. |
9.3.7 Example
Example 9.0 Methanol geometry optimization with constraints.
$comment
Methanol geom opt with constraints in bond length and bond angles.
$end
$molecule
0 1
C 0.14192 0.33268 0.00000
O 0.14192 -1.08832 0.00000
H 1.18699 0.65619 0.00000
H -0.34843 0.74268 0.88786
H -0.34843 0.74268 -0.88786
H -0.77395 -1.38590 0.00000
$end
$rem
GEOM_OPT_PRINT 6
JOBTYPE opt
EXCHANGE hf
BASIS 3-21g
$end
$opt
CONSTRAINT
stre 1 6 1.8
bend 2 1 4 110.0
bend 2 1 5 110.0
ENDCONSTRAINT
$end
9.3.8 Summary
$opt
CONSTRAINT
stre atom1 atom2 value
...
bend atom1 atom2 atom3 value
...
outp atom1 atom2 atom3 atom4 value
...
tors atom1 atom2 atom3 atom4 value
...
linc atom1 atom2 atom3 atom4 value
...
linp atom1 atom2 atom3 atom4 value
...
ENDCONSTRAINT
FIXED
atom coordinate_reference
...
ENDFIXED
DUMMY
idum type list_length defining_list
...
ENDDUMMY
CONNECT
atom list_length list
...
ENDCONNECT
$end
9.4 Potential Energy Scans
It is often useful to scan potential energy surfaces (PES).
In a SN1 chemical reaction, for example, such a scan can give an idea about
how the potential energy changes upon bond breaking.
In more complicated reactions involving multiple bond breaking/formation,
a multi-dimensional PES reveals one (or more) reaction pathway connecting the
reactant to the transition state and finally to the product.
In force-field development, 1-dimensional torsional scan is essential for
the optimization of accurate dihedral parameters.
Finally, Ramachandran plots, which are essentially 2-dimensional torsional scans,
are key tools for studying conformational changes of peptides and proteins.
Q-Chem supports 1-dimensional and 2-dimensional PES scans in which one or two
coordinates (e.g., stretching, bending, torsion) are being scanned whereas all
other degrees of freedom are being
optimized. For these calculations, JOBTYPE needs to be set to
PES_SCAN, and the
following input section (with one or two motions) should be specified:
$scan
stre atom1 atom2 value1 value2 incr
...
bend atom1 atom2 atom3 value1 value2 incr
...
tors atom1 atom2 atom3 atom4 value1 value2 incr
...
$end
The example below allows us to scan the torsional potential of butane,
which is a sequence of constrained optimizations with
the C1-C2-C3-C4 dihedral angle fixed at -180, -165, -150,
…, 165, 180 degrees.
Example 9.0 One-dimensional torsional scan of butane
$molecule
0 1
C 1.934574 -0.128781 -0.000151
C 0.556601 0.526657 0.000200
C -0.556627 -0.526735 0.000173
C -1.934557 0.128837 -0.000138
H 2.720125 0.655980 -0.000236
H 2.061880 -0.759501 -0.905731
H 2.062283 -0.759765 0.905211
H 0.464285 1.168064 -0.903444
H 0.464481 1.167909 0.903924
H -0.464539 -1.167976 0.903964
H -0.464346 -1.168166 -0.903402
H -2.062154 0.759848 0.905185
H -2.720189 -0.655832 -0.000229
H -2.061778 0.759577 -0.905748
$end
$rem
jobtype pes_scan
exchange hf
correlation none
basis sto-3g
$end
$scan
tors 1 2 3 4 -180 180 15
$end
A 2-dimensional scan of butane can be performed using the following input:
Example 9.0 Two-dimensional torsional scan of butane
$molecule
0 1
C 1.934574 -0.128781 -0.000151
C 0.556601 0.526657 0.000200
C -0.556627 -0.526735 0.000173
C -1.934557 0.128837 -0.000138
H 2.720125 0.655980 -0.000236
H 2.061880 -0.759501 -0.905731
H 2.062283 -0.759765 0.905211
H 0.464285 1.168064 -0.903444
H 0.464481 1.167909 0.903924
H -0.464539 -1.167976 0.903964
H -0.464346 -1.168166 -0.903402
H -2.062154 0.759848 0.905185
H -2.720189 -0.655832 -0.000229
H -2.061778 0.759577 -0.905748
$end
$rem
jobtype pes_scan
exchange hf
correlation none
basis sto-3g
$end
$scan
tors 1 2 3 4 -180 180 30
stre 2 3 1.5 1.6 0.05
$end
Here the first dimension is the the scan of the C1-C2-C3-C4 dihedral angle from
-180 to 180 degree at 30-degree intervals. For the second dimension, we scan the
C2-C3 bond length from 1.5 Å to 1.6 Å at 0.05 Å increments.
9.5 Intrinsic Reaction Coordinates
The concept of a reaction path, although seemingly well-defined chemically
(i.e., how the atoms in the system move to get from reactants to products),
is somewhat ambiguous mathematically because, using the usual definitions, it
depends on the coordinate system. Stationary points on a potential energy
surface are independent of coordinates, but the path connecting them is
not, and so different coordinate systems will produce different reaction paths.
There are even different definitions of what constitutes a "reaction path";
the one used in Q-Chem is based on the intrinsic reaction coordinate (IRC),
first defined in this context by Fukui [386]. This is essentially a
series of steepest descent paths going downhill from the transition state.
The reaction path is most unlikely to be a straight line and so by taking a
finite step length along the direction of the gradient you will leave the
"true" path. A series of small steepest descent steps will zig-zag along the
actual reaction path (this is known as "stitching"). Ishida et al. [387]
developed a predictor-corrector algorithm, involving a
second gradient calculation after the initial steepest descent step, followed
by a line search along the gradient bisector to get back on the path; this was
subsequently improved by Schmidt et al. [388], and is the method
we have adopted. For the first step downhill from the transition state this
approach cannot be used (as the gradient is zero); instead a step is taken
along the Hessian mode corresponding to the imaginary frequency.
The reaction path can be defined and followed in Z-matrix coordinates,
Cartesian coordinates or mass-weighted Cartesians. The latter represents the
"true" IRC as defined by Fukui [386]. However, if the main reason
for following the reaction path is simply to determine which minima a given
transition state connects (perhaps the major use), then it doesn't matter which
coordinates are used. In order to use the IRC code the transition state
geometry and the exact Hessian must be available. These must be computed via
transition state (JOBTYPE = TS) and frequency calculation
(JOBTYPE = FREQ) respectively.
9.5.1 Job Control
An IRC calculation is invoked by setting the JOBTYPE $rem to
RPATH.
RPATH_COORDS
Determines which coordinate system to use in the IRC search. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use mass-weighted coordinates. |
1 | Use Cartesian coordinates. |
2 | Use Z-matrix coordinates. |
RECOMMENDATION:
|
| RPATH_DIRECTION
Determines the direction of the eigen mode to follow. This will not usually be
known prior to the Hessian diagonalization. |
TYPE:
DEFAULT:
OPTIONS:
1 | Descend in the positive direction of the eigen mode. |
-1 | Descend in the negative direction of the eigen mode. |
RECOMMENDATION:
It is usually not possible to determine in which direction to
go a priori, and therefore both directions will need to be
considered. |
|
|
|
RPATH_MAX_CYCLES
Specifies the maximum number of points to find on the reaction path. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of cycles. |
RECOMMENDATION:
Use more points if the minimum is desired, but not reached using the default. |
|
| RPATH_MAX_STEPSIZE
Specifies the maximum step size to be taken (in thousandths of a.u.). |
TYPE:
DEFAULT:
150 | corresponding to a step size of 0.15 a.u.. |
OPTIONS:
RECOMMENDATION:
|
|
|
RPATH_TOL_DISPLACEMENT
Specifies the convergence threshold for the step. If a step size is chosen by
the algorithm that is smaller than this, the path is deemed to have reached the
minimum. |
TYPE:
DEFAULT:
5000 | Corresponding to 0.005 a.u. |
OPTIONS:
n | User-defined. Tolerance = n/1000000. |
RECOMMENDATION:
Use default. Note that this option only controls the
threshold for ending the RPATH job and does nothing to
the intermediate steps of the calculation. A smaller value will
provide reaction paths that end closer to the true minimum.
Use of smaller values without adjusting RPATH_MAX_STEPSIZE,
however, can lead to oscillations about the minimum. |
|
| RPATH_PRINT
Specifies the print output level. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default, little additional information is printed at higher levels. Most
of the output arises from the multiple single point calculations that are
performed along the reaction pathway. |
|
|
|
9.5.2 Example
Example 9.0
$molecule
0 1
C
H 1 1.20191
N 1 1.22178 2 72.76337
$end
$rem
JOBTYPE freq
BASIS sto-3g
EXCHANGE hf
$end
@@@
$molecule
read
$end
$rem
JOBTYPE rpath
BASIS sto-3g
EXCHANGE hf
SCF_GUESS read
RPATH_MAX_CYCLES 30
$end
9.6 Freezing String Method
Perhaps the most significant difficulty in locating transition states
is to obtain a good initial guess of the geometry to feed into a surface walking algorithm.
This difficulty becomes especially relevant for large systems, where the search space dimensionality is high.
Interpolation algorithms are promising methods for locating good guesses of
the minimum energy pathway connecting reactant and product states, as well as approximate saddle point geometries.
For example, the nudged elastic band method [389,[390] and the string method [391]
start from a certain initial reaction pathway connecting the reactant and the product state,
and then optimize in discretized path space towards the minimum energy pathway.
The highest energy point on the approximate minimum energy pathway
becomes a good initial guess for the saddle point configuration
that can subsequently be used with any local surface walking algorithm.
Inevitably, the performance of an interpolation method heavily relies on the choice of the initial reaction pathway,
and a poorly chosen initial pathway can cause slow convergence, or convergence to an incorrect pathway.
The freezing string [392,[393] and growing string methods [394] offer elegant solutions to this problem,
in which two string fragments (one from the reactant and the other from the product state)
are grown until the two fragments join.
The freezing string method offers a choice between Cartesian and Linear Synchronous Transit (LST) interpolation methods. It also allows
users to choose between conjugate gradient and quasi-Newton optimization techniques.
It can be invoked by (JOBTYPE = FSM) using the following $rem keyword:
FSM_NNODE
Specifies the number of nodes along the string |
TYPE:
DEFAULT:
OPTIONS:
N | number of nodes in FSM calculation |
RECOMMENDATION:
15. Use 10 to 20 nodes for a typical calculation. Reaction paths that connect multiple elementary steps should be separated into individual elementary steps, and one FSM job run for each pair of intermediates. Use a higher number when the FSM is followed by an approximate-Hessian
based transition state search (Section 9.7). |
|
| FSM_NGRAD
Specifies the number of perpendicular gradient steps used to optimize each node |
TYPE:
DEFAULT:
OPTIONS:
N | number of perpendicular gradients per node |
RECOMMENDATION:
4. Anything between 2 and 6 should work, where increasing the number is only needed for difficult reaction paths. |
|
|
|
FSM_MODE
Specifies the method of interpolation |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
2. In most cases, LST is superior to Cartesian interpolation. |
|
| FSM_OPT_MODE
Specifies the method of optimization |
TYPE:
DEFAULT:
OPTIONS:
1 | Conjugate gradients |
2 | Quasi-Newton method with BFGS Hessian update |
RECOMMENDATION:
2. The quasi-Newton method is more efficient when the number of nodes is high. |
|
|
|
References [392] and [393] provide a guide to a typical use of this
method. The following example input will be helpful for setting up the job:
Example 9.0
$molecule
0 1
Si 1.028032 -0.131573 -0.779689
H 0.923921 -1.301934 0.201724
H 1.294874 0.900609 0.318888
H -1.713989 0.300876 -0.226231
H -1.532839 0.232021 0.485307
****
Si 0.000228 -0.000484 -0.000023
H 0.644754 -1.336958 -0.064865
H 1.047648 1.052717 0.062991
H -0.837028 0.205648 -1.211126
H -0.8556026 0.079077 1.213023
$end
$rem
jobtype fsm
fsm_ngrad 3
fsm_nnode 12
fsm_mode 2
fsm_opt_mode 2
exchange b3lyp
basis 6-31G
$end
The $molecule section should include geometries for two optimized intermediates separated by **** symbols.
The order of the atoms is important, as Q-Chem will assume atom X in the reaction complex moves to atom X in the product complex.
The FSM string is printed out in the file `stringfile.txt',
which is an XYZ file containing the structures connecting reactant to product.
Each node along the path is labeled with its energy.
The highest energy node can be taken from this file and used to run a TS search, as detailed in section 9.1.
If the string returns a pathway that is unreasonable,
double check whether the atoms in the two input geometries are in the correct order.
9.7 Hessian-Free Transition State Search
Once a guess structure to the transition state is obtained, standard eigenvector-following methods such as the Partitioned-Rational Function Optimization (P-RFO) [395] can be employed to refine the guess to the exact transition state. The reliability of P-RFO depends on the quality of the Hessian input, which enables the method to distinguish between the reaction coordinate (characterized by a negative eigenvalue) and the remaining degrees of freedom. In routine calculations therefore, an exact Hessian is determined via frequency calculation prior to the P-RFO search. Since the cost of evaluating an exact Hessian typically scales one power of system size higher than the energy or the gradient, this step becomes impractical for systems containing large number of atoms.
The exact Hessian calculation can be avoided by constructing an approximate Hessian based on the output of FSM. The tangent direction at the transition state guess on the FSM string is a good approximation to the Hessian eigenvector corresponding to the reaction coordinate. The tangent is therefore used to calculate the correct eigenvalue and corresponding eigenvector by variationally minimizing the Rayleigh-Ritz ratio [396]. The reaction coordinate information is then incorporated into a guess matrix which, in turn, is obtained by transforming a diagonal matrix in delocalized internal coordinates [397] [398] to Cartesian coordinates. The resulting approximate Hessian, by design, has a single negative eigenvalue corresponding to the reaction coordinate. This matrix is then used in place of the exact Hessian as input to the P-RFO method.
An example of this one-shot, Hessian-free approach that combines the FSM and P-RFO methods in order to determine the exact transition state from reactant and product structures is shown below:
Example 9.0
$molecule
0 1
Si 1.028032 -0.131573 -0.779689
H 0.923921 -1.301934 0.201724
H 1.294874 0.900609 0.318888
H -1.713989 0.300876 -0.226231
H -1.532839 0.232021 0.485307
****
Si 0.000228 -0.000484 -0.000023
H 0.644754 -1.336958 -0.064865
H 1.047648 1.052717 0.062991
H -0.837028 0.205648 -1.211126
H -0.8556026 0.079077 1.213023
$end
$rem
jobtype fsm
fsm_ngrad 3
fsm_nnode 18
fsm_mode 2
fsm_opt_mode 2
exchange b3lyp
basis 6-31g
symmetry false
sym_ignore true
$end
@@@
$rem
jobtype ts
scf_guess read
geom_opt_hessian read
max_scf_cycles 250
geom_opt_dmax 50
geom_opt_max_cycles 100
exchange b3lyp
basis 6-31g
symmetry false
sym_ignore true
$end
$molecule
read
$end
9.8 Improved Dimer Method
Once a good approximation to the minimum energy pathway is obtained, e.g.,
with the help of an interpolation algorithm such as the growing string method,
local surface walking algorithms can be used to determine the exact location of
the saddle point. Baker's partitioned rational function optimization (P-RFO)
method, which utilizes an approximate or exact Hessian, has proven to be a very
powerful method for this purpose.
The dimer method [399] on the other hand, is a mode following
algorithm that utilizes only the curvature along one direction in configuration
space (rather than the full Hessian) and requires only gradient evaluations.
It is therefore especially applicable for large systems where a full Hessian
calculation is very time consuming, or for saddle point searches where the
eigenvector of the lowest Hessian eigenvalue of the starting configuration does
not correspond to the reaction coordinate. A recent modification of this
method has been developed [400,[401] to significantly
reduce the influence of numerical noise, as it is common in quantum chemical
methods, on the performance of the dimer algorithm, and to significantly reduce
its computational cost. This improved dimer method has recently been
implemented within Q-Chem.
9.9 Ab initio Molecular Dynamics
Q-Chem can propagate classical molecular dynamics trajectories on the
Born-Oppenheimer potential energy surface generated by a particular theoretical
model chemistry (e.g., B3LYP/6-31G*). This procedure, in which the forces on the
nuclei are evaluated on-the-fly, is known variously as "direct dynamics",
"ab initio molecular dynamics", or "Born-Oppenheimer molecular
dynamics" (BOMD). In its most straightforward form, a BOMD calculation
consists of an energy + gradient calculation at each molecular dynamics time
step, and thus each time step is comparable in cost to one geometry
optimization step. A BOMD calculation may be requested using any SCF energy +
gradient method available in Q-Chem, including excited-state gradients;
however, methods lacking analytic gradients will be prohibitively expensive,
except for very small systems.
Initial Cartesian coordinates and velocities must be specified for the
nuclei. Coordinates are specified in the $molecule section as usual,
while velocities can be specified using a $velocity section with the form:
$velocity
vx,1 vy,1 vz,1
vx,2 vy,2 vz,2
vx,N vy,N vz,N
$end
Here vx,i, vy,i, and vz,I are the x, y, and z
Cartesian velocities of the ith nucleus, specified in atomic units (bohrs per
a.u. of time, where 1 a.u. of time is approximately 0.0242 fs). The
$velocity section thus has the same form as the $molecule section, but
without atomic symbols and without the line specifying charge and multiplicity.
The atoms must be ordered in the same manner in both the $velocity and
$molecule sections.
As an alternative to a $velocity section, initial nuclear velocities can be
sampled from certain distributions (e.g., Maxwell-Boltzmann), using the
AIMD_INIT_VELOC variable described below.
AIMD_INIT_VELOC can also be set to QUASICLASSICAL, which
triggers the use of quasi-classical trajectory molecular dynamics
(QCT-MD, see below).
Although the Q-Chem output file dutifully records the progress of any
ab initio molecular dynamics job, the most useful information is
printed not to the main output file but rather to a directory called "AIMD"
that is a subdirectory of the job's scratch directory. (All ab initio
molecular dynamics jobs should therefore use the -save option
described in Section 2.7.) The AIMD
directory consists of a set of files that record, in ASCII format, one line of
information at each time step. Each file contains a few comment lines
(indicated by "#") that describe its contents and which we summarize in the
list below.
- Cost: Records the number of SCF cycles, the total cpu time, and the total
memory use at each dynamics step.
- EComponents: Records various components of the total energy (all in
Hartrees).
- Energy: Records the total energy and fluctuations therein.
- MulMoments: If multipole moments are requested, they are printed here.
- NucCarts: Records the nuclear Cartesian coordinates x1, y1,
z1, x2, y2, z2, ..., xN, yN, zN
at each time step, in either bohrs or angstroms.
- NucForces: Records the Cartesian forces on the nuclei at each time step
(same order as the coordinates, but given in atomic units).
- NucVeloc: Records the Cartesian velocities of the nuclei at each time
step (same order as the coordinates, but given in atomic units).
- TandV: Records the kinetic and potential energy, as well as fluctuations in each.
- View.xyz: An xyz-formatted version of NucCarts for viewing trajectories in an external visualization program (new in v.4.0).
For ELMD jobs, there are other output files as well:
- ChangeInF: Records the matrix norm and largest magnitude element of
∆F = F(t+δt) − F(t) in the basis of
Cholesky-orthogonalized AOs. The files ChangeInP, ChangeInL, and
ChangeInZ provide analogous information for the density matrix P
and the Cholesky orthogonalization matrices L and Z defined
in [190].
- DeltaNorm: Records the norm and largest magnitude element of the
curvy-steps rotation angle matrix ∆ defined in
Ref. . Matrix elements of ∆ are the
dynamical variables representing the electronic degrees of freedom. The
output file DeltaDotNorm provides the same information for the electronic
velocity matrix d∆/dt.
- ElecGradNorm: Records the norm and largest magnitude element of the
electronic gradient matrix FP − PF in the Cholesky basis.
- dTfict: Records the instantaneous time derivative of the fictitious kinetic
energy at each time step, in atomic units.
Ab initio molecular dynamics jobs are requested by specifying
JOBTYPE = AIMD. Initial velocities must be specified either using a
$velocity section or via the AIMD_INIT_VELOC keyword described
below. In addition, the following $rem variables must be specified for any
ab initio molecular dynamics job:
AIMD_METHOD
Selects an ab initio molecular dynamics algorithm. |
TYPE:
DEFAULT:
OPTIONS:
BOMD | Born-Oppenheimer molecular dynamics. |
CURVY | Curvy-steps Extended Lagrangian molecular dynamics. |
RECOMMENDATION:
BOMD yields exact classical molecular dynamics, provided that the energy is
tolerably conserved. ELMD is an approximation to exact classical dynamics whose
validity should be tested for the properties of interest. |
|
| TIME_STEP
Specifies the molecular dynamics time step, in atomic units (1 a.u. = 0.0242
fs). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Smaller time steps lead to better energy conservation; too large a time step
may cause the job to fail entirely. Make the time step as large as possible,
consistent with tolerable energy conservation. |
|
|
|
AIMD_STEPS
Specifies the requested number of molecular dynamics steps. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
Ab initio molecular dynamics calculations can be quite expensive, and
thus Q-Chem includes several algorithms designed to accelerate such
calculations. At the self-consistent field (Hartree-Fock and DFT) level, BOMD
calculations can be greatly accelerated by using information from previous time
steps to construct a good initial guess for the new molecular orbitals or Fock
matrix, thus hastening SCF convergence. A Fock matrix extrapolation
procedure [402], based on a suggestion by Pulay and Fogarasi [403],
is available for this purpose.
The Fock matrix elements Fμν in the atomic orbital basis are
oscillatory functions of the time t, and Q-Chem's extrapolation procedure
fits these oscillations to a power series in t:
The N+1 extrapolation coefficients cn are determined by a fit to a set of
M Fock matrices retained from previous time steps. Fock matrix extrapolation
can significantly reduce the number of SCF iterations required at each time
step, but for low-order extrapolations, or if SCF_CONVERGENCE is set
too small, a systematic drift in the total energy may be observed. Benchmark
calculations testing the limits of energy conservation can be found in
Ref. , and demonstrate that numerically exact classical
dynamics (without energy drift) can be obtained at significantly reduced cost.
Fock matrix extrapolation is requested by specifying values for N and M, as
in the form of the following two $rem variables:
FOCK_EXTRAP_ORDER
Specifies the polynomial order N for Fock matrix extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Fock matrix extrapolation. |
OPTIONS:
N | Extrapolate using an Nth-order polynomial (N > 0). |
RECOMMENDATION:
|
| FOCK_EXTRAP_POINTS
Specifies the number M of old Fock matrices that are retained for use in
extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Fock matrix extrapolation. |
OPTIONS:
M | Save M Fock matrices for use in extrapolation (M > N) |
RECOMMENDATION:
Higher-order extrapolations with more saved Fock matrices are faster and
conserve energy better than low-order extrapolations, up to a point. In many
cases, the scheme (N = 6, M = 12), in conjunction with
SCF_CONVERGENCE = 6, is found to provide about a 50% savings in
computational cost while still conserving energy. |
|
|
|
When nuclear forces are computed using underlying electronic
structure methods with non-optimized orbitals (such as MP2),
a set of response equations must be solved [404].
While these equations are linear, their dimensionality necessitates
an iterative solution [405,[406],
which, in practice, looks much like the SCF equations. Extrapolation is again useful
here [191],
and the syntax for Z-vector (response) extrapolation is similar to Fock extrapolation:
Z_EXTRAP_ORDER
Specifies the polynomial order N for Z-vector extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Z-vector extrapolation. |
OPTIONS:
N | Extrapolate using an Nth-order polynomial (N > 0). |
RECOMMENDATION:
|
| Z_EXTRAP_POINTS
Specifies the number M of old Z-vectors that are retained for use in extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform response equation extrapolation. |
OPTIONS:
M | Save M previous Z-vectors for use in extrapolation (M > N) |
RECOMMENDATION:
Using the default Z-vector convergence settings, a (4,2)=(M,N) extrapolation was shown to provide the greatest speedup. At this setting, a 2-3-fold reduction in iterations was demonstrated. |
|
|
|
Assuming decent conservation, a BOMD calculation represents exact classical
dynamics on the Born-Oppenheimer potential energy surface. In contrast,
so-called extended Lagrangian molecular dynamics (ELMD) methods make an
approximation to exact classical dynamics in order to expedite the
calculations. ELMD methods-of which the most famous is Car-Parrinello
molecular dynamics-introduce a fictitious dynamics for the electronic
(orbital) degrees of freedom, which are then propagated alongside the
nuclear degrees of freedom, rather than optimized at each time step as
they are in a BOMD calculation. The fictitious electronic dynamics is
controlled by a fictitious mass parameter μ, and the value of μ
controls both the accuracy and the efficiency of the method. In the limit of
small μ the nuclei and the orbitals propagate
adiabatically, and ELMD mimics true classical dynamics. Larger values of μ
slow down the electronic dynamics, allowing for larger time steps (and more
computationally efficient dynamics), at the expense of an ever-greater
approximation to true classical dynamics.
Q-Chem's ELMD algorithm is based upon propagating the density matrix,
expressed in a basis of atom-centered Gaussian orbitals, along
shortest-distance paths (geodesics) of the manifold of allowed density
matrices P. Idempotency of P is maintained at every time step, by
construction, and thus our algorithm requires neither density matrix
purification, nor iterative solution for Lagrange multipliers (to enforce
orthogonality of the molecular orbitals). We call this procedure "curvy
steps" ELMD [190], and in a sense it is a time-dependent
implementation of the GDM algorithm (Section 4.6) for
converging SCF single-point calculations.
The extent to which ELMD constitutes a significant approximation to BOMD
continues to be debated. When assessing the accuracy of ELMD, the primary
consideration is whether there exists a separation of time scales between
nuclear oscillations, whose time scale τnuc is set by the period
of the fastest vibrational frequency, and electronic oscillations, whose time
scale τelec may be estimated according to [190]
τelec ≥ | √
|
μ/(εLUMO −εHOMO )
|
|
| (9.2) |
A conservative estimate, suggested in Ref. , is that essentially exact
classical dynamics is attained when τnuc > 10 τelec.
In practice, we recommend careful benchmarking to insure that ELMD faithfully
reproduces the BOMD observables of interest.
Due to the existence of a fast time scale τelec, ELMD requires
smaller time steps than BOMD. When BOMD is combined with Fock matrix
extrapolation to accelerate convergence, it is no longer clear that ELMD
methods are substantially more efficient, at least in Gaussian basis
sets [402,[403].
The following $rem variables are required for ELMD jobs:
AIMD_FICT_MASS
Specifies the value of the fictitious electronic mass μ, in atomic units,
where μ has dimensions of (energy)×(time)2. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Values in the range of 50-200 a.u. have been employed in test calculations;
consult [190] for examples and discussion. |
|
Additional job control variables for ab initio molecular dynamics.
AIMD_INIT_VELOC
Specifies the method for selecting initial nuclear velocities. |
TYPE:
DEFAULT:
OPTIONS:
THERMAL | Random sampling of nuclear velocities from a Maxwell-Boltzmann |
| distribution. The user must specify the temperature in Kelvin via |
| the $rem variable AIMD_TEMP. |
ZPE | Choose velocities in order to put zero-point vibrational energy into |
| each normal mode, with random signs. This option requires that a |
| frequency job to be run beforehand. |
QUASICLASSICAL | Puts vibrational energy into each normal mode. In contrast to the |
| ZPE option, here the vibrational energies are sampled from a |
| Boltzmann distribution at the desired simulation temperature. This |
| also triggers several other options, as described below. |
RECOMMENDATION:
This variable need only be specified in the event that velocities are not
specified explicitly in a $velocity section. |
|
| AIMD_MOMENTS
Requests that multipole moments be output at each time step. |
TYPE:
DEFAULT:
0 | Do not output multipole moments. |
OPTIONS:
n | Output the first n multipole moments. |
RECOMMENDATION:
|
|
|
AIMD_TEMP
Specifies a temperature (in Kelvin) for Maxwell-Boltzmann velocity sampling. |
TYPE:
DEFAULT:
OPTIONS:
User-specified number of Kelvin. |
RECOMMENDATION:
This variable is only useful in conjunction with AIMD_INIT_VELOC =
THERMAL. Note that the simulations are run at constant energy, rather than
constant temperature, so the mean nuclear kinetic energy will fluctuate in the
course of the simulation. |
|
| DEUTERATE
Requests that all hydrogen atoms be replaces with deuterium. |
TYPE:
DEFAULT:
FALSE | Do not replace hydrogens. |
OPTIONS:
TRUE | Replace hydrogens with deuterium. |
RECOMMENDATION:
Replacing hydrogen atoms reduces the fastest vibrational frequencies by a
factor of 1.4, which allow for a larger fictitious mass and time step in ELMD
calculations. There is no reason to replace hydrogens in BOMD calculations. |
|
|
|
9.9.1 Examples
Example 9.0 Simulating thermal fluctuations of the water dimer at 298 K.
$molecule
0 1
O 1.386977 0.011218 0.109098
H 1.748442 0.720970 -0.431026
H 1.741280 -0.793653 -0.281811
O -1.511955 -0.009629 -0.120521
H -0.558095 0.008225 0.047352
H -1.910308 0.077777 0.749067
$end
$rem
JOBTYPE aimd
AIMD_METHOD bomd
EXCHANGE b3lyp
BASIS 6-31g*
TIME_STEP 20 (20 a.u. = 0.48 fs)
AIMD_STEPS 1000
AIMD_INIT_VELOC thermal
AIMD_TEMP 298
FOCK_EXTRAP_ORDER 6 request Fock matrix extrapolation
FOCK_EXTRAP_POINTS 12
$end
Example 9.0 Propagating F−(H2O)4 on its first excited-state
potential energy surface, calculated at the CIS level.
$molecule
-1 1
O -1.969902 -1.946636 0.714962
H -2.155172 -1.153127 1.216596
H -1.018352 -1.980061 0.682456
O -1.974264 0.720358 1.942703
H -2.153919 1.222737 1.148346
H -1.023012 0.684200 1.980531
O -1.962151 1.947857 -0.723321
H -2.143937 1.154349 -1.226245
H -1.010860 1.980414 -0.682958
O -1.957618 -0.718815 -1.950659
H -2.145835 -1.221322 -1.158379
H -1.005985 -0.682951 -1.978284
F 1.431477 0.000499 0.010220
$end
$rem
JOBTYPE aimd
AIMD_METHOD bomd
EXCHANGE hf
BASIS 6-31+G*
ECP SRLC
PURECART 1111
CIS_N_ROOTS 3
CIS_TRIPLETS false
CIS_STATE_DERIV 1 propagate on first excited state
AIMD_INIT_VELOC thermal
AIMD_TEMP 150
TIME_STEP 25
AIMD_STEPS 827 (500 fs)
$end
Example 9.0 Simulating vibrations of the NaCl molecule using ELMD.
$molecule
0 1
Na 0.000000 0.000000 -1.742298
Cl 0.000000 0.000000 0.761479
$end
$rem
JOBTYPE freq
EXCHANGE b3lyp
ECP sbkjc
$end
@@@
$molecule
read
$end
$rem
JOBTYPE aimd
EXCHANGE b3lyp
ECP sbkjc
TIME_STEP 14
AIMD_STEPS 500
AIMD_METHOD curvy
AIMD_FICT_MASS 360
AIMD_INIT_VELOC zpe
$end
9.9.2 AIMD with Correlated Wavefunctions
While the number of time steps required in most AIMD trajectories dictates
that economical (typically SCF-based) underlying electronic structure methods
are required, other methods are also now possible. Any method with available
analytic forces can be utilized for BOMD. Currently, Q-Chem can perform
AIMD simulations with HF, DFT, MP2, RI-MP2, CCSD, and CCSD(T). The RI-MP2 method,
especially when combined with Fock matrix and response equation extrapolation,
is particularly effective as an alternative to DFT-based dynamics.
9.9.3 Vibrational Spectra
The inherent nuclear motion of molecules is experimentally observed by the
molecules' response to impinging radiation. This response is typically calculated
within the mechanical and electrical harmonic approximations (second derivative
calculations) at critical-point structures. Spectra, including anharmonic effects,
can also be obtained
from dynamics simulations. These spectra are generated from dynamical response
functions, which involve the Fourier transform of autocorrelation functions.
Q-Chem can provide both the vibrational spectral density from the velocity
autocorrelation function
D(ω) ∝ | ⌠ ⌡
|
∞
−∞
|
dt e−iωt〈 |
→
v
|
(0)· |
→
v
|
(t)〉 |
| (9.3) |
and infrared absorption intensity from the dipole
autocorrelation function
I(ω) ∝ |
ω
2π
|
| ⌠ ⌡
|
∞
−∞
|
dt e−iωt〈 |
→
μ
|
(0)· |
→
μ
|
(t)〉 |
| (9.4) |
These two features are activated
by the AIMD_NUCL_VACF_POINTS and AIMD_NUCL_DACF_POINTS keywords,
respectively, where values indicate the number of data points to include in the
correlation function. Furthermore, the AIMD_NUCL_SAMPLE_RATE keyword
controls the frequency at which these properties are sampled (entered as number
of time steps). These spectra-generated at constant energy-should be averaged
over a suitable distribution of initial conditions. The averaging indicated in the
expressions above, for example, should be performed over a Boltzmann distribution
of initial conditions.
Note that dipole autocorrelation functions can exhibit contaminating information
if the molecule is allowed to rotate/translate. While the initial conditions in Q-Chem
remove translation and rotation, numerical noise in the forces and propagation
can lead to translation and rotation over time. The trans/rot correction in Q-Chem
is activated by the PROJ_TRANSROT keyword.
AIMD_NUCL_VACF_POINTS
Number of time points to utilize in the velocity autocorrelation function for an AIMD trajectory |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute velocity autocorrelation function. |
1 ≤ n ≤ AIMD_STEPS | Compute velocity autocorrelation function for last n |
| time steps of the trajectory. |
RECOMMENDATION:
If the VACF is desired, set equal to AIMD_STEPS. |
|
| AIMD_NUCL_DACF_POINTS
Number of time points to utilize in the dipole autocorrelation function for an AIMD trajectory |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute dipole autocorrelation function. |
1 ≤ n ≤ AIMD_STEPS | Compute dipole autocorrelation function for last n |
| timesteps of the trajectory. |
RECOMMENDATION:
If the DACF is desired, set equal to AIMD_STEPS. |
|
|
|
AIMD_NUCL_SAMPLE_RATE
The rate at which sampling is performed for the velocity and/or dipole autocorrelation function(s). Specified as a multiple of steps; i.e., sampling every step is 1. |
TYPE:
DEFAULT:
OPTIONS:
1 ≤ n ≤ AIMD_STEPS | Update the velocity/dipole autocorrelation function |
| every n steps. |
RECOMMENDATION:
Since the velocity and dipole moment are routinely calculated for ab initio methods,
this variable should almost always be set to 1 when the VACF/DACF are desired. |
|
| PROJ_TRANSROT
Removes translational and rotational drift during AIMD trajectories. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not apply translation/rotation corrections. |
TRUE | Apply translation/rotation corrections. |
RECOMMENDATION:
When computing spectra (see AIMD_NUCL_DACF_POINTS, for example), this option can be utilized to remove artificial, contaminating peaks stemming from translational and/or rotational motion. Recommend setting to TRUE for all dynamics-based spectral simulations. |
|
|
|
9.9.4 Quasi-Classical Molecular Dynamics
Molecular dynamics simulations based on quasi classical trajectories
(QCT-MD) [407,[408,[409]
put vibrational energy into each mode in the initial
velocity setup step. We (as well as others [410]) have found that this can improve
on the results of purely classical simulations, for example in the calculation
of photoelectron [411] or infrared spectra [412].
Improvements include better agreement of spectral linewidths with experiment at
lower temperatures or better agreement of vibrational frequencies with
anharmonic calculations.
The improvements at low temperatures can be understood by recalling that
even at low temperature
there
is nuclear motion
due to zero-point motion. This is
included in the quasi-classical initial velocities, thus leading to finite peak
widths even at low temperatures. In contrast to that the classical simulations
yield zero peak width in the low temperature limit, because the thermal kinetic
energy goes to zero as temperature decreases. Likewise, even at room temperature
the quantum vibrational energy for high-frequency modes is often significantly
larger than the classical kinetic energy. QCT-MD therefore
typically samples regions of the potential energy surface that are higher in
energy and thus more anharmonic than the low-energy regions accessible to
classical simulations. These two effects can lead to improved peak widths as well as
a more realistic sampling of the anharmonic parts of the potential energy
surface. However, the QCT-MD method also has important limitations which
are described below and that the user
has to monitor for carefully.
In our QCT-MD implementation the initial vibrational quantum numbers are
generated as random numbers sampled from a vibrational Boltzmann distribution
at the desired simulation temperature.
In order to enable reproducibility of the results, each trajectory (and
thus its set of vibrational quantum numbers)
is denoted by a unique number using the AIMD_QCT_WHICH_TRAJECTORY variable.
In order to loop over different initial conditions, run trajectories with
different choices for AIMD_QCT_WHICH_TRAJECTORY. It is also
possible to assign initial velocities corresponding to an average over a
certain number of trajectories by choosing a negative value.
Further technical details of our QCT-MD implementation are described in detail in
Appendix A of Ref. .
| AIMD_QCT_WHICH_TRAJECTORY
Picks a set of vibrational quantum numbers from a random distribution. |
TYPE:
DEFAULT:
OPTIONS:
n | Picks the nth set of random initial velocities. |
−n | Uses an average over n random initial velocities.
|
|
RECOMMENDATION:
Pick a positive number if you want the initial velocities to correspond
to a particular set of vibrational occupation numbers and choose a
different number for each of your trajectories. If initial velocities
are desired that corresponds to an average over n trajectories, pick a
negative number.
|
|
|
|
Below is a simple example input for running a QCT-MD simulation of the
vibrational spectrum of water. Most input variables are the same as for
classical MD as described above. The use of quasi-classical initial conditions is
triggered by setting the AIMD_INIT_VELOC variable to QUASICLASSICAL.
Example 9.0 Simulating the IR spectrum of water using QCT-MD.
$comment
Don't forget to run a frequency calculation prior to this job.
$end
$molecule
0 1
O 0.000000 0.000000 0.520401
H -1.475015 0.000000 -0.557186
H 1.475015 0.000000 -0.557186
$end
$rem
jobtype aimd
input_bohr true
exchange hf
basis 3-21g
scf_convergence 6
! AIMD input
time_step 20 (in atomic units)
aimd_steps 12500 6 ps total simulation time
aimd_temp 12
aimd_print 2
fock_extrap_order 6 Use a 6th-order extrapolation
fock_extrap_points 12 of the previous 12 Fock matrices
! IR spectral sampling
aimd_moments 1
aimd_nucl_sample_rate 5
aimd_nucl_vacf_points 1000
! QCT-specific settings
aimd_init_veloc quasiclassical
aimd_qct_which_trajectory 1 Loop over several values to get
the correct Boltzmann distribution.
$end
Other types of spectra can be calculated by calculating spectral
properties along the trajectories. For example, we observed that photoelectron
spectra can be approximated quite well by calculating vertical detachment energies (VDEs)
along the trajectories and generating the spectrum as a histogram of the
VDEs [411].
We have included several simple scripts in the $QC/aimdman/tools subdirectory
that we hope the user will find helpful and
that may serve as the basis for developing more sophisticated tools. For
example, we include scripts that allow to perform
calculations along a trajectory
(md_calculate_along_trajectory)
or to
calculate vertical detachment energies along a trajectory
(calculate_rel_energies).
Another application of the QCT code is to generate random geometries
sampled from the vibrational wavefunction via a Monte Carlo algorithm.
This is triggered by setting both the AIMD_QCT_INITPOS
and AIMD_QCT_WHICH_TRAJECTORY
variables to negative numbers, say −m and −n, and setting
AIMD_STEPS to zero. This will generate m random geometries
sampled from the vibrational wavefunction corresponding to an average
over n trajectories at the user-specified simulation temperature.
| AIMD_QCT_INITPOS
Chooses the initial geometry in a QCT-MD simulation. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use the equilibrium geometry. |
n | Picks a random geometry according to the harmonic vibrational wavefunction. |
−n | Generates n random geometries sampled from |
| the harmonic vibrational wavefunction. |
|
RECOMMENDATION:
|
|
|
For systems that are described
well within the harmonic oscillator model and for properties that rely
mainly on the ground-state
dynamics, this simple MC approach may yield qualitatively
correct spectra. In fact, one may argue that it is preferable over
QCT-MD
for describing vibrational effects at very low temperatures, since the
geometries are sampled from a true quantum distribution (as opposed to classical
and quasiclassical MD). We have included another script in the aimdman/tools directory
to help with the calculation of vibrationally averaged properties (monte_geom).
Example 9.0 MC sampling of the vibrational wavefunction for HCl.
$comment
Generates 1000 random geometries for HCl based on the harmonic vibrational
wavefunction at 1 Kelvin. The wavefunction is averaged over 1000
sets of random vibrational quantum numbers (\ie{}, the ground state in
this case due to the low temperature).
$end
$molecule
0 1
H 0.000000 0.000000 -1.216166
Cl 0.000000 0.000000 0.071539
$end
$rem
jobtype aimd
exchange B3LYP
basis 6-311++G**
scf_convergence 1
SKIP_SCFMAN 1
maxscf 0
xc_grid 1
time_step 20 (in atomic units)
aimd_steps 0
aimd_init_veloc quasiclassical
aimd_qct_vibseed 1
aimd_qct_velseed 2
aimd_temp 1 (in Kelvin)
! set aimd_qct_which_trajectory to the desired
! trajectory number
aimd_qct_which_trajectory -1000
aimd_qct_initpos -1000
$end
It is also possible make some modes inactive, i.e., to put vibrational energy into a subset of modes
(all other are set to zero).
The list of active modes can be specified using the $qct_active_modes section.
Furthermore, the vibrational quantum numbers for each mode can be
specified explicitly using the $qct_vib_distribution keyword. It is
also possible to set the phases using $qct_vib_phase (allowed values
are 1 and -1).
Below is a simple sample input:
Example 9.0 User control over the QCT variables.
$comment
Makes the 1st vibrational mode QCT-active; all other ones receive zero
kinetic energy. We choose the vibrational ground state and a positive
phase for the velocity.
$end
...
$qct_active_modes
1
$end
$qct_vib_distribution
0
$end
$qct_vib_phase
1
$end
...
Finally we turn to a brief description of the limitations of QCT-MD.
Perhaps the most severe limitation stems from
the so-called "kinetic energy spilling problem"
(see, e.g., Ref. ), which
means that there can be an artificial transfer of kinetic energy between modes.
This can happen because the initial velocities are chosen according to
quantum energy levels, which are usually much higher than those of the
corresponding classical systems. Furthermore, the classical equations of
motion also allow for the transfer of non-integer multiples of the
zero-point energy between the modes, which leads to different selection
rules for the transfer of kinetic energy. Typically, energy spills from
high-energy into low-energy modes, leading to spurious "hot" dynamics.
A second problem is that QCT-MD is actually based on classical Newtonian
dynamics, which
means that the probability distribution at low temperatures can be qualitatively
wrong compared to the true quantum distribution [411].
We have implemented a routine that monitors the kinetic energy within each
normal mode along the trajectory and that is automatically switched on for quasiclassical
simulations. It is thus possible to monitor for trajectories in which
the kinetic energy in one or more modes becomes significantly larger
than the initial energy. Such trajectories should be discarded (see
Ref. for a different approach to the zero-point leakage
problem). Furthermore, this monitoring routine prints the squares of the
(harmonic) vibrational wavefunction along the trajectory. This makes it
possible to weight low-temperature results with the harmonic quantum
distribution to alleviate the failure of classical dynamics for low
temperatures.
9.10 Ab initio Path Integrals
Even in cases where the Born-Oppenheimer separation is valid, solving the
electronic Schroedinger equation- Q-Chem's main purpose-is still
only half the battle. The remainder involves the solution of the
nuclear Schroedinger equation for its resulting eigenvalues/functions.
This half is typically treated by the harmonic approximation at critical points,
but anharmonicity, tunneling, and low-frequency "floppy" motion can
lead to extremely delocalized nuclear distributions, particularly for
protons and non-covalently bonded systems.
While the Born-Oppenheimer separation allows for a local solution of the
electronic problem (in nuclear space), the nuclear half of the Schroedinger
equation is entirely non-local and requires the computation of potential
energy surfaces over large regions of configuration space. Grid-based methods,
therefore, scale exponentially with the number of degrees of freedom, and
are quickly rendered useless for all but very small molecules.
For thermal, equilibrium distributions, the path integral (PI) formalism
of Feynman provides both an elegant and computationally feasible alternative.
The equilibrium partition function, for example, may be written as a trace
of the thermal, configuration-space density matrix:
Solving for the partition function directly in this form is equally difficult,
as it still requires the eigenvalues / eigenstates of ∧H.
By inserting N−1 resolutions of the identity, however, this integral
may be converted to the following form
Z = | ⌠ ⌡
|
dx1 | ⌠ ⌡
|
dx2 … | ⌠ ⌡
|
dxN ρ | ⎛ ⎝
|
x1,x2; |
β
N
| ⎞ ⎠
|
ρ | ⎛ ⎝
|
x2,x3; |
β
N
| ⎞ ⎠
|
…ρ | ⎛ ⎝
|
xN,x1; |
β
N
| ⎞ ⎠
|
|
| (9.6) |
While this additional integration appears to be a detriment, the ability to
use a high-temperature ([(β)/N]) form of the density matrix
ρ | ⎛ ⎝
|
x,x′; |
β
N
| ⎞ ⎠
|
= |
⎛ √
|
|
e−[mN/(2βħ2)](x−x′)2 − [(β)/N]([(V(x)+V(x′))/2]) |
| (9.7) |
renders this path-integral formulation a net win. By combining the N time slices,
the partition function takes the following form (in 1-D):
| |
|
| ⎛ ⎝
|
mN
2πβħ2
| ⎞ ⎠
|
[N/2]
|
| ⌠ ⌡
|
dx1 | ⌠ ⌡
|
dx2 … | ⌠ ⌡
|
dxN e−[(β)/N][ [(mN2)/(2β2ħ2)]∑iN (xi − xi+1)2 + ∑iN V(xi)] |
| |
| |
|
| | (9.8) |
|
with the implied cyclic condition xN+1 = x1. Here, V(x) is the potential function
on which the "beads" move (the electronic potential generated by Q-Chem).
The resulting integral, as shown in the last line above, is nothing more
than a classical configuration integral in an N-dimensional space. The
effective potential appearing above describes an N-bead "ring polymer," of which
neighboring beads are harmonically coupled.
The exponentially scaling, non-local nuclear problem has, therefore, been mapped
onto an entirely classical problem, which is amenable to standard treatments
of configuration sampling. These methods typically involve (thermostated)
molecular dynamics or Monte Carlo sampling; only the latter is currently
implemented in Q-Chem. Importantly, N is reasonably small when the
temperature is not too low: room-temperature systems involving H atoms typically
are converged with roughly 30 beads. Therefore, fully quantum mechanical nuclear
distributions may be obtained at a cost only roughly 30 times a classical simulation.
Path integral Monte Carlo (PIMC) is an entirely new job type in Q-Chem and
is activated by setting JOBTYP to PIMC.
9.10.1 Classical Sampling
The 1-bead limit of the above expressions is simply classical configuration sampling.
When the temperature (controlled by the PIMC_TEMP keyword) is high or only heavy atoms are involved, the classical limit
is often appropriate. The path integral machinery (with 1 "bead") may be utilized
to perform classical Boltzmann sampling. The 1D partition function, for example,
is simply
9.10.2 Quantum Sampling
Using more beads includes more quantum mechanical delocalization (at a cost
of roughly N times the classical analog). This main
input variable-the number of time slices (beads)-is controlled by the
PIMC_NBEADSPERATOM keyword. The ratio of the inverse temperature
to beads ([(β)/N]) dictates convergence with respect
to the number of beads, so as the temperature is lowered, a concomitant
increase in the number of beads is required.
Integration over configuration space is performed by Metropolis Monte Carlo (MC). The
number of MC steps is controlled by the PIMC_MCMAX keyword and should
typically be at least ≈ 105, depending on the desired level of statistical
convergence. A warmup run, in which the ring polymer is allowed to equilibrate
(without accumulating statistics) can be performed by setting the
PIMC_WARMUP_MCMAX keyword.
Much like ab initio molecular dynamics simulations, the main results
of PIMC jobs in Q-Chem are not in the job output file. Rather, they are
compiled in the "PIMC" subdirectory of the user's scratch directory
($QCSCRATCH/PIMC). Therefore,
PIMC jobs should always be run with the -save option. The output files do contain some useful information, however, including a basic data analysis of
the simulation. Average energies (thermodynamic estimator), bond lengths (less than 5Å), bond length standard deviations and errors are printed at the end of the
output file. The $QCSCRATCH/PIMC directory additionally contains the following files:
- BondAves: running average of bond lengths for convergence testing.
- BondBins: normalized distribution of significant bond lengths, binned within 5 standard deviations of the average bond length.
- ChainCarts: human-readable file of configuration coordinates, likely to be used for further, external statistical analysis. This file can get quite large, so be sure to provide enough scratch space!
- ChainView.xyz: xyz-formatted file for viewing the ring-polymer sampling in an external visualization program. (The sampling is performed such that the center of mass of the ring polymer system remains centered.)
- Vcorr: potential correlation function for the assessment of statistical correlations in the sampling.
In each of the above files, the first few lines contain a description of the ordering of the data.
One of the unfortunate rites of passage in PIMC usage is the realization of the ramifications of the stiff bead-bead interactions as convergence (with respect to N) is approached. Nearing convergence-where quantum mechanical results are correct-the length of statistical correlations grows enormously, and special sampling techniques are required to avoid long (or non-convergent) simulations. Cartesian displacements or normal-mode displacements of the ring polymer lead to this severe stiffening. While both of these naive sampling schemes are available in Q-Chem, they are not recommended. Rather, the free-particle (harmonic bead-coupling) terms in the path integral action can be sampled directly. Several schemes are available for this purpose. Q-Chem currently utilizes the simplest of these options: Levy flights. An n-bead snippet (n < N) of the ring polymer is first chosen at random, with the length controlled by the PIMC_SNIP_LENGTH keyword. Between the endpoints of this snippet, a free-particle path is generated by a Levy construction, which exactly samples the free-particle part of the action. Subsequent Metropolis testing of the resulting potential term-for which only the potential on the moved beads is required-then dictates acceptance.
Two measures of the sampling efficiency are provided in the job output file. The lifetime of the potential autocorrelation function 〈V0Vτ〉 is provided in terms of the number of MC steps, τ. This number indicates the number of configurations that are statically correlated. Similarly, the mean-square displacement between MC configurations is also provided. Maximizing this number and/or minimizing the statistical lifetime leads to efficient sampling. Note that the optimally efficient acceptance rate may not be 50% in MC simulations. In Levy flights, the only variable controlling acceptance and sampling efficiency is the length of the snippet. The statistical efficiency can be obtained from relatively short runs, during which the length of the Levy snippet should be optimized by the user.
PIMC_NBEADSPERATOM
Number of path integral time slices ("beads") used on each atom of a PIMC simulation. |
TYPE:
DEFAULT:
OPTIONS:
1 | Perform classical Boltzmann sampling. |
> 1 | Perform quantum-mechanical path integral sampling. |
RECOMMENDATION:
This variable controls the inherent convergence of the path integral simulation. The 1-bead limit is purely classical sampling; the infinite-bead limit is exact quantum mechanical sampling. Using 32 beads is reasonably converged for room-temperature simulations of molecular systems. |
|
| PIMC_TEMP
Temperature, in Kelvin (K), of path integral simulations. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of Kelvin for PIMC or classical MC simulations. |
RECOMMENDATION:
|
|
|
PIMC_MCMAX
Number of Monte Carlo steps to sample. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of steps to sample. |
RECOMMENDATION:
This variable dictates the statistical convergence of MC/PIMC simulations. Recommend setting to at least 100000 for converged simulations. |
|
| PIMC_WARMUP_MCMAX
Number of Monte Carlo steps to sample during an equilibration period of MC/PIMC simulations. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of steps to sample. |
RECOMMENDATION:
Use this variable to equilibrate the molecule/ring polymer before collecting production statistics. Usually a short run of roughly 10% of PIMC_MCMAX is sufficient. |
|
|
|
PIMC_MOVETYPE
Selects the type of displacements used in MC/PIMC simulations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Cartesian displacements of all beads, with occasional (1%) center-of-mass moves. |
1 | Normal-mode displacements of all modes, with occasional (1%) center-of-mass moves. |
2 | Levy flights without center-of-mass moves. |
RECOMMENDATION:
Except for classical sampling (MC) or small bead-number quantum sampling (PIMC), Levy flights should be utilized. For Cartesian and normal-mode moves, the maximum displacement is adjusted
during the warmup run to the desired acceptance rate (controlled by PIMC_ACCEPT_RATE). For Levy flights, the acceptance is solely controlled by PIMC_SNIP_LENGTH. |
|
| PIMC_ACCEPT_RATE
Acceptance rate for MC/PIMC simulations when Cartesian or normal-mode displacements are utilized. |
TYPE:
DEFAULT:
OPTIONS:
0 < n < 100 | User-specified rate, given as a whole-number percentage. |
RECOMMENDATION:
Choose acceptance rate to maximize sampling efficiency, which is typically signified by the mean-square displacement (printed in the job output). Note that the maximum displacement is adjusted during the warmup run to achieve roughly this acceptance rate. |
|
|
|
PIMC_SNIP_LENGTH
Number of "beads" to use in the Levy flight movement of the ring polymer. |
TYPE:
DEFAULT:
OPTIONS:
3 ≤ n ≤ PIMC_NBEADSPERATOM | User-specified length of snippet. |
RECOMMENDATION:
Choose the snip length to maximize sampling efficiency. The efficiency can be estimated by the mean-square displacement between configurations, printed at the end of the output file. This efficiency will typically, however, be a trade-off between the mean-square displacement (length of statistical correlations) and the number of beads moved. Only the moved beads require recomputing the potential, i.e., a call to Q-Chem for the electronic energy.
(Note that the endpoints of the snippet remain fixed during a single move, so n−2 beads are actually moved for a snip length of n. For 1 or 2 beads in the simulation, Cartesian moves should be used instead.) |
|
9.10.3 Examples
Example 9.0 Path integral Monte Carlo simulation of H2 at room temperature
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE pimc
EXCHANGE hf
BASIS sto-3g
PIMC_TEMP 298
PIMC_NBEADSPERATOM 32
PIMC_WARMUP_MCMAX 10000 !Equilibration run
PIMC_MCMAX 100000 !Production run
PIMC_MOVETYPE 2 !Levy flights
PIMC_SNIP_LENGTH 10 !Moves 8 beads per MC step (10-endpts)
$end
Example 9.0 Classical Monte Carlo simulation of a water molecule at 500K
$molecule
0 1
H
O 1 1.0
H 2 1.0 1 104.5
$end
$rem
JOBTYPE pimc
EXCHANGE hf
CORRELATION rimp2
BASIS cc-pvdz
AUX_BASIS rimp2-cc-pvdz
PIMC_TEMP 500
PIMC_NBEADSPERATOM 1 !1 bead is classical sampling
PIMC_WARMUP_MCMAX 10000 !Equilibration run
PIMC_MCMAX 100000 !Production run
PIMC_MOVETYPE 0 !Cartesian displacements (ok for 1 bead)
PIMC_ACCEPT_RATE 40 !During warmup, adjusts step size to 40% acceptance
$end
9.11 Q-CHEM / CHARMM Interface
Q-Chem can perform hybrid quantum mechanics/molecular mechanics (QM/MM)
calculations either as a stand-alone program or in conjunction with the
CHARMM package [414]. In the latter case, which is described
in this section, both software
packages are required to perform the calculations, but all the code required for
communication between the programs is incorporated in the released versions.
Stand-alone QM / MM calculations are described in Section 9.12.
QM/MM jobs that utilize the CHARMM interface are
controlled using the following $rem keywords:
QM_MM
Turns on the Q-Chem/ CHARMM interface. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Do QM/MM calculation through the Q-Chem/ CHARMM interface. |
FALSE | Turn this feature off. |
RECOMMENDATION:
Use default unless running calculations with CHARMM. |
|
| QMMM_PRINT
Controls the amount of output printed from a QM/MM job. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Limit molecule, point charge, and analysis printing. |
FALSE | Normal printing. |
RECOMMENDATION:
Use default unless running calculations with CHARMM. |
|
|
|
QMMM_CHARGES
Controls the printing of QM charges to file. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Writes a charges.dat file with the Mulliken charges from the QM
region. |
FALSE | No file written. |
RECOMMENDATION:
Use default unless running calculations with CHARMM where charges on the QM
region need to be saved. |
|
| IGDEFIELD
Triggers the calculation of the electrostatic potential and/or the electric field
at the positions of the MM charges. |
TYPE:
DEFAULT:
OPTIONS:
O | Computes ESP. |
1 | Computes ESP and EFIELD. |
2 | Computes EFIELD. |
RECOMMENDATION:
Must use this $rem when IGDESP is specified. |
|
|
|
GEOM_PRINT
Controls the amount of geometric information printed at each step. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Prints out all geometric information; bond distances, angles, torsions. |
FALSE | Normal printing of distance matrix. |
RECOMMENDATION:
Use if you want to be able to quickly examine geometric parameters at the
beginning and end of optimizations. Only prints in the beginning of single
point energy calculations. |
|
| QMMM_FULL_HESSIAN
Trigger the evaluation of the full QM/MM Hessian. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluates full Hessian. |
FALSE | Hessian for QM-QM block only. |
RECOMMENDATION:
|
|
|
LINK_ATOM_PROJECTION
Controls whether to perform a link-atom projection |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Performs the projection |
FALSE | No projection |
RECOMMENDATION:
Necessary in a full QM/MM Hessian evaluation on a system with link atoms |
|
| HESS_AND_GRAD
Enables the evaluation of both analytical gradient and Hessian in a single job |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluates both gradient and Hessian. |
FALSE | Evaluates Hessian only. |
RECOMMENDATION:
Use only in a frequency (and thus Hessian) evaluation. |
|
|
|
GAUSSIAN_BLUR
Enables the use of Gaussian-delocalized external charges in a QM/MM calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Delocalizes external charges with Gaussian functions. |
FALSE | Point charges |
RECOMMENDATION:
|
Example 9.0 Do a basic QM/MM optimization of the water dimer. You need
CHARMM to do this but this is the Q-Chem file that is needed to test the
QM/MM functionality.
These are the bare necessities for a Q-Chem/ CHARMM QM/MM calculation.
$molecule
0 1
O -0.91126 1.09227 1.02007
H -1.75684 1.51867 1.28260
H -0.55929 1.74495 0.36940
$end
$rem
EXCHANGE hf ! HF Exchange
BASIS cc-pvdz ! Correlation Consistent Basis
QM_MM true ! Turn on QM/MM calculation
JOBTYPE force ! Need this for QM/MM optimizations
$end
$external_charges
1.20426 -0.64330 0.79922 -0.83400
1.01723 -1.36906 1.39217 0.41700
0.43830 -0.06644 0.91277 0.41700
$end
The Q-Chem/ CHARMM interface is unique in that:
- The external point charges can be replaced with Gaussian-delocalized charges
with a finite width [415].
This is an empirical way to include the delocalized character of the electron density of atoms in the MM region.
This can be important for the electrostatic interaction of the QM region with nearby atoms in the MM region.
- We allow the evaluation of the full QM/MM Hessian [416].
When link atoms are inserted to saturate the QM region,
all Hessian elements associated with link atoms are automatically
projected onto their QM and MM host atoms.
- For systems with a large number of MM atoms, one can define blocks consisting of multiple MM atoms
(i.e., mobile blocks) and efficiently evaluate the corresponding mobile-block Hessian (MBH) for normal mode analysis.
9.12 Stand-Alone QM / MM calculations
Q-Chem 4.0 introduces the capability of performing MM and QM/MM calculations internally,
without the need for a separate MM program. The features provided with this implementation
are limited at present, but are expected to grow in future releases.
9.12.1 Available QM / MM Methods and Features
Three modes of operation are available:
- MM calculations only (no QM)
- QM/MM calculations using a two-layer ONIOM model with mechanical embedding
- QM/MM calculations using the Janus model for electronic embedding
Q-Chem can carry out purely MM calculations, wherein the entire molecular
system is described by a MM force field and no electronic structure calculation is
performed. The MM force fields available are present are AMBER [417],
CHARMM [418], and OPLSAA [419].
As implemented in Q-Chem, the ONIOM model [420] is a mechanical embedding
scheme that partitions a molecular system into two subsystems (layers): an MM subsystem and
a QM subsystem. The total energy of an ONIOM system is given by
Etotal = EtotalMM − EQMMM + EQMQM |
| (9.10) |
where EtotalMM is the MM energy of the total system (i.e., QM + MM
subsystems), EQMMM is the MM energy of the QM subsystem, and
EQMQM is the QM energy of the QM subsystem. MM energies are computed via
a specified MM force field, and QM energies are computed
via a specified electronic structure calculation.
The advantage of the ONIOM model is its simplicity, which allows for straightforward application
to a wide variety of systems. A disadvantage of this approach, however, is that
QM subsystem does not interact directly with the MM subsystem. Instead, such
interactions are incorporated indirectly, in the EtotalMM contribution to the total
energy. As a result, the QM electron density is not polarized by the
electrostatic charges of the MM subsystem.
If the QM/MM interface partitions the two subsystems across a chemical bond, a link atom
(hydrogen) must be introduced to act as a cap for the QM subsystem. Currently, Q-Chem
supports only carbon link atoms, of atom type 26, 35, and 47 in the CHARMM27 force field.
The Janus model [421] is an electronic embedding scheme that also partitions
the system into MM and QM subsystems, but is more versatile than the ONIOM model. The Janus
model in Q-Chem is based upon the "YinYang atom" model of Shao and Kong [422].
In this approach, the total energy of the system is simply the sum of the subsystem
energies,
The MM subsystem energy, EMM, includes van der Waals interactions
between QM and MM atoms but not QM/MM Coulomb interactions.
Rather, EQM includes the direct Coulomb potential between
QM atoms and MM atoms as external charges during the QM calculation, thus
allowing the QM electron density to be polarized by the MM atoms. Because of this,
Janus is particularly well suited (as compared to ONIOM) for carrying out excited-state
QM/MM calculations, for excited states of a QM model system embedded within the
electrostatic environment of the MM system. Within a Janus calculation, Q-Chem first
computes EMM with the specified force field and then computes EQM
with the specified electronic structure theory.
When the Janus QM/MM partition cuts across a chemical bond, a YinYang atom [422]
is automatically introduced by Q-Chem. This atom acts as a hydrogen cap in the QM calculation, yet
also participates in MM interactions. To retain charge neutrality of the total
system, the YinYang atom has a single electron and a modified nuclear charge in
the QM calculation, equal to qnuclear = 1 + qMM (i.e., the charge of a proton plus the
charge on the YinYang atom in the MM subsystem).
Because this modified charge will affect the bond containing the YinYang
atom, an additional repulsive Coulomb potential is applied between the YinYang
atom and its connecting QM atom to maintain a desirable bond length. The
additional repulsive Coulomb energy is added to EMM.
The YinYang atom can be an atom of any kind, but it is highly recommended to
use carbon atoms as YinYang atoms.
Q-Chem's stand-alone QM/MM capabilities also include the following features:
- Analytic QM/MM gradients are available for QM subsystems described with
density functional theory (DFT) or Hartree-Fock (HF) electronic structure theory,
allowing for geometry optimizations and QM/MM molecular dynamics.
- Single-point QM/MM energy evaluations are available for QM subsystems described with
most post-HF correlated wavefunctions.
- Single-point QM/MM calculations are available for excited states of the QM subsystem,
where the latter may be described
using CIS, TDDFT, or correlated wavefunction models. Analytic gradients for excited
states are available for QM / MM calculations if the QM subsystem is described using CIS.
- Implicit solvation for both Janus QM/MM calculations as well as MM-only calculations
is available using the Polarizable Continuum Models (PCMs) discussed in
Section 10.2.2.
- Gaussian blurring of MM external charges is available for Janus QM / MM calculations.
- The user may add new MM atoms types and MM parameters.
- The user may define his/her own force field.
9.12.2 Using the Stand-Alone QM / MM Features
9.12.2.1 $molecule section
To perform QM/MM calculations, the user must assign MM atom types for
each atom in the $molecule section. The format for this specification is modeled upon
that used by the Tinker molecular modeling package [423], although the Tinker program is
not required to perform QM/MM calculations using Q-Chem. Force field parameters and MM atom
type numbers used within Q-Chem are identical to those used Tinker for the
AMBER99, CHARMM27, and OPLSAA force fields, and the format
of the force field parameters files is also the same.
The $molecule section must use Cartesian coordinates to define the molecular geometry
for internal QM/MM calculations; the Z-matrix format is not valid. MM atom types are
specified in the $molecule section immediately after the Cartesian coordinates on a line
so that the general format for the $molecule section is
$molecule
<Charge> <Multiplicity>
<Atom> <X> <Y> <Z> <MM atom type>
. . .
$end
For example, one can define a TIP3P water molecule using AMBER99 atom types, as follows:
$molecule
0 1
O -0.790909 1.149780 0.907453 2001
H -1.628044 1.245320 1.376372 2002
H -0.669346 1.913705 0.331002 2002
$end
When the input is specified as above, Q-Chem will determine the MM bond connectivity
based on the distances between atoms; if two atoms are sufficiently close,
they are considered to be bonded. Occasionally this approach can
lead to problems when non-bonded atoms are in close proximity of one another, in which
case Q-Chem might classify them as bonded regardless of whether the appropriate MM bond
parameters are available. To avoid such a scenario, the user can specify the
bonds explicitly by setting the $rem variable USER_CONNECT = TRUE,
in which case the $molecule section must have the following format
$molecule
<Charge> <Multiplicity>
<Atom> <X> <Y> <Z> <MM atom type> <Bond 1> <Bond 2> <Bond 3> <Bond 4>
. . .
$end
Each <Bond #> is the index of an atom to which <Atom> is bonded. Four bonds
must be specified for each atom, even if that atom is connected to fewer than four other
atoms. (For non-existent bonds, use zero as a placeholder.) Currently, Q-Chem
supports no more than four MM bonds per atom.
After setting USER_CONNECT = TRUE, a TIP3P water molecule in the AMBER99
force field could be specified as follows:
$molecule
0 1
O -0.790909 1.149780 0.907453 2001 2 3 0 0
H -1.628044 1.245320 1.376372 2002 1 0 0 0
H -0.669346 1.913705 0.331002 2002 1 0 0 0
$end
Explicitly defining the bonds in this way is highly recommended.
9.12.2.2 $force_field_params section
In many cases, all atoms types (within both the QM and MM subsystems) will be
defined by a given force field. In certain cases, however, a particular atom type
may not be defined in a given force field. For example, a QM/MM calculation on the
propoxide anion might consist of a QM subsystem containing an alkoxide functional group,
for which MM parameters do not exist. Even though the alkoxide moiety is described using
quantum mechanics, van der Waals parameters are nominally required for atoms within the QM
subsystem, which interact with the MM atoms via Lennard-Jones-type interactions.
In such cases, there are four possible options, the choice of which is left to the
user's discretion:
- Use a similar MM atom type as a substitute for the missing atom type.
- Ignore the interactions associated with the missing atom type.
- Define a new MM atom type and associated parameters.
- Define a new force field.
These options should be applied with care.
Option 1 involves selecting an atom type that closely resembles the
undefined MM atom. For example, the oxygen atom of an alkoxide moiety
could perhaps use the MM atom type corresponding to the oxygen atom of a neutral
hydroxyl group. Alternatively, the atom type could be ignored altogether (option 2) by
specifying MM atom type 0 (zero). Setting the atom type to zero should be accompanied
with setting all four explicit bond connections to placeholders if USER_CONNECT
= TRUE. An atom type of zero will cause all MM energies involving that atom to be zero.
The third option in the list above requires the user to specify a $force_field_params section in
the Q-Chem input file. This input section can be used to add new MM atom type definitions
to one of Q-Chem's built-in force fields. At a minimum, the user must specify
the atomic charge and two Lennard-Jones parameters (radius and well
depth, ϵ), for each new MM atom type. Bond, angle, and torsion parameters
for stretches, bends, and torsions involving the new atom type may also be specified, if desired.
The format for the $force_field_params input section is
$force_field_params
NumAtomTypes <n>
AtomType -1 <Charge> <LJ Radius> <LJ Epsilon>
AtomType -2 <Charge> <LJ Radius> <LJ Epsilon>
. . .
AtomType -n <Charge> <LJ Radius> <LJ Epsilon>
Bond <a> <b> <Force constant> <Equilibrium Distance>
. . .
Angle <a> <b> <c> <Force constant> <Equilibrium Angle>
. . .
Torsion <a> <b> <c> <d> <Force constant> <Phase Angle> <Multiplicity>
. . .
$end
The first line in this input section specifies how many new MM atom types appear in this
section (<n>). These are specified on the following lines labeled with the
AtomType tag. The atom type numbers are required to be negative and to appear in
the order −1, −2, −3, …, −n. The $molecule section for a water molecule, with
user-defined MM parameters for both oxygen and hydrogen, might appear as follows:
$molecule
0 1
O -0.790909 1.149780 0.907453 -1 2 3 0 0
H -1.628044 1.245320 1.376372 -2 1 0 0 0
H -0.669346 1.913705 0.331002 -2 1 0 0 0
$end
The remainder of each AtomType line in the $force_field_params section consists of a charge
(in elementary charge units), a Lennard-Jones radius (in Å), and a Lennard-Jones well
depth (ϵ, in kcal/mol).
Each (optional) Bond line in the $force_field_params section defines bond-stretching parameters
for a bond that contains a new MM atom type. The bond may consist of both atoms <a> and
<b> defined an AtomType line, or else <a> may be defined with an
AtomType line and <b> defined as a regular atom type for the force field.
In the latter case, the label for <b> should be the number of its general van der Waals
type. For example, the atom type for a TIP3P oxygen in AMBER99 is 2001, but its van der Waals
type is 21, so the latter would be specified in the Bond line.
The remaining entries of each Bond line are the harmonic force constant, in
kcal/mol/Å2, and the equilibrium distance, in Å.
Similar to the Bond lines, each (optional) Angle line consists of one or more
new atom types along with existing van der Waals types.
The central atom of the angle is <b>. The harmonic force constant (in units of
kcal/mol/degree) and equilibrium bond angle (in degrees) are the final entries in each
Bond line.
Each (optional) Torsion line consists of one or more new MM atom types
along with regular van der Waals types. The connectivity of the four atoms that constitute
the dihedral angle is <a>-<b>-<c>-<d>, and the torsional
potential energy function is
Etorsion(θ) = ktorsion [ 1 + cos( m θ− ϕ)] |
| (9.12) |
The force constant (ktorsion) is specified in kcal/mol and the phase angle (ϕ) in
degrees. The multiplicity (m) is an integer.
9.12.2.3 User-defined force fields
Option 4 in the list on page pageref is the most versatile, and allows the user to define
a completely new force field. This option is selected by setting FORCE_FIELD = READ,
which tells Q-Chem to read force field parameters from a text file whose name is specified in the
$force_field_params section as follows:
$force_field_params
Filename <path/filename>
$end
Here, <path/filename> is the full (absolute) path and name of the file that Q-Chem will attempt to read
for the MM force field. For example, if the user has a file named MyForceField.prm that
resides in the path /Users/me/parameters/, then this would be specified as
$force_field_params
Filename /Users/me/parameters/MyForceField.prm
$end
Within the force field file, the user should first declare various rules
that the force field will use, including how van der Waals interactions will be treated,
scaling of certain interactions, and the type
of improper torsion potential. The rules are declared in the file as follows:
RadiusRule <option>
EpsilonRule <option>
RadiusSize <option>
ImptorType <option>
vdw-14-scale <x>
chg-14-scale <x>
torsion-scale <x>
Currently, only a Lennard-Jones potential is available for van der Waals interactions.
RadiusRule and EpsilonRule control how to average σ and
ϵ, respectively, between atoms A and B in their Lennard-Jones potential.
The options available for both of these rules are Arithmetic [e.g.,
σAB = (σA + σB)/2] or Geometric
[e.g., σAB = (σA σB)1/2]. RadiusSize has options
Radius or Diameter, which specify whether the parameter σ is
the van der Waals radius or diameter in the Lennard-Jones potential.
ImptorType controls the type of potential to be used for improper torsion
(out-of-plane bending) energies, and has two options: Trigonometric or Harmonic.
These options are described in more detail below.
The scaling rules takes a floating point argument <x>. The vdw-14-scale and
chg-14-scale rules only affect van der Waals and Coulomb interactions, respectively,
between atoms that are separated by three consecutive bonds (atoms 1 and 4 in the
chain of bonds). These interaction energies will be scaled by <x>. Similarly,
torsion-scale scales dihedral angle torsion energies.
After declaring the force field rules, the number of MM atom types and van der Waals types
in the force field must be specified using:
where <n> is a positive integer.
Next, the atom types, van der Waals types, bonds, angles,
dihedral angle torsion, improper torsions, and Urey-Bradley parameters can be
declared in the following format:
Atom 1 <Charge> <vdw Type index> <Optional description>
Atom 2 <Charge> <vdw Type index> <Optional description>
. . .
Atom <NAtom> <Charge> <vdw Type index> <Optional description>
. . .
vdw 1 <Sigma> <Epsilon> <Optional description>
vdw 2 <Sigma> <Epsilon> <Optional description>
. . .
vdw <Nvdw> <Sigma> <Epsilon> <Optional description>
. . .
Bond <a> <b> <Force constant> <Equilibrium Distance>
. . .
Angle <a> <b> <c> <Force constant> <Equilibrium Angle>
. . .
Torsion <a> <b> <c> <d> <Force constant 1> <Phase Angle 1> <Multiplicity 1>
. . .
Improper <a> <b> <c> <d> <Force constant> <Equilibrium Angle> <Multiplicity>
. . .
UreyBrad <a> <b> <c> <Force constant> <Equilibrium Distance>
The parameters provided in the force field parameter file correspond to a basic MM energy
functional of the form
EMM = ECoul + EvdW + Ebond + Eangle + Etorsion + Eimptor + EUreyBrad |
| (9.13) |
Coulomb and van der Waals interactions are computed for all non-bonded pairs of atoms
that are at least three consecutive bonds apart (i.e., 1-4 pairs and more distant pairs).
The Coulomb energy between atom types 1 and 2 is simply
where q1 and q2 are the respective charges on the atoms (specified with <Charge> in
elementary charge units) and r12 is the distance between the two atoms. For 1-4 pairs,
fscale is defined with chg-14-scale but is unity for all other valid pairs.
The van der Waals energy between two atoms with van der Waals types a and b, and
separated by distance rab, is given by a "6-12" Lennard-Jones potential:
EvdW(rab) = fscale ϵab | ⎡ ⎣
| ⎛ ⎝
|
σab
rab
| ⎞ ⎠
|
12
|
− 2 | ⎛ ⎝
|
σab
rab
| ⎞ ⎠
|
6
| ⎤ ⎦
|
|
| (9.15) |
Here, fscale is the scaling factor for 1-4 interactions defined with vdw-14-scale
and is unity for other valid interactions.
The quantities ϵab and σab are the averages of the parameters of atoms a and b
as defined with EpsilonRule and RadiusRule, respectively (see above). The units
of <Sigma> are Å , and the units of <Epsilon> are kcal/mol. Hereafter, we
refer to atoms' van der Waals types with a, b, c, ... and atoms' charges with 1,2, 3, ....
The bond energy is a harmonic potential,
Ebond(rab) = kbond (rab − req)2 |
| (9.16) |
where kbond is provided by <Force Constant> in kcal/mol/Å2 and req
by <Equilibrium Distance> in Å. Note that <a> and <b> in the Bond definition
correspond to the van der Waals type indices from the vdw definitions, not the Atom
indices.
The bending potential between two adjacent bonds connecting three different atoms
(<a>-<b>-<c>) is also taken to
be harmonic,
Eangle(θabc) = kangle (θabc − θeq)2 |
| (9.17) |
Here, kangle is provided by <Force Constant> in kcal/mol/degrees and θeq
by <Equilibrium Angle> in degrees. Again, <a>, <b>, and <c> correspond to
van der Waals types defined with vdw.
The energy dependence of the <a>-<b>-<c>-<d> dihedral torsion angle, where <a>, <b>,
<c>, and <d> are van der Waals types, is defined by
Etorsion(θabcd) = fscale |
∑
m
|
kabcd [1 + cos(m θabcd − ϕ)] |
| (9.18) |
Here, fscale is the scaling factor defined by torsion-scale.
The force constant kabcd is defined with <Force constant> in kcal/mol, and the
phase angle ϕ is defined with <Phase Angle> in degrees.
The summation is over multiplicities, m, and Q-Chem supports up to three different values of
m per dihedral angle. The force constants and phase angles may depend on m, so if more than
one multiplicity is used, then <Force constant> <Phase Angle> <Multiplicity> should be
specified for each multiplicity. For example, to specify a dihedral torsion between van der Waals
types 2-1-1-2, with multiplicities m=2 and m=3, we might have:
Torsion 2 1 1 2 2.500 180.0 2 1.500 60.0 3
Improper torsion angle energies for four atoms <a>-<b>-<c>-<d>, where <c> is the central
atom, can be computed in one of two ways, as controlled by the ImptorType rule. If
ImptorType is set to Trigonometric, then the improper torsion energy has a functional form
similar to that used for dihedral angle torsions:
Eimptor(θabcd) = |
kabcd
Nequiv
|
[1 + cos(m θabcd − ϕ)] |
| (9.19) |
Here, θabcd is the out-of-plane angle of atom <c>, in degrees, and kabcd is the force
constant defined with <Force Constant>, in kcal/mol. The phase ϕ and multiplicity
m need to be specified in the Improper declaration, although the definition of an improper
torsion suggests that these values should be set to ϕ = 0 and m = 2.
The quantity Nequiv accounts for the number of equivalent permutations
of atoms <a>, <b>, and <d>, so that the improper torsion angle is only computed once.
If ImptorType is set to Harmonic, then in place of Eq. , the following
energy function is used:
Eimptor(θabcd) = |
kabcd
Nequiv
|
θabcd2 |
| (9.20) |
The Urey-Bradley energy, which accounts for a non-bonded interaction between atoms
<a> and <c> that are separated by two bonds (i.e., a 1-3 interaction through
<a>-<b>-<c>), is given by
EUreyBrad(rac) = kabc (rac − req)2 |
| (9.21) |
The distance in Å between atoms <a> and <c> is rac, the equilibrium distance
req is provided by <Equilibrium Distance> in Å, and the force constant kabc
is provided by <Force Constant> in kcal/mol/Å2.
A short example of a valid text-only file defining a force field for a flexible TIP3P water could be as follows:
//-- Force Field Example --//
// -- Rules -- //
RadiusRule Geometric
RadiusSize Radius
EpsilonRule Geometric
ImptorType Trigonometric
vdw-14-scale 1.0
chg-14-scale 0.8
torsion-scale 0.5
// -- Number of atoms and vdw to expect -- //
NAtom 2
Nvdw 2
// -- Atoms -- //
Atom 1 -0.8340 2 TIP3P Oxygen
Atom 2 0.4170 1 TIP3P Hydrogen
// -- vdw -- //
vdw 1 0.0000 0.0000 H parameters
vdw 2 1.7682 0.1521 O parameters
// -- Bond -- //
Bond 1 2 553.0 0.9572
// -- Angle -- //
Angle 1 2 1 100.0 104.52
Lines that do not begin with one of the keywords will be ignored, and have been used here
as comments.
9.12.2.4 $qm_atoms and $forceman sections
For QM/MM calculations (but not for purely MM calculations) the user must specify
the QM subsystem using a $qm_atoms input section, which assumes the following format:
$qm_atoms
<QM atom 1 index> <QM atom 2 index> . . .
. . .
<QM atom n index>
$end
Multiple indices can appear on a single line and the input can be split across multiple lines.
Each index is an integer corresponding to one of the atoms
in the $molecule section, beginning at 1 for the first atom
in the $molecule section. Link atoms for the ONIOM model and YinYang atoms for
the Janus model are not specified in the $qm_atoms section, as these are inserted
automatically whenever a bond connects a QM atom and an MM atom.
For Janus QM/MM calculations, there are several ways of dealing with
van der Waals interactions between the QM and MM atoms. By default,
van der Waals interactions are computed for all QM-MM and MM-MM atom pairs but not
for QM-QM atom pairs. In some cases, the user may prefer not to neglect the van der Waals
interactions between QM-QM atoms, or the user may prefer to neglect any van der Waals
interaction that involves a QM atom. Q-Chem allows the user this control via
two options in the $forceman section. To turn on QM-QM atom van der Waals interactions,
the user should include the following in their input:
Similarly, to turn off all van der Waals interactions with QM atoms, the following should be included:
$forceman
NoQM-QMorQM-MMvdw
$end
9.12.3 Additional Job Control Variables
A QM/MM job is requested by setting the $rem variables QM_MM_INTERFACE
and FORCE_FIELD.
Also required are a $qm_atoms input section and appropriate modifications to the $molecule
section, as described above. Additional job control variables are detailed here.
QM_MM_INTERFACE
Enables internal QM/MM calculations. |
TYPE:
DEFAULT:
OPTIONS:
MM | Molecular mechanics calculation (i.e., no QM region) |
ONIOM | QM/MM calculation using two-layer mechanical embedding |
JANUS | QM/MM calculation using electronic embedding
|
RECOMMENDATION:
The ONIOM model and Janus models are described above. Choosing MM
leads to no electronic structure calculation. However, when using MM, one
still needs to define the $rem variables BASIS and EXCHANGE
in order for Q-Chem to proceed smoothly.
|
|
| FORCE_FIELD
Specifies the force field for MM energies in QM/MM calculations. |
TYPE:
DEFAULT:
OPTIONS:
AMBER99 | AMBER99 force field |
CHARMM27 | CHARMM27 force field |
OPLSAA | OPLSAA force field |
RECOMMENDATION:
|
|
|
CHARGE_CHARGE_REPULSION
The repulsive Coulomb interaction parameter for YinYang atoms. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The repulsive Coulomb potential maintains bond lengths involving YinYang
atoms with the potential V(r) = Q/r. The default is parameterized for carbon atoms. |
|
| GAUSSIAN_BLUR
Enables the use of Gaussian-delocalized external charges in a QM/MM calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Delocalizes external charges with Gaussian functions. |
FALSE | Point charges |
RECOMMENDATION:
|
|
|
GAUSS_BLUR_WIDTH
Delocalization width for external MM Gaussian charges in a Janus calculations. |
TYPE:
DEFAULT:
OPTIONS:
n | Use a width of n ×10−4 Å. |
RECOMMENDATION:
Blur all MM external charges in a QM/MM calculation with the specified width.
Gaussian blurring is currently incompatible with PCM calculations. Values of
1.0-2.0 Å are recommended in Ref. .
|
|
| MODEL_SYSTEM_CHARGE
Specifies the QM subsystem charge if different from the $molecule section. |
TYPE:
DEFAULT:
OPTIONS:
n | The charge of the QM subsystem. |
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system)
has a charge that is different from the total system charge. |
|
|
|
MODEL_SYSTEM_MULT
Specifies the QM subsystem multiplicity if different from the $molecule section. |
TYPE:
DEFAULT:
OPTIONS:
n | The multiplicity of the QM subsystem. |
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system)
has a multiplicity that is different from the total system multiplicity.
ONIOM calculations must be closed shell. |
|
| USER_CONNECT
Enables explicitly defined bonds. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Bond connectivity is read from the $molecule section |
FALSE | Bond connectivity is determined by atom proximity |
RECOMMENDATION:
Set to TRUE if bond connectivity is known, in which case this connectivity must be
specified in the $molecule section. This greatly accelerates MM calculations.
|
|
|
|
9.12.4 QM / MM Examples
- QM/MM Example 1
Features of this job:
- Geometry optimization using ONIOM mechanical embedding.
- MM region (water 1) described using OPLSAA.
- QM region (water 2) described using PBE0/6-31G*.
- $molecule input section contains user-defined MM bonds.
A zero is used as a placeholder if there are no more connections.
Example 9.0 ONIOM optimization of water dimer.
$rem
exchange pbe0
basis 6-31G*
qm_mm_interface oniom
force_field oplsaa
user_connect true
jobtype opt
molden_format true
$end
$qm_atoms
4 5 6
$end
$molecule
0 1
O -0.790909 1.149780 0.907453 186 2 3 0 0
H -1.628044 1.245320 1.376372 187 1 0 0 0
H -0.669346 1.913705 0.331002 187 1 0 0 0
O 1.178001 -0.686227 0.841306 186 5 6 0 0
H 0.870001 -1.337091 1.468215 187 4 0 0 0
H 0.472696 -0.008397 0.851892 187 4 0 0 0
$end
- QM/MM Example 2
Features of this job:
- Janus electronic embedding with a YingYang link atom
(the glycosidic carbon at the C1′ position of the deoxyribose).
- MM region (deoxyribose) is described using AMBER99.
- QM region (adenine) is described using HF/6-31G*.
- The first 5 electronically excited states are computed with CIS.
MM energy interactions between a QM atom and an MM atom
(e.g., van der Waals interactions, as well as angles involving a single QM atom)
are assumed to be the same in the excited states as in the ground state.
- $molecule input section contains user-defined MM bonds.
- Gaussian-blurred charges are used on all MM atoms, with a width set to 1.5 Å.
Example 9.0 Excited-state single-point QM/MM calculation on deoxyadenosine.
$rem
exchange hf
basis 6-31G*
qm_mm_interface janus
user_connect true
force_field amber99
gaussian_blur true
gauss_blur_width 15000
cis_n_roots 5
cis_triplets false
molden_format true
print_orbitals true
$end
$qm_atoms
18 19 20 21 22 23 24 25 26 27 28 29 30 31
$end
$molecule
0 1
O 0.000000 0.000000 0.000000 1244 2 9 0 0
C 0.000000 0.000000 1.440000 1118 1 3 10 11
C 1.427423 0.000000 1.962363 1121 2 4 6 12
O 1.924453 -1.372676 1.980293 1123 3 5 0 0
C 2.866758 -1.556753 0.934073 1124 4 7 13 18
C 2.435730 0.816736 1.151710 1126 3 7 8 14
C 2.832568 -0.159062 0.042099 1128 5 6 15 16
O 3.554295 1.211441 1.932365 1249 6 17 0 0
H -0.918053 0.000000 -0.280677 1245 1 0 0 0
H -0.520597 -0.885828 1.803849 1119 2 0 0 0
H -0.520597 0.885828 1.803849 1120 2 0 0 0
H 1.435560 0.337148 2.998879 1122 3 0 0 0
H 3.838325 -1.808062 1.359516 1125 5 0 0 0
H 1.936098 1.681209 0.714498 1127 6 0 0 0
H 2.031585 -0.217259 -0.694882 1129 7 0 0 0
H 3.838626 0.075227 -0.305832 1130 7 0 0 0
H 4.214443 1.727289 1.463640 1250 8 0 0 0
N 2.474231 -2.760890 0.168322 1132 5 19 27 0
C 1.538394 -2.869204 -0.826353 1136 18 20 28 0
N 1.421481 -4.070993 -1.308051 1135 19 21 0 0
C 2.344666 -4.815233 -0.582836 1134 20 22 27 0
C 2.704630 -6.167666 -0.619591 1140 21 23 24 0
N 2.152150 -7.057611 -1.455273 1142 22 29 30 0
N 3.660941 -6.579606 0.239638 1139 22 25 0 0
C 4.205243 -5.691308 1.066416 1138 24 26 31 0
N 3.949915 -4.402308 1.191662 1137 25 27 0 0
C 2.991769 -4.014545 0.323275 1133 18 21 26 0
H 0.951862 -2.033257 -1.177884 1145 19 0 0 0
H 2.449361 -8.012246 -1.436882 1143 23 0 0 0
H 1.442640 -6.767115 -2.097307 1144 23 0 0 0
H 4.963977 -6.079842 1.729564 1141 25 0 0 0
$end
- QM/MM Example 3
Features of this job:
- An MM-only calculation. BASIS and EXCHANGE need to be defined,
in order to prevent a crash, but no electronic structure calculation is actually performed.
- All atom types and MM interactions are defined in $force_field_params using
the CHARMM27 force field. Atomic charges, equilibrium
bond distances, and equilibrium angles have been extracted from a
HF/6-31G* calculation, but the force constants and van der Waals parameters are
fictitious values invented for this example.
- Molecular dynamics is propagated for 10 steps within a microcanonical ensemble
(NVE), which is the only ensemble available at present. Initial
velocities are sampled from a Boltzmann distribution at 400 K.
Example 9.0 MM molecular dynamics with user-defined MM parameters.
$rem
basis sto-3g
exchange hf
qm_mm_interface MM
force_field charmm27
user_connect true
jobtype aimd
time_step 42
aimd_steps 10
aimd_init_veloc thermal
aimd_temp 400
$end
$molecule
-2 1
C 0.803090 0.000000 0.000000 -1 2 3 6 0
C -0.803090 0.000000 0.000000 -1 1 4 5 0
H 1.386121 0.930755 0.000000 -2 1 0 0 0
H -1.386121 -0.930755 0.000000 -2 2 0 0 0
H -1.386121 0.930755 0.000000 -2 2 0 0 0
H 1.386121 -0.930755 0.000000 -2 1 0 0 0
$end
$force_field_params
NumAtomTypes 2
AtomType -1 -0.687157 2.0000 0.1100
AtomType -2 -0.156422 1.3200 0.0220
Bond -1 -1 250.00 1.606180
Bond -1 -2 300.00 1.098286
Angle -2 -1 -2 50.00 115.870
Angle -2 -1 -1 80.00 122.065
Torsion -2 -1 -1 -2 2.500 180.0 2
$end
Further examples of QM / MM calculations can be found in the $QC/samples directory,
including a QM / MM / PCM example, QMMMPCM_crambin.in. This calculation consists of a
protein molecule (crambin) described using a force field, but with one tyrosine side chain
described using electronic structure theory. The entire QM / MM system is placed within an
implicit solvent model, of the sort described in Section 10.2.2.
Chapter 10 Molecular Properties and Analysis
10.1 Introduction
Q-Chem has incorporated a number of molecular properties and wavefunction
analysis tools, summarized as follows:
- Chemical solvent models
- Population analysis for ground and excited states
- Multipole moments for ground and excited states
- Calculation of molecular intracules
- Vibrational analysis (including isotopic substitution)
- Interface to the Natural Bond Orbital package
- Molecular orbital symmetries
- Orbital localization
- Localized Orbital Bonding Analysis
- Data generation for 2-D or 3-D plots
- Orbital visualization using the MolDen and MacMolPlt programs
- Natural transition orbitals for excited states
- NMR shielding tensors and chemical shifts
- Quantum transport modeling in the Landauer approximation
10.2 Chemical Solvent Models
Ab initio quantum chemistry makes possible the study of gas-phase
molecular properties from first principles. In liquid solution, however,
these properties may change significantly, especially in polar solvents.
Although it is possible to model solvation effects by including
explicit solvent molecules in the quantum-chemical calculation
(e.g. a super-molecular cluster calculation,
averaged over different configurations
of the molecules in the first solvation shell),
such calculations are very computationally demanding. Furthermore,
cluster calculations typically do not afford accurate solvation energies,
owing to the importance of long-range electrostatic interactions.
Accurate prediction of solvation free energies is, however, crucial for
modeling of chemical reactions and ligand / receptor interactions in solution.
Q-Chem contains several different implicit solvent models, which differ greatly in their
level of sophistication and realism. These are generally known as self-consistent
reaction field (SCRF) models, because the continuum solvent establishes a reaction field
(i.e., additional terms in the solute Hamiltonian) that depends upon the solute
electron density, and must therefore be updated self-consistently during the iterative convergence of the wavefunction. SCRF methods available within Q-Chem include the
Kirkwood-Onsager model [425,[426,[427],
the conductor-like screening model (known as COSMO [428],
GCOSMO [429],
or C-PCM [430]), and the "surface and simulation of volume polarization for
electrostatics" [SS(V)PE] model [431], which is also known as the "integral
equation formalism", IEF-PCM [432,[433]. A detailed description of these
models can be found in review articles by Tomasi [434,[435],
Mikkelsen [436], and Chipman [437,[438].
The C-PCM / GCOSMO and IEF-PCM / SS(V)PE models are examples of what are called "apparent surface charge"
SCRF models, although the term polarizable continuum models (PCMs), as popularized
by Tomasi and co-workers [435], is now used almost universally to refer to
this class of solvation models. Q-Chem employs a new "SWIG"
(Switching function / Gaussian)
implementation of these PCMs [439,[440,[441]. This approach
resolves a long-standing-though little-publicized-problem with standard PCMs,
namely, that the boundary-element methods used to discretize the solute/continuum interface may lead
to discontinuities in the potential energy surface for the solute molecule. These discontinuities
inhibit convergence of geometry optimizations, introduce serious artifacts in vibrational frequency
calculations, and make ab initio molecular dynamics calculations virtually
impossible [439,[440].
In contrast, Q-Chem's SWIG PCMs afford potential energy surfaces that
are rigorously continuous and smooth. Unlike earlier attempts to obtain smooth PCMs, the SWIG
approach largely preserves the properties of the underlying integral-equation solvent models,
so that solvation energies and molecular surface areas are hardly affected by the smoothing procedure.
Other solvent models available in Q-Chem include the "Langevin dipoles"
model [442,[443] and the highly empirical (but often quite accurate)
"Solvent Model 8" (SM8), developed at the University of Minnesota [444]. SM8 is based upon
the generalized Born method for electrostatics, augmented with
atomic surface tensions for non-electrostatic effects (cavitation, dispersion,
exchange repulsion, and solvent structure), which go beyond that which can be calculated
using only the bulk dielectric constant. Empirical corrections of this sort are also available
for the PCMs mentioned above, but within SM8 these parameters have been optimized to reproduce
experimental solvation energies.
10.2.1 Kirkwood-Onsager Model
Within the Kirkwood-Onsager model [425,[426,[427],
the solute is placed inside of a spherical cavity
surrounded by a continuous dielectric medium. This model is characterized by two parameters:
the cavity radius, a0, and the solvent dielectric constant, ε. The former is
typically calculated according to
where Vm is obtained from experiment (molecular weight or
density [445]) and NA is Avogadro's number. It is also common to add
0.5 Å to the value of a0 from Eq. (10.1) in order to account for the first
solvation shell [446]. Alternatively, a0 is sometimes selected as the
maximum distance between the solute center of mass and the solute atoms, plus the
relevant van der Waals radii. A third option is to set 2a0
(the cavity diameter) equal to the largest solute-solvent internuclear distance, plus
the the van der Waals radii of the relevant atoms. Unfortunately,
solvation energies are typically quite sensitive to the choice of a0.
Unlike older versions of the Kirkwood-Onsager model, in which the solute's electron
distribution was described entirely in terms of its dipole moment, Q-Chem's version
of this model can describe the electron density of the solute using an arbitrary-order multipole
expansion, including the Born (monopole) term [447] for charged solutes.
The solute-continuum electrostatic interaction energy is then computed using
analytic expressions for the interaction of the point multipoles with a dielectric continuum.
Energies and analytic gradients for the Kirkwood-Onsager solvent model are available
for Hartree-Fock, DFT, and CCSD calculations. Note that convergence of SCRF calculations
can sometimes be difficult, thus
it is often advisable to perform a gas-phase calculation first, which can serve as the initial
guess for the Kirkwood-Onsager calculation.
The $rem variables associated Kirkwood-Onsager reaction-field
calculations are documented below. The $rem variables
SOLUTE_RADIUS and SOLVENT_DIELECTRIC are required in addition to the
normal job control variables for energy and gradient calculations. The $rem
variable CC_SAVEAMPL may save some time for CCSD calculations using the
Kirkwood-Onsager model.
SOLVENT_METHOD
Sets the preferred solvent method. |
TYPE:
DEFAULT:
SCRF if SOLVENT_DIELECTRIC > 0 |
OPTIONS:
SCRF | Use the Kirkwood-Onsager SCRF model |
PCM | Use an apparent surface charge polarizable continuum model |
COSMO | USE the COSMO model |
RECOMMENDATION:
None. The PCMs are more sophisticated and may require additional input options. These models
are discussed in Section 10.2.2. |
|
| SOLUTE_RADIUS
Sets the solvent model cavity radius. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
SOLVENT_DIELECTRIC
Sets the dielectric constant of the solvent continuum. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| SOL_ORDER
Determines the order to which the multipole expansion of the solute charge density is carried out. |
TYPE:
DEFAULT:
OPTIONS:
L | Include up to L-th order multipoles. |
RECOMMENDATION:
The multipole expansion is usually converged at order L = 15 |
|
|
|
Example 10.0 HF-SCRF applied to H2O molecule
$molecule
0 1
O 0.00000000 0.00000000 0.11722303
H -0.75908339 0.00000000 -0.46889211
H 0.75908339 0.00000000 -0.46889211
$end
$rem
jobtype SP
exchange HF
basis 6-31g**
SOLVENT_METHOD SCRF
SOLUTE_RADIUS 18000 !1.8 Angstrom Solute Radius
SOLVENT_DIELECTRIC 359000 !35.9 Dielectric (Acetonitile)
SOL_ORDER 15 !L=15 Multipole moment order
$end
Example 10.0 CCSD/SCRF applied to 1,2-dichloroethane molecule
$comment
1,2-dichloroethane GAUCHE Conformation
$end
$molecule
0 1
C 0.6541334418569877 -0.3817051480045552 0.8808840579322241
C -0.6541334418569877 0.3817051480045552 0.8808840579322241
Cl 1.7322599856434779 0.0877596094659600 -0.4630557359272908
H 1.1862455146007043 -0.1665749506296433 1.7960750032785453
H 0.4889356972641761 -1.4444403797631731 0.8058465784063975
Cl -1.7322599856434779 -0.0877596094659600 -0.4630557359272908
H -1.1862455146007043 0.1665749506296433 1.7960750032785453
H -0.4889356972641761 1.4444403797631731 0.8058465784063975
$end
$rem
JOBTYPE SP
EXCHANGE HF
CORRELATION CCSD
BASIS 6-31g**
N_FROZEN_CORE FC
CC_SAVEAMPL 1 !Save CC amplitudes on disk
SOLVENT_METHOD SCRF
SOL_ORDER 15 !L=15 Multipole moment order
SOLUTE_RADIUS 36500 !3.65 Angstrom Solute Radius
SOLVENT_DIELECTRIC 89300 !8.93 Dielectric (methylene chloride)
$end
Example 10.0 SCRF applied to HF.
$molecule
0 1
H 0.000000 0.000000 -0.862674
F 0.000000 0.000000 0.043813
$end
$rem
JOBTYPE SP
EXCHANGE HF
BASIS 6-31G*
$end
@@@
$molecule
0 1
H 0.000000 0.000000 -0.862674
F 0.000000 0.000000 0.043813
$end
$rem
JOBTYPE FORCE
EXCHANGE HF
BASIS 6-31G*
SOLVENT_METHOD SCRF
SOL_ORDER 15
SOLVENT_DIELECTRIC 784000 78.4 Dielectric (water)
SOLUTE_RADIUS 25000 2.5 Angstrom Solute Radius
SCF_GUESS READ Read in Vacuum Solution as Guess
$end
10.2.2 Polarizable Continuum Models
Clearly, the Kirkwood-Onsager model is inappropriate if the solute is highly non-spherical.
Nowadays, a more general class of "apparent surface charge" SCRF solvation models
are much more popular, to the extent that the generic term
"polarizable continuum model" (PCM) is typically used to denote these methods, a
convention that we shall follow. Apparent surface charge PCMs improve upon the Kirkwood-Onsager
model in several ways. Most importantly, they provide a much more realistic description
of molecular shape, typically by constructing the solute cavity from a union of atom-centered spheres.
In addition, the exact electron density of the solute (rather than a multipole expansion) is used
to polarize the continuum. Electrostatic interactions between the solute and the continuum
manifest as a charge density on the cavity surface, which is discretized for
practical calculations. The surface charges are determined based upon the solute's
electrostatic potential at the cavity surface, hence the surface charges and the solute wavefunction
must be determined self-consistently.
The literature on PCMs is acronym-heavy, even by the standards of electronic structure theory, and
connections between various methods have not always been appreciated.
Chipman [431,[437,[438] has shown how various PCMs can be
formulated within a common theoretical framework (see also Ref. ), and
several such models are available within Q-Chem. The simplest of these models, computationally
speaking, are the conductor-like models, known in the literature as COSMO [428],
GCOSMO [429], or C-PCM [430]. The models called
GCOSMO and C-PCM are completely equivalent, whereas the original COSMO differs (slightly)
in the dielectric pre-factor that appears in the equations for the induced surface
charge [430,[441].
This distinction is negligible in high-dielectric solvents, but GCOSMO / C-PCM does a better
job of preserving Gauss' Law for the solute charge [429], so this is the
recommended version [441], although both are available in Q-Chem.
A more sophisticated model is the
"surface and simulation of volume polarization for electrostatics" [SS(V)PE]
approach [431], which provides an exact treatment of the surface polarization
(i.e., the surface charge induced by the solute charge that is contained within the solute
cavity) and also an approximate treatment of the volume polarization (solvent polarization
arising from the tail of the solute electron density that penetrates beyond the solute cavity).
The term SS(V)PE is Chipman's notation [431], but
this model is formally equivalent to the
"integral equation formalism", IEF-PCM, that was developed independently by
Cancès et al. [432,[433].
The GCOSMO / C-PCM model becomes equivalent to SS(V)PE / IEF-PCM in the limit
ε→∞ [431,[441], which means that the former models must
implicitly include some correction for volume polarization, even if this was not by design.
For ε >~50, numerical calculations reveal that there is
essentially no difference between IEF-PCM results and those obtained with
conductor-like models [441].
Computationally speaking, the conductor-like models are
less involved than SS(V)PE, which may be a factor for large QM / MM / PCM jobs.
In terms of their numerical implementation,
there is some ambiguity as to how the SS(V)PE (or IEF-PCM) integral equations should be turned
into finite-dimensional matrix equations, since discretization fails to preserve certain
exact symmetries of the integral operators [441]. Both symmetrized and
unsymmetrized forms are
available, but Q-Chem defaults to a particular asymmetric form that affords the
correct conductor (ε→∞) limit [441].
Historically, the term "SS(V)PE" (as used by
Chipman [431,[437,[438]) has referred to a symmetrized form
of the matrix equations; this symmetrized form is available but is not recommended.
Construction of the cavity surface is a crucial aspect of PCMs, and computed properties
are quite sensitive to the details of the cavity construction. Typically (and by default
in Q-Chem), solute
cavities are constructed from a union of atom-centered spheres whose radii are
≈ 1.2 times the atomic van der Waals radii. In Q-Chem's implementation, this
cavity surface is then
discretized using atom-centered Lebedev grids [144,[142,[448]
of the same sort used to perform the numerical integrations in DFT. Surface charges
are located at these grid points.
A long-standing (though not well-publicized) problem with the aforementioned discretization
procedure is that that it fails to afford continuous potential energy surfaces as the solute
atoms are displaced, because certain surface grid points may emerge from, or disappear within,
the solute cavity, as the atomic spheres that define the cavity are moved. This undesirable
behavior can inhibit
convergence of geometry optimizations and, in certain cases, lead to very large errors in
vibrational frequency calculations [439]. It is also a fundamental hindrance to
molecular dynamics calculations [440]. Recently, however, Lange and
Herbert [439,[440] (building upon earlier work by York and
Karplus [449]) developed a general scheme for implementing apparent surface
charge PCMs in a manner that affords smooth potential energy surfaces, even for bond-breaking.
Notably, this approach is faithful to the properties of the
underlying integral equation theory on which the PCMs are based, in the sense that
the smoothing procedure does not significantly perturb
solvation energies or cavity surface areas [440].
This implementation is based upon the use of a
switching function, in conjunction with Gaussian blurring of the cavity surface charge density, hence
these models are known as "Switching / Gaussian" (SWIG) PCMs.
Both single-point energies and analytic energy gradients are available for the SWIG PCMs described
in this section, when the solute is described using molecular mechanics, Hartree-Fock theory,
or DFT. Analytic Hessians are available for the C-PCM model only. (As usual, vibrational frequencies
for other models will be computed, if requested, by finite difference of analytic energy gradients.)
Single-point energy calculations
using correlated wavefunctions could be performed in conjunction with these solvent models,
in which case the
correlated wavefunction calculation will utilize Hartree-Fock molecular orbitals that are polarized
in the presence of the dielectric solvent.
Researchers who use these PCMs are asked to cite Refs. .
In addition to describing the underlying theory, these references
provide assessments of the discretization errors that can be anticipated using various PCMs and
Lebedev grids.
10.2.3 PCM Job Control
10.2.3.1 $rem section
A PCM calculation is requested by setting SOLVENT_METHOD = PCM.
Various other job control parameters for PCM calculations are specified in the $pcm
and $pcm_solvent input sections, which are described below. The only other $rem
variable germane to PCM calculations is a print level:
PCM_PRINT
Controls the printing level during PCM calculations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Prints PCM energy and basic surface grid information. Minimal additional printing. |
1 | Level 0 plus PCM solute-solvent interaction energy components and Gauss Law error. |
2 | Level 1 plus surface grid switching parameters and a .PQR file for visualization of |
| the cavity surface apparent surface charges. |
3 | Level 2 plus a .PQR file for visualization of the electrostatic potential at the surface |
| grid created by the converged solute. |
4 | Level 3 plus additional surface grid information, electrostatic potential and apparent |
| surface charges on each SCF cycle. |
5 | Level 4 plus extensive debugging information. |
RECOMMENDATION:
Use the default unless further information is desired. |
|
It is highly recommended that the user visualize their cavity surface in PCM calculations to ensure
that the cavity geometry is adequate for the application. This can be done by setting
PCM_PRINT to a value of 2 (or larger), which will cause Q-Chem to print several
".PQR" files that describes the cavity surface. The .PQR format is similar to the
common .PDB (protein data bank) format, but also contains charge and radius information for each
atom. One of the output .PQR files contains the charges computed in the PCM calculation
and radii (in Å) that are half of the square root of the surface area represented by each surface grid
point. Another .PQR file contains the solute's electrostatic potential (in atomic units), in
place of the charge information, and uses uniform radii for the grid points. These .PQR files
can be visualized using various third-party software, including the freely-available Visual
Molecular Dynamics (VMD) program [450,[451], which is particularly useful for
coloring the .PQR surface grid points according to their charge, and sizing them according to
their contribution to the molecular surface area (see, e.g., the pictures in
Ref. ).
10.2.3.2 $pcm section
Most PCM job control is done via options specified in the $pcm input section, which allows the
user to specify which flavor of PCM will be used, which algorithm will be used to solve the PCM
equations, as well as other job control options. The format of the $pcm section is analogous to
that of the $rem section:
$pcm
<Keyword> <parameter/option>
$end
NOTE: The following job control variables belong only in the $pcm section.
Do not place them in the $rem section.
Theory
Specifies the which polarizable continuum model will be used. |
TYPE:
DEFAULT:
OPTIONS:
CPCM | Conductor-like Polarizable Continuum Model (also known as GCOSMO) |
COSMO | Original conductor-like screening model (COSMO) |
SSVPE | Surface and Simulation of Volume Polarization for Electrostatics (i.e., |
| IEF-PCM)
|
RECOMMENDATION:
SS(V)PE model is a more sophisticated model than either C-PCM or COSMO,
in that it accounts for interactions that these models neglect,
but is more computationally demanding.
|
|
| Method
Specifies which surface discretization method will be used. |
TYPE:
DEFAULT:
OPTIONS:
SWIG | Switching/Gaussian method |
ISWIG | "Improved" Switching/Gaussian method with an alternative switching function |
RECOMMENDATION:
Use of SWIG is recommended only because it is slightly more efficient than
the switching function of ISWIG. On the other hand, ISWIG offers some
conceptually more appealing features and may be superior in certain cases.
Consult Refs. for a discussion of these differences.
|
|
|
|
Construction of the solute cavity is an important part of the model, as computed properties are
generally quite sensitive to this construction. The user should consult the literature in this
capacity, especially with regard to the radii used for the atomic spheres. The default values
provided here correspond to the consensus choice that has emerged over several decades, namely,
to use van der Waals radii scaled by a factor of 1.2. The most widely-used set of van der Waals radii
are those determined from crystallographic data by Bondi [452] (although the radius for hydrogen was later adjusted to 1.1 Å [453], and this later value is used in Q-Chem).
Bondi's analysis was recently extended to the whole main
group [454], and this extended set of van der Waals radii is available in Q-Chem.
Alternatively, atomic radii may be taken from the Universal Force Field [455],
whose main appeal is that it provides radii for all atoms of the periodic table, although the
quality of these radii for PCM applications is unclear. Finally, the user may specify
his or her own radii for cavity construction, using the $van_der_waals input section.
To do so, the user must set
PCM_VDW_RADII = READ in $rem section and also set Radii to READ in
the $pcm section. The actual values of the radii are then specified using the $van_der_waals section,
using a format that is discussed in detail in Section 10.2.6.2.
For certain applications, it is desirable to employ a "solvent-accessible" cavity surface,
rather than a van der Waals surface. The solvent-accessible surface is constructed from the
van der Waals surface by adding a certain value-equal to the presumed radius of a solvent
molecule-to each scaled atomic radius. This capability is also available.
Radii
Specifies which set of atomic van der Waals radii will be used to define the solute cavity. |
TYPE:
DEFAULT:
OPTIONS:
BONDI | Use the (extended) set of Bondi radii |
FF | Use Lennard-Jones radii from a molecular mechanics force field |
UFF | Universal Force Field radii |
READ | User defined radii, read from the $van_der_waals section
|
RECOMMENDATION:
Bondi radii are widely accepted. The FF option requires the user to specify an MM
force field using the FORCE_FIELD $rem variable, and also to define
the atom types in the $molecule section (see Section 9.12).
|
|
| vdwScale
Scaling factor for the atomic van der Waals radii used to define the solute cavity. |
TYPE:
DEFAULT:
OPTIONS:
f | Use a scaling factor of f > 0. |
RECOMMENDATION:
The default value is widely used in PCM calculations, although a value of 1.0 might be appropriate if
using a solvent-accessible surface.
|
|
|
|
SASrad
Form a "solvent accessible" surface with the given solvent probe radius. |
TYPE:
DEFAULT:
OPTIONS:
r | Use a solvent probe radius of r, in Å. |
RECOMMENDATION:
The solvent probe radius is added to the scaled van der Waals radii of the solute atoms.
A common solvent probe radius for water is 1.4 Å, but the user should consult the literature
regarding the use of solvent-accessible surfaces. |
|
Historically, discretization of the cavity surface has involved "tessellation" methods that
divide the cavity surface area into finite elements. (The GEPOL algorithm [456]
is the most widely-used tessellation scheme.) Tessellation methods, however, suffer not only from
discontinuities in the cavity surface area and solvation energy as a function of the nuclear
coordinates, but in addition they lead to analytic energy gradients that are formally
quite complicated. To avoid these problems,
Q-Chem's SWIG PCM implementation uses Lebedev grids to discretize the atomic spheres.
These are atom-centered grids with icosahedral symmetry, and may consist of anywhere from
26 to 5294 grid points per atomic sphere.
The default values used by Q-Chem were selected based on extensive numerical
tests [440,[441]. The default value for both MM and QM/MM jobs is
110 Lebedev points per atomic spheres, which numerical tests suggest is
sufficient to achieve rotational invariance of the
solvation energy [440]. Solvation energies computed with N=110 grid points
often lie within ∼ 1 kcal/mol of the N→∞ limit, although exceptions
(especially where charged solutes are involved) can be found [441].
For QM solutes, where it is more likely that solvation energies might be computed, the
Q-Chem default is N=590. In any case, for solvation energies the user should probably
test the N-dependence of the result.
The number of Lebedev grid points, N, is specified using the $pcm variables
described below. For QM/MM/PCM jobs (i.e., jobs where the solute is described by a QM/MM
calculation), the QM and MM atomic spheres may use different values of N. A smaller value
is probably required for representing the MM atomic spheres, since the electrostatic
potential generated by the MM point charges is likely to be less structured than that arising
from a continuous QM electron density. The full list of acceptable values
for the number of Lebedev points per sphere is N=26, 50, 110, 194, 302, 434, 590, 770, 974, 1202, 1454,
1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, 5294.
HPoints
The number of Lebedev grid points to be placed on H atoms in the QM system. |
TYPE:
DEFAULT:
OPTIONS:
Acceptable values are listed above. |
RECOMMENDATION:
The more grid points, the more exact the PCM solution but the more expensive the calculation. |
|
| HeavyPoints
The number of Lebedev grid points to be placed non-hydrogen atoms in the QM system. |
TYPE:
DEFAULT:
OPTIONS:
Acceptable values are listed above. |
RECOMMENDATION:
The more grid points, the more exact the PCM solution but the more expensive the calculation. |
|
|
|
MMHPoints
The number of Lebedev grid points to be placed on H atoms in the MM subsystem. |
TYPE:
DEFAULT:
OPTIONS:
Acceptable values are listed above. |
RECOMMENDATION:
This option applies only to QM/MM calculations.
The more grid points, the more exact the PCM solution but the more expensive the calculation. |
|
| MMHeavyPoints
The number of Lebedev grid points to be placed on non-hydrogen atoms in the MM subsystem. |
TYPE:
DEFAULT:
OPTIONS:
Acceptable values are listed above. |
RECOMMENDATION:
This option applies only to QM/MM calculations.
The more grid points, the more exact the PCM solution but the more expensive the calculation. |
|
|
|
For Q-Chem's smooth PCMs, the final aspect of cavity construction
is selection of a switching function to attenuate the contributions of grid points as they pass
into the interior of the solute cavity (see Ref. ).
SwitchThresh
The threshold for discarding grid points on the cavity surface. |
TYPE:
DEFAULT:
OPTIONS:
n | Discard grid points when the switching function is less than 10−n. |
RECOMMENDATION:
Use the default, which is found to avoid discontinuities within machine precision.
Increasing n reduces the cost of PCM calculations but can introduce discontinuities in
the potential energy surface.
|
|
The following example shows a very basic PCM job. The solvent dielectric is specified in the
$pcm_solvent section, which is described below.
Example 10.0 A basic example of using the PCMs: optimization of trifluoroethanol in water.
$rem
JOBTYPE OPT
BASIS 6-31G*
EXCHANGE B3LYP
SOLVENT_METHOD PCM
$end
$pcm
Theory CPCM
Method SWIG
Solver Inversion
HeavyPoints 194
HPoints 194
Radii Bondi
vdwScale 1.2
$end
$pcm_solvent
Dielectric 78.39
$end
$molecule
0 1
C -0.245826 -0.351674 -0.019873
C 0.244003 0.376569 1.241371
O 0.862012 -0.527016 2.143243
F 0.776783 -0.909300 -0.666009
F -0.858739 0.511576 -0.827287
F -1.108290 -1.303001 0.339419
H -0.587975 0.878499 1.736246
H 0.963047 1.147195 0.961639
H 0.191283 -1.098089 2.489052
$end
10.2.3.3 $pcm_solvent section
The solvent for PCM calculations is specified using the $pcm_solvent section, as documented below.
In addition, the $pcm_solvent section can be used to incorporate non-electrostatic interaction terms
into the solvation energy. (Note: the Theory keyword in the $pcm section, as
described above, specifies only how the electrostatic interactions are handled.) The general form
of the $pcm_solvent input section is shown below.
$pcm_solvent
NonEls <Option>
NSolventAtoms <Number unique of solvent atoms>
SolventAtom <Number1> <Number2> <Number3> <SASrad>
SolventAtom <Number1> <Number2> <Number3> <SASrad>
. . .
<Keyword> <parameter/option>
. . .
$end
The keyword SolventAtom requires multiple parameters, whereas all other keywords
require only a single parameter.
The $pcm_solvent input section
is used to specify the solvent dielectric, using the Dielectric keyword.
If non-electrostatic interactions are ignored, then this is the only keyword that is necessary
in the $pcm_solvent section.
Dielectric
The dielectric constant of the PCM solvent. |
TYPE:
DEFAULT:
OPTIONS:
ε | Use a dielectric constant of ε > 0. |
RECOMMENDATION:
The default corresponds to water at T=298 K. |
|
The non-electrostatic interactions currently available in Q-Chem are based on the work
of Cossi et al. [457], and are computed outside of the SCF procedure used to determine
the electrostatic interactions. The non-electrostatic energy is highly dependent
on the input parameters and can be extremely sensitive to the radii chosen to define the solute cavity.
Accordingly, the inclusion of non-electrostatic interactions is highly empirical and should be used
with caution. Following Ref. , the cavitation energy is computed using the same solute
cavity that is used to compute the electrostatic energy, whereas
the dispersion/repulsion energy is computed using a solvent-accessible surface.
The following keywords are used to define non-electrostatic parameters for PCM calculations.
NonEls
Specifies what type of non-electrostatic contributions to include. |
TYPE:
DEFAULT:
OPTIONS:
Cav | Cavitation energy |
Buck | Buckingham dispersion and repulsion energy from atomic number |
LJ | Lennard-Jones dispersion and repulsion energy from force field |
BuckCav | Buck + Cav |
LJCav | LJ + Cav
|
RECOMMENDATION:
A very limited set of parameters for the Buckingham potential is available at present. |
|
| NSolventAtoms
The number of different types of atoms. |
TYPE:
DEFAULT:
OPTIONS:
N | Specifies that there are N different types of atoms. |
RECOMMENDATION:
This keyword is necessary when NonEls = Buck, LJ, BuckCav, or LJCav.
Methanol (CH3OH), for example, has three types of atoms (C, H, and O).
|
|
|
|
SolventAtom
Specifies a unique solvent atom. |
TYPE:
DEFAULT:
OPTIONS:
Input (TYPE) | Description |
Number1 (INTEGER): | The atomic number of the atom |
Number2 (INTEGER): | How many of this atom are in a solvent molecule |
Number3 (INTEGER): | Force field atom type |
SASrad (FLOAT): | Probe radius (in Å) for defining the solvent accessible surface
|
RECOMMENDATION:
If not using LJ or LJCav, Number3 should be set to 0. The SolventAtom keyword is
necessary when NonEls = Buck, LJ, BuckCav, or LJCav. |
|
| Temperature
Specifies the solvent temperature. |
TYPE:
DEFAULT:
OPTIONS:
T | Use a temperature of T, in Kelvin.
|
RECOMMENDATION:
Used only for the cavitation energy. |
|
|
|
Pressure
Specifies the solvent pressure. |
TYPE:
DEFAULT:
OPTIONS:
P | Use a pressure of P, in bar. |
RECOMMENDATION:
Used only for the cavitation energy. |
|
| SolventRho
Specifies the solvent number density |
TYPE:
DEFAULT:
Determined for water, based on temperature. |
OPTIONS:
ρ | Use a density of ρ, in molecules/Å3. |
RECOMMENDATION:
Used only for the cavitation energy. |
|
|
|
SolventRadius
The radius of a solvent molecule of the PCM solvent. |
TYPE:
DEFAULT:
OPTIONS:
r | Use a radius of r, in Å. |
RECOMMENDATION:
Used only for the cavitation energy. |
|
The following example illustrates the use of the non-electrostatic interactions.
Example 10.0 Optimization of trifluoroethanol in water using both electrostatic and
non-electrostatic PCM interactions. OPLSAA parameters are used in the Lennard-Jones
potential for dispersion and repulsion.
$rem
JOBTYPE OPT
BASIS 6-31G*
EXCHANGE B3LYP
SOLVENT_METHOD PCM
FORCE_FIELD OPLSAA
$end
$pcm
Theory CPCM
Method SWIG
Solver Inversion
HeavyPoints 194
HPoints 194
Radii Bondi
vdwScale 1.2
$end
$pcm_solvent
NonEls LJCav
NSolventAtoms 2
SolventAtom 8 1 186 1.30
SolventAtom 1 2 187 0.01
SolventRadius 1.35
Temperature 298.15
Pressure 1.0
SolventRho 0.03333
Dielectric 78.39
$end
$molecule
0 1
C -0.245826 -0.351674 -0.019873 23
C 0.244003 0.376569 1.241371 22
O 0.862012 -0.527016 2.143243 24
F 0.776783 -0.909300 -0.666009 26
F -0.858739 0.511576 -0.827287 26
F -1.108290 -1.303001 0.339419 26
H -0.587975 0.878499 1.736246 27
H 0.963047 1.147195 0.961639 27
H 0.191283 -1.098089 2.489052 25
$end
10.2.4 Linear-Scaling QM / MM / PCM Calculations
Calculation of the PCM electrostatic interactions, for both the C-PCM / GCOSMO and
SS(V)PE / IEF-PCM methods, amounts to solution of a set of linear equations of the
form [437,[438,[439]
These equations are solved in order to determine the vector q of apparent
surface charges, given the solute's electrostatic
potential v, evaluated at the surface discretization points. The precise forms of
the matrices K and R depend upon the particular PCM, but in any case
they have dimension Ngrid×Ngrid, where Ngrid is the number of Lebedev
grid points used to discretize the cavity
surface. Construction of the matrix K−1R affords a numerically
exact solution to Eq. , whose cost scales as O(N3grid) in CPU time
and O(N2grid)
in memory. This cost is exacerbated by smooth PCMs, which discard fewer interior grid
points, and therefore use larger values of Ngrid for a given solute [439].
For QM solutes, the cost of inverting K is usually negligible relative to the
cost of the electronic structure calculation, but for the large values of Ngrid that are
encountered in typical MM/PCM or QM/MM/PCM jobs, the O(N3grid) cost of inverting
K may become prohibitively expensive.
To avoid this bottleneck, Lange and Herbert [458]
have developed an iterative conjugate gradient (CG) solver for Eq. whose cost
scales as O(N2grid) in CPU time and O(Ngrid) in memory. A number of other
cost-saving options are available, including
efficient pre-conditioners and matrix factorizations that speed up convergence of the CG
iterations, and a "fast multipole" algorithm for computing the electrostatic interactions.
Together, these features lend themselves to a solution of Eq. whose cost scales as
O(Ngrid) in both memory and CPU time, for sufficiently large systems [458].
Currently, these options are available only for the C-PCM / GCOSMO model.
Listed below are job control variables for the CG solver, which should be specified within the
$pcm input section. Researchers who use this feature
are asked to cite the original SWIG PCM references [440,[441] as well as the
reference for the CG solver [458].
Solver
Specifies the algorithm used to solve the PCM equations. |
TYPE:
DEFAULT:
OPTIONS:
INVERSION | Direct matrix inversion |
CG | Iterative conjugate gradient |
RECOMMENDATION:
Matrix inversion will be faster for small solutes but the CG solver is recommended for
large MM/PCM or QM/MM/PCM calculations.
|
|
| CGThresh
The threshold for convergence of the conjugate gradient solver. |
TYPE:
DEFAULT:
OPTIONS:
n | Conjugate gradient converges when the maximum residual is less than 10−n. |
RECOMMENDATION:
The default typically
affords PCM energies on par with the precision of matrix inversion for small systems. For
systems that have difficulty with SCF convergence, one should increase n or try the matrix
inversion solver. For well-behaved or very large systems, a smaller n might be permissible. |
|
|
|
DComp
Controls decomposition of matrices to reduce the matrix norm for the CG Solver. |
TYPE:
DEFAULT:
OPTIONS:
0 | Turns off matrix decomposition |
1 | Turns on matrix decomposition |
3 | Option 1 plus only stores upper half of matrix and enhances gradient evaluation
|
RECOMMENDATION:
|
| PreCond
Controls the use of the pre-conditioner for the CG solver. |
TYPE:
DEFAULT:
OPTIONS:
No options. Specify the keyword to enable pre-conditioning. |
RECOMMENDATION:
A Jacobi block-diagonal pre-conditioner is applied during the conjugate gradient
algorithm to improve the rate of convergence. This reduces the number of CG iterations,
at the expense of some overhead. Pre-conditioning is generally recommended for large systems. |
|
|
|
NoMatrix
Specifies whether PCM matrices should be explicitly constructed and stored. |
TYPE:
DEFAULT:
OPTIONS:
No options. Specify the keyword to avoid explicit construction of PCM matrices. |
RECOMMENDATION:
Storing the PCM matrices requires O(Ngrid2) memory. If this is prohibitive,
the NoMatrix option forgoes explicit construction of the PCM matrices, and
instead constructs the matrix elements as needed, reducing the memory requirement to
O(Ngrid).
|
|
| UseMultipole
Controls the use of the adaptive fast multipole method in the CG solver. |
TYPE:
DEFAULT:
OPTIONS:
No options. Specify the keyword in order to enable the fast multipole method. |
RECOMMENDATION:
The fast multipole approach formally reduces the CPU time to O(Ngrid), but
is only beneficial for spatially extended systems with several
thousand cavity grid points. Requires the use of NoMatrix. |
|
|
|
MultipoleOrder
Specifies the highest multipole order to use in the FMM. |
TYPE:
DEFAULT:
OPTIONS:
n | The highest order multipole in the multipole expansion. |
RECOMMENDATION:
Increasing the multipole order improves accuracy but also adds more computational
expense. The default yields satisfactory performance in common QM/MM/PCM applications. |
|
| Theta
The multipole acceptance criterion. |
TYPE:
DEFAULT:
OPTIONS:
n | A number between zero and one. |
RECOMMENDATION:
The default is recommended for general usage.
This variable determines when the use of a multipole expansion is valid. For a given grid point
and box center in the FMM, a multipole expansion is accepted when r / d < = Theta,
where d is the distance from the grid point to the box center and r is the radius of the box.
Setting Theta to one will accept all multipole expansions, whereas setting it to zero
will accept none. If not accepted, the grid point's interaction with each point inside the box is
computed explicitly. A low Theta is more accurate but also more expensive than a higher
Theta.
|
|
|
|
NBox
The FMM boxing threshold. |
TYPE:
DEFAULT:
OPTIONS:
n | The maximum number of grid points for a leaf box. |
RECOMMENDATION:
The default is recommended. This option is for advanced users only.
The adaptive FMM boxing algorithm divides space into smaller and smaller boxes
until each box has less than or equal to NBox grid points. Modification
of the threshold can lead to speedup or slowdown depending on the molecular system and
other FMM variables.
|
|
A sample input file for the linear-scaling QM/MM/PCM methodology can be found in the
$QC/samples directory, under the name QMMMPCM_crambin.in. This sample involves a QM/MM
description of a protein (crambin) in which a single tyrosine side chain is taken to be the QM
region. The entire protein is immersed in a dielectric using the C-PCM[SWIG] model.
10.2.5 Iso-Density Implementation of SS(V)PE
As discussed above, results obtained various types of PCMs are
quite sensitive to the details of the cavity construction. Q-Chem's implementation
of PCMs, using Lebedev grids, simplifies this construction somewhat, but leaves the radii of
the atomic spheres as empirical parameters (albeit ones for which widely-used default values are
provided). An alternative implementation of the SS(V)PE solvation model is
also available [438],
which attempts to further eliminate empiricism associated with cavity
construction by taking the cavity surface to be a specified iso-contour of the solute's
electron density. In this case, the cavity surface is discretized by projecting a
single-center Lebedev grid onto the iso-contour surface. Unlike the PCM implementation
discussed in Section 10.2.2, for which point-group symmetry is disabled, this
implementation of SS(V)PE supports full symmetry for all Abelian point groups.
The larger and/or the less spherical
the solute molecule is, the more points are needed to get satisfactory
precision in the results. Further experience will be required to develop
detailed recommendations for this parameter. Values as small as 110 points are
usually sufficient for diatomic or triatomic molecules. The default value of
1202 points is adequate to converge the energy within 0.1 kcal/mol
for solutes the size of mono-substituted benzenes.
No implementation yet exists for cavitation, dispersion, or other specific solvation effects,
within this iso-density implementation of SS(V)PE.
Analytic nuclear gradients are also not yet available for this implementation of
SS(V)PE, although they are available for the implementation described in
Section 10.2.2. As with the PCMs discussed in that section, the solute may be
described using Hartree-Fock theory or DFT; post-Hartree-Fock correlated wavefunctions can also
take advantage of molecular orbitals that are polarized using SS(V)PE.
Researchers who use this feature are asked to cite Ref. .
Basic job control variables for the iso-density implementation of SS(V)PE are given below.
More refined control over SS(V)PE jobs is obtained using the $svp input section that is
described in Section 10.2.5.1
SVP
Sets whether to perform the SS(V)PE iso-density solvation procedure. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform the SS(V)PE iso-density solvation procedure. |
TRUE | Perform the SS(V)PE iso-density solvation procedure. |
RECOMMENDATION:
|
| SVP_MEMORY
Specifies the amount of memory for use by the solvation module. |
TYPE:
DEFAULT:
OPTIONS:
n corresponds to the amount of memory in MB. |
RECOMMENDATION:
The default should be fine for medium size molecules
with the default Lebedev grid, only increase if needed. |
|
|
|
SVP_PATH
Specifies whether to run a gas phase computation prior to performing the
solvation procedure. |
TYPE:
DEFAULT:
OPTIONS:
0 | runs a gas-phase calculation and after |
| convergence runs the SS(V)PE computation. |
1 | does not run a gas-phase calculation. |
RECOMMENDATION:
Running the gas-phase calculation provides a good guess to start the solvation
stage and provides a more complete set of solvated properties. |
|
| SVP_CHARGE_CONV
Determines the convergence value for the charges on the cavity. When the
change in charges fall below this value, if the electron density is converged,
then the calculation is considered converged. |
TYPE:
DEFAULT:
OPTIONS:
n Convergence threshold set to 10−n. |
RECOMMENDATION:
The default value unless convergence problems arise. |
|
|
|
SVP_CAVITY_CONV
Determines the convergence value of the iterative iso-density cavity procedure. |
TYPE:
DEFAULT:
OPTIONS:
n Convergence threshold set to 10−n. |
RECOMMENDATION:
The default value unless convergence problems arise. |
|
| SVP_GUESS
Specifies how and if the solvation module will use a given guess for the
charges and cavity points. |
TYPE:
DEFAULT:
OPTIONS:
0 | No guessing. |
1 | Read a guess from a previous Q-Chem solvation computation. |
2 | Use a guess specified by the $svpirf section from the input |
RECOMMENDATION:
It is helpful to also set SCF_GUESS to READ when using a guess
from a previous Q-Chem run. |
|
|
|
The format for the $svpirf section of the input is the following:
$svpirf
<# point> <x point> <y point> <z point> <charge> <grid weight>
<# point> <x normal> <y normal> <z normal>
$end
10.2.5.1 The $svp input section
Now listed are a number of variables that directly access the solvation module
and therefore must be specified in the context of a FORTRAN namelist. The
format is as follows:
$svp
<KEYWORD>=<VALUE>, <KEYWORD>=<VALUE>,...
<KEYWORD>=<VALUE>
$end
For example, the section may look like this:
$svp
RHOISO=0.001, DIELST=78.39, NPTLEB=110
$end
The following keywords are supported in the $svp section:
DIELST
The static dielectric constant. |
TYPE:
DEFAULT:
OPTIONS:
real number specifying the constant. |
RECOMMENDATION:
The default value 78.39 is appropriate for water solvent. |
|
| ISHAPE
A flag to set the shape of the cavity surface. |
TYPE:
DEFAULT:
OPTIONS:
0 | use the electronic iso-density surface. |
1 | use a spherical cavity surface. |
RECOMMENDATION:
|
|
|
RHOISO
Value of the electronic iso-density contour used to specify the cavity surface.
(Only relevant for ISHAPE = 0). |
TYPE:
DEFAULT:
OPTIONS:
Real number specifying the density in electrons/bohr3. |
RECOMMENDATION:
The default value is optimal for most situations. Increasing the value
produces a smaller cavity which ordinarily increases the magnitude of the
solvation energy. |
|
| RADSPH
Sphere radius used to specify the cavity surface (Only relevant for ISHAPE=1). |
TYPE:
DEFAULT:
Half the distance between the outermost atoms plus 1.4 Angstroms. |
OPTIONS:
Real number specifying the radius in bohr (if positive) or in Angstroms (if negative). |
RECOMMENDATION:
Make sure that the cavity radius is larger than the length of the molecule. |
|
|
|
INTCAV
A flag to select the surface integration method. |
TYPE:
DEFAULT:
OPTIONS:
0 | Single center Lebedev integration. |
1 | Single center spherical polar integration. |
RECOMMENDATION:
The Lebedev integration is by far the more efficient. |
|
| NPTLEB
The number of points used in the Lebedev grid for the single-center surface
integration. (Only relevant if INTCAV=0). |
TYPE:
DEFAULT:
OPTIONS:
Valid choices are:
| 6, 18, 26, 38, 50, 86, 110, 146, 170, 194, 302, 350, 434, 590, 770, |
| 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, |
| 4802, or 5294. |
RECOMMENDATION:
The default value has been found adequate to obtain the energy to within 0.1
kcal/mol for solutes the size of mono-substituted benzenes. |
|
|
|
NPTTHE, NPTPHI
The number of (θ,ϕ) points used for single-centered surface
integration (relevant only if INTCAV=1). |
TYPE:
DEFAULT:
OPTIONS:
θ,ϕ specifying the number of points. |
RECOMMENDATION:
These should be multiples of 2 and 4 respectively, to provide symmetry
sufficient for all Abelian point groups. Defaults are too small for all but
the tiniest and simplest solutes. |
|
| LINEQ
Flag to select the method for solving the linear equations that determine the
apparent point charges on the cavity surface. |
TYPE:
DEFAULT:
OPTIONS:
0 | use LU decomposition in memory if space permits, else switch to LINEQ=2 |
1 | use conjugate gradient iterations in memory if space permits, else use LINEQ=2 |
2 | use conjugate gradient iterations with the system matrix stored externally on disk. |
RECOMMENDATION:
The default should be sufficient in most cases. |
|
|
|
CVGLIN
Convergence criterion for solving linear equations by the conjugate gradient
iterative method (relevant if LINEQ=1 or 2). |
TYPE:
DEFAULT:
OPTIONS:
Real number specifying the actual criterion. |
RECOMMENDATION:
The default value should be used unless convergence problems arise. |
|
The single-center surface integration approach may fail for certain highly
non spherical molecular surfaces. The program will automatically check for this,
aborting with a warning message if necessary. The single-center approach
succeeds only for what is called a "star surface", meaning that an observer
sitting at the center has an un-obstructed view of the entire surface. Said
another way, for a star surface any ray emanating out from the center will pass
through the surface only once. Some cases of failure may be fixed by simply
moving to a new center with the ITRNGR parameter described below. But
some surfaces are inherently non-star surfaces and cannot be treated with this
program until more sophisticated surface integration approaches are
implemented.
ITRNGR
Translation of the cavity surface integration grid. |
TYPE:
DEFAULT:
OPTIONS:
0 | No translation (i.e., center of the cavity at the origin |
| of the atomic coordinate system) |
1 | Translate to the center of nuclear mass. |
2 | Translate to the center of nuclear charge. |
3 | Translate to the midpoint of the outermost atoms. |
4 | Translate to midpoint of the outermost non-hydrogen atoms. |
5 | Translate to user-specified coordinates in Bohr. |
6 | Translate to user-specified coordinates in Angstroms. |
RECOMMENDATION:
The default value is recommended unless the single-center integrations
procedure fails. |
|
| TRANX, TRANY, TRANZ
x, y, and z value of user-specified translation (only relevant if
ITRNGR is set to 5 or 6 |
TYPE:
DEFAULT:
OPTIONS:
x, y, and z relative to the origin in the appropriate units. |
RECOMMENDATION:
|
|
|
IROTGR
Rotation of the cavity surface integration grid. |
TYPE:
DEFAULT:
OPTIONS:
0 | No rotation. |
1 | Rotate initial xyz axes of the integration grid to coincide |
| with principal moments of nuclear inertia (relevant if ITRNGR=1) |
2 | Rotate initial xyz axes of integration grid to coincide with |
| principal moments of nuclear charge (relevant if ITRNGR=2) |
3 | Rotate initial xyz axes of the integration grid through user-specified |
| Euler angles as defined by Wilson, Decius, and Cross. |
RECOMMENDATION:
The default is recommended unless the knowledgeable user has good reason otherwise. |
|
| ROTTHE ROTPHI ROTCHI
Euler angles (θ, ϕ, χ) in degrees for user-specified
rotation of the cavity surface. (relevant if IROTGR=3) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
IOPPRD
Specifies the choice of system operator form. |
TYPE:
DEFAULT:
OPTIONS:
0 | Symmetric form. |
1 | Non-symmetric form. |
RECOMMENDATION:
The default uses more memory but is generally more efficient, we recommend its
use unless there is shortage of memory available. |
|
The default behavior for Q-Chem's iso-density implementation of SS(V)PE
is to check that the single-center expansion method for cavity integration is valid
by searching for the iso-density surface in two different ways: by working inwards from a large
distance, and by working outwards from the origin. If the same result is obtained (within
tolerances) by both procedures, then the cavity is accepted.
If they don't agree, the program exits with an error message indicating that the inner
iso-density surface is found to be too far from the outer iso-density surface.
Some molecules, such as C60, can have a hole in the middle of the molecule.
Such molecules have two different "legal" iso-density surfaces, a small inner one inside the
"hole", and a large outer one that is the desired surface for solvation.
So the cavity check described above will cause the program to exit.
To avoid this, one can consider turning off the cavity check that works out from the origin,
leaving only the outer cavity determined by working in from large distances:
ICVICK
Specifies whether to perform cavity check |
TYPE:
DEFAULT:
OPTIONS:
0 | no cavity check, use only the outer cavity |
1 | cavity check, generating both the inner and outer cavities and compare. |
RECOMMENDATION:
Consider turning off cavity check only if the molecule has a hole and if a star
(outer) surface is expected. |
|
10.2.6 Langevin Dipoles Solvation Model
Q-Chem provides the option to calculate molecular properties in aqueous
solution and the magnitudes of the hydration free energies by the Langevin
dipoles (LD) solvation model developed by Jan Florián and
Arieh Warshel [442,[443], of the University of Southern California. In this
model, a solute molecule is surrounded by a sphere of point dipoles, with
centers on a cubic lattice. Each of these dipoles (called Langevin dipoles)
changes its size and orientation in the electrostatic field of the solute and
the other Langevin dipoles. The electrostatic field from the solute is
determined rigorously by the integration of its charge density, whereas for
dipole-dipole interactions, a 12 Å cutoff is used. The Q-Chem / ChemSol 1.0
implementation of the LD model is fully self-consistent in that the
molecular quantum mechanical calculation takes into account solute-solvent
interactions. Further details on the implementation and parameterization of
this model can be found in the original literature [442,[443].
10.2.6.1 Overview
The results of ChemSol calculations are printed in the standard output file.
Below is a part of the output for a calculation on the methoxide anion
(corresponding to the sample input given later on, and the sample file in the
$QC/samples directory).
Iterative Langevin Dipoles (ILD) | Results (kcal/mol) | |
LD Electrostatic energy | −86.14 |
Hydrophobic energy | 0.28 |
van der Waals energy (VdW) | −1.95 |
Bulk correction | −10.07 |
Solvation free energy dG(ILD) | −97.87 |
Table 10.1:
Results of the Langevin Dipoles solvation model, for aqueous methoxide.
The total hydration free energy, ∆G(ILD) is calculated as a sum of
several contributions. Note that the electrostatic part of ∆G is
calculated by using the linear-response approximation [442] and
contains contributions from the polarization of the solute charge
distribution due to its interaction with the solvent. This results from the
self-consistent implementation of the Langevin dipoles model within Q-Chem.
In order for an LD calculation to be carried out by the ChemSol program within
Q-Chem, the user must specify a single-point HF or DFT calculation (i.e., at
least $rem variables BASIS, EXCHANGE and CORRELATION) in
addition to setting CHEMSOL $rem variable equal to 1 in the keyword section.
CHEMSOL
Controls the use of ChemSol in Q-Chem. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use ChemSol. |
1 | Perform a ChemSol calculation. |
RECOMMENDATION:
|
| CHEMSOL_EFIELD
Determines how the solute charge distribution is approximated in evaluating
the electrostatic field of the solute. |
TYPE:
DEFAULT:
OPTIONS:
1 | Exact solute charge distribution is used. |
0 | Solute charge distribution is approximated by Mulliken atomic charges. |
| This is a faster, but less rigorous procedure. |
RECOMMENDATION:
|
|
|
CHEMSOL_NN
Sets the number of grids used to calculate the average hydration free energy. |
TYPE:
DEFAULT:
5 | ∆Ghydr will be averaged over 5 different grids. |
OPTIONS:
n | Number of different grids (Max = 20). |
RECOMMENDATION:
|
| CHEMSOL_PRINT
Controls printing in the ChemSol part of the Q-Chem output file. |
TYPE:
DEFAULT:
OPTIONS:
0 | Limited printout. |
1 | Full printout. |
RECOMMENDATION:
|
|
|
10.2.6.2 Customizing Langevin dipoles solvation calculations
Accurate calculations of hydration free energies require a judicious choice of
the solute-solvent boundary in terms of atom-type dependent parameters. The
default atomic van der Waals radii available in Q-Chem were chosen to provide
reasonable hydration free energies for most solutes and basis sets. These
parameters basically coincide with the ChemSol 2.0 radii given in
Ref. . The only difference between the Q-Chem and ChemSol 2.0
atomic radii stems from the fact that Q-Chem parameter set uses hybridization
independent radii for carbon and oxygen atoms.
User-defined atomic radii can be specified in the $van_der_waals section of the input
file after setting READ_VDW $rem variable to TRUE.
READ_VDW
Controls the input of user-defined atomic radii for ChemSol calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use default ChemSol parameters. |
TRUE | Read from the $van_der_waals section of the input file. |
RECOMMENDATION:
|
Two different (mutually exclusive) formats can be used, as shown below.
$van_der_waals
1
atomic_number radius
...
$end
$van_der_waals
2
sequential_atom_number VdW_radius
...
$end
The purpose of the second format is to permit the user to customize the radius of
specific atoms, in the order that they appear in the $molecule section,
rather than simply by atomic numbers as in format 1. The radii of atoms that
are not listed in the $van_der_waals input will be assigned default values. The atomic
radii that were used in the calculation are printed in the ChemSol part of the
output file in the column denoted rp. All radii should be given in Angstroms.
Example 10.0 A Langevin dipoles calculation on the methoxide anion. A
customized value is specified for the radius of the C atom.
$molecule
-1 1
C 0.0000 0.0000 -0.5274
O 0.0000 0.0000 0.7831
H 0.0000 1.0140 -1.0335
H 0.8782 -0.5070 -1.0335
H -0.8782 -0.5070 -1.0335
$end
$rem
EXCHANGE hf
BASIS 6-31G
SCF_CONVERGENCE 6
CHEMSOL 1
READ_VDW true
$end
$van_der_waals
2
1 2.5
$end
10.2.7 The SM8 Model
The SM8 model is described in detail in Ref. .
It may be employed with density functional theory
(with any density functional available in Q-Chem) or with Hartree-Fock theory.
As discussed further below, it is intended for use only with the 6-31G*, 6-31+G*, and 6-31+G** basis sets.
The SM8 model is a universal continuum solvation model where "universal" denotes applicable to all solvents,
and "continuum" denotes that the solvent is not represented explicitly
but rather as a dielectric fluid with surface tensions at the solute-solvent interface
("continuum" solvation models are sometimes called "implicit" solvation models).
SM8 is applicable to any charged or uncharged solute in any solvent or liquid medium
for which a few key descriptors are known, in particular:
- dielectric constant
- refractive index
- bulk surface tension
- acidity on the Abraham scale
- basicity on the Abraham scale
- carbon aromaticity, which equals the fraction of non-hydrogenic solvent atoms that are aromatic carbon atoms
- electronegative halogenicity, which equals the fraction of non-hydrogenic solvent atoms that are F, Cl, or Br).
The model separates the standard-state free energy of solvation into three components,
as discussed in the next three paragraphs.
The first component is the long-range bulk electrostatic contribution arising
from a self-consistent reaction field (SCRF) treatment using
the generalized Born approximation for electrostatics.
The cavities for the bulk electrostatics calculation are defined by superpositions of nuclear-centered spheres
whose sizes are determined by parameters called intrinsic atomic Coulomb radii.
The SM8 Coulomb radii have been optimized for H, C, N, O, F, Si, P, S, Cl, and Br.
For any other atom the SM8 model uses the van der Waals radius of Bondi for those atoms for which Bondi defined radii;
in cases where the atomic radius is not given in Bondi's paper [452], a radius of
2.0 Å is used.
This first contribution to the standard-state free energy of solvation is called
the electronic-nuclear-polarization (ENP) term (if the geometry is assumed to be the same in the gas and the liquid,
then this becomes just an electronic polarization (EP) term).
The bulk electrostatic term is sometimes called the electrostatic term,
but it should be emphasized that it is calculated from the bulk dielectric constant (bulk relative permittivity),
which is not a completely valid description of the solvent in the first solvation shell.
In SM8 the bulk electrostatic term is calculated within the generalized Born approximation
with geometry-dependent atomic radii calculated from the intrinsic Coulomb radii by a de-screening approximation.
The second contribution to the free energy of solvation is the contribution arising from
short-range interactions between the solute and solvent molecules in the first solvation shell.
This contribution is sometimes called the cavity-dispersion-solvent-structure (CDS) term,
and it is a sum of terms that are proportional (with geometry-dependent proportionality constants
called atomic surface tensions) to the solvent-accessible surface areas (SASAs) of the individual atoms of the solute.
The SASA of the solute molecule is the area of a surface generated
by the center of a spherical effective solvent molecule rolling on the van der Waals surface of the solute molecule.
The SASA is calculated with the Analytic Surface Area (ASA) algorithm [459].
The van der Waals radii of Bondi are used in this procedure when defined; in cases
where the atomic radius is not given in Bondi's paper [452] a radius of 2.0 Å is used.
The solvent radius is set to 0.40 Å for any solvent.
Note that the solvent-structure part of the CDS term include many aspects of solvent structure
that are not described by bulk electrostatics, for example, hydrogen bonding, exchange repulsion,
and the deviation of the effective dielectric constant in the first solvation shell from its bulk value.
The semi-empirical nature of the CDS term also makes up for errors due to
(i) assuming fixed and model-dependent values of the intrinsic Coulomb radii and
(ii) any systematic errors in the description of the solute-solvent electrostatic interaction
by the generalized Born approximation with the dielectric de-screening approximation and approximate partial atomic charges.
The third component is the concentration component.
This is zero if the standard state concentration of the solute is the same in the gas phase and solution
(for example, if it is one mole per liter in the gas as well as in the solution),
and it can be calculated from the ideal-gas formulas when they are not equal, as discussed further below.
Note: we use "liquid phase" and "solution phase" as synonyms in this documentation.
The SM8 model does not require the user to assign molecular-mechanics types to an atom or group;
all atomic surface tensions in the theory are unique and continuous functions of geometry defined
by the model and calculated from the geometry by the program.
In general, SM8 may be used with any level of electronic structure theory
as long as accurate partial charges can be computed for that level of theory.
The implementation of the SM8 model in Q-Chem utilizes
self-consistently polarized Charge Model 4 (CM4) class IV charges.
The self-consistent polarization is calculated by a
quantum mechanical self-consistent reaction field calculation.
The CM4 charges are obtained from population-analysis charges
by a mapping whose parameters depend on the basis set (and only on the basis set-for example, these parameters do not depend on which density functional is used).
The supported basis sets for which the charge parameters have been incorporated into
the SM8 solvation model of Q-Chem are
The charge mapping parameters are given in Ref. .
Other basis sets should not be used with the implementation of the SM8 model in Q-Chem.
The SM8 solvation free energy is output at 298 K for a standard-state concentration of 1 M
in both the gaseous and liquid-phase solution phases.
Solvation free energies in the literature are often tabulated
using a standard-state-gas phase pressure of 1 atm.
To convert 1-molar-to-1-molar solvation free energies at 298 K
to a standard state that uses a gas-phase pressure of 1 atm
and solute standard state concentration of 1 M, add +1.89 kcal/mol
to the computed solvation free energy.
Liquid-phase geometry optimizations can be carried out,
but basis sets that use spherical harmonic d functions or angular functions higher than
d (f, g, etc..)
are not supported for liquid-phase geometry optimization with SM8.
Since, by definition, the 6-31G*, 6-31+G*, and 6-31+G** basis sets have Cartesian d shells,
they are examples of basis sets that may be used for geometry optimization.
Liquid-phase Hessian calculations can be carried out by numerical differentiation of analytical gradients
or by double differentiation of energies (the former is much more stable and is also more economical).
The analytic gradients of SM8 are based on the analytical derivatives of the polarization free energy
and the analytical derivatives of the CDS terms derived in Ref. .
The $rem variables associated with running SM8 calculations are documented below.
Q-Chem requires at least the single-point energy calculation Q-Chem variables:
BASIS, EXCHANGE, and CORRELATION (if required), in addition to the
SM8-specific variables SMX_SOLVATION and SMX_SOLVENT.
SMX_SOLVATION
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform the SM8 solvation procedure |
TRUE | Perform the SM8 solvation procedure |
RECOMMENDATION:
|
| SMX_SOLVENT
TYPE:
DEFAULT:
OPTIONS:
any name from the list of solvents |
RECOMMENDATION:
|
|
|
The list of supported solvent keywords is as follows:
1,1,1-trichloroethane | bromoethane | m-ethylbenzoate |
1,1,2-trichloroethane | bromooctane | m-ethylethanoate |
1,1-dichloroethane | butanal | m-ethylmethanoate |
1,2,4-trimethylbenzene | butanoicacid | m-ethylphenylketone |
1,4-dioxane | butanone | m-ethylpropanoate |
1-bromo-2-methylpropane | butanonitrile | m-ethylbutanoate |
1-bromopentane | butylethanoate | m-ethylcyclohexane |
1-bromopropane | butylamine | m-ethylformamide |
1-butanol | butylbenzene | m-xylene |
1-chloropentane | carbon disulfide | heptane |
1-chloropropane | carbon tetrachloride | hexadecane |
1-decanol | chlorobenzene | hexane |
1-fluorooctane | chlorotoluene | nitrobenzene |
1-heptanol | cis-1,2-dimethylcyclohexane | nitroethane |
1-hexanol | decalin | nitromethane |
1-hexene | cyclohexane | methylaniline |
1-hexyne | cyclohexanone | nonane |
1-iodobutane | cyclopentane | octane |
1-iodopentene | cyclopentanol | pentane |
1-iodopropane | cyclopentanone | o-chlorotoluene |
1-nitropropane | decane | o-cresol |
1-nonanol | dibromomethane | o-dichlorobenzene |
1-octanol | dibutyl ether | o-nitrotoluene |
1-pentanol | dichloromethane | o-xylene |
1-pentene | diethyl ether | pentadecane |
1-pentyne | diethylsulfide | pentanal |
1-propanol | diethylamine | pentanoic acid |
2,2,20trifluoroethanol | diiodomethane | pentylethanoate |
2,2,4-trimethylpentane | dimethyldisulfide | pentylamine |
2,4-dimethylpentane | dimethylacetamide | perfluorobenzene |
2,4-dimethylpyridine | dimethylformamide | phenyl ether |
2,6-dimethylpyridine | dimethylpyridine | propanal |
2-bromopropane | dimethyl sulfoxide | propanoic acid |
2-chlorobutane | dipropylamine | propanonitrile |
2-heptanone | dodecane | propylethanoate |
2-hexanone | E-1,2-dichloroethene | propylamine |
2-methylpentane | E-2-pentene | p-xylene |
2-methylpyridine | ethanethiol | pyridine |
2-nitropropane | ethanol | pyrrolidine |
2-octanone | ethylethanoate | sec-butanol |
2-pentanone | ethylmethanoate | t-butanol |
2-propanol | ethylphenyl ether | t-butylbenzene |
2-propen-1-ol | ethylbenzene | tetrachloroethene |
3-methylpyridine | ethylene glycol | tetrahydrofuran |
3-pentanone | fluorobenzene | tetrahyrothiophenedioxide |
4-heptanone | formamide | tetralin |
4-methyl-2-pentanone | formic acid | thiophene |
4-methylpyridine | hexadecyliodide | thiophenol |
5-nonanone | hexanoic acid | toluene |
acetic acid | iodobenzene | trans-decalin |
acetone | iodoethane | tribromomethane |
acetonitrile | iodomethane | tributylphosphate |
aniline | isobutanol | trichloroethene |
anisole | isopropyl ether | trichloromethane |
benzaldehyde | isopropylbenzene | triethylamine |
benzene | isopropyltoluene | undecane |
benzonitrile | m-cresol | water |
benzyl alcohol | mesitylene | Z-1,2-dichloroethene |
bromobenzene | methanol | other
|
The "SMX_SOLVENT = other" specification requires an additional free-format file
called "solvent_data" that should contain the float-point values of
the following solvent descriptors: Dielec, SolN, SolA, SolB, SolG, SolC, SolH.
Dielec | dielectric constant, ϵ, of the solvent |
SolN | index of refraction at optical frequencies at 293 K, n20D |
SolA | Abraham's hydrogen bond acidity, ∑α2H |
SolB | Abraham's hydrogen bond basicity, ∑β2H |
SolG | γ = γm / γ0 (default is 0.0),
where γm is the macroscopic surface tension at air/solvent |
| interface at 298 K, and γ0 is 1 cal mol−1 Å−2
(1 dyne/cm = 1.43932 cal mol−1 Å−2) |
SolC | aromaticity, ϕ : the fraction of non-hydrogenic solvent atoms that are aromatic |
| carbon atoms |
SolH | electronegative "halogenicity", ψ : the fraction of non-hydrogenic solvent atoms that are |
| F, Cl or Br
|
For a desired solvent, these values can be derived from experiment
or from interpolation or extrapolation of data available for other solvents.
Solvent parameters for common organic solvents are tabulated in the
Minnesota Solvent Descriptor Database.
The latest version of this database is available at:
http://comp.chem.umn.edu/solvation/mnsddb.pdf
The SM8 test suite contains the following representative examples:
- single-point solvation energy and analytical gradient calculation for 2,2-dichloroethenyl dimethyl phosphate in water at the M06-2X / 6-31G* level;
- single-point solvation energy calculation for 2,2-dichloroethenyl dimethyl phosphate in benzene at the M06-2X / 6-31G* level;
- single-point solvation energy calculation for 2,2-dichloroethenyl dimethyl phosphate in ethanol at the M06-2X / 6-31G* level;
- single-point solvation energy calculation for 5-fluorouracil in water at the M06 / 6-31+G* level;
- single-point solvation energy calculation for 5-fluorouracil in octanol at the M06-L / 6-31+G* level;
- single-point solvation energy and analytical gradient calculation for 5-fluorouracil in fluorobenzene at the M06-HF / 6-31+G** level;
- geometry optimization for protonated methanol CH3OH2+ in water at the B3LYP / 6-31G* level;
- finite-difference frequency (with analytical gradient) calculation for protonated methanol CH3OH2+ in water at the B3LYP / 6-31G* level.
Users who wish to calculate solubilities can calculate them from the free energies of solvation
by the method described in Ref. .
The present model can also be used with confidence to calculate partition coefficients
(e.g., Henry's Law constants, octanol/water partition coefficients, etc..) by the method
described in Ref. .
The user should note that the free energies of solvation calculated by the SM8 model
in the current version of Q-Chem are all what may be called equilibrium free energies of solvation.
The nonequilibrium algorithm required for vertical excitation energies [464]
is not yet available in Q-Chem.
10.2.8 COSMO
Q-Chem also contains the original conductor-like screening (COSMO) model
from Klamt and Schüürmann [428].
Our energy and gradient implementations resemble the
ones in Turbomole [465]. To use the COSMO solvation model,
one can use the "COSMO" option for the SOLVENT_METHOD $rem variable and set the
SOLVENT_DIELECTRIC variable to specify the
dielectric constant for the solvent (see the cosmo.in sample job).
Users of the COSMO-RS package [466] can request special versions of Q-Chem for the generation
of σ-surface files (for their own solutes/solvents) for the use in their COSMOtherm and other calculations.
10.3 Wavefunction Analysis
Q-Chem performs a number of standard wavefunction analyses by default.
Switching the $rem variable WAVEFUNCTION_ANALYSIS to FALSE
will prevent the calculation of all wavefunction analysis features (described
in this section). Alternatively, each wavefunction analysis feature may be
controlled by its $rem variable. (The NBO package which is interfaced with
Q-Chem is capable of performing more sophisticated analyses. See later in
this chapter and the NBO manual for details).
WAVEFUNCTION_ANALYSIS
Controls the running of the default wavefunction analysis tasks. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform default wavefunction analysis. |
FALSE | Do not perform default wavefunction analysis. |
RECOMMENDATION:
|
Note:
WAVEFUNCTION_ANALYSIS has no effect on NBO, solvent
models or vibrational analyses. |
10.3.1 Population Analysis
The one-electron charge density, usually written as
ρ(r)= |
∑
μν
|
Pμν ϕμ (r)ϕν (r) |
| (10.3) |
represents the probability of finding an electron at the point r, but
implies little regarding the number of electrons associated with a given
nucleus in a molecule. However, since the number of electrons N is related to
the occupied orbitals ψi by
N = 2 |
N | / |
N 2 2 ∑
a
|
| ψa(r) |2 dr |
| (10.4) |
We can substitute the atomic orbital (AO) basis expansion of ψa into
Eq. (10.4) to obtain
N = |
∑
μυ
|
Pμυ Sμυ = |
∑
μ
|
(PS )μμ = Tr(PS) |
| (10.5) |
where we interpret (PS)μμ as the number of electrons associated
with ϕμ. If the basis functions are atom-centered, the number of
electrons associated with a given atom can be obtained by summing over all the
basis functions. This leads to the Mulliken formula for the net charge of the
atom A:
qA = ZA − |
∑
μ ∈ A
|
(PS )μμ |
| (10.6) |
where ZA is the atom's nuclear charge. This is called a Mulliken population
analysis [7]. Q-Chem performs a Mulliken population analysis by
default.
POP_MULLIKEN
Controls running of Mulliken population analysis. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not calculate Mulliken Population. |
TRUE | (or 1) Calculate Mulliken population |
2 | Also calculate shell populations for each occupied orbital. |
−1 | Calculate Mulliken charges for both the ground state and any CIS, |
| RPA, or TDDFT excited states.
|
RECOMMENDATION:
Leave as TRUE, unless excited-state charges are desired.
Mulliken analysis is a trivial additional calculation, for ground or
excited states. |
|
| LOWDIN_POPULATION
Run a Löwdin population analysis instead of a Mulliken. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate Löwdin Populations. |
TRUE | Run Löwdin Population analyses instead of Mulliken. |
RECOMMENDATION:
|
|
|
Although conceptually simple, Mulliken population analyses suffer from a heavy
dependence on the basis set used, as well as the possibility of producing
unphysical negative numbers of electrons. An alternative is that of Löwdin
Population analysis [467], which uses the Löwdin symmetrically orthogonalized basis
set (which is still atom-tagged) to assign the electron density. This shows
a reduced basis set dependence, but maintains the same essential features.
While Mulliken and Löwdin population analyses are commonly employed, and can
be used to produce information about changes in electron density and also localized
spin polarizations, they should not be interpreted as oxidation states of the atoms
in the system. For such information we would recommend a bonding analysis technique
(LOBA or NBO).
A more stable alternative to Mulliken or Löwdin charges are the so-called "CHELPG"
charges ("Charges from the Electrostatic Potential on a Grid") [468].
The CHELPG charges are computed to provide the best fit to the molecular electrostatic
potential evaluated on a grid, subject to the constraint that the sum of the CHELPG charges
must equal the molecular charge. Q-Chem's implementation of the CHELPG algorithm differs
slightly from the one originally algorithm described by Breneman and Wiberg [468],
in that Q-Chem weights the grid points with a smoothing function to ensure that the CHELPG
charges vary continuously as the nuclei are displaced. (For any particular geometry, however,
numerical values of the charges are quite similar to those obtained using the original
algorithm.) Details of the Q-Chem implementation can be found in Ref. .
CHELPG
Controls the calculation of CHELPG charges. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate CHELPG charges. |
TRUE | Compute CHELPG charges. |
RECOMMENDATION:
Set to TRUE if desired. For large molecules, there is some overhead associated with computing
CHELPG charges, especially if the number of grid points is large. |
|
| CHELPG_HEAD
Sets the "head space" for the CHELPG grid. |
TYPE:
DEFAULT:
OPTIONS:
N | Corresponding to a head space of N/10, in Å. |
RECOMMENDATION:
Use the default, which is the value recommended by Breneman and Wiberg [468]. |
|
|
|
CHELPG_DX
Sets the grid spacing for the CHELPG grid. |
TYPE:
DEFAULT:
OPTIONS:
N | Corresponding to a grid space of N/20, in Å. |
RECOMMENDATION:
Use the default (which corresponds to the "dense grid" of Breneman and Wiberg [468]),
unless the cost is prohibitive, in which case a larger value can be selected. |
|
Finally, Hirschfeld population analysis [470] provides yet another
definition of atomic charges in molecules via a Stockholder prescription.
The charge on atom A, qA, is defined by
qA = ZA − | ⌠ ⌡
|
dr |
ρ0A(r)
|
ρ(r), |
| (10.7) |
where ZA is the nuclear charge of A, ρ0B is the isolated ground-state atomic density
of atom B, and ρ is the molecular density. The sum goes over all atoms in the molecule.
Thus computing Hirshfeld charges requires a self-consistent calculation of the isolated atomic densities
(the promolecule) as well as the total molecule.
Following SCF completion, the Hirshfeld analysis proceeds by running another SCF calculation to obtain the atomic densities.
Next numerical quadrature is used to evaluate the integral in Eq. (10.7).
Neutral ground-state atoms are used, and the choice of
appropriate reference for a charged molecule is ambiguous (such jobs will crash).
As numerical integration (with default quadrature grid) is used, charges may not sum precisely to zero.
A larger XC_GRID may be used to improve the accuracy of the integration.
HIRSHFELD
Controls running of Hirshfeld population analysis. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculate Hirshfeld populations. |
FALSE | Do not calculate Hirshfeld populations.
|
RECOMMENDATION:
|
| HIRSHFELD_READ
Switch to force reading in of isolated atomic densities. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Read in isolated atomic densities from previous Hirshfeld calculation from disk. |
FALSE | Generate new isolated atomic densities. |
RECOMMENDATION:
Use default unless system is large. Note, atoms should be in the same order
with same basis set used as in the previous Hirshfeld
calculation (although coordinates can change). The previous calculation
should be run with the -save switch. |
|
|
|
HIRSHFELD_SPHAVG
Controls whether atomic densities should be spherically averaged in pro-molecule. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Spherically average atomic densities. |
FALSE | Do not spherically average.
|
RECOMMENDATION:
|
The next section discusses how to compute arbitrary electrostatic multipole moments for
an entire molecule, including both ground- and excited-state electron densities.
Occasionally, however, it is useful to decompose the electronic part of the multipole
moments into contributions from individual MOs. This decomposition is especially useful
for systems containing unpaired electrons [471], where the first-order moments
〈x〉, 〈y〉, and 〈z 〉 characterize the centroid
(mean position) of the half-filled MO, and the second-order moments determine its radius
of gyration, Rg, which characterizes the size of the MO.
Upon setting PRINT_RADII_GYRE = TRUE, Q-Chem will print
out centroids and radii of gyration for each MO and for the overall electron density.
If CIS or TDDFT excited states are requested, then this keyword will also print out the
centroids and radii of gyration for each excited-state electron density.
PRINT_RADII_GYRE
Controls printing of MO centroids and radii of gyration. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | (or 1) Calculate the centroid and radius of gyration for each MO and density. |
FALSE | (or 0) Do not calculate these quantities.
|
RECOMMENDATION:
|
10.3.2 Multipole Moments
Q-Chem can compute Cartesian multipole moments of the charge density to
arbitrary order, both for the ground state and for excited states
calculated using the CIS or TDDFT methods.
MULTIPOLE_ORDER
Determines highest order of multipole moments to print if wavefunction
analysis requested. |
TYPE:
DEFAULT:
OPTIONS:
n | Calculate moments to nth order. |
RECOMMENDATION:
Use default unless higher multipoles are required. |
|
| CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not calculate excited-state moments. |
TRUE | (or 1) Calculate moments for each excited state.
|
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial
additional calculation.) The MULTIPOLE_ORDER controls how many
multipole moments are printed. |
|
|
|
10.3.3 Symmetry Decomposition
Q-Chem's default is to write the SCF wavefunction molecular orbital
symmetries and energies to the output file. If requested, a symmetry
decomposition of the kinetic and nuclear attraction energies can also be
calculated.
SYMMETRY_DECOMPOSITION
Determines symmetry decompositions to calculate. |
TYPE:
DEFAULT:
OPTIONS:
0 | No symmetry decomposition. |
1 | Calculate MO eigenvalues and symmetry (if available). |
2 | Perform symmetry decomposition of kinetic energy and nuclear attraction |
| matrices. |
RECOMMENDATION:
|
10.3.4 Localized Orbital Bonding Analysis
Localized Orbital Bonding Analysis (LOBA) [303] is a technique developed by Dr. Alex Thom
and Eric Sundstrom at Berkeley with Prof. Martin Head-Gordon. Inspired by the work
of Rhee and Head-Gordon [472], it makes use of the fact that the post-SCF localized occupied orbitals of a
system provide a large amount of information about the bonding in the system.
While the canonical Molecular Orbitals can provide information about local reactivity
and ionization energies, their delocalized nature makes them rather uninformative when looking
at the bonding in larger molecules.
Localized orbitals in contrast provide a convenient way to visualize and account for electrons.
Transformations of the orbitals within the occupied subspace do not alter the resultant density;
if a density can be represented as orbitals localized on individual atoms, then those orbitals
can be regarded as non-bonding.
If a maximally localized set of orbitals still requires some to be delocalized between atoms,
these can be regarded as bonding electrons.
A simple example is that of He2 versus H2. In the former, the delocalized σg and σu
canonical orbitals may be transformed into 1s orbitals on each He atom, and there is no bond between them.
This is not possible for the H2 molecule, and so we can regard there being a bond between the atoms.
In cases of multiple bonding, multiple delocalized orbitals are required.
While there are no absolute definitions of bonding and oxidation state, it has been shown that the localized
orbitals match the chemically intuitive notions of core, non-bonded, single- and double-bonded electrons, etc..
By combining these localized orbitals with population analyses, LOBA allows the nature of the bonding within a molecule to be quickly determined.
In addition, it has been found that by counting localized electrons, the oxidation states of transition metals
can be easily found. Owing to polarization caused by ligands, an upper threshold is applied, populations above
which are regarded as "localized" on an atom, which has been calibrated to a range of transition metals, recovering standard oxidation states ranging from −II to VII.
LOBA
Specifies the methods to use for LOBA |
TYPE:
DEFAULT:
OPTIONS:
ab | |
a | specifies the localization method |
| 0 Perform Boys localization. |
| 1 Perform PM localization. |
| 2 Perform ER localization. |
b | specifies the population analysis method |
| 0 Do not perform LOBA. This is the default. |
| 1 Use Mulliken population analysis. |
| 2 Use Löwdin population analysis. |
RECOMMENDATION:
Boys Localization is the fastest. ER will require an auxiliary basis set. |
LOBA 12 provides a reasonable speed/accuracy compromise. |
|
| LOBA_THRESH
Specifies the thresholds to use for LOBA |
TYPE:
DEFAULT:
OPTIONS:
aabb | |
aa | specifies the threshold to use for localization |
bb | specifies the threshold to use for occupation |
Both are measured in %
|
RECOMMENDATION:
Decrease bb to see the smaller contributions to orbitals. Values of
aa between 40 and 75 have been shown to given meaningful results. |
|
|
|
On a technical note, LOBA can function of both Restricted and Unrestricted
SCF calculations. The figures printed in the bonding analysis count the number
of electrons on each atom from that orbital (i.e., up to 1 for Unrestricted
or singly occupied Restricted orbitals, and up to 2 for double occupied Restricted.)
10.3.5 Excited-State Analysis
For CIS, TDHF, and TDDFT excited-state calculations, we have already mentioned
that Mulliken population analysis of the excited-state electron densities may be requested
by setting POP_MULLIKEN = −1, and multipole moments of the excited-state
densities will be generated if CIS_MOMENTS = TRUE. Another useful
decomposition for excited states is to separate the excitation into "particle" and "hole"
components, which can then be analyzed separately [310].
To do this, we define a density matrix for the excited electron,
Dabelec = |
∑
i
|
(X+Y)†ai (X+Y)ib |
| (10.8) |
and a density matrix for the hole that is left behind in the occupied space:
Dijhole = |
∑
a
|
(X+Y)ia (X+Y)†aj |
| (10.9) |
The quantities X and Y are the transition density matrices,
i.e., the components of the TDDFT eigenvector [473]. The indices
i and j denote MOs that occupied in the ground state, whereas a and b index
virtual MOs. Note also that Delec + Dhole = ∆P, the
difference between the ground- and excited-state density matrices.
Upon transforming Delec and Dhole into the AO
basis, one can write
∆q = |
∑
μ
|
(Delec S)μμ = − |
∑
μ
|
(Dhole S)μμ |
| (10.10) |
where ∆q is the total charge that is transferred from the occupied space to the
virtual space. For a CIS calculation,
or for TDDFT within the Tamm-Dancoff approximation [349], ∆q = −1.
For full TDDFT calculations, ∆q may be slightly different than −1.
Comparison of Eq. (10.10) to Eq. (10.5) suggests that the quantities
(Delec S) and (Dhole S) are amenable to
to population analyses of precisely the same sort used to analyze the ground-state density
matrix. In particular, (Delec S)μμ represents the μth
AO's contribution to the excited electron, while (Dhole S)μμ
is a contribution to the hole. The sum of these quantities,
∆qμ = (Delec S)μμ + (Dhole S)μμ |
| (10.11) |
represents the contribution to ∆q arising from the μth AO.
For the particle/hole density matrices, both Mulliken and Löwdin population analyses
available, and are requested by setting CIS_MULLIKEN = TRUE.
CIS_MULLIKEN
Controls Mulliken and Löwdin population analyses for excited-state particle and
hole density matrices. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not perform particle/hole population analysis. |
TRUE | (or 1) Perform both Mulliken and Löwdin analysis of the particle and hole |
| density matrices for each excited state. |
RECOMMENDATION:
Set to TRUE if desired. This represents a trivial additional calculation. |
|
Although the excited-state analysis features described in this section require very
little computational effort, they are turned off by default, because they can generate
a large amount of output, especially if a large number of excited states are requested.
They can be turned on individually, or collectively by setting CIS_AMPL_ANAL =
TRUE. This collective option also requests the calculation of
natural transition orbitals (NTOs), which were introduced in Section 6.10.2.
(NTOs can also be requested without excited-state population analysis. Some
practical aspects of calculating and visualizing NTOs are discussed below, in
Section 10.9.2.)
CIS_AMPL_ANAL
Perform additional analysis of CIS and TDDFT excitation amplitudes,
including generation of natural transition orbitals, excited-state
multipole moments, and Mulliken analysis of the excited state densities
and particle/hole density matrices.
|
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform additional amplitude analysis. |
FALSE | Do not perform additional analysis.
|
RECOMMENDATION:
|
10.4 Intracules
The many dimensions of electronic wavefunctions makes them difficult to
analyze and interpret. It is often convenient to reduce this large number of
dimensions, yielding simpler functions that can more readily provide chemical
insight. The most familiar of these is the one-electron density ρ(r),
which gives the probability of an electron being found at the point r.
Analogously, the one-electron momentum density π(p) gives the
probability that an electron will have a momentum of p. However, the
wavefunction is reduced to the one-electron density much information is lost.
In particular, it is often desirable to retain explicit two-electron
information. Intracules are two-electron distribution functions and provide
information about the relative position and momentum of electrons. A
detailed account of the different type of intracules can be found in
Ref. . Q-Chem's intracule package was developed by Aaron Lee and
Nick Besley, and can compute the following intracules for or HF wavefunctions:
- Position intracules, P(u): describes the probability of finding two
electrons separated by a distance u.
- Momentum intracules, M(v): describes the probability of finding two
electrons with relative momentum v.
- Wigner intracule, W(u,v): describes the combined probability of
finding two electrons separated by u and with relative momentum
v.
10.4.1 Position Intracules
The intracule density, I(u), represents the probability for the
inter-electronic vector u = u1−u2:
I(u) = | ⌠ ⌡
|
ρ(r1 r2 ) δ(r12 − u) dr1 d r2 |
| (10.12) |
where ρ(r1,r2) is the two-electron
density. A simpler quantity is the spherically averaged intracule density,
where Ωu is the angular part of v, measures the
probability that two electrons are separated by a scalar distance u = |u|. This intracule is called a position intracule [474]. If
the molecular orbitals are expanded within a basis set
The quantity P(u) can be expressed as
P(u) = |
∑
μνλσ
|
Γμνλσ (μνλσ)P |
| (10.15) |
where Γμνλσ is the
two-particle density matrix and (μνλσ)P is
the position integral
(μνλσ)P = | ⌠ ⌡
|
ϕμ∗ (r) ϕν(r) ϕλ∗(r+u) ϕσ(r+u) dr dΩ |
| (10.16) |
and ϕμ(r), ϕν(r), ϕλ(r) and
ϕσ(r) are basis functions. For HF wavefunctions, the
position intracule can be decomposed into a Coulomb component,
PJ(u) = |
1
2
|
|
∑
μνλσ
|
Dμν Dλσ (μνλσ)P |
| (10.17) |
and an exchange component,
PK(u) = − |
1
2
|
|
∑
μνλσ
|
[ Dμλα Dνσα + Dμλβ Dνσβ ] (μνλσ)P |
| (10.18) |
where Dμν etc. are density matrix elements. The evaluation of
P(u), PJ(u) and PK(u) within Q-Chem has been described in
detail in Ref. .
Some of the moments of P(u) are physically significant [476], for example
| |
|
| | (10.19) |
| |
|
| | (10.20) |
| |
|
| | (10.21) |
| |
|
| | (10.22) |
|
where n is the number of electrons and, μ is the electronic dipole moment
and Q is the trace of the electronic quadrupole moment tensor. Q-Chem can
compute both moments and derivatives of position intracules.
10.4.2 Momentum Intracules
Analogous quantities can be defined in momentum space; ―I(v), for example,
represents the probability density for the relative momentum
v = p1 − p2:
|
-
I
|
(v) = | ⌠ ⌡
|
π(p1,p2 ) δ(p12−v)dp1 dp2 |
| (10.23) |
where π(p1,p2) momentum two-electron density. Similarly, the
spherically averaged intracule
where Ωv is the angular part of v, is a measure of relative
momentum v = |v| and is called the momentum intracule. The quantity M(v)
can be written as
M(v)= |
∑
μνλσ
|
Γμνλσ ( μνλσ)M |
| (10.25) |
where Γμνλσ is the two-particle density
matrix and (μνλσ)M is the momentum integral [477]
(μνλσ)M = |
v2
2π2
|
| ⌠ ⌡
|
ϕμ∗ (r)ϕν (r +q) ϕλ∗ (u+q) ϕσ (u)j0(q v) dr dq du |
| (10.26) |
The momentum integrals only possess four-fold permutational symmetry, i.e.,
|
(μνλσ)M = (νμλσ)M = (σλνμ)M = (λσμν)M |
| | (10.27) |
| (νμλσ)M = (μνσλ)M = (λσνμ)M = (σλμν)M |
| | (10.28) |
|
and therefore generation of M(v) is roughly twice as expensive as P(u).
Momentum intracules can also be decomposed into Coulomb MJ(v) and exchange
MK(v) components:
MJ(v) = |
1
2
|
|
∑
μνλσ
|
Dμν Dλσ(μνλσ)M |
| (10.29) |
MK(v) = − |
1
2
|
|
∑
μνλσ
|
[ Dμλα Dνσα + Dμλβ Dνσβ ](μνλσ)M |
| (10.30) |
Again, the even-order moments are physically significant [477]:
|
∞ ⌠ ⌡ 0
|
v0 M(v) dv = |
n(n−1)
2
|
|
| (10.31) |
|
∞ ⌠ ⌡ 0
|
v2 PJ(v) dv = 2 n ET |
| (10.33) |
where n is the number of electrons and ET is the total electronic
kinetic energy. Currently, Q-Chem can compute M(v), MJ(v) and
MK(v) using s and p basis functions only. Moments are generated
using quadrature and consequently for accurate results M(v) must be computed
over a large and closely spaced v range.
10.4.3 Wigner Intracules
The intracules P(u) and M(v) provide a representation of an electron distribution in
either position or momentum space but neither alone can provide a
complete description. For a combined position and momentum description
an intracule in phase space is required. Defining such an intracule is more
difficult since there is no phase space second-order reduced density. However,
the second-order Wigner distribution [478],
W2(r1,p1,r2,p2) = |
1
π6
|
| ⌠ ⌡
|
ρ2(r1+q1, r1−q1, r2+q2, r2−q2)e−2i(p1 ·q1 + p2 ·q2) dq1 dq2 |
| (10.35) |
can be interpreted as the probability of finding an electron at
r1 with momentum p1 and another electron at r2 with
momentum p2. [The quantity W2(r1,r2,p1,p2 is often referred to
as "quasi-probability distribution" since it is not positive everywhere.]
The Wigner distribution can be used in an analogous way to the second order
reduced densities to define a combined position and momentum intracule. This
intracule is called a Wigner intracule, and is formally defined as
W(u,v) = | ⌠ ⌡
|
W2(r1, p1, r2, p2)δ(r12−u) δ(p12−v)dr1 dr2 dp1 dp2 dΩu dΩv |
| (10.36) |
If the orbitals are expanded in a basis set, then W(u,v) can be written as
W(u,v) = |
∑
μνλσ
|
Γμνλσ ( μνλσ )W |
| (10.37) |
where (μνλσ)W is the Wigner integral
(μνλσ)W = |
v2
2π2
|
| ⌠ ⌡
|
| ⌠ ⌡
|
ϕμ∗ (r) ϕν(r+q)ϕλ∗(r+q+u) ϕσ(r+u)j0(q v) dr dq dΩu |
| (10.38) |
Wigner integrals are similar to momentum integrals and only have four-fold
permutational symmetry. Evaluating Wigner integrals is considerably more
difficult that their position or momentum counterparts. The fundamental
[ssss]w integral,
| |
|
|
u2v2
2π2
|
| ⌠ ⌡
|
| ⌠ ⌡
|
exp[−α|r−A|2 −β|r+q−B|2 −γ|r+q+u−C|2 −δ|r+u−D|2 ] × |
| |
| |
|
| | (10.39) |
|
can be expressed as
[ssss]W = |
πu2 v2 e−(R+λ2 u2 +μ2 v2)
2(α+δ)3/2(β+γ)3/2
|
| ⌠ ⌡
|
e−P·u j0 (|Q+ηu|v ) dΩu |
| (10.40) |
or alternatively
[ssss]W = |
2π2 u2 v2 e−(R+λ2 u2+μ2 v2)
(α+δ)3/2(β+γ)3/2
|
|
∞ ∑
n=0
|
(2n+1) in(P u) jn(ηu v) jn(Q v) Pn | ⎛ ⎝
|
P·Q
P Q
| ⎞ ⎠
|
|
| (10.41) |
Two approaches for evaluating (μνλσ)W have been
implemented in Q-Chem, full details can be found in Ref. . The
first approach uses the first form of [ssss]W and used Lebedev
quadrature to perform the remaining integrations over Ωu. For high
accuracy large Lebedev grids [144,[142,[448]
should be used, grids of up to 5294 points are available in Q-Chem.
Alternatively, the second form can be adopted and the integrals evaluated by
summation of a series. Currently, both methods have been implemented within
Q-Chem for s and p basis functions only.
When computing intracules it is most efficient to locate the loop over u
and / or v points within the loop over shell-quartets [480].
However, for W(u,v) this requires a large amount of memory to store all the
integrals arising from each (u,v) point. Consequently, an additional scheme,
in which the u and v points loop is outside the shell-quartet loop, is
available. This scheme is less efficient, but substantially reduces the memory
requirements.
10.4.4 Intracule Job Control
The following $rem variables can be used to control the calculation of
intracules.
INTRACULE
Controls whether intracule properties are calculated (see also the
$intracule section). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | No intracule properties. |
TRUE | Evaluate intracule properties. |
RECOMMENDATION:
|
| WIG_MEM
Reduce memory required in the evaluation of W(u,v). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use low memory option. |
TRUE | Use low memory option. |
RECOMMENDATION:
The low memory option is slower, use default unless memory is limited. |
|
|
|
WIG_LEB
Use Lebedev quadrature to evaluate Wigner integrals. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Evaluate Wigner integrals through series summation. |
TRUE | Use quadrature for Wigner integrals. |
RECOMMENDATION:
|
| WIG_GRID
Specify angular Lebedev grid for Wigner intracule calculations. |
TYPE:
DEFAULT:
OPTIONS:
Lebedev grids up to 5810 points. |
RECOMMENDATION:
Larger grids if high accuracy required. |
|
|
|
N_WIG_SERIES
Sets summation limit for Wigner integrals. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase n for greater accuracy. |
|
| N_I_SERIES
Sets summation limit for series expansion evaluation of in(x). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy. |
|
|
|
N_J_SERIES
Sets summation limit for series expansion evaluation of jn(x). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy. |
|
10.4.5 Format for the $intracule Section
int_type | 0 | Compute P(u) only |
| 1 | Compute M(v) only |
| 2 | Compute W(u,v) only |
| 3 | Compute P(u), M(v) and W(u,v) |
| 4 | Compute P(u) and M(v) |
| 5 | Compute P(u) and W(u,v) |
| 6 | Compute M(v) and W(u,v) |
u_points | | Number of points, start, end. |
v_points | | Number of points, start, end. |
moments | 0-4 | Order of moments to be computed (P(u) only). |
derivs | 0-4 | order of derivatives to be computed (P(u) only). |
accuracy | n | (10−n) specify accuracy of intracule interpolation table (P(u) only). |
Example 10.0 Compute HF/STO-3G P(u), M(v) and W(u,v) for Ne, using
Lebedev quadrature with 974 point grid.
$molecule
0 1
Ne
$end
$rem
EXCHANGE hf
BASIS sto-3g
INTRACULE true
WIG_LEB true
WIG_GRID 974
$end
$intracule
int_type 3
u_points 10 0.0 10.0
v_points 8 0.0 8.0
moments 4
derivs 4
accuracy 8
$end
Example 10.0 Compute HF/6-31G W(u,v) intracules for H2O using series
summation up to n=25 and 30 terms in the series evaluations of jn(x) and
in(x).
$molecule
0 1
H1
O H1 r
H2 O r H1 theta
r = 1.1
theta = 106
$end
$rem
EXCHANGE hf
BASIS 6-31G
INTRACULE true
WIG_MEM true
N_WIG_SERIES 25
N_I_SERIES 40
N_J_SERIES 50
$end
$intracule
int_type 2
u_points 30 0.0 15.0
v_points 20 0.0 10.0
$end
10.5 Vibrational Analysis
Vibrational analysis is an extremely important tool for the quantum chemist,
supplying a molecular fingerprint which is invaluable for aiding identification
of molecular species in many experimental studies. Q-Chem includes a
vibrational analysis package that can calculate vibrational frequencies and
their Raman [481] and infrared activities. Vibrational
frequencies are calculated by either using an analytic Hessian (if available;
see Table 9.1) or, numerical finite difference of the gradient. The
default setting in Q-Chem is to use the highest analytical derivative order
available for the requested theoretical method.
When calculating analytic frequencies at the HF and DFT levels of theory,
the coupled-perturbed SCF equations must be solved. This is the most
time-consuming step in the calculation, and also consumes the most memory.
The amount of memory required is O(N2M) where N is the number
of basis functions, and M the number of atoms. This is an order more memory
than is required for the SCF calculation, and is often the limiting
consideration when treating larger systems analytically. Q-Chem incorporates a
new approach to this problem that avoids this memory bottleneck by solving the
CPSCF equations in segments [482]. Instead of solving for all
the perturbations at once, they are divided into several segments, and the
CPSCF is applied for one segment at a time, resulting in a memory scaling of
O(N2M/Nseg), where Nseg is the number of segments. This
option is invoked automatically by the program.
Following a vibrational analysis, Q-Chem computes useful statistical
thermodynamic properties at standard temperature and pressure, including:
zero-point vibration energy (ZPVE) and, translational, rotational and
vibrational, entropies and enthalpies.
The performance of various ab initio theories in determining vibrational
frequencies has been well documented; see
Refs. .
10.5.1 Job Control
In order to carry out a frequency analysis users must at a minimum
provide a molecule within the $molecule keyword and define an appropriate
level of theory within the $rem keyword using the $rem variables
EXCHANGE, CORRELATION (if required) (Chapter 4)
and BASIS (Chapter 7). Since the default type of job
(JOBTYPE) is a single point energy (SP) calculation, the
JOBTYPE $rem variable must be set to FREQ.
It is very important to note that a vibrational frequency analysis must be
performed at a stationary point on the potential surface that has been
optimized at the same level of theory. Therefore a vibrational frequency
analysis most naturally follows a geometry optimization in the same input deck,
where the molecular geometry is obtained (see examples).
Users should also be aware that the quality of the quadrature grid used in DFT
calculations is more important when calculating second derivatives. The default
grid for some atoms has changed in Q-Chem 3.0 (see Section 4.3.11)
and for this reason vibrational frequencies may vary
slightly form previous versions. It is recommended that a grid larger than the
default grid is used when performing frequency calculations.
The standard output from a frequency analysis includes the following.
- Vibrational frequencies.
- Raman and IR activities and intensities (requires $rem
DORAMAN).
- Atomic masses.
- Zero-point vibrational energy.
- Translational, rotational, and vibrational, entropies and enthalpies.
Several other $rem variables are available that control the vibrational
frequency analysis. In detail, they are:
DORAMAN
Controls calculation of Raman intensities. Requires JOBTYPE to be set
to FREQ |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate Raman intensities. |
TRUE | Do calculate Raman intensities. |
RECOMMENDATION:
|
| VIBMAN_PRINT
Controls level of extra print out for vibrational analysis. |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard full information print out. |
| If VCI is TRUE, overtones and combination bands are also printed. |
3 | Level 1 plus vibrational frequencies in atomic units. |
4 | Level 3 plus mass-weighted Hessian matrix, projected mass-weighted Hessian |
| matrix. |
6 | Level 4 plus vectors for translations and rotations projection matrix. |
RECOMMENDATION:
|
|
|
CPSCF_NSEG
Controls the number of segments used to calculate the CPSCF equations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not solve the CPSCF equations in segments. |
n | User-defined. Use n segments when solving the CPSCF equations. |
RECOMMENDATION:
|
Example 10.0 An EDF1/6-31+G* optimization, followed by a vibrational
analysis. Doing the vibrational analysis at a stationary point is necessary for
the results to be valid.
$molecule
O 1
C 1 co
F 2 fc 1 fco
H 2 hc 1 hcp 3 180.0
co = 1.2
fc = 1.4
hc = 1.0
fco = 120.0
hco = 120.0
$end
$rem
JOBTYPE opt
EXCHANGE edf1
BASIS 6-31+G*
$end
@@@
$molecule
read
$end
$rem
JOBTYPE freq
EXCHANGE edf1
BASIS 6-31+G*
$end
10.6 Anharmonic Vibrational Frequencies
Computing vibrational spectra beyond the harmonic approximation has become an
active area of research owing to the improved efficiency of computer
techniques [486,[487,[488,[489]. To calculate the
exact vibrational spectrum within Born-Oppenheimer approximation, one has to
solve the nuclear Schrödinger equation completely using numerical
integration techniques, and consider the full configuration interaction of
quanta in the vibrational states. This has only been carried out on di- or
triatomic system [490,[491]. The difficulty of this
numerical integration arises because solving exact the nuclear Schrödinger
equation requires a complete electronic basis set, consideration of all the
nuclear vibrational configuration states, and a complete potential energy
surface (PES). Simplification of the Nuclear Vibration Theory (NVT) and PES
are the doorways to accelerating the anharmonic correction calculations.
There are five aspects to simplifying the problem:
- Expand the potential energy surface using a Taylor series and examine the
contribution from higher derivatives. Small contributions can be
eliminated, which allows for the efficient calculation of the Hamiltonian.
- Investigate the effect on the number of configurations employed in a
variational calculation.
- Avoid using variational theory (due to its expensive computational cost)
by using other approximations, for example, perturbation theory.
- Obtain the PES indirectly by applying a self-consistent field procedure.
- Apply an anharmonic wavefunction which is more appropriate for
describing the distribution of nuclear probability on an anharmonic
potential energy surface.
To incorporate these simplifications, new formulae combining information from
the Hessian, gradient and energy are used as a default procedure to calculate
the cubic and quartic force field of a given potential energy surface.
Here, we also briefly describe various NVT methods. In the early stage of
solving the nuclear Schrödinger equation (in the 1930s), second-order
Vibrational Perturbation Theory (VPT2) was
developed [492,[493,[494,[495,[489].
However, problems occur when resonances
exist in the spectrum. This becomes more problematic for larger molecules due
to the greater chance of accidental degeneracies occurring. To avoid this
problem, one can do a direct integration of the secular matrix using
Vibrational Configuration Interaction (VCI) theory [496]. It is the most
accurate method and also the least favored due to its computational expense.
In Q-Chem 3.0, we introduce a new approach to treating the wavefunction,
transition-optimized shifted Hermite (TOSH) theory [497], which uses
first-order perturbation theory, which avoids the degeneracy problems of VPT2,
but which incorporates anharmonic effects into the wavefunction, thus
increasing the accuracy of the predicted anharmonic energies.
10.6.1 Partial Hessian Vibrational Analysis
The computation of harmonic frequencies for systems with a very large number
of atoms can become computationally expensive. However, in many cases only
a few specific vibrational modes or vibrational modes localized in a
region of the system are of interest. A typical example is the calculation of
the vibrational modes of a molecule adsorbed on a surface. In such a case,
only the vibrational modes of the adsorbate are useful, and the vibrational modes
associated with the surface atoms are of less interest. If the vibrational modes of
interest are only weakly coupled to the vibrational modes associated with the rest of
the system, it can be appropriate to adopt a partial Hessian approach.
In this approach [498,[499],
only the part of the Hessian matrix comprising the second derivatives of a subset of the
atoms defined by the user is computed. These atoms are defined in the $alist block.
This results in a significant decrease in the cost of the calculation. Physically, this
approximation corresponds to assigning an infinite mass to all the atoms excluded from the
Hessian and will only yield sensible results if these atoms are not involved in the vibrational
modes of interest. VPT2 and TOSH anharmonic frequencies can be computed following a partial Hessian
calculation [500]. It is also possible to include a subset of the harmonic vibrational
modes with an anharmonic frequency calculation by invoking the ANHAR_SEL rem.
This can be useful to reduce the computational cost of an anharmonic frequency calculation or
to explore the coupling between specific vibrational modes.
Alternatively, vibrationally averaged interactions with the rest of the system can be folded
into a partial Hessian calculation using vibrational subsystem analysis [501,[416].
Based on an adiabatic approximation, this procedure reduces the cost of diagonalizing
the full Hessian, while providing a local probe of fragments vibrations, and providing
better than partial Hessian accuracy for the low frequency modes of large molecules [502].
Mass-effects from the rest of the system can be vibrationally averaged or excluded within this scheme.
PHESS
Controls whether partial Hessian calculations are performed. |
TYPE:
DEFAULT:
0 | Full Hessian calculation |
OPTIONS:
0 | Full Hessian calculation |
1 | Partial Hessian calculation |
2 | Vibrational subsystem analysis (massless) |
3 | Vibrational subsystem analysis (weighted) |
RECOMMENDATION:
|
| N_SOL
Specifies number of atoms included in the Hessian |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
PH_FAST
Lowers integral cutoff in partial Hessian calculation is performed. |
TYPE:
DEFAULT:
FALSE | Use default cutoffs |
OPTIONS:
TRUE | Lower integral cutoffs |
RECOMMENDATION:
|
| ANHAR_SEL
Select a subset of normal modes for subsequent anharmonic frequency analysis. |
TYPE:
DEFAULT:
FALSE | Use all normal modes |
OPTIONS:
TRUE | Select subset of normal modes |
RECOMMENDATION:
|
|
|
Example 10.0 This example shows a partial Hessian frequency
calculation of the vibrational frequencies
of acetylene on a model of the C(100) surface
$comment
acetylene - C(100)
partial Hessian calculation
$end
$molecule
0 1
C 0.000 0.659 -2.173
C 0.000 -0.659 -2.173
H 0.000 1.406 -2.956
H 0.000 -1.406 -2.956
C 0.000 0.786 -0.647
C 0.000 -0.786 -0.647
C 1.253 1.192 0.164
C -1.253 1.192 0.164
C 1.253 -1.192 0.164
C 1.297 0.000 1.155
C -1.253 -1.192 0.164
C 0.000 0.000 2.023
C -1.297 0.000 1.155
H -2.179 0.000 1.795
H -1.148 -2.156 0.654
H 0.000 -0.876 2.669
H 2.179 0.000 1.795
H -1.148 2.156 0.654
H -2.153 -1.211 -0.446
H 2.153 -1.211 -0.446
H 1.148 -2.156 0.654
H 1.148 2.156 0.654
H 2.153 1.211 -0.446
H -2.153 1.211 -0.446
H 0.000 0.876 2.669
$end
$rem
JOBTYPE freq
EXCHANGE hf
BASIS sto-3g
PHESS TRUE
N_SOL 4
$end
$alist
1
2
3
4
$end
Example 10.0 This example shows an anharmonic
frequency calculation for ethene where only the C-H
stretching modes are included in the anharmonic analysis.
$comment
ethene
restricted anharmonic frequency analysis
$end
$molecule
0 1
C 0.6665 0.0000 0.0000
C -0.6665 0.0000 0.0000
H 1.2480 0.9304 0.0000
H -1.2480 -0.9304 0.0000
H -1.2480 0.9304 0.0000
H 1.2480 -0.9304 0.0000
$end
$rem
JOBTYPE freq
EXCHANGE hf
BASIS sto-3g
ANHAR_SEL TRUE
N_SOL 4
$end
$alist
9
10
11
12
$end
10.6.2 Vibration Configuration Interaction Theory
To solve the nuclear vibrational Schrödinger equation, one can only use
direct integration procedures for diatomic molecules [490,[491].
For larger systems, a truncated version of full configuration
interaction is considered to be the most accurate approach. When one applies
the variational principle to the vibrational problem, a basis function for the
nuclear wavefunction of the nth excited state of mode i is
ψ(n)i = ϕ(n)i |
m ∏
j ≠ i
|
ϕ(0)j |
| (10.42) |
where the ϕi(n) represents the harmonic oscillator eigenfunctions for
normal mode qi. This can be expressed in terms of Hermite polynomials:
ϕ(n)i = | ⎛ ⎝
|
ωi [1/2]
π[1/2]2nn!
| ⎞ ⎠
|
[1/2]
|
e− [(ωi qi2)/2] Hn(qi | √
|
ωi
|
) |
| (10.43) |
With the basis function defined in Eq. (10.42), the nth wavefunction can
be described as a linear combination of the Hermite polynomials:
Ψ(n) = |
n1 ∑
i=0
|
|
n2 ∑
j=0
|
|
n3 ∑
k=0
|
… |
nm ∑
m=0
|
c(n)ijk…mψijk…m(n) |
| (10.44) |
where ni is the number of quanta in the ith mode. We propose the notation
VCI(n) where n is the total number of quanta, i.e.:
To determine this expansion coefficient c(n), we integrate the ∧H,
as in Eq. (4.1), with Ψ(n) to get the eigenvalues
c(n) = E(n)VCI(n) = 〈Ψ(n) | |
^
H
|
| Ψ(n) 〉 |
| (10.46) |
This gives us frequencies that are corrected for anharmonicity to n quanta accuracy
for a m-mode molecule. The size of the secular matrix on the right hand of
Eq. (10.46) is ((n+m)!/n!m!)2, and the storage of this matrix can
easily surpass the memory limit of a computer. Although this method is highly
accurate, we need to seek for other approximations for computing large
molecules.
10.6.3 Vibrational Perturbation Theory
Vibrational perturbation theory has been historically popular for calculating
molecular spectroscopy. Nevertheless, it is notorious for the inability of
dealing with resonance cases. In addition, the non-standard formulas for
various symmetries of molecules forces the users to modify inputs on a
case-by-case basis [503,[504,[505], which narrows the
accessibility of this method. VPT applies perturbation treatments on the same
Hamiltonian as in Eq. (4.1), but divides it into an unperturbed part,
∧U,
|
^
U
|
= |
m ∑
i
|
| ⎛ ⎝
|
− |
1
2
|
|
∂2
∂qi2
|
+ |
ωi2
2
|
qi2 | ⎞ ⎠
|
|
| (10.47) |
and a perturbed part, ∧V:
|
^
V
|
= |
1
6
|
|
m ∑
ijk = 1
|
ηijkqiqjqk + |
1
24
|
|
m ∑
ijkl = 1
|
ηijklqiqj qkql |
| (10.48) |
One can then apply second-order perturbation theory to get the ith excited
state energy:
E(i) = |
^
U
|
(i)
|
+ 〈Ψ(i) | |
^
V
|
| Ψ(i) 〉 + |
∑
j ≠ i
|
|
|
|
| (10.49) |
The denominator in Eq. (10.49) can be zero either because of symmetry
or accidental degeneracy. Various solutions, which depend on the type of
degeneracy that occurs, have been developed which ignore the zero-denominator
elements from the Hamiltonian [506,[503,[504,[505]. An
alternative solution has been proposed by Barone [489] which can
be applied to all molecules by changing the masses of one or more nuclei in
degenerate cases. The disadvantage of this method is that it will break the
degeneracy which results in fundamental frequencies no longer retaining their
correct symmetry. He proposed
EiVPT2 = |
∑
j
|
ωj (nj+1/2) + |
∑
i ≤ j
|
xij (ni+1/2)(nj+1/2) |
| (10.50) |
where, if rotational coupling is ignored, the anharmonic constants xij are
given by
xij = |
1
4ωiωj
|
| ⎛ ⎝
|
ηiijj − |
m ∑
k
|
|
ηiikηjjk
ωk2
|
+ |
m ∑
k
|
|
2(ωi2+ωj2−ωk2)ηijk2
[(ωi+ωj)2−ωk2] [(ωi−ωj)2−ωk2]
| ⎞ ⎠
|
|
| (10.51) |
10.6.4 Transition-Optimized Shifted Hermite Theory
So far, every aspect of solving the nuclear wave equation has been
considered, except the wavefunction. Since Schrödinger proposed his
equation, the nuclear wavefunction has traditionally be expressed in terms
of Hermite functions, which are designed for the harmonic oscillator case.
Recently [497], a modified representation has been presented. To
demonstrate how this approximation works, we start with a simple example. For
a diatomic molecule, the Hamiltonian with up to quartic derivatives can be
written as
|
^
H
|
= − |
1
2
|
|
∂2
∂q2
|
+ |
1
2
|
ω2q2 + ηiii q3 + ηiiii q4 |
| (10.52) |
and the wavefunction is expressed as in Eq. (10.43). Now, if
we shift the center of the wavefunction by σ, which is equivalent to a
translation of the normal coordinate q, the shape will still remain the same,
but the anharmonic correction can now be incorporated into the wavefunction.
For a ground vibrational state, the wavefunction is written as
ϕ(0) = | ⎛ ⎝
|
ω
π
| ⎞ ⎠
|
[1/4]
|
e −[(ω)/2] ( q − σ) 2 |
| (10.53) |
Similarly, for the first excited vibrational state, we have
ϕ(1) = | ⎛ ⎝
|
4ω3
π
| ⎞ ⎠
|
[1/4]
|
( q − σ) e[(ω)/2]( q − σ) 2 |
| (10.54) |
Therefore, the energy difference between the first vibrational excited state
and the ground state is
∆ETOSH = ω+ |
ηiiii
8ω2
|
+ |
ηiii σ
2ω
|
+ |
ηiiii σ2
4ω
|
|
| (10.55) |
This is the fundamental vibrational frequency from first-order perturbation
theory.
Meanwhile, We know from the first-order perturbation theory with an ordinary
wavefunction within a QFF PES, the energy is
The differences between these two wavefunctions are the two extra terms arising
from the shift in Eq. (10.55). To determine the shift, we compare
the energy with that from second-order perturbation theory:
∆EVPT2 = ω+ |
ηiiii
8ω2
|
− |
5ηiii 2
24ω4
|
|
| (10.57) |
Since σ is a very small quantity compared with the other variables, we
ignore the contribution of σ2 and compare ∆ETOSH with
∆EVPT2, which yields an initial guess for σ:
Because the only difference between this approach and the ordinary wavefunction
is the shift in the normal coordinate, we call it "transition-optimized
shifted Hermite" (TOSH) functions [497]. This approximation gives
second-order accuracy at only first-order cost.
For polyatomic molecules, we consider Eq. (10.55), and propose
that the energy of the ith mode be expressed as:
∆EiTOSH = ωi + |
1
8ωi
|
|
∑
j
|
|
ηiijj
ωj
|
+ |
1
2ωi
|
|
∑
j
|
ηiij σij + |
1
4ωi
|
|
∑
j,k
|
ηiijkσijσik |
| (10.59) |
Following the same approach as for the diatomic case, by comparing this with
the energy from second-order perturbation theory, we obtain the shift as
σij = |
(δij−2)(ωi+ωj)ηiij
4ωi ωj2(2ωi+ωj)
|
− |
∑
k
|
|
ηkkj
4ωkωj2
|
|
| (10.60) |
10.6.5 Job Control
The following $rem variables can be used to control the calculation of
anharmonic frequencies.
ANHAR
Performing various nuclear vibrational theory (TOSH, VPT2, VCI) calculations
to obtain vibrational anharmonic frequencies. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Carry out the anharmonic frequency calculation. |
FALSE | Do harmonic frequency calculation. |
RECOMMENDATION:
Since this calculation involves the third and fourth derivatives at the
minimum of the potential energy surface, it is recommended that the
GEOM_OPT_TOL_DISPLACEMENT, GEOM_OPT_TOL_GRADIENT and
GEOM_OPT_TOL_ENERGY tolerances are set tighter. Note that VPT2
calculations may fail if the system involves accidental degenerate resonances.
See the VCI $rem variable for more details about increasing the
accuracy of anharmonic calculations. |
|
| VCI
Specifies the number of quanta involved in the VCI calculation. |
TYPE:
DEFAULT:
OPTIONS:
User-defined. Maximum value is 10. |
RECOMMENDATION:
The availability depends on the memory of the machine.
Memory allocation for VCI calculation is the square of
2*(NVib+NVCI)/NVibNVCI with double precision.
For example, a machine with 1.5 GB memory and for molecules with fewer than 4
atoms, VCI(10) can be carried out, for molecule containing fewer than 5 atoms,
VCI(6) can be carried out, for molecule containing fewer than 6 atoms, VCI(5)
can be carried out. For molecules containing fewer than 50 atoms, VCI(2) is
available. VCI(1) and VCI(3) usually overestimated the true energy while
VCI(4) usually gives an answer close to the converged energy. |
|
|
|
FDIFF_DER
Controls what types of information are used to compute higher
derivatives. The default uses a combination of energy, gradient and Hessian
information, which makes the force field calculation faster. |
TYPE:
DEFAULT:
3 | for jobs where analytical 2nd derivatives are available. |
0 | for jobs with ECP. |
OPTIONS:
0 | Use energy information only. |
1 | Use gradient information only. |
2 | Use Hessian information only. |
3 | Use energy, gradient, and Hessian information. |
RECOMMENDATION:
When the molecule is larger than benzene with small basis set,
FDIFF_DER=2 may be faster. Note that FDIFF_DER will be set
lower if analytic derivatives of the requested order are not available. Please
refers to IDERIV. |
|
| MODE_COUPLING
Number of modes coupling in the third and fourth derivatives calculation. |
TYPE:
DEFAULT:
2 | for two modes coupling. |
OPTIONS:
n | for n modes coupling, Maximum value is 4. |
RECOMMENDATION:
|
|
|
IGNORE_LOW_FREQ
Low frequencies that should be treated as rotation can be ignored during |
anharmonic correction calculation. |
TYPE:
DEFAULT:
300 | Corresponding to 300 cm−1. |
OPTIONS:
n | Any mode with harmonic frequency less than n will be ignored. |
RECOMMENDATION:
|
| FDIFF_STEPSIZE_QFF
Displacement used for calculating third and fourth derivatives by finite difference. |
TYPE:
DEFAULT:
5291 | Corresponding to 0.1 bohr. For calculating third and fourth derivatives. |
OPTIONS:
n | Use a step size of n×10−5. |
RECOMMENDATION:
Use default, unless on a very flat potential, in which case a larger value
should be used. |
|
|
|
Example 10.0 A four-quanta anharmonic frequency calculation on formaldehyde
at the EDF2/6-31G* optimized ground state geometry, which is obtained in the
first part of the job. It is necessary to carry out the harmonic frequency
first and this will print out an approximate time for the subsequent anharmonic
frequency calculation. If a FREQ job has already been performed, the
anharmonic calculation can be restarted using the saved scratch files from the
harmonic calculation.
$molecule
0 1
C
O, 1, CO
H, 1, CH, 2, A
H, 1, CH, 2, A, 3, D
CO = 1.2
CH = 1.0
A = 120.0
D = 180.0
$end
$rem
JOBTYPE OPT
EXCHANGE EDF2
BASIS 6-31G*
GEOM_OPT_TOL_DISPLACEMENT 1
GEOM_OPT_TOL_GRADIENT 1
GEOM_OPT_TOL_ENERGY 1
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE FREQ
EXCHANGE EDF2
BASIS 6-31G*
ANHAR TRUE
VCI 4
$end
10.6.6 Isotopic Substitutions
By default Q-Chem calculates vibrational frequencies using the atomic masses
of the most abundant isotopes (taken from the Handbook of Chemistry and
Physics, 63rd Edition). Masses of other isotopes can be specified
using the $isotopes section and by setting the ISOTOPES $rem
variable to TRUE. The format of the $isotopes section is as follows:
$isotopes
number_of_isotope_loops tp_flag
number_of_atoms [temp pressure] (loop 1)
atom_number1 mass1
atom_number2 mass2
...
number_of_atoms [temp pressure] (loop 2)
atom_number1 mass1
atom_number2 mass2
...
$end
Note:
Only the atoms whose masses are to be changed from the default
values need to be specified. After each loop all masses are reset to the
default values. Atoms are numbered according to the order in the $molecule
section. |
An initial loop using the default masses is always performed first. Subsequent
loops use the user-specified atomic masses. Only those atoms whose masses are
to be changed need to be included in the list, all other atoms will adopt the
default masses. The output gives a full frequency analysis for each loop. Note
that the calculation of vibrational frequencies in the additional loops only
involves a rescaling of the computed Hessian, and therefore takes little
additional computational time.
The first line of the $isotopes section specifies the number of substitution loops
and also whether the temperature and pressure should be modified. The tp_flag
setting should be set to 0 if the default temperature and pressure are to be used (298.18 K
and 1 atm respectively), or 1 if they are to be altered. Note that the
temperatures should be specified in Kelvin (K) and pressures in atmospheres (atm).
ISOTOPES
Specifies if non-default masses are to be used in the frequency calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use default masses only. |
TRUE | Read isotope masses from $isotopes section. |
RECOMMENDATION:
|
Example 10.0 An EDF1/6-31+G* optimization, followed by a vibrational
analysis. Doing the vibrational analysis at a stationary point is necessary for
the results to be valid.
$molecule
0 1
C 1.08900 0.00000 0.00000
C -1.08900 0.00000 0.00000
H 2.08900 0.00000 0.00000
H -2.08900 0.00000 0.00000
$end
$rem
BASIS 3-21G
JOBTYPE opt
EXCHANGE hf
CORRELATION none
$end
@@@
$molecule
read
$end
$rem
BASIS 3-21G
JOBTYPE freq
EXCHANGE hf
CORRELATION none
SCF_GUESS read
ISOTOPES 1
$end
$isotopes
2 0 ! two loops, both at std temp and pressure
4
1 13.00336 ! All atoms are given non-default masses
2 13.00336
3 2.01410
4 2.01410
2
3 2.01410 ! H's replaced with D's
4 2.01410
$end
10.7 Interface to the NBO Package
Q-Chem has incorporated the Natural Bond Orbital package (v. 5.0) for molecular
properties and wavefunction analysis. The NBO package is invoked by
setting the $rem variable NBO to TRUE and is initiated after the SCF
wavefunction is obtained.
(If switched on for a geometry optimization, the NBO package will only be
invoked at the end of the last optimization step.)
Users are referred to the NBO user's manual for
options and details relating to exploitation of the features offered in this
package.
NBO analysis is also available for excited states calculated using
CIS or TDDFT. Excited-state NBO analysis is still in its infancy,
and users should be aware that the convergence of the NBO search
procedure may be less well-behaved for excited states than it is for
ground states, and may require specification of additional NBO parameters
in the $nbo section that is described below.
Consult Ref. for details and suggestions.
NBO
Controls the use of the NBO package. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not invoke the NBO package. |
1 | Do invoke the NBO package, for the ground state. |
2 | Invoke the NBO package for the ground state, and also each |
| CIS, RPA, or TDDFT excited state.
|
RECOMMENDATION:
|
The general format for passing options from Q-Chem to the NBO program is
shown below:
$nbo
{NBO program keywords, parameters and options}
$end
Note:
(1) $rem variable NBO must be set to TRUE before the
$nbo keyword is recognized.
(2) Q-Chem does not support facets of the NBO package which require
multiple job runs |
10.8 Orbital Localization
The concept of localized orbitals has already been visited in this manual in the context of perfect-pairing and methods.
As the SCF energy is independent of the partitioning of the electron density into orbitals, there is considerable flexibility as to how this may be done.
The canonical picture, where the orbitals are eigenfunctions of the Fock operator is useful in determining reactivity, for, through Koopmans theorem,
the orbital energy eigenvalues give information about the corresponding ionization energies and electron affinities.
As a consequence, the HOMO and LUMO are very informative as to the reactive sites of a molecule.
In addition, in small molecules, the canonical orbitals lead us to the chemical description of σ and π bonds.
In large molecules, however, the canonical orbitals are often very delocalized, and so information about chemical bonding is not readily available from them.
Here, orbital localization techniques can be of great value in visualizing the bonding, as localized orbitals often correspond to the chemically intuitive orbitals which might be expected.
Q-Chem has three post-SCF localization methods available. These can be performed separately over both occupied and virtual spaces. The localization scheme attributed to
Boys [508,[509] minimizes the radial extent of the localized orbitals, i.e., ∑i 〈ii| |r1−r2|2 |ii〉, and although is relatively fast, does not separate σ and π orbitals, leading to two `banana-orbitals' in the case of a double bond [510]. Pipek-Mezey localized orbitals [510] maximize the locality of Mulliken populations, and are of a similar cost to Boys localized orbitals, but maintain σ−π separation.
Edmiston-Ruedenberg localized orbitals [511] maximize the self-repulsion of the orbitals, ∑i 〈ii|[1/r]|ii〉. This is more computationally expensive to calculate as it requires a two-electron property to be evaluated, but the work of Dr. Joe
Subotnik [512] and later Prof. Young-Min Rhee and Westin Kurlancheek with Prof. Martin Head-Gordon at Berkeley has, through use of the Resolution of the Identity approximation, reduced the formal cost may be asymptotically reduced to cubic scaling with the number of occupied orbitals.
BOYSCALC
Specifies the Boys localized orbitals are to be calculated |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not perform localize the occupied space. |
1 | Allow core-valence mixing in Boys localization. |
2 | Localize core and valence separately. |
RECOMMENDATION:
|
| ERCALC
Specifies the Edmiston-Ruedenberg localized orbitals are to be calculated |
TYPE:
DEFAULT:
OPTIONS:
aabcd | |
aa | specifies the convergence threshold. |
| If aa > 3, the threshold is set to 10−aa. The default is 6. |
| If aa=1, the calculation is aborted after the guess, allowing Pipek-Mezey |
| orbitals to be extracted. |
b | specifies the guess: |
| 0 Boys localized orbitals. This is the default |
| 1 Pipek-Mezey localized orbitals. |
c | specifies restart options (if restarting from an ER calculation): |
| 0 No restart. This is the default |
| 1 Read in MOs from last ER calculation. |
| 2 Read in MOs and RI integrals from last ER calculation. |
d | specifies how to treat core orbitals |
| 0 Do not perform ER localization. This is the default. |
| 1 Localize core and valence together. |
| 2 Do separate localizations on core and valence. |
| 3 Localize only the valence electrons. |
| 4 Use the $localize section.
|
RECOMMENDATION:
ERCALC 1 will usually suffice, which uses threshold 10−6. |
|
|
|
The $localize section may be used to specify orbitals subject to ER localization if require. It contains a list of the orbitals to include in the localization. These may span multiple lines.
If the user wishes to specify separate beta orbitals to localize, include a zero before listing the beta orbitals, which acts as a separator, e.g.,
$localize
2 3 4 0
2 3 4 5 6
$end
10.9 Visualizing and Plotting Orbitals and Densities
Q-Chem can generate orbital and density data in several formats
suitable for plotting with a third-party visualization program.
10.9.1 Visualizing Orbitals Using MolDen and MacMolPlt
Upon request, Q-Chem will generate an input file for MolDen, a
freely-available molecular visualization program [513,[514].
MolDen can be used to view
ball-and-stick molecular models (including stepwise visualization of a
geometry optimization), molecular orbitals, vibrational normal modes,
and vibrational spectra.
MolDen also contains a powerful Z-matrix editor. In conjunction with
Q-Chem, orbital visualization via MolDen is
currently supported for s, p, and d functions (pure or Cartesian), as
well as pure f functions. Upon setting MOLDEN_FORMAT to
TRUE, Q-Chem will append a MolDen-formatted input file
to the end of the Q-Chem log file. As some versions of MolDen
have difficulty parsing the Q-Chem log file itself, we
recommend that the user cut
and paste the MolDen-formatted part of the Q-Chem log file into a
separate file to be read by MolDen.
MOLDEN_FORMAT
Requests a MolDen-formatted input file containing information from a Q-Chem
job. |
TYPE:
DEFAULT:
OPTIONS:
True | Append MolDen input file at the end of the Q-Chem output file. |
RECOMMENDATION:
|
MolDen-formatted files can also be read by MacMolPlt, another
freely-available visualization program [515,[516]. MacMolPlt
generates orbital iso-contour surfaces much more rapidly than
MolDen, however, within MacMolPlt these surfaces are only available
for Cartesian Gaussian basis functions, i.e.,
PURECART = 2222, which may not be the default.
Example 10.0 Generating a MolDen file for molecular orbital visualization.
$molecule
0 1
O
H 1 0.95
H 1 0.95 2 104.5
$end
$rem
EXCHANGE hf
BASIS cc-pvtz
PRINT_ORBITALS true (default is to print 5 virtual orbitals)
MOLDEN_FORMAT true
$end
For geometry optimizations and vibrational frequency calculations, one
need only set MOLDEN_FORMAT to TRUE, and the relevant
geometry
or normal mode information will automatically appear in the MolDen
section of the Q-Chem log file.
Example 10.0 Generating a MolDen file to step through a geometry optimization.
$molecule
0 1
O
H 1 0.95
H 1 0.95 2 104.5
$end
$rem
JOBTYPE opt
EXCHANGE hf
BASIS 6-31G*
MOLDEN_FORMAT true
$end
10.9.2 Visualization of Natural Transition Orbitals
For excited states calculated using the CIS, RPA, or TDDFT methods,
construction of Natural Transition Orbitals (NTOs), as described in
Section 6.10.2, is requested using the $rem variable
NTO_PAIRS. This variable also determines the number of hole/particle
NTO pairs that are output for each excited state. Although the total number of
hole/particle pairs is equal to the number of occupied MOs,
typically only a very small number of these pairs (often just one pair) have significant amplitudes.
(Additional large-amplitude NTOs may
be encountered in cases of strong electronic coupling between multiple chromophores [101].)
NTO_PAIRS
Controls the writing of hole/particle NTO pairs for excited state. |
TYPE:
DEFAULT:
OPTIONS:
N | Write N NTO pairs per excited state. |
RECOMMENDATION:
If activated (N > 0), a minimum of two NTO pairs will be printed for each state.
Increase the value of N if additional NTOs are desired.
|
|
When NTO_PAIRS > 0, Q-Chem will generate the NTOs in MolDen format.
The NTOs are state-specific, in the sense that each excited state has its own
NTOs, and therefore a separate MolDen file is required for each excited state. These files
are written to the job's scratch directory, in a sub-directory called NTOs, so to
obtain the NTOs the scratch directory must be saved using the
-save option that is described in Section 2.7. The output files
in the NTOs directory have an obvious file-naming convention.
The "hole" NTOs (which are linear combinations of
the occupied MOs) are printed to the
MolDen files in order of increasing importance, with the corresponding
excitation amplitudes replacing the canonical MO eigenvalues. (This is a formatting
convention only; the excitation amplitudes are unrelated to the MO eigenvalues.)
Following the holes are the "particle" NTOs,
which represent the excited electron and are linear combinations of the virtual MOs.
These are written in order of decreasing
amplitude. To aid in distinguishing the two sets within the MolDen files,
the amplitudes of the holes are listed with negative signs, while the corresponding
NTO for the excited electron has the same amplitude with a positive sign.
Due to the manner in which the NTOs are constructed
(see Section 6.10.2), NTO analysis is available only when the number of virtual
orbitals exceeds the number of occupied orbitals, which may not be the case for
minimal basis sets.
Example 10.0 Generating MolDen-formatted natural transition orbitals
for several excited states of uracil.
$molecule
0 1
N -2.181263 0.068208 0.000000
C -2.927088 -1.059037 0.000000
N -4.320029 -0.911094 0.000000
C -4.926706 0.301204 0.000000
C -4.185901 1.435062 0.000000
C -2.754591 1.274555 0.000000
N -1.954845 2.338369 0.000000
H -0.923072 2.224557 0.000000
H -2.343008 3.268581 0.000000
H -4.649401 2.414197 0.000000
H -6.012020 0.301371 0.000000
H -4.855603 -1.768832 0.000000
O -2.458932 -2.200499 0.000000
$end
$rem
EXCHANGE B3LYP
BASIS 6-31+G*
CIS_N_ROOTS 3
NTO_PAIRS 2
$end
10.9.3 Generation of Volumetric Data Using $plots
The simplest way to visualize the charge densities and molecular orbitals that
Q-Chem evaluates is via a graphical user interface, such as those described in
the preceding section. An alternative procedure, which is often useful
for generating high-quality images for publication, is to evaluate certain
densities and orbitals on a user-specified grid of points. This is
accomplished by invoking the $plots option, which is itself
enabled by requesting IANLTY = 200.
The format of the $plots input is documented below. It permits plotting
of molecular orbitals, the SCF ground-state density, and excited-state
densities obtained from CIS, RPA or TDDFT/TDA, or TDDFT calculations.
Also in connection with excited states, either transition densities,
attachment / detachment densities, or natural transition orbitals
(at the same levels of theory given above) can be plotted as well.
By default,
the output from the $plots command is one (or several) ASCII files in the
working directory, named plot.mo, etc..
The results then must be visualized
with a third-party program capable of making 3-D plots.
(Some suggestions are given in Section 10.9.4.)
An example of the use of the $plots option is the following input deck:
Example 10.0 A job that evaluates the H2 HOMO and LUMO on a 1×1×15
grid, along the bond axis. The plotting output is in an ASCII file called
plot.mo, which lists for each grid point, x, y, z, and the value of each
requested MO.
$molecule
0 1
H 0.0 0.0 0.35
H 0.0 0.0 -0.35
$end
$rem
EXCHANGE hf
BASIS 6-31g**
IANLTY 200
$end
$plots
Plot the HOMO and the LUMO on a line
1 0.0 0.0
1 0.0 0.0
15 -3.0 3.0
2 0 0 0
1 2
$end
General format for the $plots section of the Q-Chem input deck.
$plots
One comment line
Specification of the 3-D mesh of points on 3 lines:
Nx xmin xmax
Ny ymin ymax
Nz zmin zmax
A line with 4 integers indicating how many things to plot:
NMO NRho NTrans NDA
An optional line with the integer list of MO's to evaluate
(only if NMO > 0)
MO(1) MO(2) … MO(NMO)
An optional line with the integer list of densities to evaluate
(only if NRho > 0)
Rho(1) Rho(2) … Rho(NRho)
An optional line with the integer list of transition densities
(only if NTrans > 0)
Trans(1) Trans(2) … Trans(NTrans)
An optional line with states for detachment / attachment densities
(if NDA > 0)
DA(1) DA(2) … DA(NDA)
$end
Line 1 of the $plots keyword section is reserved for comments. Lines 2-4 list
the number of one dimension points and the range of the grid (note that
coordinate ranges are in Angstroms, while all output is in atomic units). Line
5 must contain 4 non-negative integers indicating the number of: molecular
orbitals (NMO), electron densities (NRho), transition
densities and attach / detach densities (NDA), to have mesh values
calculated.
The final lines specify which MOs, electron densities, transition densities and
CIS attach / detach states are to be plotted (the line can be left blank, or
removed, if the number of items to plot is zero). Molecular orbitals are
numbered 1…Nα,Nα+1 …Nα+Nβ; electron
densities numbered where 0= ground state, 1 = first excited state, 2 = second
excited state, etc.; and attach / detach specified from state 1→NDA.
By default,
all output data are printed to files in the working directory, overwriting any
existing file of the same name.
- Molecular orbital data is printed to a file called plot.mo;
-
densities are plotted to plots.hf;
-
restricted unrelaxed attachment / detachment analysis is sent to plot.attach.alpha
and
plot.detach.alpha;
-
unrestricted unrelaxed attachment / detachment
analysis is sent to plot.attach.alpha,
plot.detach.alpha, plot.attach.beta and plot.detach.beta;
-
restricted relaxed
attachments / detachment analysis is plotted in plot.attach.rlx.alpha and
plot.detach.rlx.alpha; and finally
-
unrestricted relaxed attachment / detachment
analysis is sent to plot.attach.rlx.alpha,
plot.detach.rlx.alpha, plot.attach.rlx.beta and plot.detach.rlx.beta.
Output is printed
in atomic units, with coordinates first followed by item value, as shown below:
x1 y1 z1 a1 a2 ... aN
x2 y1 z1 b1 b2 ... bN
...
Instead of a standard one-, two-, or three-dimensional Cartesian grid, a
user may wish to plot orbitals or densities on a set of grid points of
his or her choosing. Such points are specified using a $grid input
section whose format is simply the Cartesian coordinates of all
user-specified grid points:
The $plots section must still be specified as described above, but
if the $grid input section is present, then these user-specified
grid points will override the ones specified in the $plots section.
The Q-Chem $plots utility allows the user to plot transition densities and
detachment / attachment densities directly from amplitudes saved from a previous
calculation, without having to solve the post-SCF (CIS, RPA, TDA, or TDDFT)
equations again. To take advantage of this feature, the same Q-Chem scratch
directory must be used, and the SKIP_CIS_RPA $rem variable must be
set to TRUE. To further reduce computational time, the
SCF_GUESS $rem can be set to READ.
SKIP_CIS_RPA
Skips the solution of the CIS, RPA, TDA or TDDFT equations for wavefunction
analysis. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to true to speed up the generation of plot data if the same calculation
has been run previously with the scratch files saved. |
|
10.9.4 Direct Generation of "Cube" Files
As an alternative to the output format discussed above, all of the
$plots data may be output directly to a sub-directory named
plots in the job's scratch directory,
which must therefore be saved using the -save option described
in Section 2.7. The plotting data in this sub-directory
are not written in the plot.* format described above, but
rather in the form of so-called
"cube" file, one for each orbital or density that is requested.
The "cube" format is a standard one
for volumetric data, and consists of a small
header followed by the orbital or density values at each grid point, in
ASCII format. Because the grid coordinates themselves are not printed
(their locations are implicit from information contained in the header),
each individual cube file is much smaller than the corresponding
plot.* file would be.
Cube files can be read by many standard (and freely-available)
graphics programs, including MacMolPlt [515,[516] and
VMD [450,[451]. VMD, in particular, is recommended
for generation of high-quality images for publication.
Cube files for the MOs and densities requested in the $plots section are
requested by setting MAKE_CUBE_FILES to
TRUE, with the $plots section specified as described in
Section 10.9.3.
MAKE_CUBE_FILES
Requests generation of cube files for MOs, NTOs, or NBOs. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not generate cube files. |
TRUE | Generate cube files for MOs and densities. |
NTOS | Generate cube files for NTOs. |
NBOS | Generate cube files for NBOs.
|
RECOMMENDATION:
|
Cube files are also available for natural transition orbitals
(Sections 6.10.2 and 10.9.2) by setting
MAKE_CUBE_FILES to NTOS, although in this case the
procedure is somewhat more complicated, due to the state-specific nature
of these quantities. Cube files for the NTOs are generated only for a
single excited state, whose identity is specified using
CUBEFILE_STATE. Cube files for additional states are
readily obtained using a sequence of Q-Chem jobs, in which the second
(and subsequent) jobs read in the converged ground- and excited-state
information using SCF_GUESS and SKIP_CIS_RPA.
CUBEFILE_STATE
Determines which excited state is used to generate cube files |
TYPE:
DEFAULT:
OPTIONS:
n | Generate cube files for the nth excited state
|
RECOMMENDATION:
|
An additional complication is the manner in which to specify which
NTOs will be output as cube files. When MAKE_CUBE_FILES is
set to TRUE, this is specified in the $plots section, in the
same way that MOs would be specified for plotting. However, one must
understand the order in which the NTOs are stored. For a system with
Nα α-spin MOs, the occupied NTOs 1,2,…,Nα
are stored in order
of increasing amplitudes, so that the Nα'th occupied NTO is the most
important. The virtual NTOs are stored next, in order of
decreasing importance. According to this convention,
the highest occupied NTO (HONTO) → lowest unoccupied
NTO (LUNTO) excitation amplitude is always the most significant, for
any particular excited state. Thus, orbitals Nα and Nα+1
represent the most important NTO pair, while orbitals Nα −1 and
Nα+2 represent the second most important NTO pair, etc..
Example 10.0 Generating cube files for the HONTO-to-LUNTO excitation
of the second singlet excited state of uracil. Note that Nα = 29
for uracil.
$molecule
0 1
N -2.181263 0.068208 0.000000
C -2.927088 -1.059037 0.000000
N -4.320029 -0.911094 0.000000
C -4.926706 0.301204 0.000000
C -4.185901 1.435062 0.000000
C -2.754591 1.274555 0.000000
N -1.954845 2.338369 0.000000
H -0.923072 2.224557 0.000000
H -2.343008 3.268581 0.000000
H -4.649401 2.414197 0.000000
H -6.012020 0.301371 0.000000
H -4.855603 -1.768832 0.000000
O -2.458932 -2.200499 0.000000
$end
$plots
Plot the dominant NTO pair, N --> N+1
25 -5.0 5.0
25 -5.0 5.0
25 -5.0 5.0
2 0 0 0
29 30
$end
$rem
EXCHANGE B3LYP
BASIS 6-31+G*
CIS_N_ROOTS 2
CIS_TRIPLETS FALSE
NTO_PAIRS TRUE ! calculate the NTOs
MAKE_CUBE_FILES NTOS ! generate NTO cube files...
CUBEFILE_STATE 2 ! ...for the 2nd excited state
$end
Cube files for Natural Bond Orbitals (for either the ground state
or any CIS, RPA, of TDDFT excited states) can be generated in much the
same way, by setting MAKE_CUBE_FILES equal to NBOS,
and using CUBEFILE_STATE to select the desired electronic state.
CUBEFILE_STATE = 0 selects ground-state NBOs. The particular
NBOs to be plotted are selected using the $plots section, recognizing
that Q-Chem stores the NBOs in order of decreasing occupancies, with
all α-spin NBOs preceding any β-spin NBOs, in the case of an
unrestricted SCF calculation. (For ground states, there is typically one
strongly-occupied
NBO for each electron.) NBO cube files are saved to the
plots sub-directory of the job's scratch directory. One final
caveat: to get NBO cube files, the user must specify the AONBO
option in the $nbo section, as shown in the following example.
Example 10.0 Generating cube files for the NBOs of the first
excited state of H2O.
$rem
EXCHANGE HF
BASIS CC-PVTZ
CIS_N_ROOTS 1
CIS_TRIPLETS FALSE
NBO 2 ! ground- and excited-state NBO
MAKE_CUBE_FILES NBOS ! generate NBO cube files...
CUBEFILE_STATE 1 ! ...for the first excited state
$end
$nbo
AONBO
$end
$molecule
0 1
O
H 1 0.95
H 1 0.95 2 104.5
$end
$plots
Plot the 5 high-occupancy NBOs, one for each alpha electron
40 -8.0 8.0
40 -8.0 8.0
40 -8.0 8.0
5 0 0 0
1 2 3 4 5
$end
10.9.5 NCI Plots
We have implemented the non-covalent interaction (NCI) plots from Weitao Yang's
group [517,[518].
To generate these plots, one can set the PLOT_REDUCED_DENSITY_GRAD $rem variable
to TRUE (see the nci-c8h14.in input example in $QC/samples directory).
10.10 Electrostatic Potentials
Q-Chem can evaluate electrostatic potentials on a grid of points.
Electrostatic potential evaluation is controlled by the $rem variable
IGDESP, as documented below.
IGDESP
Controls evaluation of the electrostatic potential on a grid of points. If
enabled, the output is in an ACSII file, plot.esp, in the format x, y, z, esp
for each point. |
TYPE:
DEFAULT:
none no electrostatic potential evaluation |
OPTIONS:
−1 | read grid input via the $plots section of the input deck |
0 | Generate the ESP values at all nuclear positions. |
+n | read n grid points in bohrs (!) from the ACSII file ESPGrid. |
RECOMMENDATION:
|
The following example illustrates the evaluation of electrostatic potentials on
a grid, note that IANLTY must also be set to 200.
Example 10.0 A job that evaluates the electrostatic potential for H2 on
a 1 by 1 by 15 grid, along the bond axis. The output is in an ASCII file called
plot.esp, which lists for each grid point, x, y, z, and the electrostatic
potential.
$molecule
0 1
H 0.0 0.0 0.35
H 0.0 0.0 -0.35
$end
$rem
EXCHANGE hf
BASIS 6-31g**
IANLTY 200
IGDESP -1
$end
$plots
plot the HOMO and the LUMO on a line
1 0.0 0.0
1 0.0 0.0
15 -3.0 3.0
0 0 0 0
0
$end
We can also compute the electrostatic potential for the transition density,
which can be used, for example, to compute the Coulomb coupling in excitation energy transfer.
ESP_TRANS
Controls the calculation of the electrostatic potential of the transition density |
TYPE:
DEFAULT:
OPTIONS:
TRUE | compute the electrostatic potential of the excited state transition density |
FALSE | compute the electrostatic potential of the excited state electronic density |
RECOMMENDATION:
|
The electrostatic potential is a complicated object and it is not uncommon to
model it using a simplified representation based on atomic charges. For this
purpose it is well known that Mulliken charges perform very poorly. Several
definitions of ESP-derived atomic charges have been given in the literature,
however, most of them rely on a least-squares fitting of the ESP evaluated on
a selection of grid points. Although these grid points are usually chosen so
that the ESP is well modeled in the "chemically important" region, it still
remains that the calculated charges will change if the molecule is rotated.
Recently an efficient rotationally invariant algorithm was
proposed [519] that sought to model the ESP not by direct
fitting, but by fitting to the multipole moments. By doing so it was found
that the fit to the ESP was superior to methods that relied on direct fitting
of the ESP. The calculation requires the traceless form of the multipole
moments and these are also printed out during the course of the calculations.
To request these multipole-derived charges the following $rem option should
be set:
MM_CHARGES
Requests the calculation of multipole-derived charges (MDCs). |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculates the MDCs and also the traceless form of the multipole moments |
RECOMMENDATION:
Set to TRUE if MDCs or the traceless form of the multipole
moments are desired. The calculation does not take long. |
|
10.11 Spin and Charge Densities at the Nuclei
Gaussian basis sets violate nuclear cusp conditions [520,[521,[522].
This may lead to large errors in wavefunction at nuclei,
particularly for spin density calculations [523]. This problem
can be alleviated by using an averaging operator that compute wavefunction
density based on constraints that wavefunction must satisfy near Coulomb
singularity [524,[525]. The derivation of operators is
based on hyper virial theorem [526] and presented in
Ref. . Application to molecular spin densities for
spin-polarized [525] and DFT [527] wavefunctions show
considerable improvement over traditional delta function operator.
One of the simplest forms of such operators is based on the Gaussian weight
function exp[−(Z/r0)2(r−R)2]
that samples the vicinity of a nucleus of charge Z located at R. The parameter r0
has to be small enough to neglect two-electron contributions of the order
O(r04) but large enough for meaningful averaging. The range of values
between 0.15-0.3 a.u. is shown to be adequate, with final answer being
relatively insensitive to the exact choice of r0 [524,[525].
The value of r0 is chosen by
RC_R0 keyword in the units of 0.001 a.u. The averaging operators are
implemented for single determinant Hartree-Fock and DFT, and correlated SSG
wavefunctions. Spin and charge densities are printed for all nuclei in a
molecule, including ghost atoms.
RC_R0
Determines the parameter in the Gaussian weight function used to smooth the
density at the nuclei. |
TYPE:
DEFAULT:
OPTIONS:
0 | Corresponds the traditional delta function spin and charge densities |
n | corresponding to n×10−3 a.u. |
RECOMMENDATION:
We recommend value of 250 for a typical spit valence basis. For basis sets
with increased flexibility in the nuclear vicinity the smaller values of r0
also yield adequate spin density. |
|
10.12 NMR Shielding Tensors
NMR spectroscopy is a powerful technique to yield important information on
molecular systems in chemistry and biochemistry. Since there is no direct
relationship between the measured NMR signals and structural properties, the
necessity for a reliable method to predict NMR chemical shifts arises.
Examples for such assignments are numerous, for example, assignments of
solid-state spectra [528,[529]. The implementation
within Q-Chem uses gauge-including atomic orbitals
(GIAOs) [530,[531,[532] to calculate the NMR chemical
shielding tensors. This scheme has been proven to be reliable an accurate for
many applications [533].
The shielding tensor, σ, is a second-order property depending on the
external magnetic field, B, and the nuclear magnetic spin momentum, mk,
of nucleus k:
Using analytical derivative techniques to evaluate σ, the components
of this 3 ×3 tensor are computed as
σij = |
∑
μν
|
Pμν |
∂2hμν
∂Bi ∂mj,k
|
+ |
∑
μν
|
|
∂Pμν
∂Bi
|
|
∂hμν
∂mj,k
|
|
| (10.62) |
where i and j represent are Cartesian components.
To solve for the necessary
perturbed densities, ∂P/∂Bx,y,z, a new CPSCF method based
on a density matrix based formulation [534,[535] is used. This
formulation is related to a density matrix based CPSCF (D-CPSCF) formulation
employed for the computation of vibrational frequencies [536].
Alternatively, an MO-based CPSCF calculation of shielding tensors can be chosen
by the variable MOPROP. Features of the NMR package include:
- Restricted HF-GIAO and KS-DFT-GIAO NMR chemical shifts calculations
- LinK/CFMM support to evaluate Coulomb- and exchange-like matrices
- Density matrix-based coupled-perturbed SCF (D-CPSCF)
- DIIS acceleration
- Support of basis sets up to d functions
- Support of LSDA/GGA/Hybrid XC functionals
10.12.1 Job Control
The JOBTYPE must be set to NMR to request the NMR chemical shifts.
D_CPSCF_PERTNUM
Specifies whether to do the perturbations one at a time, or all together. |
TYPE:
DEFAULT:
OPTIONS:
0 | Perturbed densities to be calculated all together. |
1 | Perturbed densities to be calculated one at a time. |
RECOMMENDATION:
|
| D_SCF_CONV_1
Sets the convergence criterion for the level-1 iterations. This preconditions
the density for the level-2 calculation, and does not include any
two-electron integrals. |
TYPE:
DEFAULT:
4 | corresponding to a threshold of 10−4. |
OPTIONS:
n < 10 | Sets convergence threshold to 10−n. |
RECOMMENDATION:
The criterion for level-1 convergence must be less than or equal to the
level-2 criterion, otherwise the D-CPSCF will not converge. |
|
|
|
D_SCF_CONV_2
Sets the convergence criterion for the level-2 iterations. |
TYPE:
DEFAULT:
4 | Corresponding to a threshold of 10−4. |
OPTIONS:
n < 10 | Sets convergence threshold to 10−n. |
RECOMMENDATION:
|
| D_SCF_MAX_1
Sets the maximum number of level-1 iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
D_SCF_MAX_2
Sets the maximum number of level-2 iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| D_SCF_DIIS
Specifies the number of matrices to use in the DIIS extrapolation in the
D-CPSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | n = 0 specifies no DIIS extrapolation is to be used. |
RECOMMENDATION:
|
|
|
10.12.2 Using NMR Shielding Constants as an Efficient Probe of Aromaticity
Unambiguous theoretical estimates of degree of aromaticity are still on high demand.
The NMR chemical shift methodology offers one unique probe of aromaticity based
on one defining characteristics of an aromatic system-its
ability to sustain a diatropic ring current. This leads
to a response to an imposed external magnetic field
with a strong (negative) shielding at the center of the ring.
Schleyer and co. have employed this phenomenon to justify a new unique probe of
aromaticity [537]. They proposed the computed absolute magnetic shielding at ring centers
(unweighted mean of the heavy-atoms ring coordinates) as a new aromaticity criterion,
called nucleus-independent chemical shift (NICS). Aromatic rings show strong negative shielding
at the ring center (negative NICS), while anti-aromatic systems reveal positive NICS at the ring center.
As an example, a typical NICS value for benzene is about -11.5 ppm as estimated
with Q-Chem at Hartree-Fock / 6-31G* level. The same NICS value for benzene was also
reported in Ref. . The calculated NICS value for furan of −13.9 ppm with Q-Chem
is about the same as the value reported for furan in Ref. . Below is
one input example of how to the NICS of furan with Q-Chem, using the ghost atom option.
The ghost atom is placed at the center of the furan ring, and the basis set assigned to it
within the basis mix option must be the basis used for hydrogen atom.
Example 10.0 Calculation of the NMR NICS probe of furane with Hartree-Fock/6-31G* with Q-Chem.
$molecule
0 1
C -0.69480 -0.62270 -0.00550
C 0.72110 -0.63490 0.00300
C 1.11490 0.68300 0.00750
O 0.03140 1.50200 0.00230
C -1.06600 0.70180 -0.00560
H 2.07530 1.17930 0.01410
H 1.37470 -1.49560 0.00550
H -1.36310 -1.47200 -0.01090
H -2.01770 1.21450 -0.01040
GH 0.02132 0.32584 0.00034 ! ghost at the ring center
$end
$rem
JobType NMR
Exchange HF
BASIS mixed
SCF_Algorithm DIIS
PURCAR 111
SEPARATE_JK 0
LIN_K 0
CFMM_ORDER 15
GRAIN 1
CFMM_PRINT 2
CFMMSTAT 1
PRINT_PATH_TIME 1
LINK_MAXSHELL_NUMBER 1
SKIP_SCFMAN 0
IGUESS core ! Core Hamiltonian Guess
SCF_Convergence 7
ITHRSH 10 ! Threshold
IPRINT 23
D_SCF_CONVGUIDE 0 !REM_D_SCF_CONVGUIDE
D_SCF_METRIC 2 !Metric...
D_SCF_STORAGE 50 !REM_D_SCF_STORAGE
D_SCF_RESTART 0 !REM_D_SCF_RESTART
PRINT_PATH_TIME 1
SYM_IGNORE 1
NO_REORIENT 1
$end
$basis
C 1
6-31G*
****
C 2
6-31G*
****
C 3
6-31G*
****
O 4
6-31G*
****
C 5
6-31G*
****
H 6
6-31G*
****
H 7
6-31G*
****
H 8
6-31G*
****
H 9
6-31G*
****
H 10
6-31G*
****
10.13 Linear-Scaling NMR Chemical Shifts: GIAO-HF and GIAO-DFT
The importance of nuclear magnetic resonance (NMR) spectroscopy for modern
chemistry and biochemistry cannot be overestimated. Despite tremendous
progress in experimental techniques, the understanding and reliable assignment
of observed experimental spectra remains often a highly difficult task, so that
quantum chemical methods can be extremely useful both in the solution and the
solid state (e.g.,
Refs. ,
and references therein).
The cost for the computation of NMR chemical shifts within even the simplest
quantum chemical methods such as Hartree-Fock (HF) or density functional (DFT)
approximations increases conventionally with the third power of the molecular
size M, O(M3), where O(·) stands for the scaling order.
Therefore, the computation of NMR chemical shieldings has so far been limited
to molecular systems in the order of 100 atoms without molecular symmetry.
For larger systems it is crucial to reduce the increase of the computational
effort to linear, which has been recently achieved by Kussmann and
Ochsenfeld [534,[540]. In this way, the computation of NMR chemical shifts
becomes possible at both HF or DFT level for molecular systems with 1000 atoms
and more, while the accuracy and reliability of traditional methods is fully
preserved. In our formulation we use gauge-including atomic orbitals
(GIAOs) [530,[541,[531], which have proven to be
particularly successful [542]. For example, for many molecular
systems the HF (GIAO-HF) approach provides typically an accuracy of 0.2-0.4 ppm
for the computation of 1H NMR chemical shifts (e.g.
Refs. ).
NMR chemical shifts are calculated as second derivatives of the energy with
respect to the external magnetic field B and the nuclear magnetic spin
mNj of a nucleus N:
where i, j are x, y, z coordinates.
For the computation of the NMR shielding tensor it is necessary to solve for
the response of the one-particle density matrix with respect to the magnetic
field, so that the solution of the coupled perturbed SCF (CPSCF) equations
either within the HF or the DFT approach is required.
These equations can be solved within a density matrix-based formalism for the
first time with only linear-scaling effort for molecular systems with a
non-vanishing HOMO-LUMO gap [534]. The solution is even simpler
in DFT approaches without explicit exchange, since present density functionals
are not dependent on the magnetic field.
The present implementation of NMR shieldings in Q-Chem employs the LinK
(linear exchange K) method [160,[161] for the formation of exchange
contributions [534]. Since the derivative of the density matrix
with respect to the magnetic field is skew-symmetric, its Coulomb-type
contractions vanish. For the remaining Coulomb-type matrices the CFMM
method [543] is adapted [534]. In addition, a multitude of
different approaches for the solution of the CPSCF equations can be selected
within Q-Chem.
The so far largest molecular system for which NMR shieldings have been
computed, contained 1003 atoms and 8593 basis functions (GIAO-HF / 6-31G*)
without molecular symmetry [534].
10.14 Linear-Scaling Computation of Electric Properties
The search for new optical devices is a major field of materials sciences.
Here, polarizabilities and hyperpolarizabilities provide particularly important
information on molecular systems. The response of the molecular systems in the
presence of an external monochromatic oscillatory electric field is determined
by the solution of the TDSCF equations, where the perturbation is represented
as the interaction of the molecule with a single Fourier component within the
dipole approximation:
Here, E is the E-field vector, ω the corresponding frequency,
e the electronic charge and μ the dipole moment operator. Starting from
Frenkel's variational principle the TDSCF equations can be derived by standard
techniques of perturbation theory [544]. As a solution we yield
the first (Px(±ω)) and second order (e.g.
Pxy(±ω,±ω)) perturbed density matrices with which
the following properties are calculated:
- Static polarizability:
αxy(0;0) = Tr[HμxPy(ω = 0)]
- Dynamic polarizability:
αxy(±ω;±ω) = Tr[HμxPy(±ω)]
- Static hyperpolarizability:
βxyz(0;0,0) = Tr[HμxPyz(ω = 0,ω = 0)]
- Second harmonic generation:
βxyz(±2ω;±ω,±ω) = Tr[HμxPyz(±ω,±ω)]
- Electro-optical Pockels effect:
βxyz(±ω;0,±ω) = Tr[HμxPyz(ω = 0,±ω)]
- Optical rectification:
βxyz(0;±ω,±ω) = Tr[HμxPyz(±ω,±ω)]
where Hμx is the matrix representation of the x
component of the dipole moments.
The TDSCF calculation is the most time consuming step and scales asymptotically
as O(N3) because of the AO / MO transformations. The scaling behavior of
the two-electron integral formations, which dominate over a wide range because
of a larger pre-factor, can be reduced by LinK / CFMM from quadratic to linear
(O(N2)→O(N)).
Third-order properties can be calculated with the equations above after a
second-order TDSCF calculation (MOPROP: 101/102) or by use of Wigner's
(2n+1) rule [545] (MOPROP: 103/104). Since the second order TDSCF
depends on the first-order results, the convergence of the algorithm may be
problematically. So we recommend the use of 103/104 for the calculation of
first hyperpolarizabilities.
These optical properties can be computed for the first time using
linear-scaling methods (LinK/CFMM) for all integral contractions [535].
Although the present implementation available in Q-Chem still
uses MO-based time-dependent SCF (TDSCF) equations both at the HF and DFT
level, the pre-factor of this O(M3) scaling step is rather small, so that
the reduction of the scaling achieved for the integral contractions is most
important. Here, all derivatives are computed analytically.
Further specifications of the dynamic properties are done in the section $fdpfreq
in the following format:
$fdpfreq
property
frequencies
units
$end
The first line is only required for third order properties to specify the kind
of first hyperpolarizability:
- StaticHyper Static Hyperpolarizability
- SHG Second harmonic generation
- EOPockels Electro-optical Pockels effect
- OptRect Optical rectification
Line number 2 contains the values (FLOAT) of the frequencies of the
perturbations. Alternatively, for dynamic polarizabilities an equidistant
sequence of frequencies can be specified by the keyword WALK (see
example below). The last line specifies the units of the given frequencies:
- au Frequency (atomic units)
- eV Frequency (eV)
- nm Wavelength (nm) → Note that 0 nm will be treated as 0.0 a.u.
- Hz Frequency (Hertz)
- cmInv Wavenumber (cm−1)
10.14.1 Examples for Section $fdpfreq
Example 10.0 Static and Dynamic polarizabilities, atomic units:
$fdpfreq
0.0 0.03 0.05
au
$end
Example 10.0 Series of dynamic polarizabilities, starting with 0.00
incremented by 0.01 up to 0.10:
$fdpfreq
walk 0.00 0.10 0.01
au
$end
Example 10.0 Static first hyperpolarizability, second harmonic generation and
electro-optical Pockels effect, wavelength in nm:
$fdpfreq
StaticHyper SHG EOPockels
1064
nm
$end
10.14.2 Features of Mopropman
- Restricted/unrestricted HF and KS-DFT CPSCF/TDSCF
- LinK/CFMM support to evaluate Coulomb- and exchange-like matrices
- DIIS acceleration
- Support of LSDA/GGA/Hybrid XC functionals listed below
- Analytical derivatives
The following XC functionals are supported:
Exchange:
Correlation:
- Wigner
- VWN (both RPA and No. 5 parameterizations)
- Perdew-Zunger 81
- Perdew 86 (both PZ81 and VWN (No. 5) kernel)
- LYP
10.14.3 Job Control
The following options can be used:
MOPROP
Specifies the job for mopropman. |
TYPE:
DEFAULT:
OPTIONS:
1 | NMR chemical shielding tensors. |
2 | Static polarizability. |
100 | Dynamic polarizability. |
101 | First hyperpolarizability. |
102 | First hyperpolarizability, reading First order results from disk. |
103 | First hyperpolarizability using Wigner's (2n+1) rule. |
104 | First hyperpolarizability using Wigner's (2n+1) rule, reading |
| first order results from disk. |
RECOMMENDATION:
|
| MOPROP_PERTNUM
Set the number of perturbed densities that will to be treated together. |
TYPE:
DEFAULT:
OPTIONS:
0 | All at once. |
n | Treat the perturbed densities batch-wise. |
RECOMMENDATION:
|
|
|
MOPROP_CONV_1ST
Sets the convergence criteria for CPSCF and 1st order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n < 10 | Convergence threshold set to 10−n. |
RECOMMENDATION:
|
| MOPROP_CONV_2ND
Sets the convergence criterion for second-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n < 10 | Convergence threshold set to 10−n. |
RECOMMENDATION:
|
|
|
MOPROP_criteria_1ST
The maximal number of iterations for CPSCF and first-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | Set maximum number of iterations to n. |
RECOMMENDATION:
|
| MOPROP_MAXITER_2ND
The maximal number of iterations for second-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | Set maximum number of iterations to n. |
RECOMMENDATION:
|
|
|
MOPROP_DIIS
Controls the use of Pulays DIIS. |
TYPE:
DEFAULT:
OPTIONS:
0 | Turn off DIIS. |
5 | Turn on DIIS. |
RECOMMENDATION:
|
| MOPROP_DIIS_DIM_SS
Specified the DIIS subspace dimension. |
TYPE:
DEFAULT:
OPTIONS:
0 | No DIIS. |
n | Use a subspace of dimension n. |
RECOMMENDATION:
|
|
|
SAVE_LAST_GPX
Save last G[Px] when calculating dynamic
polarizabilities in order to call mopropman in a second run with MOPROP = 102. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
10.15 Atoms in Molecules
Q-Chem can output a file suitable for analysis with the Atoms in Molecules
package (AIMPAC). The source for AIMPAC can be freely downloaded from the
web site:
http://www.chemistry.mcmaster.ca/aimpac/imagemap/imagemap.htm
Users should check this site for further information about installing and
running AIMPAC. The AIMPAC input file is created by specifying a filename for
the WRITE_WFN $rem.
WRITE_WFN
Specifies whether or not a wfn file is created, which is suitable for use with
AIMPAC. Note that the output to this file is currently limited to f orbitals,
which is the highest angular momentum implemented in AIMPAC. |
TYPE:
DEFAULT:
(NULL) | No output file is created. |
OPTIONS:
filename | Specifies the output file name. The suffix .wfn will |
| be appended to this name. |
RECOMMENDATION:
|
10.16 Distributed Multipole Analysis
Distributed Multipole Analysis (DMA) [546] is a method to represent the electrostatic potential of a molecule
in terms of a multipole expansion around a set of points.
The points of expansion are the atom centers and (optionally) bond midpoints.
Current implementation performs expansion into charges, dipoles, quadrupoles and octupoles.
DO_DMA
Specifies whether to perform Distributed Multipole Analysis. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Turn off DMA. |
TRUE | Turn on DMA. |
RECOMMENDATION:
|
| DMA_MIDPOINTS
Specifies whether to include bond midpoints into DMA expansion. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not include bond midpoints. |
TRUE | Include bond midpoint. |
RECOMMENDATION:
|
|
|
10.17 Electronic Couplings for Electron Transfer and Energy Transfer
10.17.1 Eigenstate-Based Methods
For electron transfer (ET) and excitation energy transfer (EET) processes, the electronic coupling is one of the important parameters that determine their reaction rates.
For ET, Q-Chem provides the coupling values calculated with the generalized Mulliken-Hush
(GMH) [547], fragment-charge difference (FCD) [548], Boys
localization [549], and Edmiston-Ruedenbeg localization [550] schemes.
For EET, options include fragment-excitation difference (FED) [551], fragment-spin difference (FSD) [552], occupied-virtual separated Boys localization [553] or Edmiston-Ruedenberg localization [550].
In all these schemes, a vertical excitation such as CIS, RPA or TDDFT is required, and the GMH, FCD, FED, FSD, Boys or ER coupling values are calculated based on the excited state results.
10.17.1.1 Two-state approximation
Under the two-state approximation, the diabatic reactant and product states are assumed to be a linear combination of the eigenstates.
For ET, the choice of such linear combination is determined by a zero transition dipoles (GMH) or maximum charge differences (FCD).
In the latter, a 2×2 donor-acceptor charge difference matrix, ∆q,
is defined, with elements
| |
|
| |
| |
|
| ⌠ ⌡
|
r ∈ D
|
ρmn(r)dr − | ⌠ ⌡
|
r ∈ A
|
ρmn(r)dr |
| | (10.66) |
|
where ρmn(r) is the matrix element of the density operator between states
|m〉 and |n〉.
For EET, a maximum excitation difference is assumed in the FED, in which a excitation difference matrix is similarly defined with elements
| |
| |
| |
|
| ⌠ ⌡
|
r ∈ D
|
ρex(mn)(r)dr − | ⌠ ⌡
|
r ∈ A
|
ρex(mn)(r)dr |
| | (10.67) |
|
where ρex(mn)(r) is the sum of attachment and detachment densities
for transition |m〉→ |n〉, as they correspond to the electron and hole densities in an excitation.
In the FSD, a maximum spin difference is used and the corresponding spin difference matrix is defined with its elements as,
| |
|
| |
| |
|
| ⌠ ⌡
|
r ∈ D
|
σ(mn)(r)dr − | ⌠ ⌡
|
r ∈ A
|
σ(mn)(r)dr |
| | (10.68) |
|
where σmn(r) is the spin density, difference between α-spin and β-spin densities, for transition from |m〉→ |n〉.
Since Q-Chem uses a Mulliken population analysis for the integrations in
Eqs. (10.66), (10.67), and (10.68), the matrices ∆q, ∆x and ∆s are not symmetric.
To obtain a pair of orthogonal states as the diabatic reactant and product states, ∆q, ∆x and ∆s are symmetrized in Q-Chem. Specifically,
The final coupling values are obtained as listed below:
- For GMH,
VET = |
|
⎛ √
|
( |
→
μ
|
11
|
− |
→
μ
|
22
|
)2 + 4|[(μ)\vec]12|2 |
|
|
|
|
| (10.72) |
- For FCD,
VET = |
|
⎛ √
|
(∆q11 − ∆q22)2 + 4 |
∆q
|
2 12
|
|
|
|
| (10.73) |
- For FED,
VEET = |
|
⎛ √
|
(∆x11 − ∆x22)2 + 4 |
∆x
|
2 12
|
|
|
|
| (10.74) |
- For FSD,
VEET = |
|
⎛ √
|
(∆s11 − ∆s22)2 + 4 |
∆s
|
2 12
|
|
|
|
| (10.75) |
Q-Chem provides the option to control FED, FSD, FCD and GMH calculations after a single-excitation calculation, such as CIS, RPA, TDDFT/TDA and TDDFT.
To obtain ET coupling values using GMH (FCD) scheme, one should set $rem variables
STS_GMH (STS_FCD) to be TRUE.
Similarly, a FED (FSD) calculation is turned on by setting the $rem variable STS_FED (STS_FSD) to be TRUE.
In FCD, FED and FSD calculations, the donor and acceptor fragments are defined via the $rem variables STS_DONOR and STS_ACCEPTOR. It is necessary to arrange the atomic order in the $molecule section such that the atoms in the donor (acceptor) fragment is in one consecutive block. The ordering numbers of beginning and ending atoms for the donor and acceptor blocks are included in $rem variables STS_DONOR and STS_ACCEPTOR.
The couplings will be calculated between all choices of excited states with the same spin. In FSD, FCD and GMH calculations, the coupling value between the excited and reference (ground) states will be included, but in FED, the ground state is not included in the analysis.
It is important to select excited states properly, according to the distribution of charge or excitation, among other characteristics, such that the coupling obtained can properly describe the electronic coupling of the corresponding process in the two-state approximation.
STS_GMH
Control the calculation of GMH for ET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a GMH calculation. |
TRUE | Include a GMH calculation. |
RECOMMENDATION:
When set to true computes Mulliken-Hush electronic couplings. It yields
the generalized Mulliken-Hush couplings as well as the transition dipole
moments for each pair of excited states and for each excited state with
the ground state. |
|
| STS_FCD
Control the calculation of FCD for ET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform an FCD calculation. |
TRUE | Include an FCD calculation. |
RECOMMENDATION:
|
|
|
STS_FED
Control the calculation of FED for EET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a FED calculation. |
TRUE | Include a FED calculation. |
RECOMMENDATION:
|
| STS_FSD
Control the calculation of FSD for EET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a FSD calculation. |
TRUE | Include a FSD calculation. |
RECOMMENDATION:
For RCIS triplets, FSD and FED are equivalent. FSD will be automatically switched off and perform a FED calculation. |
|
|
|
STS_DONOR
Define the donor fragment. |
TYPE:
DEFAULT:
0 | No donor fragment is defined. |
OPTIONS:
i-j | Donor fragment is in the ith atom to the jth atom. |
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j. |
|
| STS_ACCEPTOR
Define the acceptor molecular fragment. |
TYPE:
DEFAULT:
0 | No acceptor fragment is defined. |
OPTIONS:
i-j | Acceptor fragment is in the ith atom to the jth atom. |
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j. |
|
|
|
STS_MOM
Control calculation of the transition moments between excited states in
the CIS and TDDFT calculations (including SF variants). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate state-to-state transition moments. |
TRUE | Do calculate state-to-state transition moments. |
RECOMMENDATION:
When set to true requests the state-to-state dipole transition moments for
all pairs of excited states and for each excited state with the ground
state. |
|
Example 10.0 A GMH & FCD calculation to analyze electron-transfer couplings
in an ethylene and a methaniminium cation.
$molecule
1 1
C 0.679952 0.000000 0.000000
N -0.600337 0.000000 0.000000
H 1.210416 0.940723 0.000000
H 1.210416 -0.940723 0.000000
H -1.131897 -0.866630 0.000000
H -1.131897 0.866630 0.000000
C -5.600337 0.000000 0.000000
C -6.937337 0.000000 0.000000
H -5.034682 0.927055 0.000000
H -5.034682 -0.927055 0.000000
H -7.502992 -0.927055 0.000000
H -7.502992 0.927055 0.000000
$end
$rem
EXCHANGE hf
BASIS 6-31+G
CIS_N_ROOTS 20
CIS_SINGLETS true
CIS_TRIPLETS false
STS_GMH true !turns on the GMH calculation
STS_FCD true !turns on the FCD calculation
STS_DONOR 1-6 !define the donor fragment as atoms 1-6 for FCD calc.
STS_ACCEPTOR 7-12 !define the acceptor fragment as atoms 7-12 for FCD calc.
MEM_STATIC 200 !increase static memory for a CIS job with larger basis set
$end
Example 10.0 An FED calculation to analyze excitation-energy transfer
couplings in a pair of stacked ethylenes.
$molecule
0 1
C 0.670518 0.000000 0.000000
H 1.241372 0.927754 0.000000
H 1.241372 -0.927754 0.000000
C -0.670518 0.000000 0.000000
H -1.241372 -0.927754 0.000000
H -1.241372 0.927754 0.000000
C 0.774635 0.000000 4.500000
H 1.323105 0.936763 4.500000
H 1.323105 -0.936763 4.500000
C -0.774635 0.000000 4.500000
H -1.323105 -0.936763 4.500000
H -1.323105 0.936763 4.500000
$end
$rem
EXCHANGE hf
BASIS 3-21G
CIS_N_ROOTS 20
CIS_SINGLETS true
CIS_TRIPLETS false
STS_FED true
STS_DONOR 1-6
STS_ACCEPTOR 7-12
$end
10.17.1.2 Multi-state treatments
When dealing with multiple charge or electronic excitation centers, diabatic states
can be constructed with Boys [549] or Edmiston-Ruedenberg [550] localization.
In this case, we construct diabatic states
{ | ΞI > } as linear combinations of adiabatic states
{ | ΦI > }
with a general rotation matrix U that is Nstate ×Nstate in size:
| ΞI > = |
Nstates ∑
J=1
|
| ΦJ > Uji I = 1 …Nstates |
| (10.76) |
The adiabatic states can be produced with any method, in principle, but the Boys / ER-localized
diabatization methods have been implemented thus far only for CIS or TDDFT methods in Q-Chem.
In analogy to orbital localization, Boys-localized diabatization corresponds to maximizing the
charge separation between diabatic state centers:
fBoys(U) = fBoys({ΞI}) = |
Nstates ∑
I,J = 1
|
| ⎢ ⎢
|
〈ΞI | |
→
μ
|
| ΞI 〉− 〈ΞJ | |
→
μ
|
| ΞJ 〉 | ⎢ ⎢
|
2
|
|
| (10.77) |
Here, →μ represents the dipole operator.
ER-localized diabatization prescribes maximizing self-interaction energy:
| |
|
| | (10.78) |
| |
|
|
Nstates ∑
I = 1
|
| ⌠ ⌡
|
d |
→
R
|
1
|
| ⌠ ⌡
|
d |
→
R
|
2
|
|
〈ΞI | |
^
ρ
|
( |
→
R
|
2
|
) | ΞI 〉〈ΞI | |
^
ρ
|
( |
→
R
|
1
|
) | ΞI 〉 |
|
|
| |
|
where the density operator at position →R is
|
^
ρ
|
( |
→
R
|
) = |
∑
j
|
δ( |
→
R
|
− |
→
r
|
(j)
|
) |
| (10.79) |
Here, →r (j) represents the position of the jth electron.
These models reflect different assumptions about the interaction of our quantum system with some fictitious
external electric field/potential: (i) if we assume a fictitious field that is linear in space, we arrive at Boys localization;
(ii) if we assume a fictitious potential energy that responds linearly to the charge density of our system,
we arrive at ER localization. Note that in the two-state limit, Boys localized diabatization reduces nearly exactly
to GMH [549].
As written down in Eq. (10.77), Boys localized diabatization applies only to charge transfer, not to
energy transfer. Within the context of CIS or TDDFT calculations, one can easily extend Boys localized
diabatization [553]
by separately localizing the occupied and virtual components of →μ, →μocc and →μvirt:
| |
|
| | (10.80) |
| |
|
|
Nstates ∑
I,J = 1
|
| ⎛ ⎝
| ⎢ ⎢
|
〈ΞI | |
→
μ
|
occ
|
| ΞI 〉− 〈ΞJ | |
→
μ
|
occ
|
| ΞJ 〉 | ⎢ ⎢
|
2
|
+ | ⎢ ⎢
|
〈ΞI | |
→
μ
|
virt
|
| ΞI 〉− 〈ΞJ | |
→
μ
|
virt
|
| ΞJ 〉 | ⎢ ⎢
|
2
| ⎞ ⎠
|
|
| |
|
where
and the occupied / virtual components are defined by
| |
|
|
δIJ |
∑
i
|
|
→
μ
|
ii
|
− |
∑
aij
|
tIai tJaj |
→
μ
|
ij
|
|
| + |
| |
| | (10.82) |
| |
|
< ΞI| |
→
μ
|
occ
|
|ΞJ > + < ΞI| |
→
μ
|
virt
|
|ΞJ > |
| |
|
Note that when we maximize the Boys OV function, we are simply
performing Boys-localized diabatization separately on the electron attachment and detachment densities.
Finally, for energy transfer, it can be helpful to understand the origin of the diabatic couplings.
To that end, we now provide the ability to decompose the diabatic coupling between diabatic states into
into Coulomb (J), Exchange (K) and one-electron (O) components [554]:
|
|
|
|
∑
iab
|
tPai tQbi Fab − |
∑
ija
|
tPai tQaj Fij |
| + |
| − |
| |
|
|
| |
|
| | (10.83) |
|
BOYS_CIS_NUMSTATE
Define how many states to mix with Boys localized diabatization. |
TYPE:
DEFAULT:
0 | Do not perform Boys localized diabatization. |
OPTIONS:
1 to N where N is the number of CIS states requested (CIS_N_ROOTS) |
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV
or a typical reorganization energy in solvent. |
|
| ER_CIS_NUMSTATE
Define how many states to mix with ER localized diabatization. |
TYPE:
DEFAULT:
0 | Do not perform ER localized diabatization. |
OPTIONS:
1 to N where N is the number of CIS states requested (CIS_N_ROOTS) |
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV
or a typical reorganization energy in solvent. |
|
|
|
LOC_CIS_OV_SEPARATE
Decide whether or not to localized the "occupied" and "virtual" components of the localized diabatization
function, i.e., whether to localize the electron attachments and detachments separately. |
TYPE:
DEFAULT:
FALSE | Do not separately localize electron attachments and detachments. |
OPTIONS:
RECOMMENDATION:
If one wants to use Boys localized diabatization for energy transfer (as opposed to electron transfer)
, this is a necessary option. ER is more rigorous technique, and does not require this OV feature, but will be somewhat slower. |
|
| CIS_DIABATH_DECOMPOSE
Decide whether or not to decompose the diabatic coupling into Coulomb, exchange, and one-electron terms. |
TYPE:
DEFAULT:
FALSE | Do not decompose the diabatic coupling. |
OPTIONS:
RECOMMENDATION:
These decompositions are most meaningful for electronic excitation transfer processes.
Currently, available only for CIS, not for TD-DFT diabatic states. |
|
|
|
Example 10.0 A calculation using ER localized diabatization to construct the diabatic Hamiltonian and
couplings between a square of singly-excited Helium atoms.
$molecule
0 1
he 0 -1.0 1.0
he 0 -1.0 -1.0
he 0 1.0 -1.0
he 0 1.0 1.0
$end
$rem
jobtype sp
exchange hf
cis_n_roots 4
cis_singles false
cis_triplets true
basis 6-31g**
scf_convergence 8
symmetry false
rpa false
sym_ignore true
sym_ignore true
loc_cis_ov_separate false ! NOT localizing attachments/detachments separately.
er_cis_numstate 4 ! using ER to mix 4 adiabatic states.
cis_diabatH_decompose true ! decompose diabatic couplings into
! Coulomb, exchange, and one-electron components.
$end
$localized_diabatization
On the next line, list which excited adiabatic states we want to mix.
1 2 3 4
$end
10.17.2 Diabatic-State-Based Methods
10.17.2.1 Electronic coupling in charge transfer
A charge transfer involves a change in the electron numbers in a pair of
molecular fragments. As an example, we will use the following reaction when
necessary, and a generalization to other cases is straightforward:
where an extra electron is localized to the donor (D) initially, and it becomes
localized to the acceptor (A) in the final state.
The two-state secular equation for the initial and final electronic states can
be written as
This is very close to an eigenvalue problem except for the non-orthogonality
between the initial and final states. A standard eigenvalue form for
Eq. (10.85) can be obtained by using the Löwdin transformation:
where the off-diagonal element of the effective Hamiltonian matrix represents the
electronic coupling for the reaction, and it is defined by
V = Heffif = |
Hif − Sif(Hii + Hff)/2
1 − S2if
|
|
| (10.87) |
In a general case where the initial and final states are not normalized, the
electronic coupling is written as
V = | √
|
SiiSff
|
× |
Hif − Sif(Hii/Sii + Hff/Sff)/2
SiiSff − S2if
|
|
| (10.88) |
Thus, in principle, V can be obtained when the matrix elements for the
Hamiltonian H and the overlap matrix S are calculated.
The direct coupling (DC) scheme calculates the electronic coupling values via
Eq. (10.88), and it is widely used to calculate the electron transfer
coupling [555,[556,[557,[558].
In the DC scheme, the coupling matrix element is calculated directly using
charge-localized determinants (the "diabatic states" in electron transfer
literatures). In electron transfer systems, it has been shown that such
charge-localized states can be approximated by symmetry-broken unrestricted
Hartree-Fock (UHF) solutions [555,[556,[559].
The adiabatic eigenstates are assumed to be the symmetric and antisymmetric
linear combinations of the two symmetry-broken UHF solutions in a DC
calculation.
Therefore, DC couplings can be viewed as a result of two-configuration
solutions that may recover the non-dynamical correlation.
The core of the DC method is based on the corresponding orbital transformation [560]
and a calculation for Slater's determinants in Hif and
Sif [557,[558].
10.17.2.2 Corresponding orbital transformation
Let |Ψa〉 and |Ψb〉 be two single Slater-determinant
wavefunctions for the initial and final states, and a and b be the
spin-orbital sets, respectively:
Since the two sets of spin-orbitals are not orthogonal, the overlap matrix S can be defined as:
We note that S is not Hermitian in general since the molecular orbitals of
the initial and final states are separately determined. To calculate the matrix
elements Hab and Sab, two sets of new orthogonal spin-orbitals can be
used by the corresponding orbital transformation [560]. In this
approach, each set of spin-orbitals a and b are linearly transformed,
where V and U are the left-singular and right-singular matrices,
respectively, in the singular value decomposition (SVD) of S:
The overlap matrix in the new basis is now diagonal
| ⌠ ⌡
|
|
^
b
|
†
|
|
^
a
|
= U† | ⎛ ⎝
| ⌠ ⌡
|
b†a | ⎞ ⎠
|
V = |
^
s
|
|
| (10.95) |
10.17.2.3 Generalized density matrix
The Hamiltonian for electrons in molecules are a sum of one-electron and
two-electron operators. In the following, we derive the expressions for the
one-electron operator Ω(1) and two-electron operator Ω(2),
where ω(i) and ω(i,j), for the molecular Hamiltonian, are
ω(i) = h(i) = − |
1
2
|
∇i2 + V(i) |
| (10.98) |
and
The evaluation of matrix elements can now proceed:
Sab = 〈Ψb | Ψa 〉 = |
det
| (U) |
det
| (V†) |
N ∏
i=1
|
|
^
s
|
ii
|
|
| (10.100) |
Ω(1)ab = 〈Ψb|Ω(1)|Ψa〉 = |
det
| (U) |
det
| (V†) |
N ∑
i=1
|
〈 |
^
b
|
i
|
|ω(1)| |
^
a
|
i
|
〉· |
N ∏
j ≠ i
|
|
^
s
|
jj
|
|
| (10.101) |
Ω(2)ab = 〈Ψb|Ω(2)|Ψa〉 = |
1
2
|
|
det
| (U) |
det
| (V†) |
N ∑
ij
|
〈 |
^
b
|
i
|
|
^
b
|
j
|
|ω(1,2)(1 −P12)| |
^
a
|
i
|
|
^
a
|
j
|
〉· |
N ∏
k ≠ i,j
|
|
^
s
|
kk
|
|
| (10.102) |
In an atomic orbital basis set, {χ}, we can expand the molecular spin
orbitals a and b,
|
a = χA, |
^
a
|
= χAV = χ |
^
A
|
|
| | (10.104) |
| b = χB, |
^
b
|
= χBU = χ |
^
B
|
|
| | (10.105) |
|
The one-electron terms, Eq. (10.100), can be expressed as
| |
|
|
N ∑
i
|
|
∑
λσ
|
|
^
A
|
λi
|
Tii |
^
B
|
† i σ
|
〈χσ|ω(1)|χλ〉 |
| |
| |
|
| | (10.106) |
|
where Tii = Sab/∧sii and define a
generalized density matrix, G:
Similarly, the two-electron terms [Eq. (10.102)] are
| |
|
|
1
2
|
|
∑
ij
|
|
∑
λσ
|
|
∑
μν
|
|
^
A
|
λi
|
|
^
A
|
σj
|
| ⎛ ⎜
⎝
|
1
| ⎞ ⎟
⎠
|
Tjj |
^
B
|
† iμ
|
|
^
B
|
† jν
|
〈χμχν|ω(1,2)|χλχσ〉 |
| |
| |
|
| | (10.108) |
|
where GR and GL are generalized density matrices as defined
in Eq. (10.107) except Tii in GL is replaced by 1/(2sii).
The α- and β-spin orbitals are treated explicitly. In terms of the
spatial orbitals, the one- and two-electron contributions can be reduced to
| |
|
|
∑
λσ
|
Gαλσωσλ+ |
∑
λσ
|
Gβλσωσλ |
| | (10.109) |
| |
|
|
∑
λσμν
|
GLαλμGRασν(〈μν|λσ〉− 〈μν|σλ〉)+ |
∑
λσμν
|
GLβλμGRασν〈μν|λσ〉 |
| |
| |
|
+ |
∑
λσμν
|
GLαλμGRβσν〈μν|λσ〉+ |
∑
λσμν
|
GLβλμGRβσν(〈μν|λσ〉− 〈μν|σλ〉) |
| | (10.110) |
|
The resulting one- and two-electron contributions, Eqs. (10.109)
and (10.110) can be easily computed in terms of generalized density matrices
using standard one- and two-electron integral routines in Q-Chem.
10.17.2.4 Direct coupling method for electronic coupling
It is important to obtain proper charge-localized initial and final states for
the DC scheme, and this step determines the quality of the coupling values.
Q-Chem provides two approaches to construct charge-localized states:
- The "1+1" approach
Since the system consists of donor and acceptor molecules or fragments, with a
charge being localized either donor or acceptor, it is intuitive to combine
wavefunctions of individual donor and acceptor fragments to form a
charge-localized wavefunction. We call this approach "1+1" since the zeroth
order wavefunctions are composed of the HF wavefunctions of the two fragments.
For example, for the case shown in Example (10.84), we can
use Q-Chem to calculate two HF wavefunctions: those of anionic donor and of
neutral acceptor and they jointly form the initial state. For the final state,
wavefunctions of neutral donor and anionic acceptor are used. Then the
coupling value is calculated via Eq. (10.88).
Example 10.0 To calculate the electron-transfer coupling for a pair of
stacked-ethylene with "1+1" charge-localized states
$molecule
-1 2
--
-1 2, 0 1
C 0.662489 0.000000 0.000000
H 1.227637 0.917083 0.000000
H 1.227637 -0.917083 0.000000
C -0.662489 0.000000 0.000000
H -1.227637 -0.917083 0.000000
H -1.227637 0.917083 0.000000
--
0 1, -1 2
C 0.720595 0.000000 4.5
H 1.288664 0.921368 4.5
H 1.288664 -0.921368 4.5
C -0.720595 0.000000 4.5
H -1.288664 -0.921368 4.5
H -1.288664 0.921368 4.5
$end
$rem
JOBTYPE SP
EXCHANGE HF
BASIS 6-31G(d)
SCF_PRINT_FRGM FALSE
SYM_IGNORE TRUE
SCF_GUESS FRAGMO
STS_DC TRUE
$end
In the $molecule subsection, the first line is for the charge and
multiplicity of the whole system. The following blocks are two inputs for the
two molecular fragments (donor and acceptor).
In each block the first line consists of the charge and spin multiplicity in
the initial state of the corresponding fragment, a comma, then the charge and
multiplicity in the final state.
Next lines are nuclear species and their positions of the fragment. For example, in
the above example, the first block indicates that the electron donor is a
doublet ethylene anion initially, and it becomes a singlet neutral species in
the final state. The second block is for another ethylene going from a singlet
neutral molecule to a doublet anion.
Note that the last three $rem variables in this example, SYM_IGNORE,
SCF_GUESS and STS_DC must be set to be the values as in
the example in order to perform DC calculation with "1+1" charge-localized
states.
An additional $rem variable, SCF_PRINT_FRGM is included. When it
is TRUE a detailed output for the fragment HF self-consistent field
calculation is given.
- The "relaxed" approach
In "1+1" approach, the intermolecular interaction is neglected in the initial
and final states, and so the final electronic coupling can be underestimated.
As a second approach, Q-Chem can use "1+1" wavefunction as an initial guess
to look for the charge-localized wavefunction by further HF self-consistent
field calculation. This approach would `relax' the wavefunction constructed by
"1+1" method and include the intermolecular interaction effects in the initial
and final wavefunctions. However, this method may sometimes fail, leading to
either convergence problems or a resulting HF wavefunction that cannot
represent the desired charge-localized states. This is more likely to be a
problem when calculations are performed with with diffusive basis functions, or
when the donor and acceptor molecules are very close to each other.
Example 10.0 To calculate the electron-transfer coupling for a pair of
stacked-ethylene with "relaxed" charge-localized states
$molecule
-1 2
--
-1 2, 0 1
C 0.662489 0.000000 0.000000
H 1.227637 0.917083 0.000000
H 1.227637 -0.917083 0.000000
C -0.662489 0.000000 0.000000
H -1.227637 -0.917083 0.000000
H -1.227637 0.917083 0.000000
--
0 1, -1 2
C 0.720595 0.000000 4.5
H 1.288664 0.921368 4.5
H 1.288664 -0.921368 4.5
C -0.720595 0.000000 4.5
H -1.288664 -0.921368 4.5
H -1.288664 0.921368 4.5
$end
$rem
JOBTYPE SP
EXCHANGE HF
BASIS 6-31G(d)
SCF_PRINT_FRGM FALSE
SYM_IGNORE TRUE
SCF_GUESS FRAGMO
STS_DC RELAX
$end
To perform `relaxed' DC calculation, set STS_DC to be RELAX.
10.18 Calculating the Population of Effectively Unpaired ("odd") Electrons with DFT
In a stretched hydrogen molecule the two electrons that are paired
at equilibrium forming a bond become un-paired and localized on the individual
H atoms. In singlet diradicals or doublet triradicals such a weak paring
exists even at equilibrium. At a single-determinant SCF level of the theory
the valence electrons of a singlet system like H2 remain perfectly paired, and one needs to include non-dynamical correlation to decouple the bond electron
pair, giving rise to a population of effectively-unpaired ("odd", radicalized)
electrons [561,[562,[563].
When the static correlation is strong, these electrons remain mostly
unpaired and can be described as being localized on individual atoms.
These phenomena can be properly described within wave-function formalism.
Within DFT, these effects can be described by broken-symmetry approach or
by using SF-TDDFT (see Section 6.3.1). Below we describe how
to derive this sort of information from pure DFT description of such low-spin
open-shell systems without relying on
spin-contaminated solutions.
The first-order reduced density
matrix (RDM-1) corresponding to a single-determinant wavefunction (e.g.,
SCF or Kohn-Sham DFT) is idempotent:
| ⌠ ⌡
|
γσscf(1;2)γσscf(2;1) dr2=ρσ(r1) , γσscf(1;2) = |
occ ∑
i
|
ψiσks(1)ψiσks(2) , |
| (10.111) |
where ρσ(1) is the electron density of spin σ
at position r1, and γσscf
is the spin-resolved RDM-1 of a single Slater determinant.
The cross product γσscf(1;2)γσscf(2;1)
reflects the Hartree-Fock exchange (or Kohn-Sham exact-exchange)
governed by the HF exchange hole:
γσscf(1;2)γσscf(2;1) = ρα(1)hXσσ(1,2) , | ⌠ ⌡
|
hXσσ(1,2) dr2 = 1 . |
| (10.112) |
When RDM-1 includes electron correlation, it becomes nonidempotent:
Dσ(1) ≡ ρσ(1)− | ⌠ ⌡
|
γσ(1;2)γσ(2;1) dr2 ≥ 0 . |
| (10.113) |
The function Dσ(1) measures the deviation from idempotency of the correlated
RDM-1 and yields the density of effectively-unpaired (odd) electrons of spin σ
at point r1 [561,[564].
The formation of effectively-unpaired electrons in
singlet systems is therefore exclusively a correlation based phenomenon.
Summing Dσ(1) over the spin components gives the total density of odd electrons, and
integrating the latter over space gives the mean total number of odd electrons ―Nu:
Du(1)=2 |
∑
σ
|
Dσ(1)dr1, |
-
N
|
u
|
= | ⌠ ⌡
|
Du(1)dr1 . |
| (10.114) |
The appearance of a factor of 2 in Eq. (10.114) above is required
for reasons discussed in reference [564]. In Kohn-Sham DFT, the SCF RDM-1
is always idempotent which impedes the analysis of odd electron formation at that level of the theory.
Ref. [565] has proposed a remedy to this situation.
It was noted that the correlated RDM-1 cross product entering Eq. (10.113) reflects an
effective exchange (also known as cumulant exchange [562]).
The KS exact-exchange hole is itself artificially too delocalized. However, the total exchange-correlation
interaction in a finite system with strong left-right (i.e., static)
correlation is normally fairly localized,
largely confined within a region of roughly atomic size [566]. The
effective exchange
described with the correlated RDM-1 cross product should be fairly localized as well. With this in mind,
the following form of the correlated RDM-1 cross product was proposed [565]:
γσ(1;2) γσ(2;1)=ρσ(1) |
-
h
|
eff Xσσ
|
(1,2) . |
| (10.115) |
where the function ―hXσσeff(1;2)
is a model DFT exchange hole of Becke-Roussel (BR) form used in Becke's B05 method [33].
The latter describes left-right static correlation effects in terms of certain
effective exchange-correlation hole [33]. The extra delocalization
of the HF exchange hole alone is compensated by certain physically motivated
real-space corrections to it [33]:
|
-
h
|
XCαα
|
(1,2)= |
-
h
|
eff Xαα
|
(1,2)+fc(1) |
-
h
|
eff Xββ
|
(1,2) , |
| (10.116) |
where the BR exchange hole ―hXσσeff
is used in B05 as an auxiliary function,
such that the potential from the relaxed BR hole equals
that of the exact-exchange hole. This results in relaxed
normalization of the auxiliary BR hole less than or equal to 1:
| ⌠ ⌡
|
|
-
h
|
eff Xσσ
|
(1;2)dr2 = NXσeff(1) ≤ 1 . |
| (10.117) |
The expression of the relaxed normalization NXσeff(r) is
quite complicated, but it is possible to represent it in closed analytic
form [37,[38]. The smaller the relaxed normalization
NXαeff(1), the more delocalized
the corresponding exact-exchange hole [33].
The α−α exchange hole is further deepened by a fraction of the β−β exchange hole,
fc(1) ―hXββeff(1,2), which
gives rise to left-right static correlation. The local correlation factor
fc in Eq.(10.116) governs this deepening
and hence the strength of the static correlation at each point [33]:
fc(r)= |
min
| (fα(r), fβ(r), 1) , 0 ≤ fc(r) ≤ 1 ,fα(r)= |
1−NXαeff(r)
NXβeff(r)
|
. |
| (10.118) |
Using Eqs. (10.118), (10.114), and (10.115),
the density of odd electrons becomes:
Dα(1)=ρα(1)(1−NXαeff(1)) ≡ ρα(1)fc(1) NXβeff(1) . |
| (10.119) |
The final formulas for the spin-summed odd electron density and the
total mean number of odd electrons read:
Du(1)=4 andop fc(1)[ρα(1) NXβeff(1)+ρβ(1) NXαeff(1)] , |
-
N
|
u
|
= | ⌠ ⌡
|
Du(r1) dr1 . |
| (10.120) |
Here acnd−opp=0.526 is the SCF-optimized linear coefficient
of the opposite-spin static correlation energy term of the B05 functional [33,[38] .
It is informative to decompose the total mean number of odd electrons into atomic contributions.
Partitioning in real space the mean total number of odd electrons ―Nu
as a sum of atomic contributions, we obtain the
atomic population of odd electrons (FAr) as:
Here ΩA is a subregion assigned to atom
A in the system. To define these atomic regions in a simple way, we use
the partitioning of the grid space into atomic subgroups within Becke's
grid-integration scheme [139]. Since the present
method does not require symmetry breaking, singlet states are calculated in
restricted Kohn-Sham (RKS) manner even at strongly stretched bonds. This
way one avoids the destructive effects that the spin contamination has on
FAr and on the Kohn-Sham orbitals. The calculation of FAr can be done fully
self-consistently only with the RI-B05 and RI-mB05 functionals. In these cases no
special keywords are needed, just the corresponding EXCHANGE rem line for these
functionals. Atomic population of odd electron can be estimated also with any other
functional in two steps: first obtaining a converged SCF calculation with the chosen
functional, then performing one single post-SCF iteration with RI-B05
or RI-mB05 functionals reading the guess from a preceding calculation, as shown
on the input example below:
Example 10.0 To calculate the odd-electron atomic population and
the correlated bond order in stretched H2, with B3LYP/RI-mB05,
and with fully SCF RI-mB05
$comment
Stretched H2: example of B3LYP calculation of
the atomic population of odd electrons
with post-SCF RI-BM05 extra iteration.
$end
$molecule
0 1
H 0. 0. 0.0
H 0. 0. 1.5000
$end
$rem
JOBTYPE SP
SCF_GUESS CORE
EXCHANGE B3LYP
BASIS G3LARGE
purcar 222
THRESH 14
MAX_SCF_CYCLES 80
PRINT_INPUT TRUE
SCF_FINAL_PRINT 1
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
SCF_CONVERGENCE 9
$end
@@@
$comment
Now one RI-B05 extra-iteration after B3LYP
to generate the odd-electron atomic population and the
correlated bond order.
$end
$molecule
READ
$end
$rem
JOBTYPE SP
SCF_GUESS READ
EXCHANGE BM05
purcar 22222
BASIS G3LARGE
AUX_BASIS riB05-cc-pvtz
THRESH 14
PRINT_INPUT TRUE
INCDFT FALSE
XC_GRID 000128000302
SYM_IGNORE TRUE
SYMMETRY FALSE
MAX_SCF_CYCLES 0
SCF_CONVERGENCE 9
dft_cutoffs 0
1415 1
$end
@@@
$comment
Finally, a fully SCF run RI-B05 using the previous output as a guess.
The following input lines are obligatory here:
purcar 22222
AUX_BASIS riB05-cc-pvtz
dft_cutoffs 0
1415 1
$end
$molecule
READ
$end
$rem
JOBTYPE SP
SCF_GUESS READ
EXCHANGE BM05
purcar 22222
BASIS G3LARGE
AUX_BASIS riB05-cc-pvtz
THRESH 14
PRINT_INPUT TRUE
INCDFT FALSE
IPRINT 3
XC_GRID 000128000302
SYM_IGNORE TRUE
SCF_FINAL_PRINT 1
SYMMETRY FALSE
MAX_SCF_CYCLES 80
SCF_CONVERGENCE 8
dft_cutoffs 0
1415 1
$end
Once the atomic population of odd electrons is obtained, a calculation of
the corresponding correlated bond order of Mayer's type follows in the code,
using certain exact relationships between FAr, FBr,
and the correlated bond order of Mayer type BAB. Both new properties are printed
at the end of the output, right after the multipoles section. It is useful to
compare the correlated bond order with Mayer's SCF bond order. To print the latter, use
SCF_FINAL_PRINT = 1.
10.19 Quantum Transport Properties via the Landauer Approximation
Quantum transport at the molecule level involves bridging two electrodes with a molecule or
molecular system, and calculating the properties of the resulting molecular electronic device,
including current-voltage curves, the effect of the electrodes on the molecular states, etc.
For a general introduction to the field, the following references are useful [567,[568].
The quantum transport code in Q-Chem is developed by Prof. Barry Dunietz (Kent State) and his group.
This package is invoked by the $rem variable TRANS_ENABLE.
TRANS_ENABLE
Decide whether or not to enable the molecular transport code. |
TYPE:
DEFAULT:
0 | Do not perform transport calculations. |
OPTIONS:
1 | Perform transport calculations in the Landauer approximation. |
−1 | Print matrices for subsquent calls for tranchem.exe as a stand-alone
post-processing |
| utility, or for generating bulk model files. |
RECOMMENDATION:
|
Output is provided in the Q-Chem output file and in the following additional files:
- transmission.txt (transmission function in the requested energy window)
- TDOS.txt
- current.txt (I-V plot if set)
- FAmat.dat (Hamiltonian matrix for follow up calculations and analysis)
- Smat.dat (Hamiltonian matrix for follow up calculations and analysis)
T-Chem requires two parameter files:
- Trans-model.para (Specifies the cluster model of the molecular device)
- Trans-method.para (Specifies the method(s) used in the transport calculation)
All the numbers must be provided. The details of file formats are given in
Table 10.3 and Table 10.4.
Input example | Explanation | |
142 Totorb | total number of atomic orbitals in the cluster model of the molecular junction |
22 lob | number of AO's representing the left electrode (the first lob functions of input) |
22 rob | number of AO's representing the right electrode (the last rob functions of input) |
22 lg | size of the repeating unit of the left electrode |
22 rg | size of the repeating unit of the right electrode |
22 ql | must be set to the lg value (later used in NEGF optimizations) |
22 qr | must be set to the rg value (later used in NEGF optimizations) |
Table 10.2: Example of a Trans-model.para file, and the meaning of each input line. All lines must be present
Key points to note about the Trans-model.para file, as documented in Table 10.3:
- WARNING: Unphysical setting of the regions is not caught by the program and might (will!) produce unphysical transmission functions. For example, the cluster model can be accidently partitioned by mistake within the orbital space of an atom. In fact, of course, It should always be partitioned between atomic layers.
- The ordering of the atoms in the $molecule section of the Q-Chem input file IS important and is assumed to be as follows: Repeating units (left) - Molecular Junction - Repeating units (right).
- The atoms are provided by sets of left electrode repeating units first, in order of most distant from the surface
layer. Then comes the the molecular region, followed by the right electrode region which starts with the surface layer to end with the most distant layer. The atom order within each layer (each repeating unit) must be consistent.
Input example | Keywords included on each line | |
1 | enable |
0 | AB |
-6.5 -8.5 -4.5 300 | efermi-emin-emax-npoints |
0 0 0.01 0.01 0.07 | method-htype-device-bulkr_smear-green_const |
0 1 1 | readinHS-tot-start |
4.0 4 100 1.0 | vmax-numres-numlinear-lpart |
1 2 100 | printDOS-printIV-ipoints |
298.15 | IVtemp |
Table 10.3: Example of a Trans-method.para file, and the keywords that correspond to each input line. The supported values of each keyword is listed below. All lines must be present.
The allowed values of the various keywords that comprise each line of the Trans-method.para file follow.
- enable: Sets the type of calculation. Allowed values are:
- 1: Landauer level
- 3: SCF GF at V=0 (not yet enabled)
- 4: Full NEGF (not yet enabled)
- AB: Determines spins. Only closed shell singlet (0) allowed at present
- emin,emax,npoints: Energy window for which transmission is calculated (eV),
and number of points
- method: Scheme to calculate the electrode GFs:
- 0: A wide band limit (WBL) with a constant parameter (greens_const)
- 1: WBL following the procedure proposed by Ke-Baranger-Yang
- 2: WBL following the procedure proposed by Lopez-Sancho (decimation). Recommended
- 3: Full TB following the procedure proposed by Ke-Baranger-Yang
- 4: Full TB following the procedure proposed by Lopez-Sancho (decimation). Recommended
- Comment: At least a single unit of the repeating layer of the electrodes is required to be included in
the junction region. Another layer has to be included if readinhs==0 and a total of two additional layers are needed if method ≠ 0.
- htype: determines the electronic coupling terms used for the self energy calculations:
- 0: all coupling integrals between the junction and electrode are set following the cluster model (no screening imposed)
- 1: only coupling between neighboring repeating units of the electrode model is allowed
- 2: set the coupling terms by the electrode models that are read in (readinhs=1) or by the sufficiently large cluster model, where lob (rob) ≥ 2 lg (rg)
- device-smear: Imaginary smearing (eV) added to the real hamiltomnian for device GF evaluation.
- bulk-smear: Imaginary smearing (eV) added to the real hamiltomnian for electrodes GF evaluation.
- readinhs:
- 0: no electrode hamiltonians are available
- 1: precalculated electrode hamiltonians will be used. Expected files are FAmat2l.dat, FBmat2l.dat, FAmat2r.dat, FBmat2r.dat, Smat2l.dat, Smat2r.dat. totorb2 is the total number of basis functions in the electrode model (same size is assumed for both electrodes), and,
start is the first basis function from which the TB integral are to be extracted from.
- vmax, numres, numlinear, lpart
- vmax = voltage should be equal to emax-emin if IV is set
- other parameters here define the integration path for the NEGF algorithm (not used in this version, place holders must be included)
- printDOS
- 0: no total dos printing
- 1: a TDOS (of the junction region) will be printed as TDOS.txt
- printIV (and ipoints, IVtemp)
- 0: no I-V information is calculated
- 1: I(V=Vmax) is printed in current.txt
- 2: I-V for bias from 0 to Vmax with a grid of ipoints at IVtemp is evaluated and printed in current.txt
As an example, the sample Q-Chem input that works with the files documented in Table 10.3 and Table 10.4 is given below.
Example 10.0 Quantum transport calculation applied to C6 between two gold electrodes.
$molecule
0 1
Au -0.2 0 0
Au 2.5 0 0
C 4.8 0 0
C 6.5 0 0
C 8.2 0 0
C 9.9 0 0
C 11.6 0 0
C 13.3 0 0
Au 15.6 0 0
Au 18.3 0 0
$end
$rem
jobtype SP
exchange B3LYP
correlation none
BASIS lanl2dz
ECP lanl2dz
GEOM_OPT_MAXCYC 200
INCDFT FALSE
mem_static 8000
max_scf_cycles 400
MEM_TOTAL 32000
MOLDEN_FORMAT TRUE
scf_convergence 10
scf_algorithm diis
trans_enable 1
$end
Chapter 11 Effective Fragment Potential Method
The effective fragment potential (EFP) method is a computationally inexpensive way of modeling
intermolecular interactions in non-covalently bound systems. The EFP approach can be viewed as
a QM/MM scheme with no empirical parameters. Originally, EFP was developed by
Gordon's group [569,[570],
and was implemented in GAMESS [571].
The review of EFP theory and applications
as well as the complete details of our implementation can be found in Ref. .
11.1 Theoretical Background
The total energy of the system consists of the interaction energy of the effective fragments (Eef−ef)
and the energy of the ab initio (i.e., QM) region in the field of the fragments. The former includes
electrostatics, polarization, dispersion and exchange-repulsion contributions (the charge-transfer term,
which is important for description of the ionic and highly polar species, is omitted in the current
implementation):
Eef−ef = Eelec + Epol + Edisp + Eex−rep. |
| (11.1) |
The QM-EF interactions are computed as follows. The electrostatics and polarization parts of
the EFP potential contribute to the quantum Hamiltonian via one-electron terms,
H′pq = Hpq + 〈p | |
^
V
|
elec
|
+ |
^
V
|
pol
|
| q 〉 |
| (11.2) |
whereas
dispersion and exchange-repulsion QM-EF interactions are treated as additive corrections to the total
energy, i.e., similarly to the fragment-fragment interactions.
The electrostatic component of the EFP energy accounts for Coulomb interactions. In molecular systems with
hydrogen bonds or polar molecules, this is the leading contribution to the total inter-molecular
interaction energy [573]. An accurate representation of the electrostatic potential is achieved by using multipole expansion (obtained from the
Stone's distributed multipole analysis) around atomic centers and bond midpoints (i.e., the points with
high electronic density) and truncating this expansion at octupoles [574,[546,[569,[570].
The fragment-fragment electrostatic interactions consist of charge-charge, charge-dipole,
charge-quadrupole, charge-octupole, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole terms, as
well as terms describing interactions of electronic multipoles with the nuclei
and nuclear repulsion energy.
Electrostatic interaction between an effective fragment and the QM part is described by perturbation
∧V of the ab initio Hamiltonian:
The perturbation enters the one-electron part of the Hamiltonian as a
sum of contributions from the expansion points of the effective fragments.
Contribution from each expansion point consists of four terms
originating from the electrostatic potential of the corresponding multipole (charge, dipole, quadrupole, and octupole).
The multipole representation of the electrostatic density of a fragment
breaks down when the fragments are too close. The multipole interactions become too repulsive due
to significant overlap of the electronic densities and the charge-penetration effect. The magnitude of the
charge-penetration effect is usually around 15% of the total electrostatic energy in polar systems,
however, it can be as large as 200% in systems with weak electrostatic
interactions [575]. To account for the charge-penetration effect, the simple
exponential damping of the charge-charge term is used [576,[575].
The charge-charge screened energy between the expansion
points k and l is given by the following expression, where αk and αl are the damping
parameters associated with the corresponding expansion points:
| |
|
[ 1 − (1 + αk Rkl/2) e−αk Rkl ] qk ql/Rkl , if αk=αl |
| | (11.4) |
| |
|
| ⎛ ⎝
|
1 − |
αl2
αl2 − αk2
|
e−αk Rkl − |
αk2
αk2 − αl2
|
e−αl Rkl | ⎞ ⎠
|
qk ql / Rkl , if αk ≠ αl |
| |
|
Damping parameters are included in the potential of each fragment, but
ab initio / EFP electrostatic interactions are currently calculated without damping corrections.
Polarization
accounts for the intramolecular charge redistribution in response to external electric field.
It is the major component of many-body interactions responsible for cooperative molecular behavior.
EFP employs distributed polarizabilities placed at the centers of valence LMOs.
Unlike the isotropic total molecular polarizability tensor, the distributed polarizability
tensors are anisotropic.
The polarization energy of a system consisting of an ab initio and effective fragment regions is
computed as [569]
Epol = − |
1
2
|
|
∑
k
|
μk (Fmult,k +Fai,nuc,k) + |
1
2
|
|
∑
k
|
|
-
μ
|
k
|
Fai,elec,k |
| (11.5) |
where μk and ―μk are the induced dipole and the conjugated induced dipole at the distributed
point k; Fmult,k is the external field due to static multipoles
and nuclei of other fragments, and Fai,elec,k and Fai,nuc,k are the fields due to the electronic density and nuclei of the ab initio part, respectively.
The induced dipoles at each polarizability point k are computed as
where αk is the distributed polarizability tensor at k.
The total field Ftotal,k comprises from the static field and the field due to other induced
dipoles, Findk, as well as the field due to nuclei and electronic density of the ab initio region:
Fai,total,k = Fmult,k + Find,k +Fai,elec,k + Fai,nuc,k |
| (11.7) |
As follows from the above equation, the induced dipoles on a particular fragment depend on the
values of the induced dipoles of all other fragments.
Moreover, the induced dipoles on the effective fragments depend on the ab initio electronic density,
which, in turn, is affected by the field created by these induced dipoles through a one electron
contribution to the Hamiltonian:
|
^
V
|
pol
|
= − |
1
2
|
|
∑
k
|
|
x,y,z ∑
a
|
|
R3
|
|
| (11.8) |
where R and a are the
distance and its Cartesian components between an electron and the polarizability point k.
In sum, the total polarization energy is computed self-consistently using a two level iterative procedure. The objectives of the higher and lower levels are to converge
the wavefunction and induced dipoles for a given fixed wavefunction, respectively. In the absence of
the ab initio region, the induced dipoles of the EF system are iterated until
self-consistent with each other.
Self-consistent treatment of polarization accounts for many-body interaction effects. The current
implementation does not include damping of the polarization energy.
Dispersion provides a leading contribution to van der Waals and
π-stacking interactions. The dispersion interaction is expressed as the inverse R dependence:
where coefficients C6 are derived
from the frequency-dependent distributed polarizabilities with expansion points
located at the LMO centroids, i.e., at the same centers as the polarization expansion points. The higher-order dispersion terms (induced dipole-induced quadrupole, induced quadrupole / induced quadrupole, etc..) are approximated as 1/3 of the C6 term [577].
For small distances between effective fragments dispersion interactions are corrected for charge
penetration and electronic density overlap effect with the Tang-Toennies damping
formula [578] with parameter b=1.5:
C6kl → | ⎛ ⎝
|
1 − e−bR |
∞ ∑
k=0
|
|
(bR)k
k!
| ⎞ ⎠
|
C6kl |
| (11.10) |
Ab initio / EFP dispersion interactions are currently treated as EFP-EFP interactions, i.e., the QM
region should be represented as a fragment, and dispersion QM-EF interaction is evaluated using
Eq. (11.9).
Exchange-repulsion
originates from the Pauli exclusion principle, which states that the wavefunction of
two identical fermions must be anti-symmetric. In traditional classical force fields,
exchange-repulsion is introduced as a positive (repulsive) term, e.g., R−12
in the Lennard-Jones potential. In contrast, EFP uses a
wavefunction-based formalism to account for this inherently quantum effect.
Exchange-repulsion is the only
non-classical component of EFP and the only one that is repulsive.
The exchange-repulsion interaction is derived as an expansion in the intermolecular overlap, truncated at
the quadratic term [579,[580], which requires that each effective
fragment carries a basis set that is used to calculate overlap and kinetic one-electron integrals for each
interacting pair of fragments. The exchange-repulsion contribution from each pair of localized orbitals i and j belonging to fragments A and B, respectively, is:
| |
|
| | (11.11) |
| |
|
− 2 Sij | ⎛ ⎝
|
∑
k ∈ A
|
FAik Skj + |
∑
l ∈ B
|
FBjl Sil − 2 Tij | ⎞ ⎠
|
|
| |
| |
|
+ 2 S2ij | ⎛ ⎝
|
∑
J ∈ B
|
−ZJ R−1iJ + 2 |
∑
l ∈ B
|
R−1il + |
∑
I ∈ A
|
−ZI R−1Ij + 2 |
∑
k ∈ A
|
R−1kj −R−1ij | ⎞ ⎠
|
|
| |
|
where i, j, k and l are the LMOs, I and J are the nuclei, S and T are
the intermolecular overlap and kinetic energy integrals, and F is the Fock matrix element.
The expression for the Eexchij involves overlap and kinetic energy integrals between pairs of
localized orbitals. In addition, since Eq. is derived within an infinite basis set
approximation, it requires a reasonably large basis set to be accurate [6-31+G* is considered to be the
smallest acceptable basis set, 6-311++G(3df,2p) is recommended]. These factors make exchange-repulsion the
most computationally expensive part of the EFP energy calculations of moderately sized systems.
Large systems require additional considerations. Since total exchange-repulsion energy is given by
a sum of terms in Eq. (11.11) over all the fragment pairs, its computational cost formally
scales as O(N2) with the number of effective
fragments N. However, exchange-repulsion is a short-range interaction; the overlap and kinetic energy
integrals decay exponentially with the inter-fragment distance. Therefore, by employing a distance-based
screening, the number of overlap and kinetic energy integrals scales as O(N). Consequently, for large
systems exchange-repulsion may become less computationally expensive than the long-range components of EFP
(such as Coulomb interactions).
The ab initio / EFP exchange-repulsion energy is currently calculated at the
EFP / EFP level, by
representing the quantum part as an EFP and using Eq. (11.11). In this way, the quantum
Hamiltonian is not affected by the fragments' exchange potential.
11.2 Excited-State Calculations with EFP
Interface of EFP with EOM-XX-CCSD, CIS, CIS(D), and TDDFT has been
developed [581,[582].
In the EOM-CCSD / EFP calculations, the reference-state CCSD equations
for the T cluster amplitudes
are solved with the HF Hamiltonian modified by the electrostatic
and polarization contributions due to
the effective fragments, Eq. (11.2).
In the coupled-cluster calculation, the induced dipoles of the fragments are frozen at their HF values.
The transformed Hamiltonian ―H effectively includes Coulomb and polarization
contributions from the EFP part. As ―H is diagonalized in an EOM calculation, the induced dipoles of the
effective fragments are frozen at their reference state value, i.e.,
the EOM equations are solved with a constant response of the EFP environment.
To account for solvent response to electron rearrangement in the
EOM target states (i.e., excitation or ionization), a perturbative non-iterative
correction is computed for each EOM root as follows.
The one-electron density of the target EOM state (excited or ionized)
is calculated and used to re-polarize the environment, i.e., to recalculate
the induced dipoles of the EFP part in the field of an EOM state.
These dipoles are used to compute the polarization energy corresponding to
this state.
The total energy of the excited state with inclusion of the perturbative response of the EFP
polarization is:
EEOM/EFPIP = EEOM + ∆Epol |
| (11.12) |
where EEOM is the energy found from EOM-CCSD procedure and
∆Epol has the following form:
| |
|
|
1
2
|
|
∑
k
|
|
x,y,z ∑
a
|
| ⎡ ⎣
|
−(μex,ak − μgr,ak) (Famult,k+Fanuc,k) |
| | (11.13) |
| |
|
+ ( |
~
μ
|
k ex,a
|
Fex,aai,k − |
~
μ
|
k gr,a
|
Fgr,aai,k) − (μex,ak − μgr,ak + |
~
μ
|
k ex,a
|
− |
~
μ
|
k gr,a
|
) Fex,aai,k | ⎤ ⎦
|
|
| |
|
where Fgrai and Fexai are the fields due to the reference (HF) state and
excited-state electronic densities, respectively.
μgrk and ~μgrk are the induced dipole and conjugated induced dipole at the
distributed polarizability point k consistent with the reference-state density,
while μexk and ~μexk are the induced dipoles corresponding to the excited state density.
The first two terms in Eq. (11.13) provide a difference of the polarization
energy of the QM / EFP system in the ionized and ground electronic states;
the last term is the leading correction to the interaction of the ground-state-optimized
induced dipoles with the wavefunction of the excited state.
The EOM states have both direct and indirect
polarization contributions. The indirect term comes from the orbital relaxation of the solute in the field due to induced dipoles of the solvent.
The direct term given by Eq. (11.13) is the response
of the polarizable
environment to the change in solute's electronic density upon excitation.
Note that the direct polarization contribution can be very large (tenths of eV) in EOM-IP / EFP
since the electronic
densities of the neutral and the ionized species are very different.
An important advantage of the perturbative EOM / EFP scheme is that
it does not compromise multi-state nature of EOM and that the
electronic wavefunctions of the target states remain orthogonal to each other
since they are obtained with the same (reference-state) field of
the polarizable environment. For example, transition properties between these states
can be calculated.
EOM-CCSD / EFP scheme works with any type of the EOM excitation operator Rk
currently supported in Q-Chem, i.e., spin-flipping (SF), excitation energies (EE), ionization
potential (IP), electron affinity (EA) (see Section 6.6.7 for details).
Implementation of CIS / EFP, CIS(D) / EFP, and TDDFT / EFP methods is similar to the implementation of EOM / EFP.
Polarization correction as in Eq. 11.13 is calculated and added to the CIS or TDDFT excitation
energies.
11.3 Extension to Macromolecules: Fragmented EFP Scheme
Macromolecules such as proteins or DNA present a large number of
electronic structure problems (photochemistry, redox chemistry,
reactivity) that can be described within QM/EFP framework.
EFP has been extended to deal with such complex systems via
the so-called fragmented EFP scheme (fEFP). The current
Q-Chem implementation allows one to (i) compute
interaction energy between a ligand and a
macromolecule (both represented by EFP) and (ii)
to calculate the excitation energies,
ionization potentials, electronic affinities of a
QM moiety interacting with a fEFP macromolecule
using QM/EFP scheme (see Section 11.2).
In the present implementation, the ligand cannot be
covalently bound to the macromolecule.
There are multiple ways to cut a large molecule into units depending
on the position of the cut between two covalently bound residues.
An obvious way to cut a protein is to cut through peptidic bonds such that
each fragment represents one amino acid. Alternatively, one can
bonds between two atoms of the same nature (carbonyl and carbon-α or carbon-α
and the first carbon of the side chain). The use can choose the most appropriate way to
cut.
Consider a protein (P) consisting of N amino acids,
A1A2…AN, and is split into
N fragments (Ai). The fragments can be saturated by either
Hydrogen Link Atom[583] (HLA) or by mono-valent groups
of atoms from the neighboring fragment(s) called hereafter Cap Link Atom (CLA).
If fragments are capped using the HLA scheme, the hydrogen is
located along the peptidic bond axis and at the distance
corresponding to the equilibrium bond length of a CH bond:
P = A1H + |
N−1 ∑
i=2
|
HAiH + HAN |
| (11.14) |
In the CLA scheme, the cap has exactly the same geometry as the
respective neighboring group.
If the cuts are through peptidic bonds (one fragment is one amino acid),
the caps (Ci) are either an aldehyde to saturate the -N(H) end of the fragment,
or an amine to saturate the -C(=O) extremity of the fragment.
P = A1C2 + |
N−1 ∑
i=2
|
Ci−1AiCi+1 + CN−1AN |
| (11.15) |
Q-Chem provides a two-step script, .pl@, located in $QC/bin
which takes a PDB file and breaks it into capped fragments in the GAMESS
format, such that the EFP parameters for these capped fragments
can be generated, as explained in Section 11.9.
As the EFP parameters are generated for each capped fragment, the
neighboring fragments have duplicated parameter points (overlapping areas)
in both the HLA and CLA schemes due to the overlapping caps.
Since multipole expansion points and polarizability expansion
points are computed on each capped residue by the standard
procedure, the multipole (and damping terms) and polarizabilities
need to be removed (C∅) from the overlapping areas.
Equations (11.14) and (11.15) become:
P = A1C∅ + |
N−1 ∑
i=2
|
C∅AiC∅ +C∅AN |
| (11.16) |
The details concerning this removing procedure are presented in Section 11.9.
Once these duplicate parameters are removed from the EFP
parameters of the capped fragments, the EFP-EFP and QM-EFP calculations
can be conducted as usual.
Currently, fEFP includes electrostatic and polarization contributions,
which appear in EFP(ligand) / fEFP(macromolecule) and
in QM / fEFP calculations (note that the QM part is not covalently
bound to the macromolecule).
Consequently, the total interaction energy (Etot) between a ligand
(L) and a protein (P) divided into fragments is:
Etot (P−L) = Eelec (P−L) + Epol (P−L) |
| (11.17) |
The electrostatics is an additive term; its contribution to fragment-fragment and
ligand-fragment interaction is computed as follows:
Eelec (P−L) = |
N ∑
i
|
Eelec (C∅Ai=1C∅−L ) |
| (11.18) |
The polarization contribution in an EFP system (no QM) is:
Epol (P−L) = −
|
1
2
|
|
∑ k ∈ P,L
|
μk Fmult,k +
|
1
2
|
|
∑
k ∈ P
|
μk Fmult,k
|
|
(11.19)
|
The first term is the polarization energy obtained upon convergence of the
induced dipoles of the ligand (μkefp (L)) and all fragments
(μkfefp (Ai)).
The system is thus fully polarized, all fragments (Ai or L) are
polarizing each other until self-consistency.
The second term of Eq. (11.19) is the polarization
of the protein by itself; this value has to be subtracted once the
induced dipoles (Eq. 11.20) converged.
The LA scheme is available to perform QM / fEFP job. In this situation the fEFP
has to include a macromolecule (covalent bond between fragments). This scheme
is not able yet to perform QM / fEFP / EFP in which a macromolecule and solvent
molecules would be described at the EFP level of theory.
In addition to the HLA and CLA schemes, Q-Chem also features
Molecular Fragmentation with Conjugated Caps approach (MFCC)
which avoids the issue of overlapping of saturated fragments and was developed in 2003 by
Zhang[584,[585].
MFCC procedure consists of a summation over the interactions between a ligand and
capped residues (CLA scheme) and a subtraction over the interactions of
merged caps (Ci+1Ci−1), the so-called "concaps", with the ligand.
N−1 concap fragments are actually used to subtract the overlapping effect.
P = A1C2 + |
N−1 ∑
i=2
|
Ci−1AiCi+1 + CN−1AN − |
N−1 ∑
i
|
Ci+1Ci−1 |
| (11.20) |
In this scheme the contributions due to overlapping caps simply cancel out
and the EFP parameters do not need any modifications, in contrast to the
HLA or CLA procedures. However, the number of parameters that need to be generated
is larger (N capped fragments + N−1 concaps).
The MFCC electrostatic interaction energy is given as
the sum of the interaction energy between each capped fragment
(Ci−1 Ai Ci+1) and
the ligand minus the interaction energy
between each concap (Ci−1 Ci+1) and the ligand:
Eelec (P−L) = |
N ∑
i
|
Eelec (Ci−1 Ai Ci+1 − L ) − |
N−1 ∑
i
|
Eelec ( Ci−1Ci+1 − L ) |
| (11.21) |
The main advantage of MFCC is that the multipole expansion
obtained on each capped residue or concap are kept during the
Eelec(P−L) calculation.
In the present implementation, there are no polarization contributions.
The MFCC scheme is not yet available for QM / fEFP.
11.4 EFP Fragment Library
The effective fragments are rigid and their potentials are generated from a set of
ab initio calculations on each unique isolated fragment.
The EF potential includes: (i) multipoles (produced by the Stone's Distributed Multipolar
Analysis) for Coulomb and polarization terms; (ii) static polarizability tensors centered at localized
molecular orbital (LMO) centroids (obtained from coupled-perturbed Hartree-Fock calculations), which are
used for calculations of polarization; (iii) dynamic polarizability tensors centered on the LMOs that are
generated by time-dependent HF calculations and used for calculations of dispersion; and (iv) the Fock
matrix, basis set, and localized orbitals needed for the exchange-repulsion term. Additionally,
the EF potential contains coordinates of atoms, coordinates of the points of
multipolar expansion (typically, atoms and bond mid-points), coordinates of the LMO centroids, electrostatic screening parameters, and atomic labels of the EF atoms.
Q-Chem provides a library of standard fragments with precomputed effective fragment
potentials. Currently, the library includes common organic solvents and nucleobases; see
Table 11.1.
Table 11.1: Standard fragments available in Q-Chem
|
acetone | ACETONE_L |
carbon tetrachloride | CCL4_L |
dichloromethane | DCM_L |
methane | METHANE_L |
methanol | METHANOL_L |
ammonia | AMMONIA_L |
acetonitrile | ACETONITRILE_L |
water | WATER_L |
dimethyl sulfoxide | DMSO_L |
benzene | BENZENE_L |
phenol | PHENOL_L |
toluene | TOLUENE_L |
thymine | THYMINE_L |
adenine | ADENINE_L |
cytosine C1 | CYTOSINE_C1_L |
cytosine C2a | CYTOSINE_C2A_L |
cytosine C2b | CYTOSINE_C2B_L |
cytosine C3a | CYTOSINE_C3A_L |
cytosine C3b | CYTOSINE_C3B_L |
guanine enol N7 | GUANINE_EN7_L |
guanine enol N9 | GUANINE_EN9_L |
guanine enol N9RN7 | GUANINE_EN9RN7_L |
guanine keton N7 | GUANINE_KN7_L |
guanine keton N9 | GUANINE_KN9_L | |
Note:
The fragments from Q-Chem fragment library have _L added to their names to
distinguish them from user-defined fragments. |
The parameters for the standard fragments were computed as follows.
The geometries of the solvent molecules were optimized by MP2 / cc-pVTZ; geometries of nucleobases
were optimized with RI-MP2 / cc-pVTZ.
The EFP parameters were obtained
in GAMESS. To generate the electrostatic multipoles and electrostatic
screening parameters, analytic DMA procedure was used, with 6-31+G* basis for non-aromatic
compounds and 6-31G* for aromatic compounds and nucleobases. The rest of the potential, i.e.,
static and dynamic polarizability tensors, wavefunction, Fock matrix, etc., were
obtained with 6-311++G(3df,2p) basis set.
11.5 EFP Job Control
The current version supports single point calculations in systems consisting of (i) ab initio
and EFP regions (QM / MM); or (ii) EFP region only (pure MM).
The ab initio region can be described by conventional quantum methods like HF,
DFT, or correlated methods including methods for the excited states
[CIS, CIS(D), TDDFT, EOM-XX-CCSD methods]. Theoretical details on the interface of EFP with EOM-CCSD
and CIS(D) can be found in Refs. .
Note:
Currently, only correlated methods handled by CCMAN and CCMAN2 are interfaced with EFP. |
Note:
EFP provides both implicit (through orbital response) and explicit
(as instantaneous response of the polarizable EFP fragments) corrections
to the electronic excited states. EFP-modified excitation energies are printed in the property
section of the output. |
Electrostatic, polarization, exchange repulsion, and dispersion contributions are calculated between
EFs; only electrostatic and polarization terms are evaluated between ab initio and EF regions.
To obtain dispersion and exchange-repulsion ab initio-EF energies, a separate calculation
with the ab initio region being represented by EFP is required.
The ab initio region is specified by regular Q-Chem input using $molecule and $rem sections. In calculations with no QM part, the $molecule section should contain a dummy atom (for example, helium).
Positions of EFs are specified in the $efp_fragments section.
Each line in this section contains the information on an individual fragment: fragment's name and
its position. The position of the fragment is specified by center-of-mass coordinates (x, y, z)
and the Euler rotation angles (α, β, γ) relative to the fragment frame,
i.e., the coordinates of the standard fragment provided in the fragment library.
For user-defined fragments, a $efp_params section should be present. This section contains EFP
parameters for each unique effective fragment.
EFP
Specifies that EFP calculation is requested |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The keyword should be present if excited state calculation is requested |
|
| EFP_FRAGMENTS_ONLY
Specifies whether there is a QM part |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Only MM part is present: all fragments are treated by EFP |
FALSE | QM part is present: do QM/MM EFP calculation |
RECOMMENDATION:
|
|
|
EFP_INPUT
Specifies the EFP fragment input format |
TYPE:
DEFAULT:
FALSE | Old format with dummy atom in $molecule section |
OPTIONS:
TRUE | New format without dummy atom in $molecule section |
FALSE | Old format |
RECOMMENDATION:
|
| FEFP_EFP
Specifies that fEFP_EFP calculation is requested to compute the total
interaction energies between a ligand (the last fragment in the $efp_fragments
section) and the protein (represented by fEFP) |
TYPE:
DEFAULT:
OPTIONS:
OFF | disables fEFP |
LA | enables fEFP with the Link Atom (HLA or CLA) scheme (only
electrostatics and polarization) |
MFCC | enables fEFP with MFCC (only
electrostatics) |
RECOMMENDATION:
The keyword should be invoked if EFP / fEFP is requested (interaction
energy calculations). This keyword has to be employed with EFP_FRAGMENT_ONLY =
TRUE.
To switch on / off electrostatics or polarzation interactions, the usual EFP
controls are employed. |
|
|
|
FEFP_QM
Specifies that fEFP_QM calculation is requested to perform a
QM/fEFPcompute computation. The fEFP part is a fractionated macromolecule. |
TYPE:
DEFAULT:
OPTIONS:
OFF | disables fEFP_QM and performs a QM/EFP calculation |
LA | enables fEFP_QM with the Link Atom scheme |
RECOMMENDATION:
|
The keyword should be invoked if QM / fEFP is requested. This keyword has to be
employed with efp_fragment_only false. Only electrostatics is available.
11.6 Examples
Example 11.0 Old format : EFP energy computation of benzene dimer (both fragments are treated by EFP).
All the EFP parameters are read from the EFP library ($QCAUX/efp).
Note that a dummy atom (He) is placed in a $molecule section for a pure MM/EFP calculation.
$molecule
0 1
He 5.0 5.0 5.0
$end
$rem
exchange hf
basis 6-31G(d)
jobtype sp
purecart 2222
efp_fragments_only true
EFP 1
$end
$efp_fragments
BENZENE_L -0.30448173 -2.24210052 -0.29383131 -0.642499 1.534222 -0.568147
BENZENE_L -0.60075437 1.36443336 0.78647823 3.137879 1.557344 -2.568550
$end
Example 11.0 New format : EFP energy computation of benzene dimer (both fragments are treated by EFP).
All the EFP parameters are read from the EFP library ($QCAUX/efp).
$molecule
0 1
--
0 1
-0.30448173 -2.24210052 -0.29383131 -0.642499 1.534222 -0.568147
--
0 1
-0.60075437 1.36443336 0.78647823 3.137879 1.557344 -2.568550
$end
$rem
exchange hf
jobtype sp
purecart 2222
efp_fragments_only true
efp_input true
$end
$efp_fragments
BENZENE_L
BENZENE_L
$end
Example 11.0 New format : EFP energy and gradient computation of benzene dimer (both fragments are treated by EFP).
All the EFP parameters are read from the EFP library ($QCAUX/efp).
$molecule
0 1
--
0 1
-0.30448173 -2.24210052 -0.29383131 -0.642499 1.534222 -0.568147
--
0 1
-0.60075437 1.36443336 0.78647823 3.137879 1.557344 -2.568550
$end
$rem
exchange hf
jobtype force
purecart 2222
efp_fragments_only true
efp_input true
$end
$efp_fragments
BENZENE_L
BENZENE_L
$end
Example 11.0 New format : EFP optimization of water dimer (both fragments are treated by EFP).
All the EFP parameters are read from the EFP library ($QCAUX/efp). Note that
for EFP optimization only the new format (without dummy atom) is supported.
$molecule
0 1
--
0 1
-2.57701480 1.41334560 3.40415522 0.000000 1.000000 0.000000
--
0 1
-6.07701480 1.41334560 3.40415522 0.000000 0.000000 0.000000
$end
$rem
exchange hf
jobtype opt
purecart 2222
efp_fragments_only true
efp_input true
$end
$efp_fragments
WATER_L
WATER_L
$end
Example 11.0 QM/MM computation of one water molecule in the QM part and
one water + two ammonia molecules in the MM part.
The EFP parameters will be taken from the EFP library ($QCAUX/efp).
Note that for QM/MM calculation this is the only format that is supported.
$molecule
0 1
O1 0.47586 0.56326 0.53843
H2 0.77272 1.00240 1.33762
H3 0.04955 -0.23147 0.86452
$end
$rem
exchange hf
basis 6-31G(d)
jobtype sp
purecart 2222
EFP 1
$end
$efp_fragments
WATER_L -2.12417561 1.22597097 -0.95332054 -2.902133 1.734999 -1.953647
AMMONIA_L 1.04358758 1.90477190 2.88279926 -1.105309 2.033306 -1.488582
AMMONIA_L -4.16795656 -0.98129149 -1.27785935 2.526442 1.658262 -2.742084
$end
Example 11.0 Excited states of formaldehyde with 6 EFP water molecules by
CIS(D).
$molecule
0 1
C1 1.0632450881806 2.0267971791743 0.4338879750526
O2 1.1154451117032 1.0798728186948 1.1542424552747
H3 1.0944666250874 3.0394904220684 0.8360468907200
H4 0.9836601903170 1.9241779934791 -0.6452234478151
$end
$rem
basis 6-31+G*
exchange hf
efp_fragments_only false
purecart 2222
scf_convergence 8
correlation cis(d)
eom_ee_singlets 2
eom_ee_triplets 2
EFP = 1
$end
$efp_fragments
WATER_L 1.45117729 -1.31271387 -0.39790305 -1.075756 2.378141 1.029199
WATER_L 1.38370965 0.22282733 -2.74327999 2.787663 1.446660 0.168420
WATER_L 4.35992117 -1.31285676 0.15919381 -1.674869 2.547933 -2.254831
WATER_L 4.06184149 2.79536141 0.05055916 -1.444143 0.750463 -2.291224
WATER_L 4.09898096 0.83731430 -1.93049301 2.518412 1.592607 -2.199818
WATER_L 3.96160175 0.71581837 2.05653146 0.825946 1.414384 0.966187
$end
Example 11.0 EOM-IP-CCSD/EFP calculation of CN radical hydrated by 6 waters.
EOM_FAKE_IPEA keyword can be either or .
QM_EFP and EFP_EFP exchange-repulsion should be turned off.
$molecule
-1 1
C 1.0041224092 2.5040921109 -0.3254633433
N 0.8162211575 2.3197739512 0.7806258675
$end
$rem
basis 6-31+G*
exchange hf
efp_fragments_only false
purecart 2222
scf_convergence 8
correlation ccsd
eom_ip_states = 4
EFP = 1
eom_fake_ipea true
efp_exrep 0
efp_qm_exrep 0
$end
$efp_fragments
WATER_L 1.12736608 -1.43556954 -0.73517708 -1.45590530 2.99520330 0.11722720
WATER_L 1.25577919 0.62068648 -2.69876653 2.56168924 1.26470722 0.33910203
WATER_L 3.76006184 -1.03358049 0.45980636 -1.53852111 2.58787281 -1.98107746
WATER_L 4.81593067 2.87535152 -0.24524178 -1.86802100 0.73283467 -2.17837806
WATER_L 4.07402278 0.74020006 -1.92695949 2.21177738 1.69303397 -2.30505848
WATER_L 3.60104027 1.35547341 1.88776964 0.43895304 1.25442317 1.07742578
$end
11.7 Calculation of User-Defined EFP Potentials
User-defined EFP parameters can be generated in MAKEFP job in GAMESS
(see the GAMESS manual for details). The GAMESS EFP parameters
can be converted to Q-Chem
library format using a special script located in $QCAUX/efp directory. Alternatively,
the output can be converted into Q-Chem input format ($efp_params)
to be used as a user-defined potential.
The EFP potential generation begins by determining an accurate structure
for the fragment (EFP is the frozen-geometry potential, so the fragment
geometry will remain the same in all subsequent calculations). We recommend
MP2 / cc-PVTZ level of theory.
11.7.1 Generating EFP Parameters in GAMESS
EFP parameters can be generated in GAMESS using MAKEFP job (RUNTYP=MAKEFP).
For EFP parameters calculations, 6-311++G(3df,2p) basis set is recommended.
Original Stone's distributed multipole analysis (bigexp=0 in the group $stone is recommended
for non-aromatic compound; optionally, one may decrease the basis set to 6-31G* or 6-31+G*
for generation of electrostatic multipoles and screening parameters. (To prepare such a "mixed"
potential, one has to run two separate MAKEFP calculations in larger and smaller bases, and combine
the corresponding parts of the potential).
In aromatic compounds, one must either use numerical grid for generation of multipoles (bigexp=4.0)
or use 6-31G* basis with standard analytic DMA, which is recommended.
The MAKEFP job produces (usually in the scratch directory) the .efp file
containing all the necessary EFP parameters. See GAMESS manual for further details.
Example of the RUNTYP=MAKEFP file for water.
$control units=angs local=boys runtyp=makefp coord=cart icut=11 $end
$system timlim=99999 mwords=200 $end
$scf soscf=.f. diis=.t. CONV=1.0d-06 $end
$basis gbasis=n311 ngauss=6 npfunc=2 ndfunc=3 nffunc=1
diffs=.t. diffsp=.t. $end
$stone
bigexp=0.0
$end
$DAMP IFTTYP(1)=3,2 IFTFIX(1)=1,1 thrsh=500.0 $end
$dampgs
H3=H2
BO31=BO21
$end
$data
Water H2O (GEOMETRY: MP2/cc-pVTZ)
C1
O1 8.0 0.0000 0.0000 0.1187
H2 1.0 0.0000 0.7532 -0.4749
H3 1.0 0.0000 -0.7532 -0.4749
$end
Example of the RUNTYP=MAKEFP file for benzene.
$contrl units=bohr local=boys runtyp=makefp coord=cart icut=11 $end
$system timlim=99999 mwords=200 $end
$scf soscf=.f. diis=.t. CONV=1.0d-06 $end
$basis gbasis=n311 ngauss=6 npfunc=2 ndfunc=3 nffunc=1
diffs=.t. diffsp=.t. $end
$stone
bigexp=4.0
$end
$DAMP IFTTYP(1)=3,2 IFTFIX(1)=1,1 thrsh=500.0 $end
$dampgs
C6=C5
C2=C1
C3=C1
C4=C1
C5=C1
C6=C1
H8=H7
H9=H7
H10=H7
H11=H7
H12=H7
BO32=BO21
BO43=BO21
BO54=BO21
BO61=BO21
BO65=BO21
BO82=BO71
BO93=BO71
BO104=BO71
BO115=BO71
BO126=BO71
$end
$DATA
Benzene C6H6 (GEOMETRY: MP2/cc-pVTZ)
C1
C1 6.0 1.3168 -2.2807 0.0000
C2 6.0 2.6336 0.0000 0.0000
C3 6.0 1.3168 2.2807 0.0000
C4 6.0 -1.3168 2.2807 0.0000
C5 6.0 -2.6336 -0.0000 0.0000
C6 6.0 -1.3168 -2.2807 0.0000
H7 1.0 2.3386 -4.0506 0.0000
H8 1.0 4.6772 0.0000 0.0000
H9 1.0 2.3386 4.0506 0.0000
H10 1.0 -2.3386 4.0506 0.0000
H11 1.0 -4.6772 0.0000 0.0000
H12 1.0 -2.3386 -4.0506 0.0000
$END
11.7.2 Converting EFP Parameters to the Q-Chem Library Format
To convert GAMESS-generated *.efp files to the Q-Chem library
format, one can use the script
_g2qlib.pl@ located in the $QC/bin directory.
The script takes three command line arguments:
- the name of the .efp file generated by GAMESS;
- the basis set name used in RUNTYP = MAKEFP;
- fragment name for the Q-Chem library.
In this example, the script $QC/bin/efp_g2qlib.pl is used to
convert _gms.efp@ file generated in RUNTYP = MAKEFP run
with 6-311++G(3df,2p) basis set:
efp_g2qlib.pl mywater_gms.efp '6-311++G(3df,2p)' MYWATER_L
Note:
Do not forget to place the second command-line argument-namely, the basis
set name used for the calculation of the wavefunction and Fock matrix-in
single quotes: "6-311++G(3df,2p)" |
Note:
Remember to add "_L" at the end of the file
to distinguish the library fragments from the user-defined ones. |
Note:
Keep the .efp extension for files in the Q-Chem EFP library. If you change the extension
of the file with EFP parameters, Q-Chem will not be able to recognize the file. |
Move the produced _L.efp@ file to the $QCAUX/efp directory.
Now the EFP parameters for the fragment with name MYWATER_L are accessible via Q-Chem EFP library
and can be requested in the EFP input:
$efp_fragments
...
MYWATER_L -0.61278300 -3.71417606 3.79298003 2.366406 1.095665 3.136731
...
$end
11.7.3 Converting EFP Parameters to the Q-Chem Input Format
GAMESS EFP input file can be converted into Q-Chem input format using the script
_g2qinp.pl@
located in the $QC/bin directory. The script takes the name of the GAMESS input file as
a command line argument and produces the Q-Chem input file for a EFP calculation with
user-defined fragments.
The script $QC/bin/efp_g2qinp.pl is used to convert
mywater_gms.inp GAMESS input file into Q-Chem input file
mywater.in:
efp_g2qinp.pl mywater_gms.in > mywater.in
Input (GAMESS EFP input file):
$efrag
FRAGNAME=WATER
A01O1 -3.4915850115 1.6130035943 6.3466249474
A02H2 -2.6802987071 1.8846442441 5.9133847256
A03H3 -4.1772171823 2.0219163332 5.8152398722
FRAGNAME=WATER
A01O1 -2.6611610192 2.9115390164 4.0238365827
A02H2 -1.9087196288 3.4457621856 4.2848131325
A03H3 -2.5308218271 2.7856450856 3.0821181464
$END
$WATER
….
…EFP Parameters….
….
$END
Output (Q-Chem EFP input):
$efp_fragments
WATER -3.48455373 1.65108563 6.29264698 -0.182579 0.622332 -1.458796
WATER -2.61176287 2.93438808 3.98574400 -1.137556 0.960145 0.226277
$end
$efp_params
fragment WATER
….
…EFP Parameters….
er_basis
….
$end
After conversion, the $efp_params section of the Q-Chem input file contains the line
starting with the er_basis keyword. This line has to be edited manually. The basis set used to
produce EFP parameters (MAKEEFP job) has to be specified. For example, if the EFP parameters
were produced using 6-311++G(3df,2p) basis set, the line in the above example should be edited
as follows:
$efp_params
fragment WATER
….
…EFP Parameters….
er_basis 6-311++G(3df,2p)
….
$end
The script reads Cartesian coordinates of fragments and EFP parameters only and creates
$efp_fragments and $efp_params sections of the Q-Chem input file. All other required Q-Chem
input sections ($rem, $molecule, etc..) need to be added manually.
11.8 Converting PDB Coordinates into Q-Chem EFP Input Format
The commonly used format for 3D structural data of biological molecules is Protein Data
Bank (PDB) file. The Cartesian coordinates of molecules from PDB file can be converted to
Q-Chem format for EFP fragments. The script _pdb2qinp.pl@ located in $QC/bin
is designed to convert coordinates of solvent molecules from PDB file to the Q-Chem format.
However, only EFP fragments available in the Q-Chem EFP library located in $QCAUX/efp directory
will be recognized and converted by the script. Since PDB format has many variants some
adjustments are required to original PBD file to make it suitable for conversion.
Consider example of PDB file which contains four water molecules:
Example: Water.pdb
REMARK TIPS3P model
REMARK
ATOM 1 OH2 TIP 1 2.950 -0.088 2.278 1.00 0.00 WAT
ATOM 2 H1 TIP 1 3.539 -0.465 1.623 1.00 0.00 WAT
ATOM 3 H2 TIP 1 2.635 -0.852 2.764 1.00 0.00 WAT
ATOM 4 OH2 TIP 2 -4.195 -1.030 -2.858 1.00 0.00 WAT
ATOM 5 H1 TIP 2 -4.507 -1.544 -3.606 1.00 0.00 WAT
ATOM 6 H2 TIP 2 -3.809 -1.689 -2.279 1.00 0.00 WAT
ATOM 7 OH2 TIP 3 0.489 3.999 2.164 1.00 0.00 WAT
ATOM 8 H1 TIP 3 0.125 4.885 2.215 1.00 0.00 WAT
ATOM 9 H2 TIP 3 -0.239 3.441 2.441 1.00 0.00 WAT
ATOM 10 OH2 TIP 4 -4.118 1.169 2.350 1.00 0.00 WAT
ATOM 11 H1 TIP 4 -4.186 0.349 1.858 1.00 0.00 WAT
ATOM 12 H2 TIP 4 -4.667 1.776 1.851 1.00 0.00 WAT
The PDB file contains 4 water molecules with tag "TIP". Each molecule has 3 atoms with
tags "OH1", "H1", and "H2". However, the Q-Chem EFP library ($QCAUX/efp) contains EFP
fragment with name WATER_L that has 3 atoms with tags "A01O1", "A02H2", "A03H3"
(see section "labels" at the bottom of the library file WATER_L.efp). In order to match the
water molecule in PBD file and EFP library, the following lines should be added to the PDB file:
Example: Modified Water.pdb
REMARK TIPS3P model
REMARK
REMARK EFPF TIP WATER_L
REMARK EFPA TIP OH2 A01O1
REMARK EFPA TIP H1 A02H2
REMARK EFPA TIP H2 A02H3
ATOM 1 OH2 TIP 1 2.950 -0.088 2.278 1.00 0.00 WAT
ATOM 2 H1 TIP 1 3.539 -0.465 1.623 1.00 0.00 WAT
ATOM 3 H2 TIP 1 2.635 -0.852 2.764 1.00 0.00 WAT
ATOM 4 OH2 TIP 2 -4.195 -1.030 -2.858 1.00 0.00 WAT
ATOM 5 H1 TIP 2 -4.507 -1.544 -3.606 1.00 0.00 WAT
ATOM 6 H2 TIP 2 -3.809 -1.689 -2.279 1.00 0.00 WAT
ATOM 7 OH2 TIP 3 0.489 3.999 2.164 1.00 0.00 WAT
ATOM 8 H1 TIP 3 0.125 4.885 2.215 1.00 0.00 WAT
ATOM 9 H2 TIP 3 -0.239 3.441 2.441 1.00 0.00 WAT
ATOM 10 OH2 TIP 4 -4.118 1.169 2.350 1.00 0.00 WAT
ATOM 11 H1 TIP 4 -4.186 0.349 1.858 1.00 0.00 WAT
ATOM 12 H2 TIP 4 -4.667 1.776 1.851 1.00 0.00 WAT
New lines start with keyword "REMARK" in order to prevent reading of these lines by
other programs. The first line "REMARK EFPF TIP WATER_L" contains keyword "EFPF"
which is recognized by the script and tells to match a PDB molecule with tag "TIP"
and a Q-Chem EFP library fragment with tag "WATER_L". The general format
for this line is as follows:
REMARK EFPF <PDB_TAG> <EFP_TAG>
where <PDB_TAG> is a tag of the molecule in PDB file and <EFP\_TAG> is a name of the
corresponding Q-Chem EFP library fragment.
The next three lines contain keyword "EFPA" which maps each atom type from PDB file to
corresponding atom in the Q-Chem EFP fragment. For example, the line "REMARK EFPA TIP OH2 A01O1"
maps the oxygen atom "OH1" from molecule with tag "TIP" in PDB file to atom "A01O1" of the
corresponding EFP fragment. The general format for these lines is as follows:
REMARK EFPA <PDB_TAG> <PDB_ATOM> <EFP_ATOM>
where <PDB_TAG> is a tag for the molecule, <PDB_ATOM> is a tag for the atom in the PDB file, and
<EFP_ATOM> is the name of the corresponding atom in the EFP fragment from the Q-Chem EFP library.
Note:
The names of all EFP fragments can be found in the directory $QCAUX/efp. The names of atoms
in EFP fragments are listed in the last section. "labels" at the bottom of *.efp files. |
Example Section "labels" from "$QCAUX/efp/WATER_L.efp":
fragment WATER_L
……
labels
A01O1 O 0.0000000000 0.0000000000 0.0664326840
A02H2 H 0.0000000000 0.7532000000 -0.5271673160
A03H3 H 0.0000000000 -0.7532000000 -0.5271673160
$end
Finally, when each atom of the solvent molecule is matched to the corresponding atom of
EFP fragment the script converts the coordinates and produces the $efp_fragments section of the
Q-Chem EFP input file.
Example: Converted Water.pdb file
$efp_fragments
WATER_L -3.48394754 1.65103763 6.29199349 -0.183326 0.621595 -1.458096
WATER_L -2.61157557 2.93379532 3.98490930 -1.138655 0.960180 0.226963
WATER_L 0.42784805 4.01733780 2.18237025 1.279456 1.850972 -1.727770
WATER_L -4.15254211 1.15707665 2.29452048 1.903173 0.582438 1.562171
$end
Note:
Since the solvent molecules converted from the PBD file are present in the Q-Chem EFP library,
the section $efp_fragments is not required in the input. |
11.9 fEFP Input Structure
A two-step script, .pl@ located in
$QC/bin, allows users to break molecular structures
from a PDB file into the capped fragments in the GAMESS format, such
that parameters for fEFP calculations can be generated.
To use the .pl@ scripts you need a PDB file, a MAP file, and a
directory with all your .efp files.
Run the following commands to: (1) obtain the N input file generating the
N EFP parameters for the N capped fragments, and (2) create the
GAMESS EFP input file.
perl prefp.pl 1 <PDB file> <MAP file>
perl prefp.pl 2 <PDB file> <.efp path> <MAP file> <GMS input file name>
To convert the GAMESS input file into Q-Chem format,
see the previous section.
At the first step the script splits the biomolecule (PDB format) into N
fragments generating N MAKEFP input files (GAMESS) with the
help of a MAP file.
At the second step the .efp file from GAMESS MAKEFP is analyzed and
is auto-edited using the same MAP file to create the final GAMESS EFP
input.
The MAP file is required as an input for the script.
It defines groups of atoms belonging to each EFP
fragment both for the MAKEFP calculation and for the consequent EFP jobs.
Here is a description of the MAP file:
Each fragment described using section $RESIDUE followed by closing $end
In this example the Lys2 is extracted cutting through the peptidic bond,
the cut bond is saturated with hydrogen atom. The explanation of each
variable is given below.
$Residue
Name = lys2
PreAtoms = 14-35
NH = 14,12
CH = 34,36
PostAtoms = 14-35
Rescharge = +1
USEFP = lys2
$end
The four first lines are required for the first step of the script
(GAMESS MAKEFP job); the next ones are necessary for the actual
EFP job.
Name:
Residue name
PreAtoms:
Atoms which belongs to the residue for GAMESS MAKEFP
calculation.
CH, NH, or OH:
In the case of broken bonds a hydrogen atom is added so
that in X-Y bond (X belongs to the Lys2 residue and Y belongs to the
previous or next residue) the Y atom is replaced by H along the X-Y axis.
The default equilibrium distance for the X-H bond is set to 1.08 Å for
a C-H bond, to 1.00 Å for a N-H bond, and to 0.94 Å for a O-H bond.
It required to
specify the atom number of the X and Y atoms.
PostAtoms:
Atoms which belong to the residue after removing the
overlapping fragment atoms or caps when the HLA or the CLA
scheme is used. This important step removes multipoles and polarizability
expansion points of those
atoms according to the cutoff procedure (set by default to 1.3 Å and
1.2 Å for multipoles and polarizability expansion points, respectively).
Multipole expansion at duplicated points are eliminated but to
maintain the net integer charge on each amino acid the monopole expansion
of the caps is redistributed on the natural
fragment. This method is called
Expand-Remove-Redistribute. Concerning the polarizability expansion
points, only one polarizability expansion point is removed when a hydrogen atom
saturates the dangling bond, whereas 6 or 5 polarizability points are
removed when the cap is an amine or an aldehyde, respectively.
ResCharge: The net charge of the residue after removing the
overlapping fragment atoms (cfr. LA scheme).
USEFP:
Name of the EFP fragment (and .efp file) to use with this fragment in the
actual EFP calculation.
Note:
In the MFCC scheme, the two first letters of the concap fragment
have to be 'CC'. |
Note:
If the PostAtoms keyword is not present, the second script will
generate an EFP job file without any modification of the parameters, which is
useful for the MFCC
scheme. |
11.10 Advanced EFP options
Example 11.0
Calculation of EFP parameters for water (developers only!)
$molecule
0 1
O 0.000000 0.000000 0.121795
H -0.346602 -0.400000 -0.477274
H 0.346602 0.400000 -0.477274
$end
$rem
exchange hf
basis 6-311G
jobtype makeefp
print_general_basis true
DAMP_GRID_STEP 50
DAMP_GEN_GAUSS_FIT 1
DAMP_GEN_EXP_FIT 1
DMA_MIDPOINTS 1
$end
The following keywords control calculation of selected energy components in EFP job.
EFP_ELEC
Controls fragment-fragment electrostatics in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off electrostatics |
RECOMMENDATION:
|
| EFP_POL
Controls fragment-fragment polarization in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off polarization |
RECOMMENDATION:
|
|
|
EFP_EXREP
Controls fragment-fragment exchange repulsion in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off exchange repulsion |
RECOMMENDATION:
|
| EFP_DISP
Controls fragment-fragment dispersion in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off dispersion |
RECOMMENDATION:
|
|
|
EFP_QM_ELEC
Controls QM-EFP electrostatics |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off electrostatics |
RECOMMENDATION:
|
| EFP_QM_POL
Controls QM-EFP polarization |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off polarization |
RECOMMENDATION:
|
|
|
EFP_QM_EXREP
Controls QM-EFP exchange-repulsion |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off exchange-repulsion |
RECOMMENDATION:
|
| EFP_ELEC_DAMP
Controls fragment-fragment electrostatic screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off electrostatic screening |
1 | use overlap-based damping correction |
2 | use exponential damping correction if screening parameters are provided in the EFP potential. |
| If the parameters are not provided damping will be automatically disabled. |
RECOMMENDATION:
|
|
|
EFP_DISP_DAMP
Controls fragment-fragment dispersion screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off dispersion screening |
1 | use Tang-Toennies screening, with fixed parameter b=1.5 |
RECOMMENDATION:
|
| EFP_QM_ELEC_DAMP
Controls QM-EFP electrostatics screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off electrostatic screening |
1 | use overlap based damping correction |
RECOMMENDATION:
|
|
|
Example 11.0 EFP gradient computation for adenine dimer using library parameters.
$molecule
0 1
--
0 1
-1.37701480 1.41334560 3.40415522 2.504756 3.030456 -3.010720
--
0 1
-1.34215671 0.11877860 6.80424961 -2.807868 3.052053 -2.714643
$end
$rem
jobtyp force
exchange hf
purecart 2222
EFP_INPUT true
EFP_FRAGMENTS_ONLY 1
MAX_SCF_CYCLES 0
print_input false
sym_ignore true
$end
$efp_fragments
ADENINE_L
ADENINE_L
$end
Example 11.0 EFP energy computation for benzene dimer.
This is an old-style input for EFP-only
job which uses He atom trick. It will work but is no longer necessary
(see the example above).
$molecule
0 1
He 5.0 5.0 5.0
$end
$rem
exchange hf
basis 6-31G(d)
jobtype sp
purecart 2222
efp_fragments_only true
EFP_ELEC 1 ! EFP-EFP electrostatics is ON
EFP_POL 1 ! EFP-EFP polarization is ON
EFP_EXREP 1 ! EFP-EFP exchange-repulsion is ON
EFP_DISP 1 ! EFP-EFP dispersion is ON
EFP_QM_ELEC 1 ! QM-EFP electrostatics is ON
EFP_QM_POL 1 ! QM-EFP polarization is ON
EFP_QM_EXREP 0 ! QM-EFP exchange-repulsion is OFF
EFP_ELEC_DAMP 2 ! EFP-EFP electrostatic damping is EXPONENTIAL
EFP_DISP_DAMP 1 ! EFP-EFP dispersion damping is ON
EFP_QM_ELEC_DAMP 0 ! QM-EFP electrostatic damping is OFF
$end
$efp_fragments
BENZENE_L -0.30448173 -2.24210052 -0.29383131 -0.642499 1.534222 -0.568147
BENZENE_L -0.60075437 1.36443336 0.78647823 3.137879 1.557344 -2.568550
$end
Chapter 12 Methods Based on Absolutely-Localized Molecular Orbitals
12.1 Introduction
Molecular complexes and molecular clusters represent a broad class of systems with interesting chemical and physical properties.
Such systems can be naturally partitioned into fragments each representing a molecule or several molecules. Q-Chem contains a set of methods designed to use such partitioning either for physical or computational advantage. Some of these methods were developed and implemented by Dr. Rustam Z. Khaliullin at the University of California-Berkeley, with Profs. Martin Head-Gordon and Alexis Bell; the open shell versions were developed by
Paul Horn at Berkeley, working with Martin Head-Gordon.
Others were developed by Dr. Leif D. Jacobson at Ohio State University, with Prof. John M. Herbert.
The list of methods that use partitioning includes:
- Initial guess at the MOs as a superposition of the converged MOs on the isolated fragments (FRAGMO guess) [586].
- Constrained (locally-projected) SCF methods for molecular interactions (SCF MI methods) between closed shell fragments [586], and also open shell fragments [587].
- Single Roothaan-step (RS) correction methods that improve FRAGMO and SCF MI description of molecular systems [586,[587].
- Automated calculation of the BSSE with counterpoise correction method (full SCF and RS implementation).
- Energy decomposition analysis and charge transfer analysis [588,[589,[587].
- Analysis of intermolecular bonding in terms of complementary occupied-virtual pairs [589,[590,[587].
- The variational explicit polarization (XPol) method, a self-consistent,
charge-embedded, monomer-based SCF calculation [591,[592,[469]
- Symmetry-adapted perturbation theory (SAPT), a monomer-based method for computing
intermolecular interaction energies and
decomposing them into physically-meaningful components [593,[594]
- XPol+SAPT, which extends the SAPT methodology to systems consisting of more than two
monomers [592,[469]
12.2 Specifying Fragments in the $molecule Section
To request any of the methods mentioned above one must specify how system is partitioned into fragments. All atoms and all electrons in the systems should be assigned to a fragment. Each fragment must contain an integer number of electrons. In the current implementation, both open and closed-shell fragments are allowed. In order to specify fragments, the fragment descriptors must be inserted into the $molecule section of the Q-Chem input file. A fragment descriptor consists of two lines: the first line must start with two hyphens followed by optional comments, the second line must contain the charge and the multiplicity of the fragment. At least two fragments must be specified. Fragment descriptors in the $molecule section does not affect jobs that are not designed to use fragmentation.
Example 12.0
Fragment descriptors in the $molecule section.
$molecule
0 1
-- water molecule - proton donor
0 1
O1
H2 O1 0.96
H3 O1 0.96 H2 105.4
-- water molecule - proton acceptor
0 1
O4 O1 ROO H2 105.4 H3 0.0
X5 O4 2.00 O1 120.0 H2 180.0
H6 O4 0.96 X5 55.6 O1 90.0
H7 O4 0.96 X5 55.6 01 -90.0
ROO = 2.4
$end
Open shell systems must have a number of alpha electrons greater than the number of beta electrons. However, individual fragments in the system can be made to contain excess beta electrons by specifying a negative multiplicity. For instance, a multiplicity of −2 indicates one excess beta electron, as in the second fragment of the following example.
Example 12.0
Open shell fragment descriptors in the $molecule section.
$molecule
0 1
-- An alpha spin H atom
0 2
H1
-- A beta spin H atom
0 -2
H2 H1 1.50
$end
12.3 FRAGMO Initial Guess for SCF Methods
An accurate initial guess can be generated for molecular systems by superimposing converged molecular orbitals on isolated fragments. This initial guess is requested by specifying FRAGMO option for SCF_GUESS keyword and can be used for both the conventional SCF methods and the locally-projected SCF methods. The number of SCF iterations can be greatly reduced when FRAGMO is used instead of SAD. This can lead to significant time savings for jobs on multifragment systems with large basis sets [588]. Unlike the SAD guess, the FRAGMO guess is idempotent.
To converge molecular orbitals on isolated fragments, a child Q-Chem job is executed for each fragment. $rem variables of the child jobs are inherited from the $rem section of the parent job. If SCF_PRINT_FRGM is set to TRUE the output of the child jobs is redirected to the output file of the parent job. Otherwise, the output is suppressed.
Additional keywords that control child Q-Chem processes can be set in the $rem_frgm section of the parent input file. This section has the same structure as the $rem section. Options in the $rem_frgm section override options of the parent job. $rem_frgm is intended to specify keywords that control the SCF routine on isolated fragments. Please be careful with the keywords in $rem_frgm section. $rem variables FRGM_METHOD, FRGM_LPCORR, JOBTYPE, BASIS, PURECART, ECP are not allowed in $rem_frgm and will be ignored. $rem variables FRGM_METHOD, FRGM_LPCORR, JOBTYPE, SCF_GUESS, MEM_TOTAL, MEM_STATIC are not inherited from the parent job.
Example 12.0
FRAGMO guess can be used with the conventional SCF calculations. $rem_frgm keywords in this example specify that the SCF on isolated fragments does not have to be converged tightly.
$molecule
0 1
--
0 1
O -0.106357 0.087598 0.127176
H 0.851108 0.072355 0.136719
H -0.337031 1.005310 0.106947
--
0 1
O 2.701100 -0.077292 -0.273980
H 3.278147 -0.563291 0.297560
H 2.693451 -0.568936 -1.095771
--
0 1
O 2.271787 -1.668771 -2.587410
H 1.328156 -1.800266 -2.490761
H 2.384794 -1.339543 -3.467573
--
0 1
O -0.518887 -1.685783 -2.053795
H -0.969013 -2.442055 -1.705471
H -0.524180 -1.044938 -1.342263
$end
$rem
JOBTYPE SP
EXCHANGE EDF1
CORRELATION NONE
BASIS 6-31(2+,2+)g(df,pd)
SCF_GUESS FRAGMO
SCF_PRINT_FRGM FALSE
$end
$rem_frgm
SCF_CONVERGENCE 2
THRESH 5
$end
12.4 Locally-Projected SCF Methods
Constrained locally-projected SCF is an efficient method for removing the SCF diagonalization bottleneck in calculations for systems of weakly interacting components such as molecular clusters and molecular complexes [586,[587]. The method is based on the equations of the locally-projected SCF for molecular interactions (SCF MI) [595,[596,[597,[586,[587]. In the SCF MI method, the occupied molecular orbitals on a fragment can be expanded only in terms of the atomic orbitals of the same fragment. Such constraints produce non-orthogonal MOs that are localized on fragments and are called absolutely-localized molecular orbitals (ALMOs). The ALMO approximation excludes charge-transfer from one fragment to another. It also prevents electrons on one fragment from borrowing the atomic orbitals of other fragments to compensate for incompleteness of their own AOs and, therefore, removes the BSSE from the interfragment binding energies. The locally-projected SCF methods perform an iterative minimization of the SCF energy with respect to the ALMOs coefficients. The convergence of the algorithm is accelerated with the locally-projected modification of the DIIS extrapolation method [586].
The ALMO approximation significantly reduces the number of variational degrees of freedom of the wavefunction. The computational advantage of the locally-projected SCF methods over the conventional SCF method grows with both basis set size and number of fragments. Although still cubic scaling, SCF MI effectively removes the diagonalization step as a bottleneck in these calculations, because it contains such a small prefactor. In the current implementation, the SCF MI methods do not speed up the evaluation of the Fock matrix and, therefore, do not perform significantly better than the conventional SCF in the calculations dominated by the Fock build.
Two locally-projected schemes are implemented. One is based on the locally-projected equations of
Stoll et al. [595], the other utilizes the locally-projected equations of
Gianinetti et al. [596] These methods have comparable performance. The Stoll iteration is only slightly faster than the Gianinetti iteration but the Stoll equations might be a little bit harder to converge. The Stoll equations also produce ALMOs that are orthogonal within a fragment. The type of the locally-projected SCF calculations is requested by specifying either STOLL or GIA for the FRGM_METHOD keyword.
Example 12.0
Locally-projected SCF method of Stoll
$molecule
0 1
--
-1 1
B 0.068635 0.164710 0.123580
F -1.197609 0.568437 -0.412655
F 0.139421 -1.260255 -0.022586
F 1.118151 0.800969 -0.486494
F 0.017532 0.431309 1.531508
--
+1 1
N -2.132381 -1.230625 1.436633
H -1.523820 -1.918931 0.977471
H -2.381590 -0.543695 0.713005
H -1.541511 -0.726505 2.109346
H -2.948798 -1.657993 1.873482
$end
$rem
JOBTYPE SP
EXCHANGE B
CORRELATION P86
BASIS 6-31(+,+)G(d,p)
FRGM_METHOD STOLL
$end
$rem_frgm
SCF_CONVERGENCE 2
THRESH 5
$end
12.4.1 Locally-Projected SCF Methods with Single Roothaan-Step Correction
Locally-projected SCF cannot quantitatively reproduce the full SCF intermolecular interaction energies for systems with significant charge-transfer between the fragments (e.g., hydrogen bonding energies in water clusters). Good accuracy in the intermolecular binding energies can be achieved if the locally-projected SCF MI iteration scheme is combined with a charge-transfer perturbative correction [586]. To account for charge-transfer, one diagonalization of the full Fock matrix is performed after the locally-projected SCF equations are converged and the final energy is calculated as infinite-order perturbative correction to the locally-projected SCF energy. This procedure is known as single Roothaan-step (RS) correction [586,[180,[598]. It is performed if FRGM_LPCORR is set to RS. To speed up evaluation of the charge-transfer correction, second-order perturbative correction to the energy can be evaluated by solving the linearized single-excitation amplitude equations. This algorithm is called the approximate Roothaan-step correction and can be requested by setting FRGM_LPCORR to ARS.
Both ARS and RS corrected energies are very close to the full SCF energy for systems of weakly interacting fragments but are less computationally expensive than the full SCF calculations. To test the accuracy of the ARS and RS methods, the full SCF calculation can be done in the same job with the perturbative correction by setting FRGM_LPCORR to RS_EXACT_SCF or to ARS_EXACT_SCF. It is also possible to evaluate only the full SCF correction by setting FRGM_LPCORR to EXACT_SCF.
The iterative solution of the linear single-excitation amplitude equations in the ARS method is controlled by a set of NVO keywords described below.
Restrictions. Only single point HF and DFT energies can be evaluated with the locally-projected methods. Geometry optimization can be performed using numerical gradients. Wavefunction correlation methods (MP2, CC, etc..) are not implemented for the absolutely-localized molecular orbitals. SCF_ALGORITHM cannot be set to anything but DIIS, however, all SCF convergence algorithms can be used on isolated fragments (set SCF_ALGORITHM in the $rem_frgm section).
Example 12.0
Comparison between the RS corrected energies and the conventional SCF energies can be made by calculating both energies in a single run.
$molecule
0 1
--
0 1
O -1.56875 0.11876 0.00000
H -1.90909 -0.78106 0.00000
H -0.60363 0.02937 0.00000
--
0 1
O 1.33393 -0.05433 0.00000
H 1.77383 0.32710 -0.76814
H 1.77383 0.32710 0.76814
$end
$rem
JOBTYPE SP
EXCHANGE HF
CORRELATION NONE
BASIS AUG-CC-PVTZ
FRGM_METHOD GIA
FRGM_LPCORR RS_EXACT_SCF
$end
$rem_frgm
SCF_CONVERGENCE 2
THRESH 5
$end
12.4.2 Roothaan-Step Corrections to the FRAGMO Initial Guess
For some systems good accuracy for the intermolecular interaction energies can be achieved without converging SCF MI calculations and applying either the RS or ARS charge-transfer correction directly to the FRAGMO initial guess. Set FRGM_METHOD to NOSCF_RS or NOSCF_ARS to request the single Roothaan correction or approximate Roothaan correction, respectively. To get a somewhat better energy estimate set FRGM_METHOD to NOSCF_DRS and NOSCF_RS_FOCK. In the case of NOSCF_RS_FOCK, the same steps as in the NOSCF_RS method are performed followed by one more Fock build and calculation of the proper SCF energy. In the case of the double Roothaan-step correction, NOSCF_DRS, the same steps as in NOSCF_RS_FOCK are performed followed by one more diagonalization. The final energy in the NOSCF_DRS method is evaluated as a perturbative correction, similar to the single Roothaan-step correction.
Charge-transfer corrections applied directly to the FRAGMO guess are included in Q-Chem to test accuracy and performance of the locally-projected SCF methods. However, for some systems they give a reasonable estimate of the binding energies at a cost of one (or two) SCF step(s).
12.4.3 Automated Evaluation of the Basis-Set Superposition Error
Evaluation of the basis-set superposition error (BSSE) is automated in Q-Chem. To calculate BSSE-corrected binding energies, specify fragments in the $molecule section and set JOBTYPE to BSSE. The BSSE jobs are not limited to the SCF energies and can be evaluated for multifragment systems at any level of theory. Q-Chem separates the system into fragments as specified in the $molecule section and performs a series of jobs on (a) each fragment, (b) each fragment with the remaining atoms in the system replaced by the ghost atoms, and (c) on the entire system. Q-Chem saves all calculated energies and prints out the uncorrected and the BSSE corrected binding energies. The $rem_frgm section can be used to control calculations on fragments, however, make sure that the fragments and the entire system are treated equally. It means that all numerical methods and convergence thresholds that affect the final energies (such as SCF_CONVERGENCE, THRESH, PURECART, XC_GRID) should be the same for the fragments and for the entire system. Avoid using $rem_frgm in the BSSE jobs unless absolutely necessary.
Important. It is recommended to include PURECART keyword in all BSSE jobs. GENERAL basis cannot be used for the BSSE calculations in the current implementation. Use MIXED basis instead.
Example 12.0
Evaluation of the BSSE corrected intermolecular interaction energy
$molecule
0 1
--
0 1
O -0.089523 0.063946 0.086866
H 0.864783 0.058339 0.103755
H -0.329829 0.979459 0.078369
--
0 1
O 2.632273 -0.313504 -0.750376
H 3.268182 -0.937310 -0.431464
H 2.184198 -0.753305 -1.469059
--
0 1
O 0.475471 -1.428200 -2.307836
H -0.011373 -0.970411 -1.626285
H 0.151826 -2.317118 -2.289289
$end
$rem
JOBTYPE BSSE
EXCHANGE HF
CORRELATION MP2
BASIS 6-31(+,+)G(d,p)
$end
12.5 Energy Decomposition and Charge-Transfer Analysis
12.5.1 Energy Decomposition Analysis
The strength of intermolecular binding is inextricably connected to the fundamental nature of interactions between the molecules. Intermolecular complexes can be stabilized through weak dispersive forces, electrostatic effects (e.g., charge-charge, charge-dipole, and charge-induced dipole interactions) and donor-acceptor type orbital interactions such as forward and back-donation of electron density between the molecules. Depending on the extent of these interactions, the intermolecular binding could vary in strength from just several kJ/mol (van der Waals complexes) to several hundred kJ/mol (metal-ligand bonds in metal complexes). Understanding the contributions of various interaction modes enables one to tune the strength of the intermolecular binding to the ideal range by designing materials that promote desirable effects. One of the most powerful techniques that modern first principles electronic structure methods provide to study and analyze the nature of intermolecular interactions is the decomposition of the total molecular binding energy into the physically meaningful components such as dispersion, electrostatic, polarization, charge transfer, and geometry relaxation terms.
Energy decomposition analysis based on absolutely-localized molecular orbitals (ALMO EDA) is implemented in Q-Chem [588], including the open shell generalization [587]. In ALMO EDA, the total intermolecular binding energy is decomposed into the "frozen density" component (FRZ), the polarization (POL) term, and the charge-transfer (CT) term. The "frozen density" term is defined as the energy change that corresponds to bringing infinitely separated fragments together without any relaxation of their MOs. The FRZ term is calculated as a difference between the FRAGMO guess energy and the sum of the converged SCF energies on isolated fragments. The polarization (POL) energy term is defined as the energy lowering due to the intrafragment relaxation of the frozen occupied MOs on the fragments. The POL term is calculated as a difference between the converged SCF MI energy and the FRAGMO guess energy. Finally, the charge-transfer (CT) energy term is due to further interfragment relaxation of the MOs. It is calculated as a difference between the fully converged SCF energy and the converged SCF MI energy.
The total charge-transfer term includes the energy lowering due to electron transfer from the occupied orbitals on one molecule (more precisely, occupied in the converged SCF MI state) to the virtual orbitals of another molecule as well as the further energy change caused by induction that accompanies such an occupied / virtual mixing. The energy lowering of the occupied-virtual electron transfer can be described with a single non-iterative Roothaan-step correction starting from the converged SCF MI solution. Most importantly, the mathematical form of the SCF MI(RS) energy expression allows one to decompose the occupied-virtual mixing term into bonding and back-bonding components for each pair of molecules in the complex. The remaining charge-transfer energy term (i.e., the difference between SCF MI(RS) energy and the full SCF energy) includes all induction effects that accompany occupied-virtual charge transfer and is generally small. This last term is called higher order (HO) relaxation. Unlike the RS contribution, the higher order term cannot be divided naturally into forward and back-donation terms. The BSSE associated with each charge-transfer term (forward donation, back-bonding, and higher order effects) can be corrected individually.
To perform energy decomposition analysis, specify fragments in the $molecule section and set JOBTYPE to EDA. For a complete EDA job, Q-Chem
- performs the SCF on isolated fragments (use the $rem_frgm section if convergence issues arise but make sure that keywords in this section do not affect the final energies of the fragments),
- generates the FRAGMO guess to obtain the FRZ term,
- converges the SCF MI equations to evaluate the POL term,
- performs evaluation of the perturbative (RS or ARS) variational correction to calculate the forward donation and back-bonding components of the CT term for each pair of molecules in the system,
- converges the full SCF procedure to evaluate the higher order relaxation component of the CT term.
The FRGM_LPCORR keyword controls evaluation of the CT term in an EDA job. To evaluate all of the CT components mentioned above set this keyword to RS_EXACT_SCF or ARS_EXACT_SCF. If the HO term in not important then the final step (i.e., the SCF calculation) can be skipped by setting FRGM_LPCORR to RS or ARS. If only the total CT term is required then set FRGM_LPCORR to EXACT_SCF.
ALMO charge transfer analysis (ALMO CTA) is performed together with ALMO EDA [589]. The ALMO charge transfer scale, Delta Q, provides a measure of the distortion of the electronic clouds upon formation of an intermolecular bond and is such that all CT terms (i.e., forward-donation, back-donation, and higher order relaxation) have well defined energetic effects (i.e., ALMO CTA is consistent with ALMO EDA).
To remove the BSSE from the CT term (both on the energy and charge scales), set EDA_BSSE to TRUE. Q-Chem generates an input file for each fragment with MIXED basis set to perform the BSSE correction. As with all jobs with MIXED basis set and d or higher angular momentum basis functions on atoms, the PURECART keyword needs to be initiated. If EDA_BSSE=TRUE GENERAL basis sets cannot be used in the current implementation.
Please note that the energy of the geometric distortion of the fragments is not included into the total binding energy calculated in an EDA job. The geometry optimization of isolated fragments must be performed to account for this term.
Example 12.0
Energy decomposition analysis of the binding energy between the water molecules in a tetramer. ALMO CTA results are also printed out.
$molecule
0 1
--
0 1
O -0.106357 0.087598 0.127176
H 0.851108 0.072355 0.136719
H -0.337031 1.005310 0.106947
--
0 1
O 2.701100 -0.077292 -0.273980
H 3.278147 -0.563291 0.297560
H 2.693451 -0.568936 -1.095771
--
0 1
O 2.271787 -1.668771 -2.587410
H 1.328156 -1.800266 -2.490761
H 2.384794 -1.339543 -3.467573
--
0 1
O -0.518887 -1.685783 -2.053795
H -0.969013 -2.442055 -1.705471
H -0.524180 -1.044938 -1.342263
$end
$rem
JOBTYPE EDA
EXCHANGE EDF1
CORRELATION NONE
BASIS 6-31(+,+)g(d,p)
PURECART 1112
FRGM_METHOD GIA
FRGM_LPCORR RS_EXACT_SCF
EDA_BSSE TRUE
$end
Example 12.0
An open shell EDA example of Na+ interacting with the methyl radical.
$molecule
1 2
--
0 2
C -1.447596 -0.000023 0.000019
H -1.562749 0.330361 -1.023835
H -1.561982 0.721445 0.798205
H -1.561187 -1.052067 0.225866
--
1 1
Na 1.215591 0.000036 -0.000032
$end
$rem
JOBTYPE EDA
EXCHANGE B3LYP
BASIS 6-31G*
UNRESTRICTED TRUE
SCF_GUESS FRAGMO
FRGM_METHOD STOLL
FRGM_LPCORR RS_EXACT_SCF
EDA_BSSE TRUE
DIIS_SEPARATE_ERRVEC 1
$end
12.5.2 Analysis of Charge-Transfer Based on Complementary Occupied / Virtual Pairs
In addition to quantifying the amount and energetics of intermolecular charge transfer, it is often useful to have a simple description of orbital interactions in intermolecular
complexes. The polarized ALMOs obtained from the SCF MI procedure and used as a reference basis set in the decomposition analysis do not directly show which occupied-virtual orbital pairs are of most importance in forming intermolecular bonds. By performing rotations of the polarized ALMOs within a molecule, it is possible to find a "chemist's basis set" that represents bonding between molecules in terms of just a few localized orbitals called complementary occupied-virtual pairs (COVPs). This orbital interaction model validates existing conceptual descriptions of intermolecular bonding. For example, in the modified ALMO basis, hydrogen bonding in water dimer is represented as an electron pair localized on an oxygen atom donating electrons to the O-H σ-antibonding orbital on the other molecule [590], and the description of synergic bonding in metal complexes agrees well with simple Dewar-Chatt-Duncanson model [599,[589,[600].
Set EDA_COVP to TRUE to perform the COVP analysis of the CT term in an EDA job. COVP analysis is currently implemented only for systems of two fragments. Set EDA_PRINT_COVP to TRUE to print out localized orbitals that form occupied-virtual pairs. In this case, MOs obtained in the end of the run (SCF MI orbitals, SCF MI(RA) orbitals, converged SCF orbitals) are replaced by the orbitals of COVPs. Each orbital is printed with the corresponding CT energy term in kJ/mol (instead of the energy eigenvalues in hartrees). These energy labels make it easy to find correspondence between an occupied orbital on one molecule and the virtual orbital on the other molecule. The examples below show how to print COVP orbitals. One way is to set $rem variable PRINT_ORBITALS, the other is to set IANLTY to 200 and use the $plots section in the Q-Chem input. In the first case the orbitals can be visualized using MOLDEN (set MOLDEN_FORMAT to TRUE), in the second case use VMD or a similar third party program capable of making 3D plots.
Example 12.0 COVP analysis of the CT term. The COVP orbitals are printed in the Q-Chem and MOLDEN formats.
$molecule
0 1
--
0 1
O -1.521720 0.129941 0.000000
H -1.924536 -0.737533 0.000000
H -0.571766 -0.039961 0.000000
--
0 1
O 1.362840 -0.099704 0.000000
H 1.727645 0.357101 -0.759281
H 1.727645 0.357101 0.759281
$end
$rem
JOBTYPE EDA
BASIS 6-31G
PURECART 1112
EXCHANGE B3LYP
CORRELATION NONE
FRGM_METHOD GIA
FRGM_LPCORR RS_EXACT_SCF
EDA_COVP TRUE
EDA_PRINT_COVP TRUE
PRINT_ORBITALS 16
MOLDEN_FORMAT TRUE
$end
Example 12.0
COVP analysis of the CT term. Note that it is not necessary to run a full EDA job. It is suffice to set FRGM_LPCORR to RS or ARS and EDA_COVP to TRUE to perform the COVP analysis. The orbitals of the most significant occupied-virtual pair are printed into an ASCII file called plot.mo which can be converted into a cube file and visualized in VMD.
$molecule
0 1
--
0 1
O -1.521720 0.129941 0.000000
H -1.924536 -0.737533 0.000000
H -0.571766 -0.039961 0.000000
--
0 1
O 1.362840 -0.099704 0.000000
H 1.727645 0.357101 -0.759281
H 1.727645 0.357101 0.759281
$end
$rem
JOBTYPE SP
BASIS 6-31G
PURECART 1112
EXCHANGE B3LYP
CORRELATION NONE
FRGM_METHOD GIA
FRGM_LPCORR RS
IANLTY 200
EDA_COVP TRUE
EDA_PRINT_COVP TRUE
$end
$plots
MOs
80 -4.0 4.0
60 -3.0 3.0
60 -3.0 3.0
2 0 0 0
6 11
$end
12.6 Job Control for Locally-Projected SCF Methods
FRGM_METHOD
Specifies a locally-projected method. |
TYPE:
DEFAULT:
OPTIONS:
STOLL | Locally-projected SCF equations of Stoll are solved. |
GIA | Locally-projected SCF equations of Gianinetti are solved. |
NOSCF_RS | Single Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_ARS | Approximate single Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_DRS | Double Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_RS_FOCK | Non-converged SCF energy of the single Roothaan-step MOs. |
RECOMMENDATION:
STOLL and GIA are for variational optimization of the ALMOs. NOSCF options are for computationally fast corrections of the FRAGMO initial guess. |
|
| FRGM_LPCORR
Specifies a correction method performed after the locally-projected equations are converged. |
TYPE:
DEFAULT:
OPTIONS:
ARS | Approximate Roothaan-step perturbative correction. |
RS | Single Roothaan-step perturbative correction. |
EXACT_SCF | Full SCF variational correction. |
ARS_EXACT_SCF | Both ARS and EXACT_SCF in a single job. |
RS_EXACT_SCF | Both RS and EXACT_SCF in a single job. |
RECOMMENDATION:
For large basis sets use ARS, use RS if ARS fails. |
|
|
|
SCF_PRINT_FRGM
Controls the output of Q-Chem jobs on isolated fragments. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | The output is printed to the parent job output file. |
FALSE | The output is not printed. |
RECOMMENDATION:
Use TRUE if details about isolated fragments are important. |
|
| EDA_BSSE
Calculates the BSSE correction when performing the energy decomposition analysis. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE unless a very large basis set is used. |
|
|
|
EDA_COVP
Perform COVP analysis when evaluating the RS or ARS charge-transfer correction. COVP analysis is currently implemented only for systems of two fragments. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE to perform COVP analysis in an EDA or SCF MI(RS) job. |
|
| EDA_PRINT_COVP
Replace the final MOs with the CVOP orbitals in the end of the run. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE to print COVP orbitals instead of conventional MOs. |
|
|
|
NVO_LIN_MAX_ITE
Maximum number of iterations in the preconditioned conjugate gradient solver of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of iterations. |
RECOMMENDATION:
|
| NVO_LIN_CONVERGENCE
Target error factor in the preconditioned conjugate gradient solver of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Solution of the single-excitation amplitude equations is considered converged if the maximum residual is less than 10−n multiplied by the current DIIS error. For the ARS correction, n is automatically set to 1 since the locally-projected DIIS error is normally several orders of magnitude smaller than the full DIIS error. |
|
|
|
NVO_METHOD
Sets method to be used to converge solution of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Experimental option. Use default. |
|
| NVO_UVV_PRECISION
Controls convergence of the Taylor series when calculating the Uvv block from the single-excitation amplitudes. Series is considered converged when the maximum element of the term is less than 10−n. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
NVO_UVV_PRECISION must be the same as or larger than THRESH. |
|
|
|
NVO_UVV_MAXPWR
Controls convergence of the Taylor series when calculating the Uvv block from the single-excitation amplitudes. If the series is not converged at the nth term, more expensive direct inversion is used to calculate the Uvv block. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| NVO_TRUNCATE_DIST
Specifies which atomic blocks of the Fock matrix are used to construct the preconditioner. |
TYPE:
DEFAULT:
OPTIONS:
n > 0 | If distance between a pair of atoms is more than n angstroms |
| do not include the atomic block. |
-2 | Do not use distance threshold, use NVO_TRUNCATE_PRECOND instead. |
-1 | Include all blocks. |
0 | Include diagonal blocks only. |
RECOMMENDATION:
This option does not affect the final result. However, it affects the rate of the PCG algorithm convergence. For small systems use default. |
|
|
|
NVO_TRUNCATE_PRECOND
Specifies which atomic blocks of the Fock matrix are used to construct the preconditioner. This variable is used only if NVO_TRUNCATE_DIST is set to −2. |
TYPE:
DEFAULT:
OPTIONS:
n | If the maximum element in an atomic block is less than 10−n do not include |
| the block. |
RECOMMENDATION:
Use default. Increasing n improves convergence of the PCG algorithm but overall may slow down calculations. |
|
12.7 The Explicit Polarization (XPol) Method
XPol is an approximate, fragment-based molecular orbital method that was developed as a
"next-generation" force field [601,[602,[591,[603].
The basic idea of the method is to treat
a molecular liquid, solid, or cluster as a collection of fragments, where each fragment is a molecule. Intra-molecular interactions are treated with a self-consistent field method (Hartree-Fock
or DFT), but each fragment is embedded in a field of point charges
that represent electrostatic interactions with the other fragments. These charges are updated
self-consistently by collapsing each fragment's electron density onto a set of atom-centered point
charges, using charge analysis procedures (Mulliken, Löwdin, or CHELPG, for example;
see Section 10.3.1). This approach incorporates many-body polarization, at a cost
that scales linearly with the number of fragments, but neglects the antisymmetry
requirement of the total electronic wavefunction. As a result, intermolecular exchange-repulsion
is neglected, as is dispersion since the latter is an electron correlation effect. As such,
the XPol treatment of polarization must be augmented with empirical, Lennard-Jones-type intermolecular
potentials in order to obtain meaningful optimized geometries, vibrational frequencies or
dynamics.
The XPol method is based upon an ansatz
in which the supersystem wavefunction is written as a direct product of fragment
wavefunctions,
where Nfrag is the number of fragments. We assume here that the fragments are molecules and that covalent
bonds remain intact. The fragment wavefunctions are antisymmetric with respect to exchange of
electrons within a fragment, but not to exchange between fragments.
For closed-shell fragments described by Hartree-Fock theory, the XPol total energy
is [591,[592]
EXPol = |
∑
A
|
| ⎡ ⎣
|
2 |
∑
a
|
ca† ( hA+ JA − \tfrac12KA) ca +EnucA | ⎤ ⎦
|
+ Eembed . |
| (12.2) |
The term in square brackets is the ordinary Hartree-Fock energy expression for fragment A. Thus,
ca is a vector of occupied MO expansion coefficients (in the AO basis) for the occupied MO
a ∈ A; hA consists of the one-electron integrals; and JA and
KA are the Coulomb and exchange matrices, respectively, constructed from the density matrix
for fragment A. The additional terms in Eq. ,
Eembed = \tfrac12 |
∑
A
|
|
∑
B ≠ A
|
|
∑
J ∈ B
|
| ⎛ ⎝
|
−2 |
∑
a
|
ca† IJ ca + |
∑
I ∈ A
|
LIJ | ⎞ ⎠
|
qJ , |
| (12.3) |
arise from the electrostatic embedding.
The matrix IJ is defined by its AO matrix elements,
( IJ )μν = | 〈
|
μ | ⎢ ⎢
⎢
|
1
| ⎢ ⎢
⎢
|
ν | 〉
|
, |
| (12.4) |
and LIJ is given by
According to Eqs. and , each fragment is embedded in the electrostatic
potential arising from a set of point charges, {qJ}, on all of the other fragments; the factor of
1/2 in Eq. avoids
double-counting. Exchange interactions between fragments are ignored, and the electrostatic interactions
between fragments are approximated by interactions between the charge density of one fragment and point
charges on the other fragments.
Crucially, the vectors ca are constructed within the ALMO
ansatz [586], so that MOs for each fragment
are represented in terms of only those AOs that are centered on atoms in the same fragment. This
choice affords a method whose cost grows
linearly with respect to Nfrag, and where basis set superposition
error is excluded by construction. In compact basis sets, the ALMO ansatz excludes inter-
fragment charge transfer as well.
The original XPol method of Xie et al. [602,[591,[603]
uses Mulliken charges for the embedding charges qJ in Eq. ,
though other charge schemes could be envisaged. In non-minimal basis sets, the use of Mulliken
charges is beset by severe convergence problems [592], and
Q-Chem's implementation of XPol offers the alternative of using either
Löwdin charges or "CHELPG" charges [468], the latter being.
derived from the electrostatic potential as discussed in
Section 10.3.1. The CHELPG charges are found to be stable
and robust, albeit with a somewhat larger computational cost as compared to Mulliken or
Löwdin charges [592,[469].
Researchers who use Q-Chem's XPol code are asked to cite
Refs. , and .
12.7.1 Supplementing XPol with Empirical Potentials
In order to obtain physical results, one must either supplement the XPol energy expression with
either empirical intermolecular potentials or else with an ab initio treatment of
intermolecular interactions. The latter approach is described in Section 12.9. Here,
we describe how to add Lennard-Jones or Buckingham potentials to the XPol energy, using
the $xpol_mm and $xpol_params sections described below.
The Lennard-Jones potential is
VLJ(Rij) = 4 ϵij | ⎡ ⎣
| ⎛ ⎝
|
σij
Rij
| ⎞ ⎠
|
12
|
− | ⎛ ⎝
|
σij
Rij
| ⎞ ⎠
|
6
| ⎤ ⎦
|
, |
| (12.6) |
where Rij represents the distance between atoms i and j. This potential is characterized
by two parameters, a well depth ϵij and a length scale σij. Although
quite common, the R−12 repulsion is unrealistically steep. The Buckingham potential
replaces this with an exponential function,
VBuck(Rij) = ϵij | ⎡ ⎣
|
A e−B [(Rij)/(σij)] − C | ⎛ ⎝
|
σij
Rij
| ⎞ ⎠
|
6
| ⎤ ⎦
|
, |
| (12.7) |
Here, A, B, and C are additional (dimensionless) constants, independent of atom type.
In both Eq. and Eq. , the parameters ϵij
and σij are determined using the geometric mean of atomic well-depth and length-scale
parameters. For example,
The atomic parameters σi and ϵi must be specified using a $xpol_mm
section in the Q-Chem input file. The format is a molecular mechanics-like
specification of atom types and connectivities. All atoms specified in the $molecule section
must also be specified in the $xpol_mm section. Each line must contain an atom number, atomic symbol, Cartesian coordinates, integer atom type,
and any connectivity data. The $xpol_params section specifies, for each atom type, a value for ϵ in kcal/mol and a
value for σ in Angstroms. A Lennard-Jones potential is used by default; if
a Buckingham potential is desired, then
the first line of the $xpol_params section should contain the string BUCKINGHAM
followed by values for the A, B, and C parameters.
The use of these sections is shown in two examples below using Lennard-Jones and Buckingham
potentials, respectively.
Example 12.0
An XPol single point calculation on the water dimer using a Lennard-Jones potential.
$molecule
0 1
-- water 1
0 1
O -1.364553 .041159 .045709
H -1.822645 .429753 -.713256
H -1.841519 -.786474 .202107
-- water 2
0 1
O 1.540999 .024567 .107209
H .566343 .040845 .096235
H 1.761811 -.542709 -.641786
$end
$rem
EXCHANGE HF
CORRELATION NONE
BASIS 3-21G
XPOL TRUE
XPOL_CHARGE_TYPE QLOWDIN
$end
$xpol_mm
1 O -1.364553 .041159 .045709 1 2 3
2 H -1.822645 .429753 -.713256 2 1
3 H -1.841519 -.786474 .202107 2 1
4 O 1.540999 .024567 .107209 1 5 6
5 H .566343 .040845 .096235 2 4
6 H 1.761811 -.542709 -.641786 2 4
$end
$xpol_params
1 0.16 3.16
2 0.00 0.00
$end
Example 12.0
An XPol single point calculation on the water dimer using a Buckingham potential.
$molecule
0 1
-- water 1
0 1
O -1.364553 .041159 .045709
H -1.822645 .429753 -.713256
H -1.841519 -.786474 .202107
-- water 2
0 1
O 1.540999 .024567 .107209
H .566343 .040845 .096235
H 1.761811 -.542709 -.641786
$end
$rem
EXCHANGE HF
CORRELATION NONE
BASIS 3-21G
XPOL TRUE
XPOL_CHARGE_TYPE QLOWDIN
$end
$xpol_mm
1 O -1.364553 .041159 .045709 1 2 3
2 H -1.822645 .429753 -.713256 2 1
3 H -1.841519 -.786474 .202107 2 1
4 O 1.540999 .024567 .107209 1 5 6
5 H .566343 .040845 .096235 2 4
6 H 1.761811 -.542709 -.641786 2 4
$end
$xpol_params
BUCKINGHAM 500000.0 12.5 2.25
1 0.16 3.16
2 0.00 0.00
$end
12.7.2 Job Control Variables for XPol
XPol calculations are enabled by setting the $rem variable XPOL to TRUE.
These calculations can be used in combination with Hartree-Fock theory and with most density
functionals, a notable exception being that XPol is not yet implemented for meta-GGA
functionals (Section 4.3.3). Combining XPol with solvation models
(Section 10.2) or external charges ($external_charges) is also not available.
Analytic gradients are available when Mulliken or Löwdin embedding charges are used, but
not yet available for CHELPG embedding charges.
XPOL
Perform a self-consistent XPol calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform an XPol calculation. |
FALSE | Do not perform an XPol calculation. |
RECOMMENDATION:
|
| XPOL_CHARGE_TYPE
Controls the type of atom-centered embedding charges for XPol calculations. |
TYPE:
DEFAULT:
OPTIONS:
QLOWDIN | Löwdin charges. |
QMULLIKEN | Mulliken charges. |
QCHELPG | CHELPG charges.
|
RECOMMENDATION:
Problems with Mulliken charges in extended basis sets can lead to XPol convergence failure.
Löwdin charges tend to be more stable, and CHELPG charges are both robust and provide
an accurate electrostatic embedding. However, CHELPG charges are more expensive to compute,
and analytic energy gradients are not yet available for this choice.
|
|
|
|
XPOL_MPOL_ORDER
Controls the order of multipole expansion that describes electrostatic interactions. |
TYPE:
DEFAULT:
OPTIONS:
GAS | No electrostatic embedding; monomers are in the gas phase. |
CHARGES | Charge embedding.
|
RECOMMENDATION:
Should be set to GAS to do a dimer SAPT calculation (see
Section 12.8).
|
|
| XPOL_PRINT
Print level for XPol calculations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Higher values prints more information |
|
|
|
12.8 Symmetry-Adapted Perturbation Theory (SAPT)
12.8.1 Theory
Symmetry-adapted perturbation theory (SAPT) is a theory of intermolecular interactions.
When computing intermolecular interaction energies one typically computes the energy
of two molecules infinitely separated and in contact, then computes the interaction energy by
subtraction. SAPT, in contrast, is a perturbative expression for the interaction energy itself.
The various terms in the perturbation series are physically meaningful, and this decomposition of
the interaction energy can aid in the interpretation of the results.
A brief overview of the theory is given below; for additional technical details, the reader is
referred to Jeziorski et al. [604,[593]. Additional context can be found
in a pair of more recent review articles [594,[605].
In SAPT, the Hamiltonian for the A…B dimer is written as
\HatH = \HatFA + \HatFB + ξ\HatWA + η\HatWB + ζ\HatV , |
| (12.9) |
where \HatWA and \HatWB are Møller-Plesset fluctuation operators for fragments A and B,
whereas ∧V consists of the intermolecular Coulomb operators. This part of the perturbation is
conveniently expressed as
|
^
V
|
= |
∑
i ∈ A
|
|
∑
j ∈ B
|
\Hatv(ij) |
| (12.10) |
with
\Hatv(ij) = |
1
|
+ |
NA
|
+ |
NB
|
+ |
V0
NA NB
|
. |
| (12.11) |
The quantity V0 is the nuclear interaction energy between the two fragments and
describes the interaction of electron j ∈ B with nucleus I ∈ A.
Starting from a zeroth-order Hamiltonian \HatH0 = \HatFA + \HatFB and zeroth-order
wavefunctions that are direct products of monomer wavefunctions,
|Ψ0 〉 = |ΨA〉|ΨB〉, the SAPT
approach is based on a symmetrized Rayleigh-Schrödinger perturbation
expansion [604,[593] with respect to the perturbation parameters
ξ, η, and ζ in Eq. . The resulting interaction energy can
be expressed as[604,[593]
Eint = |
∞ ∑
i=1
|
|
∞ ∑
j=0
|
(E(ij)pol + E(ij)exch) . |
| (12.13) |
Because it makes no sense to treat \HatWA and \HatWB at different orders of perturbation
theory, there are only two indices in this expansion: j for the monomer fluctuations potentials
and i for the intermolecular perturbation.
The terms E(ij)pol are known collectively as the polarization expansion, and these
are precisely the same terms that would appear in ordinary Rayleigh-Schrödinger
perturbation theory, which is valid when the monomers are well-separated.
The polarization expansion contains
electrostatic, induction and dispersion interactions, but in the symmetrized
Rayleigh-Schrödinger expansion, each term E(ij)pol has a
corresponding exchange term, E(ij)exch, that
arises from an antisymmetrizer \HatAAB that is introduced in order to project away
the Pauli-forbidden components of the interaction energy that would otherwise
appear [593].
The version of SAPT that is implemented in Q-Chem assumes that ξ = η = 0, an approach that
is usually called SAPT(0) [605].
Within the SAPT(0) formalism, the interaction energy is formally
expressed by the following symmetrized Rayleigh-Schrödinger
expansion [604,[593]:
Eint(ζ) = |
〈Ψ0|ζ\HatV\HatAAB|Ψ(ζ)〉
〈Ψ0|\HatAAB|Ψ(ζ)〉
|
, |
| (12.14) |
The antisymmetrizer \HatAAB in this expression can be written as
\HatAAB = |
NA! NB!
(NA+NB)!
|
\HatAA \HatAB | ⎛ ⎝
|
\Hat1 + \HatPAB + \HatP′ | ⎞ ⎠
|
, |
| (12.15) |
where \HatAA and \HatAB are antisymmetrizers for the two monomers and
\HatPAB is a sum of all one-electron exchange operators between the two monomers. The operator \HatP′ in
Eq.
denotes all of the three-electron and higher-order exchanges. This operator is neglected
in what
is known as the "single-exchange" approximation [604,[593], which
is expected to be quite accurate at typical van der Waals and larger intermolecular separations,
but sometimes breaks down at smaller intermolecular separations [606].
Only terms up to ζ = 2 in Eq. -that is,
second order in the intermolecular interaction-have been implemented
in Q-Chem. It is common to relabel these low-order terms in
the following way [cf. Eq. ]:
EintSAPT(0) = E(1)elst + E(1)exch + Epol(2) + Eexch(2) . |
| (12.16) |
The electrostatic part of the first-order energy correction is denoted E(1)elst and
represents the Coulomb interaction between the two monomer electron densities [593].
The quantity E(1)exch is the corresponding first-order
(i.e., Hartree-Fock) exchange correction. Explicit formulas for these corrections can be found
in Ref. . The second-order term from the polarization expansion,
denoted Epol(2) in Eq. , consists of a
dispersion contribution (which arises for the first time at second order) as well as a second-order
correction for induction. The latter can be written
Eind(2) = Eind(2)(A ← B) + Eind(2)(B ← A) , |
| (12.17) |
where the notation A ← B, for example, indicates that the frozen charge density of B
polarizes the density of A. In detail,
Eind(2)(A ← B) = 2 |
∑
ar
|
tar (wB)ra |
| (12.18) |
where
(wB)ar = ( |
^
v
|
B
|
)ar + |
∑
b
|
(ar|bb) |
| (12.19) |
and tar = (wB)ar/(ϵa − ϵr).
The second term in Eq. , in which A polarizes B, is obtained by
interchanging labels [592].
The second-order dispersion correction has a form reminiscent of the MP2 correlation energy:
Edisp(2) = 4 |
∑
abrs
|
|
(ar|bs) (ra|sb)
ϵa + ϵb − ϵr − ϵs
|
. |
| (12.20) |
The induction and dispersion corrections both have accompanying exchange corrections
(exchange-induction and exchange-dispersion) [604,[593].
The similarity between Eq. and the MP2 correlation energy means that
SAPT jobs, like MP2 calculations, can be greatly accelerated using resolution-of-identity (RI)
techniques, and an RI version of SAPT is available in Q-Chem.
To use it, one must specify an auxiliary basis set. The same ones used for RI-MP2 work equally
well for RI-SAPT, but one should always select the auxiliary basis set that is tailored for use
with the primary basis of interest, as in the RI-MP2 examples in Section 5.5.1.
It is common to replace Eind(2) and Eexch-ind(2) in
Eq. with their "response" (resp) analogues,
which are the infinite-order
correction for polarization arising from a frozen partner
density [604,[593]. Operationally, this
substitution involves replacing the second-order induction amplitudes, tar
in Eq. ,
with amplitudes obtained from solution of the coupled-perturbed Hartree-Fock
equations [405].
(The perturbation is simply the electrostatic potential of the other monomer.) In addition,
it is common to correct the SAPT(0) binding energy for higher-order polarization effects
by adding a correction term of the form [593,[605]
δEintHF = EintHF − | ⎛ ⎝
|
Eelst(1) + Eexch(1) + Eind,resp(2) + Eexch−ind,resp(2) | ⎞ ⎠
|
|
| (12.21) |
to the interaction energy. Here,
EintHF is the counterpoise-corrected Hartree-Fock binding energy for A…B.
Both the response corrections and the δEHFint
correction have been implemented as options in Q-Chem's implementation
of SAPT.
It is tempting to replace Hartree-Fock MOs and eigenvalues in the SAPT(0) formulas with their
Kohn-Sham counterparts, as a low-cost means of introducing monomer electron correlation. The
resulting procedure is known as SAPT(KS) [607]. Unfortunately, SAPT(KS)
results are generally in poor agreement with benchmark dispersion energies, owing to incorrect
asymptotic behavior of approximate exchange-correlation potentials, and this realization led
to the development of a method called SAPT(DFT) [608],
in which the sum-over-states dispersion formula in
Eq. is replaced with a generalized
Casimir-Polder-type expression based on frequency-dependent density susceptibilities for the
monomers. The latter are computed by solving time-dependent coupled Hartree-Fock or Kohn-Sham
equations [608]. Although it is possible to perform SAPT(KS) calculations
with Q-Chem, SAPT(DFT) is not implemented in Q-Chem.
Finally, some discussion of basis sets is warranted. Typically, SAPT calculations are performed in the
so-called dimer-centered basis set (DCBS) [609], which means that the combined A + B
basis set is used to calculate the zeroth-order wavefunctions for both A and B.
This leads to the unusual situation that there are more MOs than basis functions: one set of occupied
and virtual MOs for each monomer, both expanded in the same (dimer) AO basis. As an alternative to
the DCBS, one might calculate |ΨA〉 using only A's basis functions (similarly
for B), in which case the SAPT calculation is said to employ the monomer-centered basis set
(MCBS) [609]. However, MCBS results are generally of poorer quality.
As an efficient alternative to the DCBS, Jacobson and Herbert [592]
introduced a projected ("proj") basis set, borrowing
an idea from dual-basis MP2 calculations [181]. In this approach, the SCF iterations
are performed in the MCBS but then Fock matrices for
fragments A and B are constructed in the dimer (A+B) basis set and
"psuedo-canonicalized", meaning that the occupied-occupied and
virtual-virtual blocks of these matrices are diagonalized. This procedure does not mix occupied
and virtual orbitals, and thus
leaves the fragment densities and and zeroth-order fragment energies unchanged. However, it
does provide a larger set
of virtual orbitals that extend over the partner fragment. This larger virtual space is then
used to evaluate the
perturbative corrections. All three of these basis options (MCBS, DCBS, and projected
basis) are available in Q-Chem.
12.8.2 Job Control for SAPT Calculations
Q-Chem's implementation of SAPT(0) was designed from the start as a correction for XPol
calculations, a functionality that is described in Section 12.9. As such, a SAPT
calculation is requested by setting both of the $rem variable SAPT and XPOL to
TRUE. (Alternatively, one may set RISAPT=TRUE to use the RI version of
SAPT.) If one wishes to perform a traditional
SAPT calculation based on gas-phase SCF monomer wavefunctions rather than XPol monomer
wavefunctions, then the $rem variable XPOL_MPOL_ORDER should be set to GAS.
SAPT energy components are printed separately at the end of a SAPT job.
If EXCHANGE = HF, then the calculation corresponds to SAPT(0), whereas a SAPT(KS)
calculation is requested by specifying the desired density functional. [Note that meta-GGAs are
not yet available for SAPT(KS) calculations in Q-Chem.] At present, only single-point energies
for closed-shell (restricted) calculations are possible. Frozen orbitals are also unavailable.
Researchers who use Q-Chem's SAPT code are asked to cite
Refs. and .
SAPT
Requests a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Run a SAPT calculation. |
FALSE | Do not run SAPT.
|
RECOMMENDATION:
If SAPT is set to TRUE, one should also specify XPOL=TRUE and
XPOL_MPOL_ORDER=GAS.
|
|
| RISAPT
Requests an RI-SAPT calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Compute four-index integrals using the RI approximation. |
FALSE | Do not use RI.
|
RECOMMENDATION:
Set to TRUE if an appropriate auxiliary basis set is available, as
RI-SAPT is much faster and affords negligible errors (as compared to ordinary SAPT)
if the auxiliary
basis set is matched to the primary basis set. (The former must be specified using
AUX_BASIS.)
|
|
|
|
SAPT_ORDER
Selects the order in perturbation theory for a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
SAPT1 | First order SAPT. |
SAPT2 | Second order SAPT. |
ELST | First-order Rayleigh-Schrödinger perturbation theory. |
RSPT | Second-order Rayleigh-Schrödinger perturbation theory.
|
RECOMMENDATION:
SAPT2 is the most meaningful. |
|
SAPT_EXCHANGE
Selects the type of first-order exchange that is used in a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
S_SQUARED | Compute first order exchange in the single-exchange ("S2") approximation. |
S_INVERSE | Compute the exact first order exchange.
|
RECOMMENDATION:
The single-exchange approximation is expected to be adequate except possibly at very short
intermolecular distances, and is somewhat faster to compute.
|
|
| SAPT_BASIS
Controls the MO basis used for SAPT corrections. |
TYPE:
DEFAULT:
OPTIONS:
MONOMER | Monomer-centered basis set (MCBS). |
DIMER | Dimer-centered basis set (DCBS). |
PROJECTED | Projected basis set.
|
RECOMMENDATION:
The DCBS is more costly than the MCBS and can only be used with
XPOL_MPOL_ORDER=GAS (i.e., it is not available for use with XPol).
The PROJECTED choice is an efficient compromise that is available for use with
XPol. |
|
|
|
SAPT_CPHF
Requests that the second-order corrections Eind(2) and Eexch-ind(2)
be replaced by their infinite-order "response" analogues, Eind,resp(2) and
Eexch-ind,resp(2).
|
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluate the response corrections and use Eind,resp(2) and
Eexch-ind,resp(2) |
FALSE | Omit these corrections and use
Eind(2) and Eexch-ind(2).
|
RECOMMENDATION:
Computing the response corrections requires solving CPHF equations for pair
of monomers, which is somewhat expensive but may improve the accuracy when
the monomers are polar.
|
|
| SAPT_DSCF
Request the δEintHF correction |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluate this correction. |
FALSE | Omit this correction.
|
RECOMMENDATION:
Evaluating the δEintHF correction requires an SCF calculation on
the entire (super)system. This corrections effectively yields a
"Hartree-Fock plus dispersion"
estimate of the interaction energy. |
|
|
|
SAPT_PRINT
Controls level of printing in SAPT. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Larger values generate additional output. |
|
Example 12.0
Example showing a SAPT(0) calculation using the RI approximation in a DCBS.
$rem
BASIS AUG-CC-PVDZ
AUX_BASIS RIMP2-AUG-CC-PVDZ
EXCHANGE HF
RISAPT TRUE
XPOL TRUE
XPOL_MPOL_ORDER GAS ! gas-phase monomer wave functions
SAPT_BASIS DIMER
SYM_IGNORE TRUE
$end
$molecule
0 1
-- formamide
0 1
C -2.018649 0.052883 0.000000
O -1.452200 1.143634 0.000000
N -1.407770 -1.142484 0.000000
H -1.964596 -1.977036 0.000000
H -0.387244 -1.207782 0.000000
H -3.117061 -0.013701 0.000000
-- formamide
0 1
C 2.018649 -0.052883 0.000000
O 1.452200 -1.143634 0.000000
N 1.407770 1.142484 0.000000
H 1.964596 1.977036 0.000000
H 0.387244 1.207782 0.000000
H 3.117061 0.013701 0.000000
$end
12.9 The XPol+SAPT Method
XPol+SAPT was proposed by Jacobson and Herbert [592,[469] as a
low-order-scaling, systematically-improveable method applicable to large systems. The
idea is simply to replace the need for empirical parameters in the XPol method with on-the-fly
evaluation of exchange-repulsion and dispersion interactions via pairwise-additive
SAPT [592]. From another point of view, this approach uses XPol to evaluate
many-body (non-pairwise-additive) polarization effects, but then assumes that dispersion and
exchange-repulsion interactions are pairwise additive, and evaluates them via
pairwise SAPT(0) or SAPT(KS) calculations.
In particular, the zeroth-order Hamiltonian for XPol+SAPT is taken by the the sum of
fragment Fock operators defined by the XPol procedure, and the perturbation is the usual
SAPT intermolecular perturbation [Eq. ] less the intermolecular
interactions contained in the XPol fragment Fock operators.
A standard SAPT(0) correction is then computed for each pair of monomers,
using Eq. in
conjunction with the modified perturbation, to obtain dimer interaction energy EintAB.
The total energy for XPol+SAPT is then
EXPol+SAPT = |
∑
A
|
| ⎛ ⎝
|
∑
a
|
| ⎡ ⎣
|
2 ϵaA − ca† ( JA − \tfrac12KA)ca | ⎤ ⎦
|
+ EnucA + |
∑
B > A
|
EintAB | ⎞ ⎠
|
. |
| (12.22) |
In this expression, we have removed the
over-counting of two-electron interactions present in Hartree-Fock theory, effectively taking the
intrafragment perturbation to first order. The generalization to a Kohn-Sham description
of the monomers is straightforward, and is available in Q-Chem.
The inclusion of many-body polarization within the zeroth-order Hamiltonian makes the subsequent SAPT
corrections less meaningful in terms of energy decomposition analysis. For instance, the
first-order electrostatic correction in XPS is not the total electrostatic energy,
since the former
corrects for errors in the approximate electrostatic treatment at zeroth order (i.e.,
the electrostatic embedding).
The dispersion correction may be less contaminated, since
all of the XPS modifications to the traditional SAPT perturbation are one-electron operators
and therefore the pairwise dispersion correction differs
from its traditional SAPT analogue only insofar as the MOs are perturbed by the electrostatic
embedding. This should be kept in mind when interpreting the output of an XPol+SAPT calculation.
A combined XPol+SAPT calculation is requested by setting the $rem variables
XPOL and SAPT equal to TRUE and also
setting XPOL_MPOL_ORDER = CHARGES.
As with XPol, XPol+SAPT will not function with a solvation model or external changes.
Currently, only single-point energies are available (i.e., gradients would need to be evaluated
via finite difference), and only for closed-shell (restricted) calculations with no frozen
orbitals.
Researchers who use Q-Chem's XPol+SAPT code are asked to cite
Refs. and . The latter contains a
thorough discussion of the theory; a briefer summary can be found in
Ref. .
Example 12.0
Example showing an XPol+SAPT(0) calculation using CHELPG charges and CPHF.
$rem
BASIS CC-PVDZ
EXCHANGE HF
XPOL TRUE
XPOL_MPOL_ORDER CHARGES
XPOL_CHARGE_TYPE QCHELPG
SAPT TRUE
SAPT_CPHF TRUE
SYM_IGNORE TRUE
$end
$molecule
0 1
-- formic acid
0 1
C -1.888896 -0.179692 0.000000
O -1.493280 1.073689 0.000000
O -1.170435 -1.166590 0.000000
H -2.979488 -0.258829 0.000000
H -0.498833 1.107195 0.000000
-- formic acid
0 1
C 1.888896 0.179692 0.000000
O 1.493280 -1.073689 0.000000
O 1.170435 1.166590 0.000000
H 2.979488 0.258829 0.000000
H 0.498833 -1.107195 0.000000
$end
Appendix A Geometry Optimization with Q-Chem
A.1 Introduction
Geometry optimization refers to the determination of stationary points,
principally minima and transition states, on molecular potential energy
surfaces. It is an iterative process, requiring the repeated calculation of
energies, gradients and (possibly) Hessians at each optimization cycle until
convergence is attained. The optimization step involves modifying the current
geometry, utilizing current and previous energy, gradient and Hessian
information to produce a revised geometry which is closer to the target
stationary point than its predecessor was. The art of geometry optimization
lies in calculating the step h, the displacement from the starting
geometry on that cycle, so as to converge in as few cycles as possible.
There are four main factors that influence the rate of convergence. These are:
- Initial starting geometry.
- Algorithm used to determine the step h.
- Quality of the Hessian (second derivative) matrix.
- Coordinate system chosen.
The first of these factors is obvious: the closer the initial geometry is to
the final converged geometry the fewer optimization cycles it will take to
reach it. The second factor is again obvious: if a poor step h is
predicted, this will obviously slow down the rate of convergence. The third
factor is related to the second: the best algorithms make use of second
derivative (curvature) information in calculating h, and the better this
information is, the better will be the predicted step. The importance of the
fourth factor (the coordinate system) has been generally appreciated
later on: a good choice of coordinates can enhance the convergence
rate by an order of magnitude (a factor of 10) or more, depending on the
molecule being optimized.
Q-Chem includes a powerful suite of algorithms for geometry optimization
written by Jon Baker and known collectively as Optimize. These algorithms have
been developed and perfected over the past ten years and the code is robust and
has been well tested. Optimize is a general geometry optimization package for
locating both minima and transition states. It can optimize using Cartesian,
Z-matrix coordinates or delocalized internal coordinates. The last of these
are generated automatically from the Cartesian coordinates and are often found
to be particularly effective. It also handles fixed constraints on distances,
angles, torsions and out-of-plane bends, between any atoms in the molecule,
whether or not the desired constraint is satisfied in the starting geometry.
Finally it can freeze atomic positions, or any x, y, z Cartesian atomic
coordinates.
Optimize is designed to operate with minimal user input. All that is required
is the initial guess geometry, either in Cartesian coordinates (e.g., from a
suitable model builder such as HyperChem) or as a Z-matrix, the type of
stationary point being sought (minimum or transition state) and details of any
imposed constraints. All decisions as to the optimization strategy (what
algorithm to use, what coordinate system to choose, how to handle the
constraints) are made by Optimize.
Note particularly, that although the starting geometry is input in a particular
coordinate system (as a Z-matrix, for example) these coordinates are not
necessarily used during the actual optimization. The best coordinates for the
majority of geometry optimizations are delocalized internals, and these will be
tried first. Only if delocalized internals fail for some reason, or if
conditions prevent them being used (e.g., frozen atoms) will other coordinate
systems be tried. If all else fails the default is to switch to Cartesian
coordinates. Similar defaults hold for the optimization algorithm, maximum step
size, convergence criteria, etc. You may of course override the default
choices and force a particular optimization strategy, but it is not normally
necessary to provide Optimize with anything other than the minimal information
outlined above.
The heart of the Optimize package (for both minima and transition states) is
Baker's eigenvector-following (EF) algorithm [395]. This was
developed following the work of Cerjan and Miller [611] and
Simons and co-workers [612,[613]. The Hessian
mode-following option incorporated into this algorithm is capable of locating
transition states by walking uphill from the associated minima. By following
the lowest Hessian mode, the EF algorithm can locate transition states starting
from any reasonable input geometry and Hessian.
An additional option available for minimization is Pulay's GDIIS
algorithm [614], which is based on the well known DIIS technique for
accelerating SCF convergence [174]. GDIIS must be specifically
requested, as the EF algorithm is the default.
Although optimizations can be carried out in Cartesian or Z-matrix
coordinates, the best choice, as noted above, is usually delocalized internal
coordinates. These coordinates were developed by Baker
et al. [397], and can be considered as a further extension of the natural
internal coordinates developed by Pulay et al. [615,[398]
and the redundant optimization method of Pulay and Fogarasi [616].
Optimize incorporates a very accurate and efficient Lagrange multiplier
algorithm for constrained optimization. This was originally developed for use
with Cartesian coordinates [617,[618] and can handle
constraints that are not satisfied in the starting geometry. The
Lagrange multiplier approach has been modified for use with delocalized
internals [619]; this is much more efficient and is now the
default. The Lagrange multiplier code can locate constrained transition states
as well as minima.
A.2 Theoretical Background
Consider the energy, E(x0) at some point x0 on a potential energy
surface. We can express the energy at a nearby point x = x0 + h
by means of the Taylor series
E(x0 +h) = E(x0) + ht |
d E(x0)
dx
|
+ |
1
2
|
ht |
d2E(x0)
dx1 dx2
|
h +… |
| (A.1) |
If we knew the exact form of the energy functional E(x) and all its
derivatives, we could move from the current point x0 directly to a
stationary point, (i.e., we would know exactly what the step h ought to
be). Since we typically know only the lower derivatives of E(x) at best,
then we can estimate the step h by differentiating the Taylor series with
respect to h, keeping only the first few terms on the right hand side,
and setting the left hand side, d E(x0+h)/dh, to zero, which
is the value it would have at a genuine stationary point. Thus
|
d E(x0 + h)
dh
|
= |
dE(x0)
dx
|
+ |
d2E(x0)
dx1 dx2
|
h+higher terms (ignored) |
| (A.2) |
from which
where
|
dE
dx
|
≡ g (gradient vector), |
d2E
dx1 dx2
|
≡ H (Hessian matrix) |
| (A.4) |
Equation (A.3) is known as the Newton-Raphson step. It is the major component
of almost all geometry optimization algorithms in quantum chemistry.
The above derivation assumed exact first (gradient) and second (Hessian)
derivative information. Analytical gradients are available for all
methodologies supported in Q-Chem; however analytical second derivatives are
not. Furthermore, even if they were, it would not necessarily be advantageous
to use them as their evaluation is usually computationally demanding, and,
efficient optimizations can in fact be performed without an exact Hessian. An
excellent compromise in practice is to begin with an approximate Hessian
matrix, and update this using gradient and displacement information generated
as the optimization progresses. In this way the starting Hessian can be
"improved" at essentially no cost. Using Eq. (A.3) with an approximate
Hessian is called the quasi Newton-Raphson step.
The nature of the Hessian matrix (in particular its eigenvalue structure) plays
a crucial role in a successful optimization. All stationary points on a
potential energy surface have a zero gradient vector; however the character of
the stationary point (i.e., what type of structure it corresponds to) is
determined by the Hessian. Diagonalization of the Hessian matrix can be
considered to define a set of mutually orthogonal directions on the energy
surface (the eigenvectors) together with the curvature along those directions
(the eigenvalues). At a local minimum (corresponding to a well in the potential
energy surface) the curvature along all of these directions must be positive,
reflecting the fact that a small displacement along any of these directions
causes the energy to rise. At a transition state, the curvature is negative
(i.e., the energy is a maximum) along one direction, but positive along all
the others. Thus, for a stationary point to be a transition state the Hessian
matrix at that point must have one and only one negative eigenvalue, while for
a minimum the Hessian must have all positive eigenvalues. In the latter case
the Hessian is called positive definite. If searching for a minimum it
is important that the Hessian matrix be positive definite; in fact, unless the
Hessian is positive definite there is no guarantee that the step predicted by
Eq. (A.3) is even a descent step (i.e., a direction that will actually
lower the energy). Similarly, for a transition state search, the Hessian must
have one negative eigenvalue. Maintaining the Hessian eigenvalue structure is
not difficult for minimization, but it can be a difficulty when trying to find
a transition state.
In a diagonal Hessian representation the Newton-Raphson step can be written
where ui and bi are the eigenvectors and eigenvalues of the Hessian
matrix H and Fi = uitg is the component of g along the
local direction (eigenmode) ui. As discussed by Simons et al. [612],
the Newton-Raphson step can be considered as minimizing
along directions ui which have positive eigenvalues and maximizing along
directions with negative eigenvalues. Thus, if the user is searching for a
minimum and the Hessian matrix is positive definite, then the Newton-Raphson
step is appropriate since it is attempting to minimize along all directions
simultaneously. However, if the Hessian has one or more negative eigenvalues,
then the basic Newton-Raphson step is not appropriate for a minimum search,
since it will be maximizing and not minimizing along one or more directions.
Exactly the same arguments apply during a transition state search except that
the Hessian must have one negative eigenvalue, because the user has to maximize
along one direction. However, there must be only one negative
eigenvalue. A positive definite Hessian is a disaster for a transition state
search because the Newton-Raphson step will then lead towards a minimum.
If firmly in a region of the potential energy surface with the right Hessian
character, then a careful search (based on the Newton-Raphson step) will
almost always lead to a stationary point of the correct type. However, this is
only true if the Hessian is exact. If an approximate Hessian is being improved
by updating, then there is no guarantee that the Hessian eigenvalue structure
will be retained from one cycle to the next unless one is very careful during
the update. Updating procedures that "guarantee" conservation of a positive
definite Hessian do exist (or at least warn the user if the update is likely to
introduce negative eigenvalues). This can be very useful during a minimum
search; but there are no such guarantees for preserving the Hessian character
(one and only one negative eigenvalue) required for a transition state.
In addition to the difficulties in retaining the correct Hessian character,
there is the matter of obtaining a "correct" Hessian in the first instance.
This is particularly acute for a transition state search. For a minimum search
it is possible to "guess" a reasonable, positive-definite starting Hessian (for
example, by carrying out a molecular mechanics minimization initially and using
the mechanics Hessian to begin the ab initio optimization) but this
option is usually not available for transition states. Even if the user
calculates the Hessian exactly at the starting geometry, the guess for the
structure may not be sufficiently accurate, and the expensive, exact Hessian
may not have the desired eigenvalue structure.
Consequently, particularly for a transition state search, an alternative to the
basic Newton-Raphson step is clearly needed, especially when the Hessian matrix
is inappropriate for the stationary point being sought.
One of the first algorithms that was capable of taking corrective action during
a transition state search if the Hessian had the wrong eigenvalue structure,
was developed by Poppinger [620], who suggested that, instead
of taking the Newton-Raphson step, if the Hessian had all positive
eigenvalues, the lowest Hessian mode be followed uphill; whereas, if there were
two or more negative eigenvalues, the mode corresponding to the least negative
eigenvalue be followed downhill. While this step should lead the user back into
the right region of the energy surface, it has the disadvantage that the user
is maximizing or minimizing along one mode only, unlike the Newton-Raphson
step which maximizes/minimizes along all modes simultaneously. Another drawback
is that successive such steps tend to become linearly dependent, which degrades
most of the commonly used Hessian updates.
A.3 Eigenvector-Following (EF) Algorithm
The work of Cerjan and Miller [611], and later Simons and
co-workers [612,[613], showed that there was a better step
than simply directly following one of the Hessian eigenvectors. A simple
modification to the Newton-Raphson step is capable of guiding the search away
from the current region towards a stationary point with the required
characteristics. This is
in which λ can be regarded as a shift parameter on the Hessian
eigenvalue bi. Scaling the Newton-Raphson step in this manner effectively
directs the step to lie primarily, but not exclusively (unlike Poppinger's
algorithm [620]), along one of the local eigenmodes, depending
on the value chosen for λ.
References all utilize the same basic
approach of Eq. (A.6) but differ in the means of determining the value of
λ.
The EF algorithm [395] utilizes the rational function approach
presented in Refs. , yielding an eigenvalue equation of the form
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
=λ | ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
|
| (A.7) |
from which a suitable λ can be obtained. Expanding Eq. (A.7) yields
and
In terms of a diagonal Hessian representation, Eq. (A.8) rearranges to
Eq. (A.6), and substitution of Eq. (A.6) into the diagonal form of
Eq. (A.9) gives
which can be used to evaluate λ iteratively.
The eigenvalues, λ, of the RFO equation Eq. (A.7) have the
following important properties [613]:
- The (n+1) values of λ bracket the n eigenvalues of the
Hessian matrix λi < bi < λi+1.
- At a stationary point, one of the eigenvalues, λ, of Eq. (A.7) is
zero and the other n eigenvalues are those of the Hessian at the
stationary point.
- For a saddle point of order m, the zero eigenvalue separates the m
negative and the (n−m) positive Hessian eigenvalues.
This last property, the separability of the positive and negative Hessian
eigenvalues, enables two shift parameters to be used, one for modes along which
the energy is to be maximized and the other for which it is minimized. For a
transition state (a first-order saddle point), in terms of the Hessian
eigenmodes, we have the two matrix equations
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
=λp | ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
|
| (A.11) |
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
=λn | ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
|
| (A.12) |
where it is assumed that we are maximizing along the lowest Hessian mode
u1. Note that λp is the highest eigenvalue of Eq. (A.11), which
is always positive and approaches zero at convergence, and λn is
the lowest eigenvalue of Eq. (A.12), which it is always negative and again
approaches zero at convergence.
Choosing these values of λ gives a step that attempts to maximize along
the lowest Hessian mode, while at the same time minimizing along all the other
modes. It does this regardless of the Hessian eigenvalue structure (unlike the
Newton-Raphson step). The two shift parameters are then used in
Eq. (A.6) to give the final step
h = |
−F1
(b1 −λp )
|
u1 − |
n ∑
i=2
|
|
−Fi
(bi −λn )
|
ui |
| (A.13) |
If this step is greater than the maximum allowed, it is scaled down. For
minimization only one shift parameter, λn, would be used which would
act on all modes.
In Eq. (A.11) and Eq. (A.12) it was assumed that the step would maximize along the
lowest Hessian mode, b1, and minimize along all the higher modes. However,
it is possible to maximize along modes other than the lowest, and in this way
perhaps locate transition states for alternative rearrangements / dissociations
from the same initial starting point. For maximization along the kth
mode (instead of the lowest mode), Eq. (A.11) is replaced by
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
| ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
=λp | ⎛ ⎜
⎜ ⎝
|
|
| ⎞ ⎟
⎟ ⎠
|
|
| (A.14) |
and Eq. (A.12) would now exclude the kth mode but include the lowest.
Since what was originally the kth mode is the mode along which
the negative eigenvalue is required, then this mode will eventually become the
lowest mode at some stage of the optimization. To ensure that the original mode
is being followed smoothly from one cycle to the next, the mode that is
actually followed is the one with the greatest overlap with the mode followed
on the previous cycle. This procedure is known as mode following. For
more details and some examples, see Ref. .
A.4 Delocalized Internal Coordinates
The choice of coordinate system can have a major influence on the rate of
convergence during a geometry optimization. For complex potential energy
surfaces with many stationary points, a different choice of coordinates can
result in convergence to a different final structure.
The key attribute of a good set of coordinates for geometry optimization is the
degree of coupling between the individual coordinates. In general, the less
coupling the better, as variation of one particular coordinate will then have
minimal impact on the other coordinates. Coupling manifests itself primarily as
relatively large partial derivative terms between different coordinates. For
example, a strong harmonic coupling between two different coordinates, i and
j, results in a large off-diagonal element, Hij, in the Hessian (second
derivative) matrix. Normally this is the only type of coupling that can be
directly "observed" during an optimization, as third and higher derivatives
are ignored in almost all optimization algorithms.
In the early days of computational quantum chemistry geometry optimizations
were carried out in Cartesian coordinates. Cartesians are an obvious choice as
they can be defined for all systems and gradients and second derivatives are
calculated directly in Cartesian coordinates. Unfortunately, Cartesians
normally make a poor coordinate set for optimization as they are heavily
coupled. Cartesians have been returning to favor later on because of their
very general nature, and because it has been clearly demonstrated that if
reliable second derivative information is available (i.e., a good starting
Hessian) and the initial geometry is reasonable, then Cartesians can be as
efficient as any other coordinate set for small to medium-sized
molecules [621,[618]. Without good Hessian data, however, Cartesians
are inefficient, especially for long chain acyclic systems.
In the 1970s Cartesians were replaced by Z-matrix coordinates. Initially the
Z-matrix was utilized simply as a means of geometry input; it is far easier
to describe a molecule in terms of bond lengths, bond angles and dihedral
angles (the natural way a chemist thinks of molecular structure) than to
develop a suitable set of Cartesian coordinates. It was subsequently found that
optimization was generally more efficient in Z-matrix coordinates than in
Cartesians, especially for acyclic systems. This is not always the case, and
care must be taken in constructing a suitable Z-matrix. A good general rule
is ensure that each variable is defined in such a way that changing its value
will not change the values of any of the other variables. A brief discussion
concerning good Z-matrix construction strategy is given by Schlegel [622].
In 1979 Pulay et al. published a key paper, introducing what were termed
natural internal coordinates into geometry optimization [615].
These coordinates involve the use of individual bond displacements as
stretching coordinates, but linear combinations of bond angles and torsions as
deformational coordinates. Suitable linear combinations of bends and torsions
(the two are considered separately) are selected using group theoretical
arguments based on local pseudo symmetry. For example, bond angles around an
sp3 hybridized carbon atom are all approximately tetrahedral, regardless
of the groups attached, and idealized tetrahedral symmetry can be used to
generate deformational coordinates around the central carbon atom.
The major advantage of natural internal coordinates in geometry optimization is
their ability to significantly reduce the coupling, both harmonic and
anharmonic, between the various coordinates. Compared to natural internals,
Z-matrix coordinates arbitrarily omit some angles and torsions (to prevent
redundancy), and this can induce strong anharmonic coupling between the
coordinates, especially with a poorly constructed Z-matrix. Another advantage
of the reduced coupling is that successful minimizations can be carried out in
natural internals with only an approximate (e.g., diagonal) Hessian provided
at the starting geometry. A good starting Hessian is still needed for a
transition state search.
Despite their clear advantages, natural internals have only become used widely
at a later stage. This is because, when used in the early programs, it was
necessary for the user to define them. This situation changed in 1992 with the
development of computational algorithms capable of automatically generating
natural internals from input Cartesians [398]. For minimization,
natural internals have become the coordinates of first
choice [398,[618].
There are some disadvantages to natural internal coordinates as they are
commonly constructed and used:
- Algorithms for the automatic construction of natural internals are
complicated. There are a large number of structural possibilities, and to
adequately handle even the most common of them can take several thousand
lines of code.
- For the more complex molecular topologies, most assigning algorithms
generate more natural internal coordinates than are required to
characterize all possible motions of the system (i.e., the generated
coordinate set contains redundancies).
- In cases with a very complex molecular topology (e.g., multiply fused
rings and cage compounds) the assigning algorithm may be unable to
generate a suitable set of coordinates.
The redundancy problem has been addressed in an excellent paper by
Pulay and Fogarasi [616], who have developed a scheme for carrying
out geometry optimization directly in the redundant coordinate space.
Baker et al. have developed a set of delocalized internal
coordinates [397] which eliminate all of the above-mentioned
difficulties. Building on some of the ideas in the redundant optimization
scheme of Pulay and Fogarasi [616], delocalized internals form a
complete, non-redundant set of coordinates which are as good as, if not
superior to, natural internals, and which can be generated in a simple and
straightforward manner for essentially any molecular topology, no matter how
complex.
Consider a set of n internal coordinates q = (q1,q2,…qn)t
Displacements ∆q in q are related to the corresponding
Cartesian displacements ∆X by means of the usual B-matrix [623],
If any of the internal coordinates q are redundant, then the rows
of the B-matrix will be linearly dependent.
Delocalized internal coordinates are obtained simply by constructing and
diagonalizing the matrix G=BBt. Diagonalization of G
results in two sets of eigenvectors; a set of m (typically 3N−6, where N
is the number of atoms) eigenvectors with eigenvalues λ > 0, and a set
of nm eigenvectors with eigenvalues λ = 0 (to numerical precision).
In this way, any redundancies present in the original coordinate set q
are isolated (they correspond to those eigenvectors with zero eigenvalues).
The eigenvalue equation of G can thus be written
where U is the set of non-redundant eigenvectors of G (those with
λ > 0) and R is the corresponding redundant set.
The nature of the original set of coordinates q is unimportant, as long
as it spans all the degrees of freedom of the system under consideration. We
include in q, all bond stretches, all planar bends and all proper
torsions that can be generated based on the atomic connectivity. These
individual internal coordinates are termed primitives. This blanket
approach generates far more primitives than are necessary, and the set q
contains much redundancy. This is of little concern, as solution of
Eq. (A.16) takes care of all redundancies.
Note that eigenvectors in both U and R will each be linear
combinations of potentially all the original primitives. Despite this apparent
complexity, we take the set of non-redundant vectors U as our working
coordinate set. Internal coordinates so defined are much more delocalized than
natural internal coordinates (which are combinations of a relatively small
number of bends or torsions) hence, the term delocalized internal coordinates.
It may appear that because delocalized internals are such a complicated
mixing of the original primitive internals, they are a poor choice for use
in an actual optimization. On the contrary, arguments can be made that
delocalized internals are, in fact, the "best" possible choice, certainly at
the starting geometry. The interested reader is referred to the original
literature for more details [397].
The situation for geometry optimization, comparing Cartesian, Z-matrix and
delocalized internal coordinates, and assuming a "reasonable" starting
geometry, is as follows:
- For small or very rigid medium-sized systems (up to about 15 atoms),
optimizations in Cartesian and internal coordinates ("good" Z-matrix or
delocalized internals) should perform similarly.
- For medium-sized systems (say 15-30 atoms) optimizations in Cartesians
should perform as well as optimizations in internal coordinates, provided
a reliable starting Hessian is available.
- For large systems (30+ atoms), unless these are very rigid, neither
Cartesian nor Z-matrix coordinates can compete with delocalized
internals, even with good quality Hessian information. As the system
increases, and with less reliable starting geometries, the advantage of
delocalized internals can only increase.
There is one particular situation in which Cartesian coordinates may be the
best choice. Natural internal coordinates (and by extension delocalized
internals) show a tendency to converge to low energy structures [618].
This is because steps taken in internal coordinate space
tend to be much larger when translated into Cartesian space, and, as a result,
higher energy local minima tend to be "jumped over", especially if there is
no reliable Hessian information available (which is generally not needed for a
successful optimization). Consequently, if the user is looking for a local
minimum (i.e., a metastable structure) and has both a good starting geometry
and a decent Hessian, the user should carry out the optimization in Cartesian
coordinates.
A.5 Constrained Optimization
Constrained optimization refers to the optimization of molecular structures in
which certain parameters (e.g., bond lengths, bond angles or dihedral angles)
are fixed. In quantum chemistry calculations, this has traditionally been
accomplished using Z-matrix coordinates, with the desired parameter set in
the Z-matrix and simply omitted from the optimization space. In 1992, Baker
presented an algorithm for constrained optimization directly in Cartesian
coordinates [617]. Baker's algorithm used both penalty functions
and the classical method of Lagrange multipliers [624], and was
developed in order to impose constraints on a molecule obtained from a
graphical model builder as a set of Cartesian coordinates. Some improvements
widening the range of constraints that could be handled were made in 1993 [618].
Q-Chem includes the latest version of this algorithm,
which has been modified to handle constraints directly in delocalized internal
coordinates [619].
The essential problem in constrained optimization is to minimize a function of,
for example, n variables F(x) subject to a series of m constraints of
the form Ci(x) = 0, i=l. m. Assuming m < n, then perhaps the best
way to proceed (if this were possible in practice) would be to use the m
constraint equations to eliminate m of the variables, and then solve the
resulting unconstrained problem in terms of the ((n−m) independent variables.
This is exactly what occurs in a Z-matrix optimization. Such an approach
cannot be used in Cartesian coordinates as standard distance and angle
constraints are non-linear functions of the appropriate coordinates. For
example a distance constraint (between atoms i and j in a molecule) is
given in Cartesians by (Rij − R0) = 0, with
Rij = | √
|
( xi −xj )2+( yi −yj )2+( zi −zj )2
|
|
| (A.17) |
and R0 the constrained distance. This obviously cannot be satisfied by
elimination. What can be eliminated in Cartesians are the individual x, y
and z coordinates themselves and in this way individual atoms can be totally
or partially frozen.
Internal constraints can be handled in Cartesian coordinates by introducing the
Lagrangian function
L(x,λ) = F(x)− |
m ∑
i=1
|
λi Ci(x) |
| (A.18) |
which replaces the function F(x) in the unconstrained case. Here, the
λi are the so-called Lagrange multipliers, one for each constraint
Ci(x). Differentiating Eq. (A.18) with respect to x and λ affords
| |
|
|
dF(x)
dxj
|
− |
m ∑
i=1
|
λi |
dCi(x)
dxj
|
|
| | (A.19) |
| |
|
| | (A.20) |
|
At a stationary point of the Lagrangian we have ∇L = 0, i.e., all
dL/dxj = 0 and all dL/dλi = 0. This latter condition means that all
Ci(x) = 0 and thus all constraints are satisfied. Hence, finding a set
of values (x, λ) for which ∇L = 0 will give a
possible solution to the constrained optimization problem in exactly the same
way as finding an x for which g = ∇F = 0 gives a solution
to the corresponding unconstrained problem.
The Lagrangian second derivative matrix, which is the analogue of the Hessian matrix
in an unconstrained optimization, is given by
where
| |
|
|
d2F(x)
dxj dxk
|
− |
m ∑
i=1
|
λi |
d2Ci(x)
dxj dxk
|
|
| | (A.22) |
| |
|
| | (A.23) |
| |
|
| | (A.24) |
|
Thus, in addition to the standard gradient vector and Hessian matrix for the
unconstrained function F(x), we need both the first and second
derivatives (with respect to coordinate displacement) of the constraint
functions. Once these quantities are available, the corresponding Lagrangian
gradient, given by Eq. (A.19), and Lagrangian second derivative matrix, given by
Eq. (A.21), can be formed, and the optimization step calculated in a similar
manner to that for a standard unconstrained optimization [617].
In the Lagrange multiplier method, the unknown multipliers, λi, are an
integral part of the parameter set. This means that the optimization space
consists of all n variables x plus all m Lagrange multipliers
λ, one for each constraint. The total dimension of the constrained
optimization problem, nm, has thus increased by m compared to the
corresponding unconstrained case. The Lagrangian Hessian matrix,
∇2L, has m extra modes compared to the standard (unconstrained)
Hessian matrix, ∇2F. What normally happens is that these additional
modes are dominated by the constraints (i.e., their largest components
correspond to the constraint Lagrange multipliers) and they have negative
curvature (a negative Hessian eigenvalue). This is perhaps not surprising when
one realizes that any motion in the parameter space that breaks the constraints
is likely to lower the energy.
Compared to a standard unconstrained minimization, where a stationary point is
sought at which the Hessian matrix has all positive eigenvalues, in the
constrained problem we are looking for a stationary point of the Lagrangian
function at which the Lagrangian Hessian matrix has as many negative
eigenvalues as there are constraints (i.e., we are looking for an mth-order
saddle point). For further details and practical applications of
constrained optimization using Lagrange multipliers in Cartesian coordinates,
see [617].
Eigenvector following can be implemented in a constrained optimization in a
similar way to the unconstrained case. Considering a constrained minimization
with m constraints, then Eq. (A.11) is replaced by
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
=λp | ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
|
| (A.25) |
and Eq. (A.12) by
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
=λn | ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
|
| (A.26) |
where now the bi are the eigenvalues of ∇2L, with corresponding
eigenvectors ui, and Fi = uit∇L. Here Eq. (A.25)
includes the m constraint modes along which a negative Lagrangian Hessian
eigenvalue is required, and Eq. (A.26) includes all the other modes.
Equations (A.25) and (A.26) implement eigenvector following
for a constrained minimization. Constrained transition state searches can be
carried out by selecting one extra mode to be maximized in addition to the m
constraint modes, i.e., by searching for a saddle point of the Lagrangian
function of order m+l.
It should be realized that, in the Lagrange multiplier method, the desired
constraints are only satisfied at convergence, and not necessarily at
intermediate geometries. The Lagrange multipliers are part of the optimization
space; they vary just as any other geometrical parameter and, consequently the
degree to which the constraints are satisfied changes from cycle to cycle,
approaching 100% satisfied near convergence. One advantage this brings is
that, unlike in a standard Z-matrix approach, constraints do not have to be
satisfied in the starting geometry.
Imposed constraints can normally be satisfied to very high accuracy, 10−6
or better. However, problems can arise for both bond and dihedral angle
constraints near 0° and 180° and, instead of attempting to
impose a single constraint, it is better to split angle constraints near these
limiting values into two by using a dummy atom Baker:1993b, exactly
analogous to splitting a 180° bond angle into two 90° angles
in a Z-matrix.
Note:
Exact 0° and 180° single angle constraints cannot be
imposed, as the corresponding constraint normals, ∇Ci, are zero,
and would result in rows and columns of zeros in the Lagrangian Hessian
matrix. |
A.6 Delocalized Internal Coordinates
We do not give further details of the optimization algorithms available in
Q-Chem for imposing constraints in Cartesian coordinates, as it is far
simpler and easier to do this directly in delocalized internal coordinates.
At first sight it does not seem particularly straightforward to impose any
constraints at all in delocalized internals, given that each coordinate is
potentially a linear combination of all possible primitives. However, this
is deceptive, and in fact all standard constraints can be imposed by a
relatively simple Schmidt orthogonalization procedure. In this instance
consider a unit vector with unit component corresponding to the primitive
internal (stretch, bend or torsion) that one wishes to keep constant. This
vector is then projected on to the full set, U, of active
delocalized coordinates, normalized, and then all n, for example, delocalized
internals are Schmidt orthogonalized in turn to this normalized, projected
constraint vector. The last coordinate taken in the active space should drop
out (since it will be linearly dependent on the other vectors and the
constraint vector) leaving (n−1) active vectors and one constraint vector.
In more detail, the procedure is as follows (taken directly from
Ref. ). The initial (usually unit) constraint vector C is
projected on to the set U of delocalized internal coordinates according to
Cproj= |
∑
| 〈 C | C Uk Uk 〉 Uk |
| (A.27) |
where the summation is over all n active coordinates Uk. The
projected vector Cproj is then normalized and an (n+l)
dimensional vector space V is formed, comprising the normalized,
projected constraint vector together with all active delocalized coordinates
This set of vectors is Schmidt orthogonalized according to the standard procedure,
|
~
V
|
k
|
=αk | ⎛ ⎝
|
Vk − |
k−1 ∑
l=1
|
〈 Vk | Vk ~Vl ~Vl 〉 |
~
V
|
l
| ⎞ ⎠
|
|
| (A.29) |
where the first vector taken, V1, is Cproj. The αk
in Eq. (A.29) is a normalization factor. As noted above, the last vector
taken, Vn+1 = Uk, will drop out, leaving a fully orthonormal set
of (n−1) active vectors and one constraint vector.
After the Schmidt orthogonalization the constraint vector will contain all the
weight in the active space of the primitive to be fixed, which will have a zero
component in all of the other (n−1) vectors. The fixed primitive has thus
been isolated entirely in the constraint vector which can now be removed from
the active subspace for the geometry optimization step.
Extension of the above procedure to multiple constraints is straightforward.
In addition to constraints on individual primitives, it is also possible to
impose combinatorial constraints. For example, if, instead of a unit vector,
one started the constraint procedure with a vector in which two components were
set to unity, then this would impose a constraint in which the sum of the two
relevant primitives were always constant. In theory any desired linear
combination of any primitives could be constrained.
Note further that imposed constraints are not confined to those primitive
internals generated from the initial atomic connectivity. If we wish to
constrain a distance, angle or torsion between atoms that are not formally
connected, then all we need to do is add that particular coordinate to our
primitive set. It can then be isolated and constrained in exactly the same way
as a formal connectivity constraint.
Everything discussed thus far regarding the imposition of constraints in
delocalized internal coordinates has involved isolating each constraint in one
vector which is then eliminated from the optimization space. This is very
similar in effect to a Z-matrix optimization, in which constraints are
imposed by elimination. This, of course, can only be done if the desired
constraint is satisfied in the starting geometry. We have already seen that the
Lagrange multiplier algorithm, used to impose distance, angle and torsion
constraints in Cartesian coordinates, can be used even when the constraint is
not satisfied initially. The Lagrange multiplier method can also be used with
delocalized internals, and its implementation with internal coordinates brings
several simplifications and advantages.
In Cartesians, as already noted, standard internal constraints (bond distances,
angles and torsions) are somewhat complicated non-linear functions of the x,
y and z coordinates of the atoms involved. A torsion, for example, which
involves four atoms, is a function of twelve different coordinates. In
internals, on the other hand, each constraint is a coordinate in its own right
and is therefore a simple linear function of just one coordinate (itself).
If we denote a general internal coordinate by R, then the constraint function
Ci(R) is a function of one coordinate, Ri, and it and its
derivatives can be written
dCi(Ri)/dRi = 1; dCi(Ri)/dRj = 0 |
| (A.31) |
where R0 is the desired value of the constrained coordinate, and
Ri is its current value. From Eq. (A.31) we see that the constraint normals,
dCi(R)/dRi, are simply unit vectors and the Lagrangian Hessian matrix,
Eq. (A.21), can be obtained from the normal Hessian matrix by adding m
columns (and m rows) of, again, unit vectors.
A further advantage, in addition to the considerable simplification, is the
handling of 0° and 180° dihedral angle constraints. In
Cartesian coordinates it is not possible to formally constrain bond angles and
torsions to exactly 0° or 180° because the corresponding
constraint normal is a zero vector. Similar difficulties do not arise in
internal coordinates, at least for torsions, because the constraint normals are
unit vectors regardless of the value of the constraint; thus 0° and
180° dihedral angle constraints can be imposed just as easily as any
other value. 180° bond angles still cause difficulties, but
near-linear arrangements of atoms require special treatment even in
unconstrained optimizations; a typical solution involves replacing a near
180° bond angle by two special linear co-planar and perpendicular
bends [625], and modifying the torsions where necessary. A
linear arrangement can be enforced by constraining the co-planar and
perpendicular bends.
One other advantage over Cartesians is that in internals the constraint
coordinate can be eliminated once the constraint is satisfied to the desired
accuracy (the default tolerance is 10−6 in atomic units: Bohrs and
radians). This is not possible in Cartesians due to the functional form of the
constraint. In Cartesians, therefore, the Lagrange multiplier algorithm must be
used throughout the entire optimization, whereas in delocalized internal
coordinates it need only be used until all desired constraints are satisfied;
as constraints become satisfied they can simply be eliminated from the
optimization space and once all constraint coordinates have been eliminated
standard algorithms can be used in the space of the remaining unconstrained
coordinates. Normally, unless the starting geometry is particularly poor in
this regard, constraints are satisfied fairly early on in the optimization (and
at more or less the same time for multiple constraints), and Lagrange
multipliers only need to be used in the first half-dozen or so cycles of a
constrained optimization in internal coordinates.
A.7 GDIIS
Direct inversion in the iterative subspace (DIIS) was originally developed by
Pulay for accelerating SCF convergence [174]. Subsequently, Csaszar
and Pulay used a similar scheme for geometry optimization, which they termed
GDIIS [614]. The method is somewhat different from the usual
quasi-Newton type approach and is included in Optimize as an alternative to the
EF algorithm. Tests indicate that its performance is similar to EF, at least
for small systems; however there is rarely an advantage in using GDIIS in
preference to EF.
In GDIIS, geometries xi generated in previous optimization cycles are
linearly combined to find the "best" geometry on the current cycle
where the problem is to find the best values for the coefficients ci.
If we express each geometry, xi, by its deviation from the sought-after
final geometry, xf, i.e., xf = xi + ei, where
ei is an error vector, then it is obvious that if the conditions
and
are satisfied, then the relation
also holds.
The true error vectors ei are, of course, unknown. However, in the case
of a nearly quadratic energy function they can be approximated by
where gi is the gradient vector corresponding to the geometry xi
and H is an approximation to the Hessian matrix. Minimization of the
norm of the residuum vector r, Eq. (A.34), together with the constraint
equation, Eq. (A.35), leads to a system of (m+l) linear equations
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
| ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
= | ⎛ ⎜ ⎜ ⎜
⎜ ⎜ ⎝
|
|
| ⎞ ⎟ ⎟ ⎟
⎟ ⎟ ⎠
|
|
| (A.38) |
where Bij = 〈ei|ej〉 is the scalar product of the error
vectors ei and ej, and λ is a Lagrange multiplier.
The coefficients ci determined from Eq. (A.38) are used to calculate an
intermediate interpolated geometry
and its corresponding interpolated gradient
A new, independent geometry is generated from the interpolated geometry and
gradient according to
Note:
Convergence is theoretically guaranteed regardless of the quality of the
Hessian matrix (as long as it is positive definite), and the original GDIIS
algorithm used a static Hessian (i.e., the original starting Hessian, often a
simple unit matrix, remained unchanged during the entire optimization).
However, updating the Hessian at each cycle generally results in more rapid
convergence, and this is the default in Optimize. |
Other modifications to the original method include limiting the number of
previous geometries used in Eq. (A.33) and, subsequently, by neglecting
earlier geometries, and eliminating any geometries more than a certain distance
(default: 0.3 a.u.) from the current geometry.
Appendix B AOINTS
B.1 Introduction
Within the Q-Chem program, an Atomic Orbital INTegralS (AOINTS) package has
been developed which, while relatively invisible to the user, is one of the
keys to the overall speed and efficiency of the Q-Chem program.
"Ever since Boys' introduction of Gaussian basis sets to quantum chemistry in
1950, the calculation and handling of the notorious two-electron repulsion
integrals (ERIs) over Gaussian functions has been an important avenue of
research for practicing computational chemists. Indeed, the emergence of
practically useful computer programs has been fueled in no small part by
the development of sophisticated algorithms to compute the very large number of
ERIs that are involved in calculations on molecular systems of even modest
size" [626].
The ERI engine of any competitive quantum chemistry software package will be
one of the most complicated aspects of the package as whole. Coupled with the
importance of such an engine's efficiency, a useful yardstick of a program's
anticipated performance can be quickly measured by considering the components
of its ERI engine. In recent times, developers at Q-Chem, Inc. have made
significant contributions to the advancement of ERI algorithm technology (for
example, see Refs. ), and it is not
surprising that Q-Chem's AOINTS package is considered the most advanced of its kind.
B.2 Historical Perspective
Prior to the 1950s, the most difficult step in the systematic application of
Schrödinger wave mechanics to chemistry was the calculation of the
notorious two-electron integrals that measure the repulsion between electrons.
Boys [635] showed that this step can be made easier (although still
time consuming) if Gaussian, rather than Slater, orbitals are used in the basis
set. Following the landmark paper of computational chemistry [636]
(again due to Boys) programs were constructed that could calculate all the ERIs
that arise in the treatment of a general polyatomic molecule with s and p
orbitals. However, the programs were painfully slow and could only be applied
to the smallest of molecular systems.
In 1969, Pople constructed a breakthrough ERI algorithm, a hundred time faster
than its predecessors. The algorithm remains the fastest available for its
associated integral classes and is now referred to as the Pople-Hehre
axis-switch method [637].
Over the two decades following Pople's initial development, an enormous amount
of research effort into the construction of ERIs was documented, which built on
Pople's original success. Essentially, the advances of the newer algorithms
could be identified as either better coping with angular momentum (L) or,
contraction (K); each new method increasing the speed and application of
quantum mechanics to solving real chemical problems.
By 1990, another barrier had been reached. The contemporary programs had become
sophisticated and both academia and industry had begun to recognize and use the
power of ab initio quantum chemistry, but the software was struggling
with "dusty deck syndrome" and it had become increasingly difficult for it to
keep up with the rapid advances in hardware development. Vector processors,
parallel architectures and the advent of the graphical user interface were all
demanding radically different approaches to programming and it had become clear
that a fresh start, with a clean slate, was both inevitable and desirable.
Furthermore, the integral bottleneck had re-emerged in a new guise and the
standard programs were now hitting the N2 wall. Irrespective of the speed at
which ERIs could be computed, the unforgiving fact remained that the number of
ERIs required scaled quadratically with the size of the system.
The Q-Chem project was established to tackle this problem and to seek new
methods that circumvent the N2 wall. Fundamentally new approaches to
integral theory were sought and the ongoing advances that have
resulted [543,[204,[205,[638,[639]
have now placed Q-Chem firmly at the vanguard of the field. It should be
emphasized, however, that the O(N) methods that we have developed still
require short-range ERIs to treat interactions between nearby electrons, thus
the importance of contemporary ERI code remains.
The chronological development and evolution of integral methods can be
summarized by considering a time line showing the years in which important new
algorithms were first introduced. These are best discussed in terms of the type
of ERI or matrix elements that the algorithm can compute efficiently.
1950 | Boys | | ERIs with low L and low K |
1969 | Pople | | ERIs with low L and high K |
1976 | Dupuis | | Integrals with any L and low K |
1978 | McMurchie | | Integrals with any L and low K |
1982 | Almlöf | | Introduction of the direct SCF approach |
1986 | Obara | | Integrals with any L and low K |
1988 | Head-Gordon | | Integrals with any L and low K |
1991 | Gill | | Integrals with any L and any K |
1994 | White | | J matrix in linear work |
1996 | Schwegler | | HF exchange matrix in linear work |
1997 | Challacombe | | Fock matrix in linear work |
B.3 AOINTS: Calculating ERIs with Q-Chem
The area of molecular integrals with respect to Gaussian basis functions has
recently been reviewed [627] and the user is referred to this review
for deeper discussions and further references to the general area. The purpose
of this short account is to present the basic approach, and in particular, the
implementation of ERI algorithms and aspects of interest to the user in the
AOINTS package which underlies the Q-Chem program.
We begin by observing that all of the integrals encountered in an ab
initio calculation, of which overlap, kinetic energy, multipole moment,
internuclear repulsion, nuclear-electron attraction and inter electron
repulsion are the best known, can be written in the general form
( ab|cd )= | ⌠ ⌡
|
ϕa (r1 )ϕb (r1 )θ(r12 )ϕc (r2 )ϕd (r2 )dr1 dr2 |
| (B.1) |
where the basis functions are contracted Gaussian's (CGTF)
ϕa (r)=( x−Ax )ax ( y−Ay )ay ( z−Az )az |
Ka ∑
i=1
|
Dai e−αi | r−A |2 |
| (B.2) |
and the operator θ is a two-electron operator. Of the two-electron
operators (Coulomb, CASE, anti-Coulomb and delta-function) used in the Q-Chem
program, the most significant is the Coulomb, which leads us to the ERIs.
An ERI is the classical Coulomb interaction (θ(x) = 1/x in B.1) between
two charge distributions referred to as bras (ab| and kets |cd).
B.4 Shell-Pair Data
It is common to characterize a bra, a ket and a bra-ket by their degree of
contraction and angular momentum. In general, it is more convenient to compile
data for shell-pairs rather than basis-function pairs. A shell is defined as
that sharing common exponents and centers. For example, in the case of a number
of Pople derived basis sets, four basis functions, encompassing a range of
angular momentum types (i.e., s, px, py, pz on the same atomic
center sharing the same exponents constitute a single shell.
The shell-pair data set is central to the success of any modern integral
program for three main reasons. First, in the formation of shell-pairs, all
pairs of shells in the basis set are considered and categorized as either
significant or negligible. A shell-pair is considered negligible if the shells
involved are so far apart, relative to their diffuseness, that their overlap is
negligible. Given the rate of decay of Gaussian basis functions, it is not
surprising that most of the shell-pairs in a large molecule are negligible,
that is, the number of significant shell-pairs increases linearly with the
size of the molecule. Second, a number of useful intermediates which are
frequently required within ERI algorithms should be computed once in
shell-pair formation and stored as part of the shell-pair information,
particularly those which require costly divisions. This prevents re-evaluating
simple quantities. Third, it is useful to sort the shell-pair information by
type (i.e., angular momentum and degree of contraction). The reasons for this
are discussed below.
Q-Chem's shell-pair formation offers the option of two basic integral
shell-pair cutoff criteria; one based on the integral threshold ($rem
variable THRESH) and the other relative to machine precision.
Intelligent construction of shell-pair data scales linearly with the size of
the basis set, requires a relative amount of CPU time which is almost entirely
negligible for large direct SCF calculations, and for small jobs, constitutes
approximately 10% of the job time.
B.5 Shell-Quartets and Integral Classes
Given a sorted list of shell-pair data, it is possible to construct all
potentially important shell-quartets by pairing of the shell-pairs with one
another. Because the shell-pairs have been sorted, it is possible to deal with
batches of integrals of the same type or class (e.g., (ss|ss),
(sp|sp), (dd|dd), etc.) where an integral class is
characterized by both angular momentum (L) and degree of contraction (K).
Such an approach is advantageous for vector processors and for semi-direct
integral algorithms where the most expensive (high K or L integral classes
can be computed once, stored in memory (or disk) and only less expensive
classes rebuilt on each iteration.
While the shell-pairs may have been carefully screened, it is possible for a
pair of significant shell-pairs to form a shell-quartet which need not be
computed directly. Three cases are:
- The quartet is equivalent, by point group symmetry, to another quartet
already treated.
- The quartet can be ignored on the basis of cheaply computed ERI bounds [631]
on the largest quartet bra-ket.
- On the basis of an incremental Fock matrix build, the largest density
matrix element which will multiply any of the bra-kets associated with
the quartet may be negligibly small.
Note:
Significance and negligibility is always based on the level
of integral threshold set by the $rem variable THRESH. |
B.6 Fundamental ERI
The fundamental ERI [627] and the basis of all ERI algorithms is
usually represented
| |
|
| |
| |
|
DA DB DC DD | ⌠ ⌡
|
e−α| r1 −A |2e−β| r1 −B |2 | ⎡ ⎣
|
1
r12
| ⎤ ⎦
|
e−γ| r2 −C |2e−δ| r2 −D |2dr1 dr2 |
| | (B.3) |
|
which can be reduced to a one-dimensional integral of the form
[0](0)=U(2 ϑ2)1/2 | ⎛ ⎝
|
2
π
| ⎞ ⎠
|
1/2
|
|
1 ⌠ ⌡ 0
|
e−Tu2du |
| (B.4) |
and can be efficiently computed using a modified Chebyshev interpolation
scheme [629]. Equation (B.4) can also be adapted for the
general case [0](m) integrals required for most calculations.
Following the fundamental ERI, building up to the full bra-ket ERI (or
intermediary matrix elements, see later) are the problems of angular momentum
and contraction.
Note:
Square brackets denote primitive integrals and parentheses denote
fully-contracted integrals. |
B.7 Angular Momentum Problem
The fundamental integral is essentially an integral without angular momentum
(i.e., it is an integral of the type [ss|ss]). Angular momentum,
usually depicted by L, has been problematic for efficient ERI formation,
evident in the above time line. Initially, angular momentum was calculated by
taking derivatives of the fundamental ERI with respect to one of the Cartesian
coordinates of the nuclear center. This is an extremely inefficient route, but
it works and was appropriate in the early development of ERI methods. Recursion
relations [643,[644] and the newly developed tensor
equations [158] are the basis for the modern approaches.
B.8 Contraction Problem
The contraction problem may be described by considering a general contracted
ERI of s-type functions derived from the STO-3G basis set. Each basis
function has degree of contraction K = 3. Thus, the ERI may be written
| |
|
|
3 ∑
i=1
|
|
3 ∑
j=1
|
|
3 ∑
k=1
|
|
3 ∑
l=1
|
DAi DBj DCk DDl |
| |
| |
|
× | ⌠ ⌡
|
e−αi | r1 −A |2e−βj | r1 −B |2 | ⎡ ⎣
|
1
r12
| ⎤ ⎦
|
e−γk | r2 −C |2e−δl | r2 −D |2dr1 dr2 |
| |
| |
|
|
3 ∑
i=1
|
|
3 ∑
j=1
|
|
3 ∑
k=1
|
|
3 ∑
l=1
|
[si sj |sk sl ] |
| | (B.5) |
|
and requires 81 primitive integrals for the single ERI. The problem escalates
dramatically for more highly contracted sets (STO-6G, 6-311G) and has been the
motivation for the development of techniques for shell-pair modeling [645],
in which a second shell-pair is constructed with fewer
primitives that the first, but introduces no extra error relative to the
integral threshold sought.
The Pople-Hehre axis-switch method [637] is excellent for high
contraction low angular momentum integral classes.
B.9 Quadratic Scaling
The success of quantitative modern quantum chemistry, relative to its
primitive, qualitative beginnings, can be traced to two sources: better
algorithms and better computers. While the two technologies continue to improve
rapidly, efforts are heavily thwarted by the fact that the total number of ERIs
increases quadratically with the size of the molecular system. Even large
increases in ERI algorithm efficiency yield only moderate increases in
applicability, hindering the more widespread application of ab initio
methods to areas of, perhaps, biochemical significance where semi-empirical
techniques [646,[647] have already proven so valuable.
Thus, the elimination of quadratic scaling algorithms has been the theme of
many research efforts in quantum chemistry throughout the 1990s and has seen
the construction of many alternative algorithms to alleviate the problem.
Johnson was the first to implement DFT exchange / correlation functionals whose
computational cost scaled linearly with system size [648]. This
paved the way for the most significant breakthrough in the area with the linear
scaling CFMM algorithm [543] leading to linear scaling DFT
calculations [154]. Further breakthroughs have been made with
traditional theory in the form of the
QCTC [638,[649,[650] and
ONX [639,[162] algorithms, while more radical
approaches [204,[205] may lead to entirely new approaches
to ab initio
calculations. Investigations into the quadratic Coulomb problem has not only
yielded linear scaling algorithms, but is also providing large insights into
the significance of many molecular energy components.
Linear scaling Coulomb and SCF exchange / correlation algorithms are not the end
of the story as the O(N3) diagonalization step has been rate limiting
in semi-empirical techniques and, been predicted [651] to become
rate limiting in ab initio approaches in the medium term. However,
divide-and-conquer techniques [652,[653,[654,[655]
and the recently developed quadratically convergent SCF algorithm [536]
show great promise for reducing this problem.
B.10 Algorithm Selection
No single ERI algorithm is available to efficiently handle all integral
classes; rather, each tends to have specific integral classes where the
specific algorithm outperforms the alternatives. The PRISM algorithm [630]
is an intricate collection of pathways and steps in which the
path chosen is that which is the most efficient for a given class. It appears
that the most appropriate path for a given integral class depends on the
relative position of the contraction step (lowly contracted bra-kets prefer
late contraction, highly contracted bra-kets are most efficient with early
contraction steps).
Careful studies have provided FLOP counts which are the current basis of
integral algorithm selection, although care must be taken to ensure that
algorithms are not rate limited by MOPs [628]. Future algorithm
selection criteria will take greater account of memory, disk, chip
architecture, cache size, vectorization and parallelization characteristics of
the hardware, many of which are already exist within Q-Chem.
B.11 More Efficient Hartree-Fock Gradient and Hessian Evaluations
Q-Chem combines the Head-Gordon-Pople (HGP) method [632] and the
COLD prism method [158] for Hartree-Fock gradient and Hessian
evaluations. All two-electron four-center integrals are classified according
to their angular momentum types and degrees of contraction. For each type of
integrals, the program chooses one with a lower cost. In practice, the HGP
method is chosen for most integral classes in a gradient or Hessian
calculation, and thus it dominates the total CPU time.
Recently the HGP codes within Q-Chem were completely rewritten for the
evaluation of the P IIx P term in the gradient evaluation, and the P
IIxy P term in the Hessian evaluation. Our emphasis is to improve code
efficiency by reducing cache misses rather than by reducing FLOP counts. Some
timing results from a Hartree-Fock calculation on azt are shown below.
Basis Set | AIX | Linux |
| Gradient Evaluation: P IIx P Term |
| Old | New | New/Old | Old | New | New/Old |
3-21G | 34 s | 20 s | 0.58 | 25 s | 14 s | 0.56 |
6-31G** | 259 s | 147 s | 0.57 | 212 s | 120 s | 0.57 |
DZ | 128 s | 118 s | 0.92 | 72 s | 62 s | 0.86 |
cc-pVDZ | 398 s | 274 s | 0.69 | 308 s | 185 s | 0.60 |
|
| Hessian Evaluation: P IIxy P term |
| Old | New | New/Old | Old | New | New/Old |
3-21G | 294 s | 136 s | 0.46 | 238 s | 100 s | 0.42 |
6-31G** | 2520 s | 976 s | 0.39 | 2065 s | 828 s | 0.40 |
DZ | 631 s | 332 s | 0.53 | 600 s | 230 s | 0.38 |
cc-pVDZ | 3202 s | 1192 s | 0.37 | 2715 s | 866 s | 0.32 |
|
Table B.1: The AIX timings were obtained on an IBM RS/6000 workstation with AIX4
operating system, and the Linux timings on an Opteron cluster where the
Q-Chem executable was compiled with an intel 32-bit compiler.
B.12 User-Controllable Variables
AOINTS has been optimally constructed so that the fastest integral algorithm
for ERI calculation is chosen for the given integral class and batch. Thus, the
user has not been provided with the necessary variables for overriding the
program's selection process. The user is, however, able to control the accuracy
of the cutoff used during shell-pair formation (METECO) and the
integral threshold (THRESH). In addition, the user can force the use
of the direct SCF algorithm (DIRECT_SCF) and increase the default
size of the integrals storage buffer (INCORE_INTS_BUFFER).
Currently, some of Q-Chem's linear scaling algorithms, such as QCTC and ONX
algorithms, require the user to specify their use. It is anticipated that
further research developments will lead to the identification of situations in
which these, or combinations of these and other algorithms, will be selected
automatically by Q-Chem in much the same way that PRISM algorithms choose the
most efficient pathway for given integral classes.
Appendix C Q-Chem Quick Reference
C.1 Q-Chem Text Input Summary
Keyword | Description | |
$molecule | Contains the molecular coordinate input (input file requisite). |
$rem | Job specification and customization parameters (input file requisite). |
$end | Terminates each keyword section. |
$basis | User-defined basis set information (see Chapter 7). |
$comment | User comments for inclusion into output file. |
$ecp | User-defined effective core potentials (see Chapter 8). |
$empirical_dispersion | User-defined van der Waals parameters for DFT dispersion |
| correction. |
$external_charges | External charges and their positions. |
$force_field_params | Force field parameters for QM / MM calculations
(see Section`9.12). |
$intracule | Intracule parameters (see Chapter 10). |
$isotopes | Isotopic substitutions for vibrational calculations (see
Chapter 10). |
$localized_diabatization | Information for mixing together multiple adiabatic states into |
| diabatic states (see Chapter 10). |
$multipole_field | Details of a multipole field to apply. |
$nbo | Natural Bond Orbital package. |
$occupied | Guess orbitals to be occupied. |
$opt | Constraint definitions for geometry optimizations. |
$pcm | Special parameters for polarizable continuum models (see Section |
| 10.2.3). |
$pcm_solvent | Special parameters for polarizable continuum models (see Section |
| 10.2.3). |
$plots | Generate plotting information over a grid of points (see |
| Chapter 10). |
$qm_atoms | Specify the QM region for QM / MM calculations
(see Section 9.12). |
$svp | Special parameters for the SS(V)PE module. |
$svpirf | Initial guess for SS(V)PE) module. |
$van_der_waals | User-defined atomic radii for Langevin dipoles solvation (see |
| Chapter 10). |
$xc_functional | Details of user-defined DFT exchange-correlation functionals. |
$cdft | Special options for the constrained DFT method as implemented. |
Table C.1: Q-Chem user input section keywords. See the $QC/samples
directory with your release for specific examples of Q-Chem input using these
keywords.
Note:
(1) Users are able to enter keyword sections in any order.
(2) Each keyword section must be terminated with the $end keyword.
(3) Not all keywords have to be entered, but $rem and $molecule are compulsory.
(4) Each keyword section will be described below.
(5) The entire Q-Chem input is case-insensitive.
(6) Multiple jobs are separated by the string @@@ on a single line. |
C.1.1 Keyword: $molecule
Four methods are available for inputing geometry information:
- Z-matrix (Angstroms and degrees):
$molecule
{Z-matrix}
{blank line, if parameters are being used}
{Z-matrix parameters, if used}
$end
- Cartesian Coordinates (Angstroms):
$molecule
{Cartesian coordinates}
{blank line, if parameter are being used}
{Coordinate parameters, if used}
$end
- Read from a previous calculation:
$molecule
read
$end
- Read from a file:
$molecule
read filename
$end
C.1.2 Keyword: $rem
See also the list of $rem variables at the end of this Appendix. The general
format is:
$rem
REM_VARIABLE VALUE [optional comment]
$end
C.1.3 Keyword: $basis
The format for the user-defined basis section is as follows:
$basis | | | | | |
| X | 0 | | | |
| L | K | scale | | |
| α1 | C1Lmin | C1Lmin+1 | … | C1Lmax |
| α2 | C2Lmin | C2Lmin+1 | … | C2Lmax |
| : | : | : | ··· | : |
| αK | CKLmin | CKLmin+1 | … | CKLmax |
**** | | | | | |
$end | | | | | |
where
X | Atomic symbol of the atom (atomic number not accepted) |
L | Angular momentum symbol (S, P, SP, D, F, G) |
K | Degree of contraction of the shell (integer) |
scale | Scaling to be applied to exponents (default is 1.00) |
ai | Gaussian primitive exponent (positive real number) |
CiL | Contraction coefficient for each angular momentum (non-zero real numbers). |
Atoms are terminated with **** and the complete basis set is
terminated with the $end keyword terminator. No blank lines can be
incorporated within the general basis set input. Note that more than one
contraction coefficient per line is one required for compound shells like SP.
As with all Q-Chem input deck information, all input is case-insensitive.
C.1.4 Keyword: $comment
Note that the entire input deck is echoed to the output file, thus making the
$comment keyword largely redundant.
$comment
User comments - copied to output file
$end
C.1.5 Keyword: $ecp
$ecp
For each atom that will bear an ECP
Chemical symbol for the atom
ECP name; the L value for the ECP; number of core electrons removed
For each ECP component (in the order unprojected, ∧P0 , ∧P1 , , ∧PL−1
The component name
The number of Gaussians in the component
For each Gaussian in the component
The power of r; the exponent; the contraction coefficient
****
$end
Note:
(1) All of the information in the $ecp block is case-insensitive.
(2) The L value may not exceed 4. That is, nothing beyond G projectors is allowed.
(3) The power of r (which includes the Jacobian r2 factor) must be 0, 1 or 2. |
C.1.6 Keyword: $empirical_dispersion
$empirical_dispersion
S6 S6_value
D D_value
C6 element_1 C6_value_for_element_1 element_2 C6_value_for_element_2
VDW_RADII element_1 radii_for_element_1 element_2 radii_for_element_2
$end
Note:
This section is only for values that the user wants to change from the default values recommended by Grimme. |
C.1.7 Keyword: $external_charges
All input should be given in atomic units.
Update: While charges should indeed be listed in atomic units, the units for distances depend on the user input. If the structure is specified in Angstroms (the default), the coordinates for external charges should also be in Angstroms. If the structure is specified in atomic units, the coordinates for external charges should also be in atomic units. (See INPUT_BOHR.)
$external_charges
x-coord1 y-coord1 z-coord1 charge1
x-coord2 y-coord2 z-coord2 charge2
$end
C.1.8 Keyword: $intracule
$intracule
int_type | 0 | Compute P(u) only |
| 1 | Compute M(v) only |
| 2 | Compute W(u,v) only |
| 3 | Compute P(u), M(v) and W(u,v) |
| 4 | Compute P(u) and M(v) |
| 5 | Compute P(u) and W(u,v) |
| 6 | Compute M(v) and W(u,v) |
u_points | | Number of points, start, end. |
v_points | | Number of points, start, end. |
moments | 0-4 | Order of moments to be computed (P(u) only). |
derivs | 0-4 | order of derivatives to be computed (P(u) only). |
accuracy | n | (10−n) specify accuracy of intracule interpolation table (P(u) only). |
$end
C.1.9 Keyword: $isotopes
Note that masses should be given in atomic units.
$isotopes
number_extra_loops tp_flag
number_of_atoms [temp pressure]
atom_number1 mass1
atom_number2 mass2
...
$end
C.1.10 Keyword: $multipole_field
Multipole fields are all in atomic units.
$multipole_field
field_component1 value1
field_component2 value2
...
$end
C.1.11 Keyword: $nbo
Refer to Chapter 10 and the NBO manual for further
information. Note that the NBO $rem variable must be set to
ON to initiate the NBO package.
$nbo
[ NBO options ]
$end
C.1.12 Keyword: $occupied
$occupied
1 2 3 4 ... nalpha
1 2 3 4 ... nbeta
$end
C.1.13 Keyword: $opt
Note that units are in Angstroms and degrees. Also see the summary in the
next section of this Appendix.
$opt
CONSTRAINT
stre atom1 atom2 value
...
bend atom1 atom2 atom3 value
...
outp atom1 atom2 atom3 atom4 value
...
tors atom1 atom2 atom3 atom4 value
...
linc atom1 atom2 atom3 atom4 value
...
linp atom1 atom2 atom3 atom4 value
...
ENDCONSTRAINT
FIXED
atom coordinate_reference
...
ENDFIXED
DUMMY
idum type list_length defining_list
...
ENDDUMMY
CONNECT
atom list_length list
...
ENDCONNECT
$end
C.1.14 Keyword: $svp
$svp
<KEYWORD>=<VALUE>, <KEYWORD>=<VALUE>,...
<KEYWORD>=<VALUE>
$end
For example, the section may look like this:
$svp
RHOISO=0.001, DIELST=78.39, NPTLEB=110
$end
C.1.15 Keyword: $svpirf
$svpirf
<# point> <x point> <y point> <z point> <charge> <grid weight>
<# point> <x normal> <y normal> <z normal>
$end
C.1.16 Keyword: $plots
$plots
One comment line
Specification of the 3-D mesh of points on 3 lines:
Nx xmin xmax
Ny ymin ymax
Nz zmin zmax
A line with 4 integers indicating how many things to plot:
NMO NRho NTrans NDA
An optional line with the integer list of MO's to evaluate
(only if NMO > 0)
MO(1) MO(2) … MO(NMO)
An optional line with the integer list of densities to evaluate
(only if NRho > 0)
Rho(1) Rho(2) … Rho(NRho)
An optional line with the integer list of transition densities
(only if NTrans > 0)
Trans(1) Trans(2) … Trans(NTrans)
An optional line with states for detachment/attachment densities
(if NDA > 0)
DA(1) DA(2) … DA(NDA)
$end
C.1.17 Keyword: $localized_diabatization
$plots
One comment line.
One line with an an array of adiabatic states to mix together.
< adiabat1 > < adiabat2 > < adiabat3 > …
$end
Note: We count adiabatic states such that the first excited state is < adiabat > = 1, the fifth is < adiabat > = 5, and so forth.
C.1.18 Keyword $van_der_waals
Note: all radii are given in angstroms.
$van_der_waals
1
atomic_number VdW_radius
$end
(alternative format)
$van_der_waals
2
sequential_atom_number VdW_radius
$end
C.1.19 Keyword: $xc_functional
$xc_functional
X exchange_symbol coefficient
X exchange_symbol coefficient
...
C correlation_symbol coefficient
C correlation_symbol coefficient
...
K coefficient
$end
C.2 Geometry Optimization with General Constraints
CONSTRAINT and ENDCONSTRAINT define the beginning and end,
respectively, of the constraint section of $opt within which users may
specify up to six different types of constraints:
interatomic distances
Values in angstroms; value > 0:
stre atom1 atom2 value
|
angles
Values in degrees, 0 ≤ value ≤ 180; atom2 is the middle atom
of the bend:
bend atom1 atom2 atom3 value
|
out-of-plane-bends
Values in degrees, −180 ≤ value ≤ 180 atom2; angle between
atom4 and the atom1-atom2-atom3 plane:
outp atom1 atom2 atom3 atom4 value
|
dihedral angles
Values in degrees, −180 ≤ value ≤ 180; angle the plane
atom1-atom2-atom3 makes with the plane atom2-atom3-atom4:
tors atom1 atom2 atom3 atom4 value
|
coplanar bends
Values in degrees, −180 ≤ value ≤ 180; bending of
atom1-atom2-atom3 in the plane atom2-atom3-atom4:
linc atom1 atom2 atom3 atom4 value
|
perpendicular bends
Values in degrees, −180 ≤ value ≤ 180; bending of
atom1-atom2-atom3 perpendicular to the plane
atom2-atom3-atom4:
linp atom1 atom2 atom3 atom4 value
|
C.2.1 Frozen Atoms
Absolute atom positions can be frozen with the FIXED section. The
section starts with the FIXED keyword as the first line and ends with
the ENDFIXED keyword on the last. The format to fix a coordinate or
coordinates of an atom is:
atom coordinate_reference
coordinate_reference can be any combination of up to three characters
X, Y and Z to specify the coordinate(s) to be fixed: X, Y, Z,
XY, XZ, YZ, XYZ. The fixing characters must be next
to each other. e.g.,
FIXED
2 XY
ENDFIXED
C.3 $rem Variable List
The general format of the $rem input for Q-Chem text input files is simply
as follows:
$rem
rem_variable rem_option [comment]
rem_variable rem_option [comment]
$end
This input is not case sensitive. The following sections contain the names and
options of available $rem variables for users. The format for describing each
$rem variable is as follows:
REM_VARIABLE
A short description of what the variable controls |
TYPE:
Defines the variable as either INTEGER, LOGICAL or STRING. |
DEFAULT:
Describes Q-Chem's internal default, if any exists. |
OPTIONS:
Lists options available for the user |
RECOMMENDATION:
Gives a quick recommendation. |
|
C.3.1 General
BASIS | BASIS_LIN_DEP_THRESH |
EXCHANGE | CORRELATION |
ECP | JOBTYPE |
PURECART | |
C.3.2 SCF Control
BASIS2 | BASISPROJTYPE |
DIIS_PRINT | DIIS_SUBSPACE_SIZE |
DIRECT_SCF | INCFOCK |
MAX_DIIS_CYCLES | MAX_SCF_CYCLES |
PSEUDO_CANONICAL | SCF_ALGORITHM |
SCF_CONVERGENCE | SCF_FINAL_PRINT |
SCF_GUESS | SCF_GUESS_MIX |
SCF_GUESS_PRINT | SCF_PRINT |
THRESH | THRESH_DIIS_SWITCH |
UNRESTRICTED | VARTHRESH |
C.3.3 DFT Options
CORRELATION | EXCHANGE |
FAST_XC | INC_DFT |
INCDFT_DENDIFF_THRESH | INCDFT_GRIDDIFF_THRESH |
INCDFT_DENDIFF_VARTHRESH | INCDFT_GRIDDIFF_VARTHRESH |
XC_GRID | XC_SMART_GRID |
C.3.4 Large Molecules
CFMM_ORDER | DIRECT_SCF |
EPAO_ITERATE | EPAO_WEIGHTS |
GRAIN | INCFOCK |
INTEGRAL_2E_OPR | INTEGRALS_BUFFER |
LIN_K | MEM_STATIC |
MEM_TOTAL | METECO |
OMEGA | PAO_ALGORITHM |
PAO_METHOD | THRESH |
VARTHRESH | RI_J |
RI_K | ARI |
ARI_R0 | ARI_R1 |
C.3.5 Correlated Methods
AO2MO_DISK | CD_ALGORITHM |
CORE_CHARACTER | CORRELATION |
MEM_STATIC | MEM_TOTAL |
N_FROZEN_CORE | N_FROZEN_VIRTUAL |
PRINT_CORE_CHARACTER | |
C.3.6 Correlated Methods Handled by CCMAN and CCMAN2
Most of these $rem variables that start CC_.
These are relevant for CCSD and other CC methods (OD, VOD, CCD, QCCD, etc).
CC_CANONIZE | CC_RESTART_NO_SCF |
CC_T_CONV | CC_DIIS_SIZE |
CC_DIIS_FREQ | CC_DIIS_START |
CC_DIIS_MAX_OVERLAP | CC_DIIS_MIN_OVERLAP |
CC_RESTART | CC_SAVEAMPL |
These options are only relevant to methods involving orbital optimization
(OOCD, VOD, QCCD, VQCCD):
CC_MP2NO_GUESS | CC_MP2NO_GRAD |
CC_DIIS | CC_DIIS12_SWITCH |
CC_THETA_CONV | CC_THETA_GRAD_CONV |
CC_THETA_STEPSIZE | CC_RESET_THETA |
CC_THETA_GRAD_THRESH | CC_HESS_THRESH |
CC_ED_CCD | CC_QCCD_THETA_SWITCH |
CC_PRECONV_T2Z | CC_PRECONV_T2Z_EACH |
CC_PRECONV_FZ | CC_ITERATE_OV |
CC_CANONIZE_FREQ | CC_CANONIZE_FINAL |
Properties and optimization:
CC_REF_PROP | CC_REF_PROP_TE |
CC_FULLRESPONSE | |
C.3.7 Perfect pairing, Coupled cluster valence bond, and related methods
CCVB_METHOD | CCVB_GUESS |
GVB_N_PAIRS | GVB_LOCAL |
GVB_ORB_MAX_ITER | GVB_RESTART |
GVB_ORB_CONV | GVB_ORB_SCALE |
GVB_AMP_SCALE | GVB_DO_SANO |
GVB_PRINT |
C.3.8 Excited States: CIS, TDDFT, SF-XCIS and SOS-CIS(D)
CIS_CONVERGENCE | CIS_GUESS_DISK |
CIS_GUESS_DISK_TYPE | CIS_N_ROOTS |
CIS_RELAXED_DENSITY | CIS_SINGLETS |
CIS_STATE_DERIV | CIS_TRIPLETS |
MAX_CIS_CYCLES | RPA |
XCIS | SPIN_FLIP_XCIS
|
C.3.9 Excited States: EOM-CC and CI Methods
Those are keywords relevant to EOM-CC and CI methods handled by CCMAN/CCMAN2.
Most of these $rem variables that start CC_ and EOM_.
EOM_DAVIDSON_CONVERGENCE | EOM_DAVIDSON_MAXVECTORS |
EOM_DAVIDSON_THRESHOLD | EOM_DAVIDSON_MAX_ITER |
EOM_NGUESS_DOUBLES | EOM_NGUESS_SINGLES |
EOM_DOEXDIAG | EOM_PRECONV_DOUBLES |
EOM_PRECONV_SINGLES | EOM_PRECONV_SD |
EOM_IPEA_FILTER | EOM_FAKE_IPEA |
CC_REST_AMPL | CC_REST_TRIPLES |
CC_EOM_PROP | CC_TRANS_PROP |
CC_STATE_TO_OPT | CC_EOM_PROP |
CC_EOM_PROP_TE | CC_FULLRESPONSE |
C.3.10 Geometry Optimizations
CIS_STATE_DERIV | FDIFF_STEPSIZE |
GEOM_OPT_COORDS | GEOM_OPT_DMAX |
GEOM_OPTHESSIAN | GEOM_OPT_LINEAR_ANGLE |
GEOM_OPT_MAX_CYCLES | GEOM_OPT_MAX_DIIS |
GEOM_OPT_MODE | GEOM_OPT_PRINT |
GEOM_OPTSYMFLAG | GEOM_OPT_PRINT |
GEOM_OPTTOL_ENERGY | GEOM_OPT_TOL_DISPLACEMENT |
GEOM_OPT_TOL_ENERGY | GEOM_OPT_TOL_GRADIENT |
GEOMP_OPT_UPDATE | IDERIV |
JOBTYPE | SCF_GUESS_ALWAYS |
CC_STATE_TO_OPT | |
C.3.11 Vibrational Analysis
DORAMAN | CPSCF_NSEG |
FDIFF_STEPSIZE | IDERIV |
ISOTOPES | JOBTYPE |
VIBMAN_PRINT | ANHAR |
VCI | FDIFF_DER |
MODE_COUPLING | IGNORE_LOW_FREQ |
FDIFF_STEPSIZE_QFF |
C.3.12 Reaction Coordinate Following
JOBTYPE | RPATH_COORDS |
RPATH_DIRECTION | RPATH_MAX_CYCLES |
RPATH_MAX_STEPSIZE | RPATH_PRINT |
RPATH_TOL_DISPLACEMENT | |
C.3.13 NMR Calculations
D_CPSCF_PERTNUM | D_SCF_CONV_1 |
D_SCF_CONV_2 | D_SCF_DIIS |
D_SCF_MAX_1 | D_SCF_MAX_2 |
JOBTYPE | |
C.3.14 Wavefunction Analysis and Molecular Properties
CHEMSOL | CHEMSOL_EFIELD |
CHEMSOL_NN | CHEM_SOL_PRINT |
CIS_RELAXED_DENSITY | IGDESP |
INTRACULE | MULTIPOLE_ORDER |
NBO | POP_MULLIKEN |
PRINT_DIST_MATRIX | PRINT_ORBITALS |
READ_VDW | SOLUTE_RADIUS |
SOLVENT_DIELECTRIC | STABILITY_ANALYSIS |
WAVEFUNCTION_ANALYSIS | WRITE_WFN |
C.3.15 Symmetry
CC_SYMMETRY | |
SYM_IGNORE | SYMMETRY |
SYMMETRY_DECOMPOSITION | SYM_TOL |
C.3.16 Printing Options
CC_PRINT | CHEMSOL_PRINT |
DIIS_PRINT | GEOM_OPT_PRINT |
MOM_PRINT | PRINT_CORE_CHARACTER |
PRINT_DIST_MATRIX | PRINT_GENERAL_BASIS |
PRINT_ORBITALS | RPATH_PRINT |
SCF_FINAL_PRINT | SCF_GUESS_PRINT |
SCF_PRINT | VIBMAN_PRINT |
WRITE_WFN | |
C.3.17 Resource Control
MEM_TOTAL | MEM_STATIC |
AO2MO_DISK | CC_MEMORY |
INTEGRALS_BUFFER | MAX_SUB_FILE_NUM |
DIRECT_SCF | |
C.3.18 Alphabetical Listing
| SAPT_PRINT
Controls level of printing in SAPT. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Larger values generate additional output. |
|
|
|
RISAPT
Requests an RI-SAPT calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Compute four-index integrals using the RI approximation. |
FALSE | Do not use RI.
|
RECOMMENDATION:
Set to TRUE if an appropriate auxiliary basis set is available, as
RI-SAPT is much faster and affords negligible errors (as compared to ordinary SAPT)
if the auxiliary
basis set is matched to the primary basis set. (The former must be specified using
AUX_BASIS.)
|
|
| SAPT_DSCF
Request the δEintHF correction |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluate this correction. |
FALSE | Omit this correction.
|
RECOMMENDATION:
Evaluating the δEintHF correction requires an SCF calculation on
the entire (super)system. This corrections effectively yields a
"Hartree-Fock plus dispersion"
estimate of the interaction energy. |
|
|
|
SAPT_EXCHANGE
Selects the type of first-order exchange that is used in a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
S_SQUARED | Compute first order exchange in the single-exchange ("S2") approximation. |
S_INVERSE | Compute the exact first order exchange.
|
RECOMMENDATION:
The single-exchange approximation is expected to be adequate except possibly at very short
intermolecular distances, and is somewhat faster to compute.
|
|
| SAPT_BASIS
Controls the MO basis used for SAPT corrections. |
TYPE:
DEFAULT:
OPTIONS:
MONOMER | Monomer-centered basis set (MCBS). |
DIMER | Dimer-centered basis set (DCBS). |
PROJECTED | Projected basis set.
|
RECOMMENDATION:
The DCBS is more costly than the MCBS and can only be used with
XPOL_MPOL_ORDER=GAS (i.e., it is not available for use with XPol).
The PROJECTED choice is an efficient compromise that is available for use with
XPol. |
|
|
|
SAPT_CPHF
Requests that the second-order corrections Eind(2) and Eexch-ind(2)
be replaced by their infinite-order "response" analogues, Eind,resp(2) and
Eexch-ind,resp(2).
|
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluate the response corrections and use Eind,resp(2) and
Eexch-ind,resp(2) |
FALSE | Omit these corrections and use
Eind(2) and Eexch-ind(2).
|
RECOMMENDATION:
Computing the response corrections requires solving CPHF equations for pair
of monomers, which is somewhat expensive but may improve the accuracy when
the monomers are polar.
|
|
| SAPT_ORDER
Selects the order in perturbation theory for a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
SAPT1 | First order SAPT. |
SAPT2 | Second order SAPT. |
ELST | First-order Rayleigh-Schrödinger perturbation theory. |
RSPT | Second-order Rayleigh-Schrödinger perturbation theory.
|
RECOMMENDATION:
SAPT2 is the most meaningful. |
|
|
|
SAPT
Requests a SAPT calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Run a SAPT calculation. |
FALSE | Do not run SAPT.
|
RECOMMENDATION:
If SAPT is set to TRUE, one should also specify XPOL=TRUE and
XPOL_MPOL_ORDER=GAS.
|
|
| XPOL_MPOL_ORDER
Controls the order of multipole expansion that describes electrostatic interactions. |
TYPE:
DEFAULT:
OPTIONS:
GAS | No electrostatic embedding; monomers are in the gas phase. |
CHARGES | Charge embedding.
|
RECOMMENDATION:
Should be set to GAS to do a dimer SAPT calculation (see
Section 12.8).
|
|
|
|
XPOL_PRINT
Print level for XPol calculations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Higher values prints more information |
|
| FEFP_EFP
Specifies that fEFP_EFP calculation is requested to compute the total
interaction energies between a ligand (the last fragment in the $efp_fragments
section) and the protein (represented by fEFP) |
TYPE:
DEFAULT:
OPTIONS:
OFF | disables fEFP |
LA | enables fEFP with the Link Atom (HLA or CLA) scheme (only
electrostatics and polarization) |
MFCC | enables fEFP with MFCC (only
electrostatics) |
RECOMMENDATION:
The keyword should be invoked if EFP / fEFP is requested (interaction
energy calculations). This keyword has to be employed with EFP_FRAGMENT_ONLY =
TRUE.
To switch on / off electrostatics or polarzation interactions, the usual EFP
controls are employed. |
|
|
|
FEFP_QM
Specifies that fEFP_QM calculation is requested to perform a
QM/fEFPcompute computation. The fEFP part is a fractionated macromolecule. |
TYPE:
DEFAULT:
OPTIONS:
OFF | disables fEFP_QM and performs a QM/EFP calculation |
LA | enables fEFP_QM with the Link Atom scheme |
RECOMMENDATION:
|
The keyword should be invoked if QM / fEFP is requested. This keyword has to be
employed with efp_fragment_only false. Only electrostatics is available.
XPOL_CHARGE_TYPE
Controls the type of atom-centered embedding charges for XPol calculations. |
TYPE:
DEFAULT:
OPTIONS:
QLOWDIN | Löwdin charges. |
QMULLIKEN | Mulliken charges. |
QCHELPG | CHELPG charges.
|
RECOMMENDATION:
Problems with Mulliken charges in extended basis sets can lead to XPol convergence failure.
Löwdin charges tend to be more stable, and CHELPG charges are both robust and provide
an accurate electrostatic embedding. However, CHELPG charges are more expensive to compute,
and analytic energy gradients are not yet available for this choice.
|
|
ADC_DAVIDSON_CONV
Controls the convergence criterion of the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. |
|
| ADC_DAVIDSON_MAXITER
Controls the maximum number of iterations of the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. |
|
|
|
ADC_DAVIDSON_MAXSUBSPACE
Controls the maximum subspace size for the Davidson procedure. |
TYPE:
DEFAULT:
5 × the number of excited states to be calculated. |
OPTIONS:
RECOMMENDATION:
Should be at least 2−4 × the number of excited states to calculate. The larger the value the more disk space is required. |
|
| ADC_DAVIDSON_THRESH
Controls the threshold for the norm of expansion vectors to be added
during the Davidson procedure. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless convergence problems are encountered. The threshold
value should always be smaller or equal to the convergence criterion
ADC_DAVIDSON_CONV. |
|
|
|
ADC_DIIS_ECONV
Controls the convergence criterion for the excited state energy during DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| ADC_DIIS_MAXITER
Controls the maximum number of DIIS iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase in case of slow convergence. |
|
|
|
ADC_DIIS_RCONV
Convergence criterion for the residual vector norm of the excited state during DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| ADC_DIIS_SIZE
Controls the size of the DIIS subspace. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
ADC_DIIS_START
Controls the iteration step at which DIIS is turned on. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to a large number to switch off DIIS steps. |
|
| ADC_DO_DIIS
Activates the use of the DIIS algorithm for the calculation of ADC(2) excited states. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Use DIIS algorithm. |
FALSE | Do diagonalization using Davidson algorithm. |
RECOMMENDATION:
|
|
|
ADC_EXTENDED
Activates the ADC(2)-x variant. This option is ignored unless
ADC_ORDER is set to 2. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Activate ADC(2)-x. |
FALSE | Do an ADC(2)-s calculation. |
RECOMMENDATION:
|
| ADC_NGUESS_DOUBLES
Controls the number of excited state guess vectors which are double excitations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
ADC_NGUESS_SINGLES
Controls the number of excited state guess vectors which are single excitations. If the
number of requested excited states exceeds the total number of guess vectors (singles and
doubles), this parameter is automatically adjusted, so that the number of guess vectors
matches the number of requested excited states. |
TYPE:
DEFAULT:
Equals to the number of excited states requested. |
OPTIONS:
RECOMMENDATION:
|
| ADC_ORDER
Controls the order in perturbation theory of ADC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Activate ADC(0). |
1 | Activate ADC(1). |
2 | Activate ADC(2)-s or ADC(2)-x. |
RECOMMENDATION:
|
|
|
ADC_PRINT
Controls the amount of printing during an ADC calculation. |
TYPE:
DEFAULT:
1 | Basic status information and results are printed. |
OPTIONS:
0 | Quiet: almost only results are printed. |
1 | Normal: basic status information and results are printed. |
2 | Debug1: more status information, extended timing information. |
... |
RECOMMENDATION:
|
| ADC_PROP_ES
Controls the calculation of excited state properties (currently only dipole moments). |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculate excited state properties. |
FALSE | Only calculate transition properties from the ground state. |
RECOMMENDATION:
|
|
|
ADC_SINGLETS
Controls the number of singlet excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable in case of
restricted calculation. |
|
| ADC_STATES
Controls the number of excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable to define the number of excited states in case of unrestricted or
open-shell calculations. In restricted calculations it can also be used, if the same
number of singlet and triplet states is to be requested. |
|
|
|
ADC_STATE_SYM
Controls the irreducible representations of the electronic transitions for
which excited states should be calculated. This option is ignored, unless point-group
symmetry is present in the system and CC_SYMMETRY is set to TRUE. |
TYPE:
DEFAULT:
0 | States of all irreducible representations are calculated |
| (equivalent to setting the $rem variable to 111...). |
OPTIONS:
i1 i2 ... iN
| A sequence of 0 and 1 in which each digit represents one |
| irreducible representation. |
| 1 activates the calculation of the respective electronic transitions. |
RECOMMENDATION:
The irreducible representations are ordered according to the standard ordering in Q-Chem.
For example, in a system with D2 symmetry ADC_STATE_SYM = 0101 would activate the
calculation of B1 and B3 excited states. |
|
| ADC_TRIPLETS
Controls the number of triplet excited states to calculate. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0. |
RECOMMENDATION:
Use this variable in case of restricted calculation. |
|
|
|
ADD_CHARGED_CAGE
Add a point charge cage of a given radius and total charge. |
TYPE:
DEFAULT:
OPTIONS:
0 no cage. |
1 dodecahedral cage. |
2 spherical cage. |
RECOMMENDATION:
Spherical cage is expected to yield more accurate results, especially for small radii. |
|
| AIMD_FICT_MASS
Specifies the value of the fictitious electronic mass μ, in atomic units,
where μ has dimensions of (energy)×(time)2. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Values in the range of 50-200 a.u. have been employed in test calculations;
consult [190] for examples and discussion. |
|
|
|
AIMD_INIT_VELOC
Specifies the method for selecting initial nuclear velocities. |
TYPE:
DEFAULT:
OPTIONS:
THERMAL | Random sampling of nuclear velocities from a Maxwell-Boltzmann |
| distribution. The user must specify the temperature in Kelvin via |
| the $rem variable AIMD_TEMP. |
ZPE | Choose velocities in order to put zero-point vibrational energy into |
| each normal mode, with random signs. This option requires that a |
| frequency job to be run beforehand. |
QUASICLASSICAL | Puts vibrational energy into each normal mode. In contrast to the |
| ZPE option, here the vibrational energies are sampled from a |
| Boltzmann distribution at the desired simulation temperature. This |
| also triggers several other options, as described below. |
RECOMMENDATION:
This variable need only be specified in the event that velocities are not
specified explicitly in a $velocity section. |
|
| AIMD_METHOD
Selects an ab initio molecular dynamics algorithm. |
TYPE:
DEFAULT:
OPTIONS:
BOMD | Born-Oppenheimer molecular dynamics. |
CURVY | Curvy-steps Extended Lagrangian molecular dynamics. |
RECOMMENDATION:
BOMD yields exact classical molecular dynamics, provided that the energy is
tolerably conserved. ELMD is an approximation to exact classical dynamics whose
validity should be tested for the properties of interest. |
|
|
|
AIMD_MOMENTS
Requests that multipole moments be output at each time step. |
TYPE:
DEFAULT:
0 | Do not output multipole moments. |
OPTIONS:
n | Output the first n multipole moments. |
RECOMMENDATION:
|
| AIMD_NUCL_DACF_POINTS
Number of time points to utilize in the dipole autocorrelation function for an AIMD trajectory |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute dipole autocorrelation function. |
1 ≤ n ≤ AIMD_STEPS | Compute dipole autocorrelation function for last n |
| timesteps of the trajectory. |
RECOMMENDATION:
If the DACF is desired, set equal to AIMD_STEPS. |
|
|
|
AIMD_NUCL_SAMPLE_RATE
The rate at which sampling is performed for the velocity and/or dipole autocorrelation function(s). Specified as a multiple of steps; i.e., sampling every step is 1. |
TYPE:
DEFAULT:
OPTIONS:
1 ≤ n ≤ AIMD_STEPS | Update the velocity/dipole autocorrelation function |
| every n steps. |
RECOMMENDATION:
Since the velocity and dipole moment are routinely calculated for ab initio methods,
this variable should almost always be set to 1 when the VACF/DACF are desired. |
|
| AIMD_NUCL_VACF_POINTS
Number of time points to utilize in the velocity autocorrelation function for an AIMD trajectory |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute velocity autocorrelation function. |
1 ≤ n ≤ AIMD_STEPS | Compute velocity autocorrelation function for last n |
| time steps of the trajectory. |
RECOMMENDATION:
If the VACF is desired, set equal to AIMD_STEPS. |
|
|
|
AIMD_QCT_INITPOS
Chooses the initial geometry in a QCT-MD simulation. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use the equilibrium geometry. |
n | Picks a random geometry according to the harmonic vibrational wavefunction. |
−n | Generates n random geometries sampled from |
| the harmonic vibrational wavefunction. |
|
RECOMMENDATION:
|
| AIMD_QCT_WHICH_TRAJECTORY
Picks a set of vibrational quantum numbers from a random distribution. |
TYPE:
DEFAULT:
OPTIONS:
n | Picks the nth set of random initial velocities. |
−n | Uses an average over n random initial velocities.
|
|
RECOMMENDATION:
Pick a positive number if you want the initial velocities to correspond
to a particular set of vibrational occupation numbers and choose a
different number for each of your trajectories. If initial velocities
are desired that corresponds to an average over n trajectories, pick a
negative number.
|
|
|
|
AIMD_STEPS
Specifies the requested number of molecular dynamics steps. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| AIMD_TEMP
Specifies a temperature (in Kelvin) for Maxwell-Boltzmann velocity sampling. |
TYPE:
DEFAULT:
OPTIONS:
User-specified number of Kelvin. |
RECOMMENDATION:
This variable is only useful in conjunction with AIMD_INIT_VELOC =
THERMAL. Note that the simulations are run at constant energy, rather than
constant temperature, so the mean nuclear kinetic energy will fluctuate in the
course of the simulation. |
|
|
|
ANHAR_SEL
Select a subset of normal modes for subsequent anharmonic frequency analysis. |
TYPE:
DEFAULT:
FALSE | Use all normal modes |
OPTIONS:
TRUE | Select subset of normal modes |
RECOMMENDATION:
|
| ANHAR
Performing various nuclear vibrational theory (TOSH, VPT2, VCI) calculations
to obtain vibrational anharmonic frequencies. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Carry out the anharmonic frequency calculation. |
FALSE | Do harmonic frequency calculation. |
RECOMMENDATION:
Since this calculation involves the third and fourth derivatives at the
minimum of the potential energy surface, it is recommended that the
GEOM_OPT_TOL_DISPLACEMENT, GEOM_OPT_TOL_GRADIENT and
GEOM_OPT_TOL_ENERGY tolerances are set tighter. Note that VPT2
calculations may fail if the system involves accidental degenerate resonances.
See the VCI $rem variable for more details about increasing the
accuracy of anharmonic calculations. |
|
|
|
AO2MO_DISK
Sets the scratch space size for individual program modules |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
The minimum disk requirement of RI-CIS(D) is approximately
3SOVXD. Again, the batching scheme will become more efficient with
more available disk space. There is no simple formula for SOS-CIS(D) and SOS-CIS(D0)
disk requirement. However, because the disk space is abundant in modern computers,
this should not pose any problem. Just put the available disk space size in this case.
The actual disk usage information will also be printed in the output file. |
|
| AO2MO_DISK
Sets the amount of disk space (in megabytes) available for MP2 calculations. |
TYPE:
DEFAULT:
2000 | Corresponding to 2000 Mb. |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
Should be set as large as possible, discussed in Section 5.3.1. |
|
|
|
ARI_R0
Determines the value of the inner fitting radius (in Å ngstroms) |
TYPE:
DEFAULT:
4 | A value of 4 Å will be added to the atomic van der Waals radius. |
OPTIONS:
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable. |
|
| ARI_R1
Determines the value of the outer fitting radius (in Å ngstroms) |
TYPE:
DEFAULT:
5 | A value of 5 Å will be added to the atomic van der Waals radius. |
OPTIONS:
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable. This value also determines, in part, the smoothness of the potential energy surface. |
|
|
|
ARI
Toggles the use of the atomic resolution-of-the-identity (ARI) approximation. |
TYPE:
DEFAULT:
FALSE | ARI will not be used by default for an RI-JK calculation. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaulation time. |
|
| AUX_BASIS
Specifies the type of auxiliary basis to be used in
a method that involves RI-fitting procedures. |
TYPE:
DEFAULT:
No default is assigned. Must be defined in the input |
OPTIONS:
Symbol. Choose among the auxiliary basis sets
collected in the qchem qcaux basis library |
RECOMMENDATION:
Try a few different types of aux bases first |
|
|
|
BASIS2
Sets the small basis set to use in basis set projection. |
TYPE:
DEFAULT:
No second basis set default. |
OPTIONS:
Symbol. Use standard basis sets as per Chapter 7. |
BASIS2_GEN | General BASIS2 |
BASIS2_MIXED | Mixed BASIS2 |
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using
a basis larger than a minimal basis when BASIS2 is used for initial guess purposes.
Larger, standardized BASIS2 options are available for dual-basis calculations
(see Section 4.7). |
|
| BASISPROJTYPE
Determines which method to use when projecting the density matrix of
BASIS2 |
TYPE:
DEFAULT:
FOPPROJECTION (when DUAL_BASIS_ENERGY=false) |
OVPROJECTION (when DUAL_BASIS_ENERGY=true) |
OPTIONS:
FOPPROJECTION | Construct the Fock matrix in the second basis |
OVPROJECTION | Projects MO's from BASIS2 to BASIS. |
RECOMMENDATION:
|
|
|
BASIS_LIN_DEP_THRESH
Sets the threshold for determining linear dependence in the basis set |
TYPE:
DEFAULT:
6 | Corresponding to a threshold of 10−6 |
OPTIONS:
n | Sets the threshold to 10−n |
RECOMMENDATION:
Set to 5 or smaller if you have a poorly behaved SCF and you suspect linear
dependence in you basis set. Lower values (larger thresholds) may affect the
accuracy of the calculation. |
|
| BASIS
Specifies the basis sets to be used. |
TYPE:
DEFAULT:
OPTIONS:
General, Gen | User defined ($basis keyword required). |
Symbol | Use standard basis sets as per Chapter 7. |
Mixed | Use a mixture of basis sets (see Chapter 7). |
RECOMMENDATION:
Consult literature and reviews to aid your selection. |
|
|
|
BOYSCALC
Specifies the Boys localized orbitals are to be calculated |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not perform localize the occupied space. |
1 | Allow core-valence mixing in Boys localization. |
2 | Localize core and valence separately. |
RECOMMENDATION:
|
| BOYS_CIS_NUMSTATE
Define how many states to mix with Boys localized diabatization. |
TYPE:
DEFAULT:
0 | Do not perform Boys localized diabatization. |
OPTIONS:
1 to N where N is the number of CIS states requested (CIS_N_ROOTS) |
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV
or a typical reorganization energy in solvent. |
|
|
|
CAGE_CHARGE
Defines the total charge of the cage. |
TYPE:
DEFAULT:
400 Add a cage charged +4e. |
OPTIONS:
n total charge of the cage is n / 100 a.u. |
RECOMMENDATION:
|
| CAGE_POINTS
Defines number of point charges for the spherical cage. |
TYPE:
DEFAULT:
OPTIONS:
n n point charges are used. |
RECOMMENDATION:
|
|
|
CAGE_RADIUS
Defines radius of the charged cage. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CCVB_GUESS
Specifies the initial guess for CCVB calculations |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard GVBMAN guess (orbital localization via GVB_LOCAL + Sano procedure). |
2 | Use orbitals from previous GVBMAN calculation, along with SCF_GUESS = read. |
3 | Convert UHF orbitals into pairing VB form. |
RECOMMENDATION:
Option 1 is the most useful overall. The success of GVBMAN methods is often dependent on localized orbitals, and this guess shoots for these. Option 2 is useful for comparing results to other GVBMAN methods, or if other GVBMAN methods are able to obtain a desired result more efficiently. Option 3 can be useful for bond-breaking situations when a pertinent UHF solution has been found. It works best for small systems, or if the unrestriction is a local phenomenon within a larger molecule. If the unrestriction is nonlocal and the system is large, this guess will often produce a solution that is not the global minimum. Any UHF solution has a certain number of pairs that are unrestricted, and this will be output by the program. If GVB_N_PAIRS exceeds this number, the standard GVBMAN initial-guess procedure will be used to obtain a guess for the excess pairs |
|
|
|
CCVB_METHOD
Optionally modifies the basic CCVB method |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard CCVB model |
3 | Independent electron pair approximation (IEPA) to CCVB |
4 | Variational PP (the CCVB reference energy) |
RECOMMENDATION:
Option 1 is generally recommended. Option 4 is useful for preconditioning, and for obtaining localized-orbital solutions, which may be used in subsequent calculations. It is also useful for cases in which the regular GVBMAN PP code becomes variationally unstable. Option 3 is a simple independent-amplitude approximation to CCVB. It avoids the cubic-scaling amplitude equations of CCVB, and also is able to reach the correct dissociation energy for any molecular system (unlike regular CCVB which does so only for cases in which UHF can reach a correct dissociate limit). However the IEPA approximation to CCVB is sometimes variationally unstable, which we have yet to observe in regular CCVB. |
|
| CC_CANONIZE_FINAL
Whether to semi-canonicalize orbitals at the end of the ground state
calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should not normally have to be altered. |
|
|
|
CC_CANONIZE_FREQ
The orbitals will be semi-canonicalized every n theta resets. The thetas
(orbital rotation angles) are reset every CC_RESET_THETA
iterations. The counting of iterations differs for active space (VOD, VQCCD)
calculations, where the orbitals are always canonicalized at the first
theta-reset. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Smaller values can be tried in cases that do not converge. |
|
| CC_CANONIZE
Whether to semi-canonicalize orbitals at the start of the calculation (i.e.
Fock matrix is diagonalized in each orbital subspace) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should not normally have to be altered. |
|
|
|
CC_CONVERGENCE
Overall convergence criterion for the coupled-cluster codes. This is designed
to ensure at least n significant digits in the calculated energy, and
automatically sets the other convergence-related variables
(CC_E_CONV, CC_T_CONV, CC_THETA_CONV,
CC_THETA_GRAD_CONV) [10−n]. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to 10−n convergence criterion. Amplitude convergence is set |
| automatically to match energy convergence. |
RECOMMENDATION:
|
| CC_DIIS12_SWITCH
When to switch from DIIS2 to DIIS1 procedure, or when DIIS2 procedure is
required to generate DIIS guesses less frequently. Total value of DIIS error
vector must be less than 10−n, where n is the value of this option. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CC_DIIS_FREQ
DIIS extrapolation will be attempted every n iterations. However, DIIS2 will
be attempted every iteration while total error vector exceeds
CC_DIIS12_SWITCH. DIIS1 cannot generate guesses more frequently than
every 2 iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CC_DIIS_MAX_OVERLAP
DIIS extrapolations will not begin until square root of the maximum element of
the error overlap matrix drops below this value. |
TYPE:
DEFAULT:
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
|
|
|
CC_DIIS_MIN_OVERLAP
The DIIS procedure will be halted when the square root of smallest element of
the error overlap matrix is less than 10−n, where n is the value of this
option. Small values of the B matrix mean it will become near-singular, making
the DIIS equations difficult to solve. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CC_DIIS_SIZE
Specifies the maximum size of the DIIS space. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Larger values involve larger amounts of disk storage. |
|
|
|
CC_DIIS_START
Iteration number when DIIS is turned on. Set to a large number to disable
DIIS. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Occasionally DIIS can cause optimized orbital coupled-cluster calculations to
diverge through large orbital changes. If this is seen, DIIS should be
disabled. |
|
| CC_DIIS
Specify the version of Pulay's Direct Inversion of the Iterative Subspace
(DIIS) convergence accelerator to be used in the coupled-cluster code. |
TYPE:
DEFAULT:
OPTIONS:
0 | Activates procedure 2 initially, and procedure 1 when gradients are smaller |
| than DIIS12_SWITCH. |
1 | Uses error vectors defined as differences between parameter vectors from |
| successive iterations. Most efficient near convergence. |
2 | Error vectors are defined as gradients scaled by square root of the |
| approximate diagonal Hessian. Most efficient far from convergence. |
RECOMMENDATION:
DIIS1 can be more stable. If DIIS problems are encountered in the early stages
of a calculation (when gradients are large) try DIIS1. |
|
|
|
CC_DOV_THRESH
Specifies the minimum allowed values for the coupled-cluster energy
denominators. Smaller values are replaced by this constant during
early iterations only, so the final results are unaffected, but
initial convergence is improved when the guess is poor. |
TYPE:
DEFAULT:
2502 | Corresponding to 0.25 |
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
Increase to 0.5 or 0.75 for non-convergent coupled-cluster calculations. |
|
| CC_DOV_THRESH
Specifies minimum allowed values for the coupled-cluster energy denominators.
Smaller values are replaced by this constant during early iterations only, so
the final results are unaffected, but initial convergence is improved when the
HOMO-LUMO gap is small or when non-conventional references are used. |
TYPE:
DEFAULT:
OPTIONS:
abcde | Integer code is mapped to abc×10−de, e.g.,
2502 corresponds to 0.25 |
RECOMMENDATION:
Increase to 0.25, 0.5 or 0.75 for non convergent coupled-cluster calculations. |
|
|
|
CC_DO_DYSON_EE
Whether excited state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations. |
TYPE:
DEFAULT:
FALSE (the option must be specified to run this calculation) |
OPTIONS:
RECOMMENDATION:
|
| CC_DO_DYSON
Whether ground state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations. |
TYPE:
DEFAULT:
FALSE (the option must be specified to run this calculation) |
OPTIONS:
RECOMMENDATION:
|
|
|
CC_EOM_PROP
Whether or not the non-relaxed (expectation value) one-particle EOM-CCSD
target state properties will be calculated. The properties currently include
permanent dipole moment, the second moments 〈X2〉, 〈Y2〉, and
〈Z2〉 of electron density, and the total 〈R2〉
= 〈X2〉
+〈Y2〉+〈Z2〉 (in atomic units). Incompatible with
JOBTYPE=FORCE, OPT, FREQ. |
TYPE:
DEFAULT:
FALSE (no one-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations (EOM-CCSD equations for the left eigenvectors) need to be
solved for properties, approximately doubling the cost of calculation for each
irrep. Sometimes the equations for left and right eigenvectors converge to
different sets of target states. In this case, the simultaneous iterations of
left and right vectors will diverge, and the properties for several or all the
target states may be incorrect! The problem can be solved by varying the number
of requested states, specified with EOM_XX_STATES, or the number of guess vectors
(EOM_NGUESS_SINGLES). The cost of the one-particle properties
calculation itself is low. The one-particle density of an EOM-CCSD target
state can be analyzed with NBO package by specifying the state with
CC_STATE_TO_OPT and requesting
NBO=TRUE and CC_EOM_PROP=TRUE. |
|
| CC_E_CONV
Convergence desired on the change in total energy, between iterations. |
TYPE:
DEFAULT:
OPTIONS:
n | 10−n convergence criterion. |
RECOMMENDATION:
|
|
|
CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO) |
TYPE:
DEFAULT:
OPTIONS:
range | 0000-10000 |
abcd | Corresponding to ab.cd% |
RECOMMENDATION:
|
| CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO) |
TYPE:
DEFAULT:
OPTIONS:
range | 0000-10000 |
abcd | Corresponding to ab.cd% |
RECOMMENDATION:
|
|
|
CC_FNO_USEPOP
Selection of the truncation scheme |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CC_FNO_USEPOP
Selection of the truncation scheme |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed.
The variable CC_EOM_PROP must be also set to TRUE. |
TYPE:
DEFAULT:
FALSE | (no orbital response will be calculated) |
OPTIONS:
RECOMMENDATION:
Not available for non-UHF/RHF references. Only available for EOM/CI methods for which analytic
gradients are available. |
|
| CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed.
The variable CC_REF_PROP must be also set to TRUE. |
TYPE:
DEFAULT:
FALSE | (no orbital response will be calculated) |
OPTIONS:
RECOMMENDATION:
Not available for non UHF/RHF references and for the methods that do not have
analytic gradients (e.g., QCISD). |
|
|
|
CC_HESS_THRESH
Minimum allowed value for the orbital Hessian. Smaller values are replaced by
this constant. |
TYPE:
DEFAULT:
102 | Corresponding to 0.01 |
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
|
| CC_INCL_CORE_CORR
Whether to include the correlation contribution from frozen core orbitals in
non iterative (2) corrections, such as OD(2) and CCSD(2). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless no core-valence or core correlation is desired (e.g., for
comparison with other methods or because the basis used cannot describe core
correlation). |
|
|
|
CC_ITERATE_ON
In active space calculations, use a "mixed" iteration procedure if the
value is greater than 0. Then if the RMS orbital gradient is larger than the
value of CC_THETA_GRAD_THRESH, micro-iterations will be performed
to converge the occupied-virtual mixing angles for the current active space.
The maximum number of space iterations is given by this option. |
TYPE:
DEFAULT:
OPTIONS:
n | Up to n occupied-virtual iterations per overall cycle |
RECOMMENDATION:
Can be useful for non-convergent active space calculations |
|
| CC_ITERATE_OV
In active space calculations, use a "mixed" iteration procedure if the value
is greater than 0. Then, if the RMS orbital gradient is larger than the value
of CC_THETA_GRAD_THRESH, micro-iterations will be performed to
converge the occupied-virtual mixing angles for the current active space. The
maximum number of such iterations is given by this option. |
TYPE:
DEFAULT:
OPTIONS:
n | Up to n occupied-virtual iterations per overall cycle |
RECOMMENDATION:
Can be useful for non-convergent active space calculations. |
|
|
|
CC_MAX_ITER
Maximum number of iterations to optimize the coupled-cluster energy. |
TYPE:
DEFAULT:
OPTIONS:
n | up to n iterations to achieve convergence. |
RECOMMENDATION:
|
| CC_MEMORY
Specifies the maximum size, in Mb, of the buffers for in-core storage of
block-tensors in CCMAN and CCMAN2. |
TYPE:
DEFAULT:
50% of MEM_TOTAL. If MEM_TOTAL is not set, use 1.5 Gb.
A minimum of |
192 Mb is hard-coded. |
OPTIONS:
RECOMMENDATION:
Larger values can give better I/O performance and are recommended for systems
with large memory (add to your .qchemrc file. When running
CCMAN2 exclusively on a node, CC_MEMORY should be set to
75-80% of the total available RAM. ) |
|
|
|
CC_MP2NO_GRAD
If CC_MP2NO_GUESS is TRUE, what kind of one-particle
density matrix is used to make the guess orbitals? |
TYPE:
DEFAULT:
OPTIONS:
TRUE | 1 PDM from MP2 gradient theory. |
FALSE | 1 PDM expanded to 2nd order in perturbation theory. |
RECOMMENDATION:
The two definitions give generally similar performance. |
|
| CC_MP2NO_GUESS
Will guess orbitals be natural orbitals of the MP2 wavefunction?
Alternatively, it is possible to use an effective one-particle density matrix
to define the natural orbitals. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Use natural orbitals from an MP2 one-particle density matrix (see
CC_MP2NO_GRAD). |
FALSE | Use current molecular orbitals from SCF. |
RECOMMENDATION:
|
|
|
CC_ORBS_PER_BLOCK
Specifies target (and maximum) size of blocks in orbital space. |
TYPE:
DEFAULT:
OPTIONS:
n | Orbital block size of n orbitals. |
RECOMMENDATION:
|
| CC_PRECONV_FZ
In active space methods, whether to pre-converge other wavefunction variables
for fixed initial guess of active space. |
TYPE:
DEFAULT:
OPTIONS:
0 | No pre-iterations before active space optimization begins. |
n | Maximum number of pre-iterations via this procedure. |
RECOMMENDATION:
|
|
|
CC_PRECONV_T2Z_EACH
Whether to pre-converge the cluster amplitudes before each change of the
orbitals in optimized orbital coupled-cluster methods. The maximum number of
iterations in this pre-convergence procedure is given by the value of this
parameter. |
TYPE:
DEFAULT:
OPTIONS:
0 | No pre-convergence before orbital optimization. |
n | Up to n iterations in this pre-convergence procedure. |
RECOMMENDATION:
A very slow last resort option for jobs that do not converge. |
|
| CC_PRECONV_T2Z
Whether to pre-converge the cluster amplitudes before beginning orbital
optimization in optimized orbital cluster methods. |
TYPE:
DEFAULT:
0 | (FALSE) |
10 | If CC_RESTART, CC_RESTART_NO_SCF or
CC_MP2NO_GUESS are TRUE |
OPTIONS:
0 | No pre-convergence before orbital optimization. |
n | Up to n iterations in this pre-convergence procedure. |
RECOMMENDATION:
Experiment with this option in cases of convergence failure. |
|
|
|
CC_PRINT
Controls the output from post-MP2 coupled-cluster module of Q-Chem |
TYPE:
DEFAULT:
OPTIONS:
0→7 | higher values can lead to deforestation... |
RECOMMENDATION:
Increase if you need more output and don't like trees |
|
| CC_QCCD_THETA_SWITCH
QCCD calculations switch from OD to QCCD when the rotation gradient is below
this threshold [10−n] |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CC_REF_PROP_TE
Request for calculation of non-relaxed two-particle CCSD properties. The
two-particle properties currently include 〈S2〉. The one-particle
properties also will be calculated, since the additional cost of the
one-particle properties calculation is inferior compared to the cost of
〈S2〉. The variable CC_REF_PROP must be also set to
TRUE. |
TYPE:
DEFAULT:
FALSE | (no two-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
The two-particle properties are computationally expensive, since
they require calculation and use of the two-particle density matrix (the cost
is approximately the same as the cost of an analytic gradient calculation). Do
not request the two-particle properties unless you really need them. |
|
| CC_REF_PROP
Whether or not the non-relaxed (expectation value) or full response (including
orbital relaxation terms) one-particle CCSD
properties will be calculated. The properties currently include permanent
dipole moment, the second moments 〈X2〉, 〈Y2〉, and
〈Z2〉 of electron density, and the total
〈R2〉
= 〈X2〉+〈Y2〉+〈Z2〉 (in atomic units).
Incompatible with JOBTYPE=FORCE, OPT, FREQ. |
TYPE:
DEFAULT:
FALSE | (no one-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations need to be solved (lambda CCSD equations) for properties
with the cost approximately the same as CCSD equations. Use default if you do
not need properties. The cost of the properties calculation itself is low. The
CCSD one-particle density can be analyzed with NBO package by specifying
NBO=TRUE, CC_REF_PROP=TRUE and JOBTYPE=FORCE.
|
|
|
|
CC_RESET_THETA
The reference MO coefficient matrix is reset every n iterations to help
overcome problems associated with the theta metric as theta becomes large. |
TYPE:
DEFAULT:
OPTIONS:
n | n iterations between resetting orbital rotations to zero. |
RECOMMENDATION:
|
| CC_RESTART_NO_SCF
Should an optimized orbital coupled cluster calculation begin with optimized
orbitals from a previous calculation? When TRUE, molecular orbitals are
initially orthogonalized, and CC_PRECONV_T2Z and
CC_CANONIZE are set to TRUE while other guess options are set to
FALSE |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CC_RESTART
Allows an optimized orbital coupled cluster calculation to begin with an
initial guess for the orbital transformation matrix U other than the unit
vector. The scratch file from a previous run must be available for the U
matrix to be read successfully. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use unit initial guess. |
TRUE | Activates CC_PRECONV_T2Z, CC_CANONIZE, and |
| turns off CC_MP2NO_GUESS |
RECOMMENDATION:
Useful for restarting a job that did not converge, if files were saved. |
|
| CC_RESTR_AMPL
Controls the restriction on amplitudes is there are restricted orbitals |
TYPE:
DEFAULT:
OPTIONS:
0 | All amplitudes are in the full space |
1 | Amplitudes are restricted, if there are restricted orbitals |
RECOMMENDATION:
|
|
|
CC_RESTR_TRIPLES
Controls which space the triples correction is computed in |
TYPE:
DEFAULT:
OPTIONS:
0 | Triples are computed in the full space |
1 | Triples are restricted to the active space |
RECOMMENDATION:
|
| CC_REST_AMPL
Forces the integrals, T, and R amplitudes to be determined in the full
space even though the CC_REST_OCC and CC_REST_VIR
keywords are used. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do apply restrictions |
1 | Do not apply restrictions |
RECOMMENDATION:
|
|
|
CC_REST_OCC
Sets the number of restricted occupied orbitals including frozen occupied
orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | Restrict n occupied orbitals. |
RECOMMENDATION:
|
| CC_REST_TRIPLES
Restricts R3 amplitudes to the active space, i.e., one electron should be
removed from the active occupied orbital and one electron should be added to
the active virtual orbital. |
TYPE:
DEFAULT:
OPTIONS:
1 | Applies the restrictions |
RECOMMENDATION:
|
|
|
CC_REST_VIR
Sets the number of restricted virtual orbitals including frozen virtual
orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | Restrict n virtual orbitals. |
RECOMMENDATION:
|
| CC_SCALE_AMP
If not 0, scales down the step for updating coupled-cluster amplitudes in cases of problematic convergence. |
TYPE:
DEFAULT:
OPTIONS:
abcd | Integer code is mapped to abcd×10−2, e.g.,
90 corresponds to 0.9 |
RECOMMENDATION:
Use 0.9 or 0.8 for non convergent coupled-cluster calculations. |
|
|
|
CC_STATE_TO_OPT
Specifies which state to optimize. |
TYPE:
DEFAULT:
OPTIONS:
[i,j] | optimize the jth state of the ith irrep. |
RECOMMENDATION:
|
| CC_SYMMETRY
Controls the use of symmetry in coupled-cluster calculations |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Use the point group symmetry of the molecule |
FALSE | Do not use point group symmetry (all states will be of A symmetry). |
RECOMMENDATION:
It is automatically turned off for any finite difference calculations, e.g. second derivatives. |
|
|
|
CC_THETA_CONV
Convergence criterion on the RMS difference between successive sets of
orbital rotation angles [10−n]. |
TYPE:
DEFAULT:
OPTIONS:
n | 10−n convergence criterion. |
RECOMMENDATION:
|
| CC_THETA_GRAD_CONV
Convergence desired on the RMS gradient of the energy with respect to
orbital rotation angles [10−n]. |
TYPE:
DEFAULT:
OPTIONS:
n | 10−n convergence criterion. |
RECOMMENDATION:
|
|
|
CC_THETA_GRAD_THRESH
RMS orbital gradient threshold [10−n] above which "mixed iterations"
are performed in active space calculations if CC_ITERATE_OV is
TRUE. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Can be made smaller if convergence difficulties are encountered. |
|
| CC_THETA_STEPSIZE
Scale factor for the orbital rotation step size. The optimal rotation steps
should be approximately equal to the gradient vector. |
TYPE:
DEFAULT:
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
| If the initial step is smaller than 0.5, the program will increase step |
| when gradients are smaller than the value of THETA_GRAD_THRESH, |
| up to a limit of 0.5. |
RECOMMENDATION:
Try a smaller value in cases of poor convergence and very large orbital
gradients. For example, a value of 01001 translates to 0.1 |
|
|
|
CC_TRANS_PROP
Whether or not the transition dipole moment (in atomic units) and oscillator
strength for the EOM-CCSD target states will be calculated. By default, the
transition dipole moment is calculated between the CCSD reference and the
EOM-CCSD target states. In order to calculate transition dipole moment between
a set of EOM-CCSD states and another EOM-CCSD state, the
CC_STATE_TO_OPT
must be specified for this state. |
TYPE:
DEFAULT:
FALSE (no transition dipole and oscillator strength will be calculated) |
OPTIONS:
RECOMMENDATION:
Additional equations (for the left EOM-CCSD eigenvectors plus lambda CCSD
equations in case if transition properties between the CCSD reference and
EOM-CCSD target states are requested) need to be solved for transition
properties, approximately doubling the computational cost. The cost of the
transition properties calculation itself is low. |
|
| CC_T_CONV
Convergence criterion on the RMS difference between successive sets of
coupled-cluster doubles amplitudes [10−n] |
TYPE:
DEFAULT:
OPTIONS:
n | 10−n convergence criterion. |
RECOMMENDATION:
|
|
|
CC_Z_CONV
Convergence criterion on the RMS difference between successive doubles
Z-vector amplitudes [10−n]. |
TYPE:
DEFAULT:
OPTIONS:
n | 10−n convergence criterion. |
RECOMMENDATION:
|
| CDFTCI_PRINT
Controls level of output from CDFT-CI procedure to Q-Chem output file. |
TYPE:
DEFAULT:
OPTIONS:
0 | Only print energies and coefficients of CDFT-CI final states |
1 | Level 0 plus CDFT-CI overlap, Hamiltonian, and population matrices |
2 | Level 1 plus eigenvectors and eigenvalues of the CDFT-CI population matrix |
3 | Level 2 plus promolecule orbital coefficients and energies |
RECOMMENDATION:
Level 3 is primarily for program debugging; levels 1 and 2 may be useful
for analyzing the coupling elements |
|
|
|
CDFTCI_RESTART
To be used in conjunction with CDFTCI_STOP, this variable
causes CDFT-CI to read already-converged states from disk and begin
SCF convergence on later states. Note that the same $cdft section
must be used for the stopped calculation and the restarted calculation. |
TYPE:
DEFAULT:
OPTIONS:
n | start calculations on state n+1 |
RECOMMENDATION:
Use this setting in conjunction with CDFTCI_STOP. |
|
| CDFTCI_SKIP_PROMOLECULES
Skips promolecule calculations and allows fractional charge and spin
constraints to be specified directly. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Standard CDFT-CI calculation is performed. |
TRUE | Use the given charge/spin constraints directly, with no
promolecule calculations. |
RECOMMENDATION:
Setting to TRUE can be useful for scanning over constraint values. |
|
|
|
CDFTCI_STOP
The CDFT-CI procedure involves performing independent SCF calculations
on distinct constrained states. It sometimes occurs that the same
convergence parameters are not successful for all of the states of
interest, so that a CDFT-CI calculation might converge one of these
diabatic states but not the next. This variable allows a user to
stop a CDFT-CI calculation after a certain number of states have
been converged, with the ability to restart later on the next state,
with different convergence options. |
TYPE:
DEFAULT:
OPTIONS:
n | stop after converging state n (the first state is state 1) |
0 | do not stop early |
RECOMMENDATION:
Use this setting if some diabatic states converge but others do not. |
|
| CDFTCI_SVD_THRESH
By default, a symmetric orthogonalization is performed on the CDFT-CI
matrix before diagonalization. If the CDFT-CI overlap matrix is nearly
singular (i.e., some of the diabatic states are nearly degenerate), then
this orthogonalization can lead to numerical instability. When computing
→S−1/2, eigenvalues smaller than 10−CDFTCI_SVD_THRESH
are discarded. |
TYPE:
DEFAULT:
OPTIONS:
n | for a threshold of 10−n. |
RECOMMENDATION:
Can be decreased if numerical instabilities are encountered in the
final diagonalization. |
|
|
|
CDFTCI
Initiates a constrained DFT-configuration interaction calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform a CDFT-CI Calculation |
FALSE | No CDFT-CI |
RECOMMENDATION:
Set to TRUE if a CDFT-CI calculation is desired. |
|
| CDFT_BECKE_POP
Whether the calculation should print the Becke atomic charges at convergence |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Print Populations |
FALSE | Do not print them |
RECOMMENDATION:
Use default. Note that the Mulliken populations printed at the end of an SCF run will not typically add up to the prescribed constraint value. Only the Becke populations are guaranteed to satisfy the user-specified constraints. |
|
|
|
CDFT_CRASHONFAIL
Whether the calculation should crash or not if the constraint iterations do not converge. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Crash if constraint iterations do not converge. |
FALSE | Do not crash. |
RECOMMENDATION:
|
| CDFT_LAMBDA_MODE
Allows CDFT potentials to be specified directly, instead of being
determined as Lagrange multipliers. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Standard CDFT calculations are used. |
TRUE | Instead of specifying target charge and spin constraints, use the values |
| from the input deck as the value of the Becke weight potential
|
RECOMMENDATION:
Should usually be set to FALSE. Setting to TRUE can be useful to
scan over different strengths of charge or spin localization, as
convergence properties are improved compared to regular CDFT(-CI) calculations. |
|
|
|
CDFT_POSTDIIS
Controls whether the constraint is enforced after DIIS extrapolation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Enforce constraint after DIIS |
FALSE | Do not enforce constraint after DIIS |
RECOMMENDATION:
Use default unless convergence problems arise,
in which case it may be beneficial to experiment with setting CDFT_POSTDIIS to FALSE.
With this option set to TRUE, energies should be variational after the first iteration. |
|
| CDFT_PREDIIS
Controls whether the constraint is enforced before DIIS extrapolation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Enforce constraint before DIIS |
FALSE | Do not enforce constraint before DIIS |
RECOMMENDATION:
Use default unless convergence problems arise, in which case it may be beneficial to experiment with setting
CDFT_PREDIIS to TRUE. Note that it is possible to enforce the constraint
both before and after DIIS by setting both CDFT_PREDIIS and CDFT_POSTDIIS to TRUE. |
|
|
|
CDFT_THRESH
Threshold that determines how tightly the constraint must be satisfied. |
TYPE:
DEFAULT:
OPTIONS:
N | Constraint is satisfied to within 10−N. |
RECOMMENDATION:
Use default unless problems occur. |
|
| CDFT
Initiates a constrained DFT calculation |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform a Constrained DFT Calculation |
FALSE | No Density Constraint |
RECOMMENDATION:
Set to TRUE if a Constrained DFT calculation is desired. |
|
|
|
CD_ALGORITHM
Determines the algorithm for MP2 integral transformations. |
TYPE:
DEFAULT:
OPTIONS:
DIRECT | Uses fully direct algorithm (energies only). |
SEMI_DIRECT | Uses disk-based semi-direct algorithm. |
LOCAL_OCCUPIED | Alternative energy algorithm (see 5.3.1). |
RECOMMENDATION:
Semi-direct is usually most efficient, and will normally be chosen by default. |
|
| CFMM_ORDER
Controls the order of the multipole expansions in CFMM calculation. |
TYPE:
DEFAULT:
15 | For single point SCF accuracy |
25 | For tighter convergence (optimizations) |
OPTIONS:
n | Use multipole expansions of order n |
RECOMMENDATION:
|
|
|
CHARGE_CHARGE_REPULSION
The repulsive Coulomb interaction parameter for YinYang atoms. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The repulsive Coulomb potential maintains bond lengths involving YinYang
atoms with the potential V(r) = Q/r. The default is parameterized for carbon atoms. |
|
| CHELPG_DX
Sets the grid spacing for the CHELPG grid. |
TYPE:
DEFAULT:
OPTIONS:
N | Corresponding to a grid space of N/20, in Å. |
RECOMMENDATION:
Use the default (which corresponds to the "dense grid" of Breneman and Wiberg [468]),
unless the cost is prohibitive, in which case a larger value can be selected. |
|
|
|
CHELPG_HEAD
Sets the "head space" for the CHELPG grid. |
TYPE:
DEFAULT:
OPTIONS:
N | Corresponding to a head space of N/10, in Å. |
RECOMMENDATION:
Use the default, which is the value recommended by Breneman and Wiberg [468]. |
|
| CHELPG
Controls the calculation of CHELPG charges. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate CHELPG charges. |
TRUE | Compute CHELPG charges. |
RECOMMENDATION:
Set to TRUE if desired. For large molecules, there is some overhead associated with computing
CHELPG charges, especially if the number of grid points is large. |
|
|
|
CHEMSOL_EFIELD
Determines how the solute charge distribution is approximated in evaluating
the electrostatic field of the solute. |
TYPE:
DEFAULT:
OPTIONS:
1 | Exact solute charge distribution is used. |
0 | Solute charge distribution is approximated by Mulliken atomic charges. |
| This is a faster, but less rigorous procedure. |
RECOMMENDATION:
|
| CHEMSOL_NN
Sets the number of grids used to calculate the average hydration free energy. |
TYPE:
DEFAULT:
5 | ∆Ghydr will be averaged over 5 different grids. |
OPTIONS:
n | Number of different grids (Max = 20). |
RECOMMENDATION:
|
|
|
CHEMSOL_PRINT
Controls printing in the ChemSol part of the Q-Chem output file. |
TYPE:
DEFAULT:
OPTIONS:
0 | Limited printout. |
1 | Full printout. |
RECOMMENDATION:
|
| CHEMSOL
Controls the use of ChemSol in Q-Chem. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use ChemSol. |
1 | Perform a ChemSol calculation. |
RECOMMENDATION:
|
|
|
CHOLESKY_TOL
Tolerance of Cholesky decomposition of two-electron integrals |
TYPE:
DEFAULT:
OPTIONS:
n to define tolerance of 10−n |
RECOMMENDATION:
2 - qualitative calculations, 3 - appropriate for most cases, 4 - quantitative
(error in total energy typically less than 1e-6 hartree) |
|
| CISTR_PRINT
TYPE:
DEFAULT:
OPTIONS:
TRUE | Increase output level |
RECOMMENDATION:
|
|
|
CIS_AMPL_ANAL
Perform additional analysis of CIS and TDDFT excitation amplitudes,
including generation of natural transition orbitals, excited-state
multipole moments, and Mulliken analysis of the excited state densities
and particle/hole density matrices.
|
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform additional amplitude analysis. |
FALSE | Do not perform additional analysis.
|
RECOMMENDATION:
|
| CIS_CONVERGENCE
CIS is considered converged when error is less than 10−CIS_CONVERGENCE |
TYPE:
DEFAULT:
6 | CIS convergence threshold 10−6 |
OPTIONS:
RECOMMENDATION:
|
|
|
CIS_DIABATH_DECOMPOSE
Decide whether or not to decompose the diabatic coupling into Coulomb, exchange, and one-electron terms. |
TYPE:
DEFAULT:
FALSE | Do not decompose the diabatic coupling. |
OPTIONS:
RECOMMENDATION:
These decompositions are most meaningful for electronic excitation transfer processes.
Currently, available only for CIS, not for TD-DFT diabatic states. |
|
| CIS_GUESS_DISK_TYPE
Determines the type of guesses to be read from disk |
TYPE:
DEFAULT:
OPTIONS:
0 | Read triplets only |
1 | Read triplets and singlets |
2 | Read singlets only |
RECOMMENDATION:
Must be specified if CIS_GUESS_DISK is TRUE. |
|
|
|
CIS_GUESS_DISK
Read the CIS guess from disk (previous calculation) |
TYPE:
DEFAULT:
OPTIONS:
False | Create a new guess |
True | Read the guess from disk |
RECOMMENDATION:
Requires a guess from previous calculation. |
|
| CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not calculate excited-state moments. |
TRUE | (or 1) Calculate moments for each excited state.
|
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial
additional calculation.) The MULTIPOLE_ORDER controls how many
multipole moments are printed. |
|
|
|
CIS_MULLIKEN
Controls Mulliken and Löwdin population analyses for excited-state particle and
hole density matrices. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not perform particle/hole population analysis. |
TRUE | (or 1) Perform both Mulliken and Löwdin analysis of the particle and hole |
| density matrices for each excited state. |
RECOMMENDATION:
Set to TRUE if desired. This represents a trivial additional calculation. |
|
| CIS_N_ROOTS
Sets the number of CI-Singles (CIS) excited state roots to find |
TYPE:
DEFAULT:
0 | Do not look for any excited states |
OPTIONS:
n | n > 0 Looks for n CIS excited states |
RECOMMENDATION:
|
|
|
CIS_RELAXED_DENSITY
Use the relaxed CIS density for attachment/detachment density analysis |
TYPE:
DEFAULT:
OPTIONS:
False | Do not use the relaxed CIS density in analysis |
True | Use the relaxed CIS density in analysis. |
RECOMMENDATION:
|
| CIS_SINGLETS
Solve for singlet excited states in RCIS calculations (ignored for UCIS) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for singlet states |
FALSE | Do not solve for singlet states. |
RECOMMENDATION:
|
|
|
CIS_STATE_DERIV
Sets CIS state for excited state optimizations and vibrational analysis |
TYPE:
DEFAULT:
0 | Does not select any of the excited states |
OPTIONS:
RECOMMENDATION:
Check to see that the states do no change order during an optimization |
|
| CIS_TRIPLETS
Solve for triplet excited states in RCIS calculations (ignored for UCIS) |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Solve for triplet states |
FALSE | Do not solve for triplet states. |
RECOMMENDATION:
|
|
|
CORE_CHARACTER
Selects how the core orbitals are determined in the frozen-core
approximation. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use energy-based definition. |
1-4 | Use Mulliken-based definition (see Table 5.3.2 for details). |
RECOMMENDATION:
Use default, unless performing calculations on molecules with heavy elements. |
|
| CORRELATION
Specifies the correlation level of theory, either DFT or wavefunction-based. |
TYPE:
DEFAULT:
OPTIONS:
None | No Correlation. |
VWN | Vosko-Wilk-Nusair parameterization #5 |
LYP | Lee-Yang-Parr |
PW91, PW | GGA91 (Perdew) |
PW92 | LSDA 92 (Perdew and Wang) [44] |
PK09 | LSDA (Proynov-Kong) [45] |
LYP(EDF1) | LYP(EDF1) parameterization |
Perdew86, P86 | Perdew 1986 |
PZ81, PZ | Perdew-Zunger 1981 |
PBE | Perdew-Burke-Ernzerhof 1996 |
TPSS | The correlation component of the TPSS functional |
B94 | Becke 1994 correlation in fully analytic form |
B94hyb | Becke 1994 correlation as above, but readjusted for
use only within the hybrid scheme BR89B94hyb |
PK06 | Proynov-Kong 2006 correlation (known also as "tLap" |
(B88)OP | OP correlation [74], optimized for use with B88 exchange |
(PBE)OP | OP correlation [74], optimized for use with PBE exchange |
Wigner | Wigner |
MP2 | |
Local_MP2 | Local MP2 calculations (TRIM and DIM models) |
CIS(D) | MP2-level correction to CIS for excited states |
MP3 | |
MP4SDQ | |
MP4 | |
CCD | |
CCD(2) | |
CCSD | |
CCSD(T) | |
CCSD(2) | |
QCISD | |
QCISD(T) | |
OD | |
OD(T) | |
OD(2) | |
VOD | |
VOD(2) | |
QCCD | |
VQCCD | |
RECOMMENDATION:
Consult the literature and reviews for guidance. |
|
|
|
CORRELATION
Specifies the correlation level of theory, either DFT or wavefunction-based. |
TYPE:
DEFAULT:
OPTIONS:
MP2 | Sections 5.2 and 5.3 |
Local_MP2 | Section 5.4 |
RILMP2 | Section 5.5.1 |
ATTMP2 | Section 5.6.1 |
ATTRIMP2 | Section 5.6.1 |
ZAPT2 | A more efficient restricted open-shell MP2 method [213]. |
MP3 | Section 5.2 |
MP4SDQ | Section 5.2 |
MP4 | Section 5.2 |
CCD | Section 5.7 |
CCD(2) | Section 5.8 |
CCSD | Section 5.7 |
CCSD(T) | Section 5.8 |
CCSD(2) | Section 5.8 |
CCSD(fT) | Section 5.8.3 |
CCSD(dT) | Section 5.8.3 |
QCISD | Section 5.7 |
QCISD(T) | Section 5.8 |
OD | Section 5.7 |
OD(T) | Section 5.8 |
OD(2) | Section 5.8 |
VOD | Section 5.9 |
VOD(2) | Section 5.9 |
QCCD | Section 5.7 |
QCCD(T) | |
QCCD(2) | |
VQCCD | Section 5.9 |
RECOMMENDATION:
Consult the literature for guidance. |
|
| CPSCF_NSEG
Controls the number of segments used to calculate the CPSCF equations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not solve the CPSCF equations in segments. |
n | User-defined. Use n segments when solving the CPSCF equations. |
RECOMMENDATION:
|
|
|
CUBEFILE_STATE
Determines which excited state is used to generate cube files |
TYPE:
DEFAULT:
OPTIONS:
n | Generate cube files for the nth excited state
|
RECOMMENDATION:
|
| CUDA_RI-MP2
Enables GPU implementation of RI-MP2 |
TYPE:
DEFAULT:
OPTIONS:
FALSE | GPU-enabled MGEMM off |
TRUE | GPU-enabled MGEMM on |
RECOMMENDATION:
Necessary to set to 1 in order to run GPU-enabled RI-MP2 |
|
|
|
CUTOCC
Specifies occupied orbital cutoff |
TYPE:
INTEGER: CUTOFF=CUTOCC/100 |
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| CUTVIR
Specifies virtual orbital cutoff |
TYPE:
INTEGER: CUTOFF=CUTVIR/100 |
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
CVGLIN
Convergence criterion for solving linear equations by the conjugate gradient
iterative method (relevant if LINEQ=1 or 2). |
TYPE:
DEFAULT:
OPTIONS:
Real number specifying the actual criterion. |
RECOMMENDATION:
The default value should be used unless convergence problems arise. |
|
| DEUTERATE
Requests that all hydrogen atoms be replaces with deuterium. |
TYPE:
DEFAULT:
FALSE | Do not replace hydrogens. |
OPTIONS:
TRUE | Replace hydrogens with deuterium. |
RECOMMENDATION:
Replacing hydrogen atoms reduces the fastest vibrational frequencies by a
factor of 1.4, which allow for a larger fictitious mass and time step in ELMD
calculations. There is no reason to replace hydrogens in BOMD calculations. |
|
|
|
DFPT_EXCHANGE
Specifies the secondary functional in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
| DFPT_XC_GRID
Specifies the secondary grid in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
|
|
DFTVDW_ALPHA1
Parameter in XDM calculation with higher-order terms |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| DFTVDW_ALPHA2
Parameter in XDM calculation with higher-order terms. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DFTVDW_JOBNUMBER
TYPE:
DEFAULT:
OPTIONS:
0 | Do not apply the XDM scheme. |
1 | add vdW gradient correction to SCF. |
2 | add VDW as a DFT functional and do full SCF. |
RECOMMENDATION:
This option only works with C6 XDM formula |
|
| DFTVDW_KAI
Damping factor K for C6 only damping function |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DFTVDW_METHOD
Choose the damping function used in XDM |
TYPE:
DEFAULT:
OPTIONS:
1 | use Becke's damping function including C6 term only. |
2 | use Becke's damping function with higher-order (C8,C10) terms. |
RECOMMENDATION:
|
| DFTVDW_MOL1NATOMS
The number of atoms in the first monomer in dimer calculation |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DFTVDW_PRINT
Printing control for VDW code |
TYPE:
DEFAULT:
OPTIONS:
0 no printing. |
1 | minimum printing (default) |
2 | debug printing |
RECOMMENDATION:
|
| DFTVDW_USE_ELE_DRV
Specify whether to add the gradient correction to the XDM energy.
only valid with Becke's C6 damping function
using the interpolated BR89 model. |
TYPE:
DEFAULT:
OPTIONS:
1 | use density correction when applicable (default). |
0 | do not use this correction (for debugging purpose) |
RECOMMENDATION:
|
|
|
DFT_D3_3BODY
Controls whether the three-body interaction in Grimme's DFT-D3 method should be applied
(see Eq. (14) in Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the three-body interaction term |
TRUE | Apply the three-body interaction term
|
RECOMMENDATION:
|
| DFT_D3_RS6
Controls the strength of dispersion corrections, sr6, in the Grimme's DFT-D3 method (see Table IV in Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to sr6 = n/1000. |
RECOMMENDATION:
|
|
|
DFT_D3_S6
Controls the strength of dispersion corrections, s6, in Grimme's DFT-D3 method (see Table IV in
Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to s6 = n/1000. |
RECOMMENDATION:
|
| DFT_D3_S8
Controls the strength of dispersion corrections, s8, in Grimme's DFT-D3 method (see Table IV in
Ref. ). |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to s8 = n/1000. |
RECOMMENDATION:
|
|
|
DFT_D_A
Controls the strength of dispersion corrections in the Chai-Head-Gordon DFT-D scheme in Eq.(3) of Ref. . |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a = n/100. |
RECOMMENDATION:
|
| DFT_D
Controls the application of DFT-D or DFT-D3 scheme. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the DFT-D or DFT-D3 scheme |
EMPIRICAL_GRIMME | dispersion correction from Grimme |
EMPIRICAL_CHG | dispersion correction from Chai and Head-Gordon |
EMPIRICAL_GRIMME3 | dispersion correction from Grimme's DFT-D3 method |
| (see Section 4.3.8)
|
RECOMMENDATION:
|
|
|
DH_OS
Controls the strength of the opposite-spin component of PT2 correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to cos = n/1000000 in Eq. (4.65). |
RECOMMENDATION:
|
| DH_SS
Controls the strength of the same-spin component of PT2 correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to css = n/1000000 in Eq. (4.65). |
RECOMMENDATION:
|
|
|
DH
Controls the application of DH-DFT scheme. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply the DH-DFT scheme |
TRUE | (or 1) Apply DH-DFT scheme |
RECOMMENDATION:
|
| DIELST
The static dielectric constant. |
TYPE:
DEFAULT:
OPTIONS:
real number specifying the constant. |
RECOMMENDATION:
The default value 78.39 is appropriate for water solvent. |
|
|
|
DIIS_ERR_RMS
Changes the DIIS convergence metric from the maximum to the RMS error. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default, the maximum error provides a more reliable criterion. |
|
| DIIS_PRINT
Controls the output from DIIS SCF optimization. |
TYPE:
DEFAULT:
OPTIONS:
0 | Minimal print out. |
1 | Chosen method and DIIS coefficients and solutions. |
2 | Level 1 plus changes in multipole moments. |
3 | Level 2 plus Multipole moments. |
4 | Level 3 plus extrapolated Fock matrices. |
RECOMMENDATION:
|
|
|
DIIS_SEPARATE_ERRVEC
Control optimization of DIIS error vector in unrestricted calculations. |
TYPE:
DEFAULT:
FALSE | Use a combined alpha and beta error vector. |
OPTIONS:
FALSE | Use a combined alpha and beta error vector. |
TRUE | Use separate error vectors for the alpha and beta spaces. |
RECOMMENDATION:
When using DIIS in Q-Chem a convenient optimization for unrestricted calculations is to sum
the alpha and beta error vectors into a single vector which is used for extrapolation. This
is often extremely effective, but in some pathological systems with symmetry breaking, can lead
to false solutions being detected, where the alpha and beta components of the error vector
cancel exactly giving a zero DIIS error. While an extremely uncommon occurrence, if it is suspected,
set DIIS_SEPARATE_ERRVEC to TRUE to check. |
|
| DIIS_SUBSPACE_SIZE
Controls the size of the DIIS and/or RCA subspace during the SCF. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
DIRECT_RI
Controls use of RI and Cholesky integrals in conventional (undecomposed) form |
TYPE:
DEFAULT:
OPTIONS:
FALSE - use all integrals in decomposed format |
TRUE - transform all RI or Cholesky integral back to conventional format |
RECOMMENDATION:
By default all integrals are used in decomposed format allowing significant reduction of
memory use. If all integrals are transformed back (TRUE option) no memory reduction is
achieved and decomposition error is introduced, however, the integral transformation
is performed significantly faster and conventional CC/EOM algorithms are used. |
|
| DIRECT_SCF
TYPE:
DEFAULT:
OPTIONS:
TRUE | Forces direct SCF. |
FALSE | Do not use direct SCF. |
RECOMMENDATION:
Use default; direct SCF switches off in-core integrals. |
|
|
|
DMA_MIDPOINTS
Specifies whether to include bond midpoints into DMA expansion. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not include bond midpoints. |
TRUE | Include bond midpoint. |
RECOMMENDATION:
|
| DORAMAN
Controls calculation of Raman intensities. Requires JOBTYPE to be set
to FREQ |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate Raman intensities. |
TRUE | Do calculate Raman intensities. |
RECOMMENDATION:
|
|
|
DO_DMA
Specifies whether to perform Distributed Multipole Analysis. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Turn off DMA. |
TRUE | Turn on DMA. |
RECOMMENDATION:
|
| DUAL_BASIS_ENERGY
Activates dual-basis SCF (HF or DFT) energy correction. |
TYPE:
DEFAULT:
OPTIONS:
Analytic first derivative available for HF and DFT (see JOBTYPE) |
Can be used in conjunction with MP2 or RI-MP2 |
See BASIS, BASIS2, BASISPROJTYPE |
RECOMMENDATION:
Use Dual-Basis to capture large-basis effects at smaller basis cost.
Particularly useful with RI-MP2, in which HF often dominates. Use only proper
subsets for small-basis calculation. |
|
|
|
D_CPSCF_PERTNUM
Specifies whether to do the perturbations one at a time, or all together. |
TYPE:
DEFAULT:
OPTIONS:
0 | Perturbed densities to be calculated all together. |
1 | Perturbed densities to be calculated one at a time. |
RECOMMENDATION:
|
| D_SCF_CONV_1
Sets the convergence criterion for the level-1 iterations. This preconditions
the density for the level-2 calculation, and does not include any
two-electron integrals. |
TYPE:
DEFAULT:
4 | corresponding to a threshold of 10−4. |
OPTIONS:
n < 10 | Sets convergence threshold to 10−n. |
RECOMMENDATION:
The criterion for level-1 convergence must be less than or equal to the
level-2 criterion, otherwise the D-CPSCF will not converge. |
|
|
|
D_SCF_CONV_2
Sets the convergence criterion for the level-2 iterations. |
TYPE:
DEFAULT:
4 | Corresponding to a threshold of 10−4. |
OPTIONS:
n < 10 | Sets convergence threshold to 10−n. |
RECOMMENDATION:
|
| D_SCF_DIIS
Specifies the number of matrices to use in the DIIS extrapolation in the
D-CPSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | n = 0 specifies no DIIS extrapolation is to be used. |
RECOMMENDATION:
|
|
|
D_SCF_MAX_1
Sets the maximum number of level-1 iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| D_SCF_MAX_2
Sets the maximum number of level-2 iterations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
ECP
Defines the effective core potential and associated basis set to be used |
TYPE:
DEFAULT:
OPTIONS:
General, Gen | User defined. ($ecp keyword required) |
Symbol | Use standard pseudopotentials discussed above. |
RECOMMENDATION:
Pseudopotentials are recommended for first row transition metals and heavier
elements. Consul the reviews for more details. |
|
| EDA_BSSE
Calculates the BSSE correction when performing the energy decomposition analysis. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE unless a very large basis set is used. |
|
|
|
EDA_COVP
Perform COVP analysis when evaluating the RS or ARS charge-transfer correction. COVP analysis is currently implemented only for systems of two fragments. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE to perform COVP analysis in an EDA or SCF MI(RS) job. |
|
| EDA_PRINT_COVP
Replace the final MOs with the CVOP orbitals in the end of the run. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to TRUE to print COVP orbitals instead of conventional MOs. |
|
|
|
EFP_DISP_DAMP
Controls fragment-fragment dispersion screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off dispersion screening |
1 | use Tang-Toennies screening, with fixed parameter b=1.5 |
RECOMMENDATION:
|
| EFP_DISP
Controls fragment-fragment dispersion in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off dispersion |
RECOMMENDATION:
|
|
|
EFP_ELEC_DAMP
Controls fragment-fragment electrostatic screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off electrostatic screening |
1 | use overlap-based damping correction |
2 | use exponential damping correction if screening parameters are provided in the EFP potential. |
| If the parameters are not provided damping will be automatically disabled. |
RECOMMENDATION:
|
| EFP_ELEC
Controls fragment-fragment electrostatics in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off electrostatics |
RECOMMENDATION:
|
|
|
EFP_EXREP
Controls fragment-fragment exchange repulsion in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off exchange repulsion |
RECOMMENDATION:
|
| EFP_FRAGMENTS_ONLY
Specifies whether there is a QM part |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Only MM part is present: all fragments are treated by EFP |
FALSE | QM part is present: do QM/MM EFP calculation |
RECOMMENDATION:
|
|
|
EFP_INPUT
Specifies the EFP fragment input format |
TYPE:
DEFAULT:
FALSE | Old format with dummy atom in $molecule section |
OPTIONS:
TRUE | New format without dummy atom in $molecule section |
FALSE | Old format |
RECOMMENDATION:
|
| EFP_POL
Controls fragment-fragment polarization in EFP |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off polarization |
RECOMMENDATION:
|
|
|
EFP_QM_ELEC_DAMP
Controls QM-EFP electrostatics screening in EFP |
TYPE:
DEFAULT:
OPTIONS:
0 | switch off electrostatic screening |
1 | use overlap based damping correction |
RECOMMENDATION:
|
| EFP_QM_ELEC
Controls QM-EFP electrostatics |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off electrostatics |
RECOMMENDATION:
|
|
|
EFP_QM_EXREP
Controls QM-EFP exchange-repulsion |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off exchange-repulsion |
RECOMMENDATION:
|
| EFP_QM_POL
Controls QM-EFP polarization |
TYPE:
DEFAULT:
OPTIONS:
FALSE | switch off polarization |
RECOMMENDATION:
|
|
|
EFP
Specifies that EFP calculation is requested |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The keyword should be present if excited state calculation is requested |
|
| EOM_CORR
Specifies the correlation level. |
TYPE:
DEFAULT:
None | No correction will be computed |
OPTIONS:
SD(DT) | EOM-CCSD(dT), available for EE, SF, and IP |
SD(FT) | EOM-CCSD(dT), available for EE, SF, and IP |
SD(ST) | EOM-CCSD(sT), available for IP |
RECOMMENDATION:
|
|
|
EOM_DAVIDSON_CONVERGENCE
Convergence criterion for the RMS residuals of excited state vectors |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to 10−n convergence criterion |
RECOMMENDATION:
Use default. Should normally be set to the same value as EOM_DAVIDSON_THRESHOLD. |
|
| EOM_DAVIDSON_MAXVECTORS
Specifies maximum number of vectors in the subspace for the Davidson
diagonalization. |
TYPE:
DEFAULT:
OPTIONS:
n | Up to n vectors per root before the subspace is reset |
RECOMMENDATION:
Larger values increase disk storage but accelerate and stabilize convergence. |
|
|
|
EOM_DAVIDSON_MAX_ITER
Maximum number of iteration allowed for Davidson diagonalization procedure. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of iterations |
RECOMMENDATION:
Default is usually sufficient |
|
| EOM_DAVIDSON_THRESHOLD
Specifies threshold for including a new expansion vector in the iterative
Davidson diagonalization. Their norm must be above this threshold. |
TYPE:
DEFAULT:
00105 | Corresponding to 0.00001 |
OPTIONS:
abcde | Integer code is mapped to abc×10−de |
RECOMMENDATION:
Use default unless converge problems are encountered. Should normally be set
to the same values as EOM_DAVIDSON_CONVERGENCE, if convergence problems arise
try setting to a value less than EOM_DAVIDSON_CONVERGENCE. |
|
|
|
EOM_DIP_SINGLETS
Sets the number of singlet DIP roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any singlet DIP states. |
OPTIONS:
[i,j,k…] | Find i DIP singlet states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_DIP_STATES
Sets the number of DIP roots to find. For
closed-shell reference, defaults into EOM_DIP_SINGLETS. For open-shell references,
specifies all low-lying states. |
TYPE:
DEFAULT:
0 | Do not look for any DIP states. |
OPTIONS:
[i,j,k…] | Find i DIP states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_DIP_TRIPLETS
Sets the number of triplet DIP roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any DIP triplet states. |
OPTIONS:
[i,j,k…] | Find i DIP triplet states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_DSF_STATES
Sets the number of doubly spin-flipped target states roots to find. |
TYPE:
DEFAULT:
0 | Do not look for any DSF states. |
OPTIONS:
[i,j,k…] | Find i doubly spin-flipped states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EA_ALPHA
Sets the number of attached target states derived by attaching α electron (Ms=[1/2],
default in EOM-EA). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EA_BETA
Sets the number of attached target states derived by attaching β electron (Ms=−[1/2],
EA-SF). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EA_STATES
Sets the number of attached target states roots to find. By default, α electron will be attached
(see EOM_EA_ALPHA). |
TYPE:
DEFAULT:
0 | Do not look for any EA states. |
OPTIONS:
[i,j,k…] | Find i EA states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EE_SINGLETS
Sets the number of singlet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_EE_STATES
Sets the number of excited state roots to find. For
closed-shell reference, defaults into EOM_EE_SINGLETS. For open-shell references,
specifies all low-lying states. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_EE_TRIPLETS
Sets the number of triplet excited state roots to find. Works only
for closed-shell references. |
TYPE:
DEFAULT:
0 | Do not look for any excited states. |
OPTIONS:
[i,j,k…] | Find i excited states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_FAKE_IPEA
If TRUE, calculates fake EOM-IP or EOM-EA energies and properties using the diffuse orbital trick.
Default for EOM-EA and Dyson orbital calculations. |
TYPE:
DEFAULT:
FALSE (use proper EOM-IP code) |
OPTIONS:
RECOMMENDATION:
|
| EOM_IPEA_FILTER
If TRUE, filters the EOM-IP/EA amplitudes obtained using the diffuse orbital implementation
(see EOM_FAKE_IPEA). Helps with convergence. |
TYPE:
DEFAULT:
FALSE (EOM-IP or EOM-EA amplitudes will not be filtered) |
OPTIONS:
RECOMMENDATION:
|
|
|
EOM_IP_ALPHA
Sets the number of ionized target states derived by removing α electron (Ms=−[1/2]). |
TYPE:
DEFAULT:
0 | Do not look for any IP/α states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_IP_BETA
Sets the number of ionized target states derived by removing β electron (Ms=[1/2],
default for EOM-IP). |
TYPE:
DEFAULT:
0 | Do not look for any IP/β states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
|
|
EOM_IP_STATES
Sets the number of ionized target states roots to find. By default, β electron will be removed
(see EOM_IP_BETA). |
TYPE:
DEFAULT:
0 | Do not look for any IP states. |
OPTIONS:
[i,j,k…] | Find i ionized states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EOM_NGUESS_DOUBLES
Specifies number of excited state guess vectors which are double excitations. |
TYPE:
DEFAULT:
OPTIONS:
n | Include n guess vectors that are double excitations |
RECOMMENDATION:
This should be set to the expected number of doubly excited states (see also
EOM_PRECONV_DOUBLES), otherwise they may not be found. |
|
|
|
EOM_NGUESS_SINGLES
Specifies number of excited state guess vectors that are single excitations. |
TYPE:
DEFAULT:
Equal to the number of excited states requested |
OPTIONS:
n | Include n guess vectors that are single excitations |
RECOMMENDATION:
Should be greater or equal than the number of excited states requested. |
|
| EOM_PRECONV_DOUBLES
When not zero, doubly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging doubles. |
RECOMMENDATION:
Occasionally necessary to ensure a doubly excited state is found. Also used in DSF calculations
instead of EOM_PRECONV_SINGLES |
|
|
|
EOM_PRECONV_SD
When not zero, singly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging singles and doubles. |
RECOMMENDATION:
Occasionally necessary to ensure a doubly excited state is found. Also, very useful in EOM(2,3)
calculations. |
|
None
EOM_PRECONV_SINGLES
When not zero, singly-excited vectors are converged prior to a full excited
states calculation. Sets the maximum number of iterations for pre-converging procedure |
TYPE:
DEFAULT:
OPTIONS:
0 | do not pre-converge |
N | perform N Davidson iterations pre-converging singles. |
RECOMMENDATION:
Sometimes helps with problematic convergence. |
|
| EOM_REF_PROP_TE
Request for calculation of non-relaxed two-particle EOM-CC properties. The
two-particle properties currently include 〈S2〉. The one-particle
properties also will be calculated, since the additional cost of the
one-particle properties calculation is inferior compared to the cost of
〈S2〉. The variable CC_EOM_PROP must be also set to
TRUE. Alternatively, CC_CALC_SSQ can be used to
request 〈S2〉 calculation. |
TYPE:
DEFAULT:
FALSE | (no two-particle properties will be calculated) |
OPTIONS:
RECOMMENDATION:
The two-particle properties are computationally expensive since
they require calculation and use of the two-particle density matrix (the cost
is approximately the same as the cost of an analytic gradient calculation). Do
not request the two-particle properties unless you really need them. |
|
|
|
EOM_SF_STATES
Sets the number of spin-flip target states roots to find. |
TYPE:
DEFAULT:
0 | Do not look for any spin-flip states. |
OPTIONS:
[i,j,k…] | Find i SF states in the first irrep, j states
in the second irrep etc. |
RECOMMENDATION:
|
| EPAO_ITERATE
Controls iterations for EPAO calculations (see PAO_METHOD). |
TYPE:
DEFAULT:
0 | Use uniterated EPAOs based on atomic blocks of SPS. |
OPTIONS:
n | Optimize the EPAOs for up to n iterations. |
RECOMMENDATION:
Use default. For molecules that are not too large, one can test the
sensitivity of the results to the type of minimal functions by the use of
optimized EPAOs in which case a value of n=500 is reasonable. |
|
|
|
EPAO_WEIGHTS
Controls algorithm and weights for EPAO calculations (see PAO_METHOD). |
TYPE:
DEFAULT:
115 | Standard weights, use 1st and 2nd order optimization |
OPTIONS:
15 | Standard weights, with 1st order optimization only. |
RECOMMENDATION:
Use default, unless convergence failure is encountered. |
|
| ERCALC
Specifies the Edmiston-Ruedenberg localized orbitals are to be calculated |
TYPE:
DEFAULT:
OPTIONS:
aabcd | |
aa | specifies the convergence threshold. |
| If aa > 3, the threshold is set to 10−aa. The default is 6. |
| If aa=1, the calculation is aborted after the guess, allowing Pipek-Mezey |
| orbitals to be extracted. |
b | specifies the guess: |
| 0 Boys localized orbitals. This is the default |
| 1 Pipek-Mezey localized orbitals. |
c | specifies restart options (if restarting from an ER calculation): |
| 0 No restart. This is the default |
| 1 Read in MOs from last ER calculation. |
| 2 Read in MOs and RI integrals from last ER calculation. |
d | specifies how to treat core orbitals |
| 0 Do not perform ER localization. This is the default. |
| 1 Localize core and valence together. |
| 2 Do separate localizations on core and valence. |
| 3 Localize only the valence electrons. |
| 4 Use the $localize section.
|
RECOMMENDATION:
ERCALC 1 will usually suffice, which uses threshold 10−6. |
|
|
|
ER_CIS_NUMSTATE
Define how many states to mix with ER localized diabatization. |
TYPE:
DEFAULT:
0 | Do not perform ER localized diabatization. |
OPTIONS:
1 to N where N is the number of CIS states requested (CIS_N_ROOTS) |
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV
or a typical reorganization energy in solvent. |
|
| ESP_TRANS
Controls the calculation of the electrostatic potential of the transition density |
TYPE:
DEFAULT:
OPTIONS:
TRUE | compute the electrostatic potential of the excited state transition density |
FALSE | compute the electrostatic potential of the excited state electronic density |
RECOMMENDATION:
|
|
|
EXCHANGE
Specifies the exchange functional or exchange-correlation functional for hybrid. |
TYPE:
DEFAULT:
No default exchange functional |
OPTIONS:
HF | Fock exchange |
Slater, S | Slater (Dirac 1930) |
Becke86, B86 | Becke 1986 |
Becke, B, B88 | Becke 1988 |
muB88 | Short-range Becke exchange, as formulated by Song et al. [105] |
Gill96, Gill | Gill 1996 |
GG99 | Gilbert and Gill, 1999 |
Becke(EDF1), B(EDF1) | Becke (uses EDF1 parameters) |
PW86, | Perdew-Wang 1986 |
rPW86, | Refitted PW86 for use in vdW-DF-10 and VV10 |
PW91, PW | Perdew-Wang 1991 |
PBE | Perdew-Burke-Ernzerhof 1996 |
TPSS | The nonempirical exchange-correlation scheme of Tao, |
| Perdew, Staroverov, and Scuseria (requires also that the user |
| specify "TPSS" for correlation) |
TPSSH | The hybrid version of TPSS (with no input line for correlation) |
PBE0, PBE1PBE | PBE hybrid with 25% HF exchange |
PBEOP | PBE exchange + one-parameter progressive correlation |
wPBE | Short-range ωPBE exchange, as formulated by
Henderson et al. [106] |
muPBE | Short-range μPBE exchange, due to Song et al. [105] |
B97 | Becke97 XC hybrid |
B97-1 | Becke97 re-optimized by Hamprecht et al. |
B97-2 | Becke97-1 optimized further by Wilson et al. |
B3PW91, Becke3PW91, B3P | B3PW91 hybrid |
B3LYP, Becke3LYP | B3LYP hybrid |
B3LYP5 | B3LYP based on correlation functional #5 of |
HCTH | HCTH hybrid |
HCTH-120 | HCTH-120 hybrid |
HCTH-147 | HCTH-147 hybrid |
HCTH-407 | HCTH-407 hybrid |
| Vosko, Wilk, and Nusair rather than their functional #3 |
BOP | B88 exchange + one-parameter progressive correlation |
EDF1 | EDF1 |
EDF2 | EDF2 |
VSXC | VSXC meta-GGA, not a hybrid |
BMK | BMK hybrid |
M05 | M05 hybrid |
M052X | M05-2X hybrid |
M06L | M06-L hybrid |
M06HF | M06-HF hybrid |
M06 | M06 hybrid |
M062X | M06-2X hybrid |
M08HX | M08-HX hybrid |
M08SO | M08-SO hybrid |
M11L | M11-L hybrid |
M11 | M11 long-range corrected hybrid |
SOGGA | SOGGA hybrid |
SOGGA11 | SOGGA11 hybrid |
SOGGA11X | SOGGA11-X hybrid |
BR89 | Becke-Roussel 1989 represented in analytic form |
omegaB97 | ωB97 long-range corrected hybrid |
omegaB97X | ωB97X long-range corrected hybrid |
omegaB97X-D | ωB97X-D long-range corrected hybrid with dispersion corrections |
omegaB97X-2(LP) | ωB97X-2(LP) long-range corrected double-hybrid |
omegaB97X-2(TQZ) | ωB97X-2(TQZ) long-range corrected double-hybrid |
MCY2 | The MCY2 hyper-GGA exchange-correlation |
| (with no input line for correlation) |
B05 | The hyper-GGA exchange-correlation functional |
| B05 with RI approximation for the exact-exchange energy |
BM05 | MB05 is based on RI-B05 but made it simpler, |
| and slightly more accurate. |
PSTS | The hyper-GGA exchange-correlation functional |
| PSTS with RI approximation for the exact-exchange energy |
| density (with no input line for correlation) |
General, Gen | User defined combination of K, X and C (refer next |
| section). |
RECOMMENDATION:
Consult the literature to guide your selection. |
|
| FAST_XC
Controls direct variable thresholds to accelerate exchange correlation (XC) in
DFT. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Turn FAST_XC on. |
FALSE | Do not use FAST_XC. |
RECOMMENDATION:
Caution: FAST_XC improves the speed of a DFT calculation, but
may occasionally cause the SCF calculation to diverge. |
|
|
|
FDIFF_DER
Controls what types of information are used to compute higher
derivatives. The default uses a combination of energy, gradient and Hessian
information, which makes the force field calculation faster. |
TYPE:
DEFAULT:
3 | for jobs where analytical 2nd derivatives are available. |
0 | for jobs with ECP. |
OPTIONS:
0 | Use energy information only. |
1 | Use gradient information only. |
2 | Use Hessian information only. |
3 | Use energy, gradient, and Hessian information. |
RECOMMENDATION:
When the molecule is larger than benzene with small basis set,
FDIFF_DER=2 may be faster. Note that FDIFF_DER will be set
lower if analytic derivatives of the requested order are not available. Please
refers to IDERIV. |
|
| FDIFF_STEPSIZE_QFF
Displacement used for calculating third and fourth derivatives by finite difference. |
TYPE:
DEFAULT:
5291 | Corresponding to 0.1 bohr. For calculating third and fourth derivatives. |
OPTIONS:
n | Use a step size of n×10−5. |
RECOMMENDATION:
Use default, unless on a very flat potential, in which case a larger value
should be used. |
|
|
|
FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference. |
TYPE:
DEFAULT:
100 | Corresponding to 0.001 Å. For calculating second derivatives. |
OPTIONS:
n | Use a step size of n×10−5. |
RECOMMENDATION:
Use default, unless on a very flat potential, in which case a larger
value should be used. See FDIFF_STEPSIZE_QFF for third and fourth derivatives. |
|
| FOCK_EXTRAP_ORDER
Specifies the polynomial order N for Fock matrix extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Fock matrix extrapolation. |
OPTIONS:
N | Extrapolate using an Nth-order polynomial (N > 0). |
RECOMMENDATION:
|
|
|
FOCK_EXTRAP_POINTS
Specifies the number M of old Fock matrices that are retained for use in
extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Fock matrix extrapolation. |
OPTIONS:
M | Save M Fock matrices for use in extrapolation (M > N) |
RECOMMENDATION:
Higher-order extrapolations with more saved Fock matrices are faster and
conserve energy better than low-order extrapolations, up to a point. In many
cases, the scheme (N = 6, M = 12), in conjunction with
SCF_CONVERGENCE = 6, is found to provide about a 50% savings in
computational cost while still conserving energy. |
|
| FORCE_FIELD
Specifies the force field for MM energies in QM/MM calculations. |
TYPE:
DEFAULT:
OPTIONS:
AMBER99 | AMBER99 force field |
CHARMM27 | CHARMM27 force field |
OPLSAA | OPLSAA force field |
RECOMMENDATION:
|
|
|
FRGM_LPCORR
Specifies a correction method performed after the locally-projected equations are converged. |
TYPE:
DEFAULT:
OPTIONS:
ARS | Approximate Roothaan-step perturbative correction. |
RS | Single Roothaan-step perturbative correction. |
EXACT_SCF | Full SCF variational correction. |
ARS_EXACT_SCF | Both ARS and EXACT_SCF in a single job. |
RS_EXACT_SCF | Both RS and EXACT_SCF in a single job. |
RECOMMENDATION:
For large basis sets use ARS, use RS if ARS fails. |
|
| FRGM_METHOD
Specifies a locally-projected method. |
TYPE:
DEFAULT:
OPTIONS:
STOLL | Locally-projected SCF equations of Stoll are solved. |
GIA | Locally-projected SCF equations of Gianinetti are solved. |
NOSCF_RS | Single Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_ARS | Approximate single Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_DRS | Double Roothaan-step correction to the FRAGMO initial guess. |
NOSCF_RS_FOCK | Non-converged SCF energy of the single Roothaan-step MOs. |
RECOMMENDATION:
STOLL and GIA are for variational optimization of the ALMOs. NOSCF options are for computationally fast corrections of the FRAGMO initial guess. |
|
|
|
FSM_MODE
Specifies the method of interpolation |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
2. In most cases, LST is superior to Cartesian interpolation. |
|
| FSM_NGRAD
Specifies the number of perpendicular gradient steps used to optimize each node |
TYPE:
DEFAULT:
OPTIONS:
N | number of perpendicular gradients per node |
RECOMMENDATION:
4. Anything between 2 and 6 should work, where increasing the number is only needed for difficult reaction paths. |
|
|
|
FSM_NNODE
Specifies the number of nodes along the string |
TYPE:
DEFAULT:
OPTIONS:
N | number of nodes in FSM calculation |
RECOMMENDATION:
15. Use 10 to 20 nodes for a typical calculation. Reaction paths that connect multiple elementary steps should be separated into individual elementary steps, and one FSM job run for each pair of intermediates. Use a higher number when the FSM is followed by an approximate-Hessian
based transition state search (Section 9.7). |
|
| FSM_OPT_MODE
Specifies the method of optimization |
TYPE:
DEFAULT:
OPTIONS:
1 | Conjugate gradients |
2 | Quasi-Newton method with BFGS Hessian update |
RECOMMENDATION:
2. The quasi-Newton method is more efficient when the number of nodes is high. |
|
|
|
FTC_CLASS_THRESH_MULT
Together with FTC_CLASS_THRESH_ORDER, determines the cutoff
threshold for included a shell-pair in the dd class, i.e., the class that
is expanded in terms of plane waves. |
TYPE:
DEFAULT:
5 | Multiplicative part of the FTC classification threshold. Together with |
| the default value of the FTC_CLASS_THRESH_ORDER this leads to |
| the 5×10−5 threshold value. |
OPTIONS:
RECOMMENDATION:
Use the default. If diffuse basis sets are used and the molecule is relatively
big then tighter FTC classification threshold has to be used. According to our
experiments using Pople-type diffuse basis sets, the default 5×10−5 value provides accurate result for an alanine5 molecule while 1×10−5 threshold value for alanine10 and 5×10−6 value for
alanine15 has to be used. |
|
| FTC_CLASS_THRESH_ORDER
Together with FTC_CLASS_THRESH_MULT, determines the cutoff
threshold for included a shell-pair in the dd class, i.e., the class that
is expanded in terms of plane waves. |
TYPE:
DEFAULT:
5 | Logarithmic part of the FTC classification threshold. Corresponds to 10−5 |
OPTIONS:
RECOMMENDATION:
|
|
|
FTC_SMALLMOL
Controls whether or not the operator is evaluated on a large grid and stored in
memory to speed up the calculation. |
TYPE:
DEFAULT:
OPTIONS:
1 | Use a big pre-calculated array to speed up the FTC calculations |
0 | Use this option to save some memory |
RECOMMENDATION:
Use the default if possible and use 0 (or buy some more memory) when
needed. |
|
| FTC
Controls the overall use of the FTC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use FTC in the Coulomb part |
1 | Use FTC in the Coulomb part |
RECOMMENDATION:
Use FTC when bigger and/or diffuse basis sets are used. |
|
|
|
GAUSSIAN_BLUR
Enables the use of Gaussian-delocalized external charges in a QM/MM calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Delocalizes external charges with Gaussian functions. |
FALSE | Point charges |
RECOMMENDATION:
|
| GAUSS_BLUR_WIDTH
Delocalization width for external MM Gaussian charges in a Janus calculations. |
TYPE:
DEFAULT:
OPTIONS:
n | Use a width of n ×10−4 Å. |
RECOMMENDATION:
Blur all MM external charges in a QM/MM calculation with the specified width.
Gaussian blurring is currently incompatible with PCM calculations. Values of
1.0-2.0 Å are recommended in Ref. .
|
|
|
|
GEOM_OPT_COORDS
Controls the type of optimization coordinates. |
TYPE:
DEFAULT:
OPTIONS:
0 | Optimize in Cartesian coordinates. |
1 | Generate and optimize in internal coordinates, if this fails abort. |
-1 | Generate and optimize in internal coordinates, if this fails at any stage of the |
| optimization, switch to Cartesian and continue. |
2 | Optimize in Z-matrix coordinates, if this fails abort. |
-2 | Optimize in Z-matrix coordinates, if this fails during any stage of the |
| optimization switch to Cartesians and continue. |
RECOMMENDATION:
Use the default; delocalized internals are more efficient. |
|
| GEOM_OPT_DMAX
Maximum allowed step size. Value supplied is multiplied by 10−3. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
GEOM_OPT_HESSIAN
Determines the initial Hessian status. |
TYPE:
DEFAULT:
OPTIONS:
DIAGONAL | Set up diagonal Hessian. |
READ | Have exact or initial Hessian. Use as is if Cartesian, or transform |
| if internals. |
RECOMMENDATION:
An accurate initial Hessian will improve the performance of the optimizer, but is
expensive to compute. |
|
| GEOM_OPT_LINEAR_ANGLE
Threshold for near linear bond angles (degrees). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
GEOM_OPT_MAX_CYCLES
Maximum number of optimization cycles. |
TYPE:
DEFAULT:
OPTIONS:
n | User defined positive integer. |
RECOMMENDATION:
The default should be sufficient for most cases. Increase if the initial
guess geometry is poor, or for systems with shallow potential wells. |
|
| GEOM_OPT_MAX_DIIS
Controls maximum size of subspace for GDIIS. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use GDIIS. |
-1 | Default size = min(NDEG, NATOMS, 4) NDEG = number of molecular |
| degrees of freedom. |
n | Size specified by user. |
RECOMMENDATION:
Use default or do not set n too large. |
|
|
|
GEOM_OPT_MODE
Determines Hessian mode followed during a transition state search. |
TYPE:
DEFAULT:
OPTIONS:
0 | Mode following off. |
n | Maximize along mode n. |
RECOMMENDATION:
Use default, for geometry optimizations. |
|
| GEOM_OPT_PRINT
Controls the amount of Optimize print output. |
TYPE:
DEFAULT:
3 | Error messages, summary, warning, standard information and gradient print
out. |
OPTIONS:
0 | Error messages only. |
1 | Level 0 plus summary and warning print out. |
2 | Level 1 plus standard information. |
3 | Level 2 plus gradient print out. |
4 | Level 3 plus Hessian print out. |
5 | Level 4 plus iterative print out. |
6 | Level 5 plus internal generation print out. |
7 | Debug print out. |
RECOMMENDATION:
|
|
|
GEOM_OPT_SYMFLAG
Controls the use of symmetry in Optimize. |
TYPE:
DEFAULT:
OPTIONS:
1 | Make use of point group symmetry. |
0 | Do not make use of point group symmetry. |
RECOMMENDATION:
|
| GEOM_OPT_TOL_DISPLACEMENT
Convergence on maximum atomic displacement. |
TYPE:
DEFAULT:
1200 ≡ 1200 ×10−6 tolerance on maximum atomic displacement. |
OPTIONS:
n | Integer value (tolerance = n ×10−6). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
|
|
GEOM_OPT_TOL_ENERGY
Convergence on energy change of successive optimization cycles. |
TYPE:
DEFAULT:
100 ≡ 100 ×10−8 tolerance on maximum gradient component. |
OPTIONS:
n Integer value (tolerance = value n ×10−8). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
| GEOM_OPT_TOL_GRADIENT
Convergence on maximum gradient component. |
TYPE:
DEFAULT:
300 | ≡ 300×10−6 tolerance on maximum gradient component. |
OPTIONS:
n | Integer value (tolerance = n ×10−6). |
RECOMMENDATION:
Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY
must be satisfied. |
|
|
|
GEOM_OPT_UPDATE
Controls the Hessian update algorithm. |
TYPE:
DEFAULT:
OPTIONS:
-1 | Use the default update algorithm. |
0 | Do not update the Hessian (not recommended). |
1 | Murtagh-Sargent update. |
2 | Powell update. |
3 | Powell/Murtagh-Sargent update (TS default). |
4 | BFGS update (OPT default). |
5 | BFGS with safeguards to ensure retention of positive definiteness |
| (GDISS default). |
RECOMMENDATION:
|
| GEOM_PRINT
Controls the amount of geometric information printed at each step. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Prints out all geometric information; bond distances, angles, torsions. |
FALSE | Normal printing of distance matrix. |
RECOMMENDATION:
Use if you want to be able to quickly examine geometric parameters at the
beginning and end of optimizations. Only prints in the beginning of single
point energy calculations. |
|
|
|
GRAIN
Controls the number of lowest-level boxes in one dimension for CFMM. |
TYPE:
DEFAULT:
-1 | Program decides best value, turning on CFMM when useful |
OPTIONS:
-1 | Program decides best value, turning on CFMM when useful |
1 | Do not use CFMM |
n ≥ 8 | Use CFMM with n lowest-level boxes in one dimension |
RECOMMENDATION:
This is an expert option; either use the default, or use a value of 1 if CFMM
is not desired. |
|
| GVB_AMP_SCALE
Scales the default orbital amplitude iteration step size by n/1000 for IP/RCC.
PP amplitude equations are solved analytically, so this parameter does not
affect PP. |
TYPE:
DEFAULT:
1000 | Corresponding to 100% |
OPTIONS:
RECOMMENDATION:
Default is usually fine, but in some highly-correlated
systems it can help with convergence to use smaller values. |
|
|
|
GVB_DO_ROHF
Sets the number of Unrestricted-in-Active Pairs to be kept restricted. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If n is the same value as GVB_N_PAIRS returns the ROHF solution
for GVB, only works with the UNRESTRICTED=TRUE
implementation of GVB with GVB_OLD_UPP=0 (it's default value) |
|
| GVB_DO_SANO
Sets the scheme used in determining the active virtual orbitals
in a Unrestricted-in-Active Pairs GVB calculation. |
TYPE:
DEFAULT:
OPTIONS:
0 | No localization or Sano procedure |
1 | Only localizes the active virtual orbitals |
2 | Uses the Sano procedure |
RECOMMENDATION:
Different initial guesses can sometimes lead to different
solutions. Disabling sometimes can aid in finding more non-local solutions for the orbitals. |
|
|
|
GVB_GUESS_MIX
Similar to SCF_GUESS_MIX, it breaks alpha / beta symmetry for UPP by
mixing the alpha HOMO and LUMO orbitals according to the user-defined fraction
of LUMO to add the HOMO. 100 corresponds to a 1:1 ratio of HOMO and LUMO in
the mixed orbitals. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined, 0 ≤ n ≤ 100 |
RECOMMENDATION:
25 often works well to break symmetry without overly
impeding convergence. |
|
| GVB_LOCAL
Sets the localization scheme used in the initial guess wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
0 | No Localization |
1 | Boys localized orbitals |
2 | Pipek-Mezey orbitals |
RECOMMENDATION:
Different initial guesses can sometimes lead to different solutions.
It can be helpful to try both to ensure the global minimum has been found. |
|
|
|
GVB_N_PAIRS
Alternative to CC_REST_OCC and CC_REST_VIR for setting
active space size in GVB and valence coupled cluster methods. |
TYPE:
DEFAULT:
PP active space (1 occ and 1 virt for each valence electron pair) |
OPTIONS:
RECOMMENDATION:
Use the default unless one wants to study a special active space. When using
small active spaces, it is important to ensure that the proper orbitals are
incorporated in the active space. If not, use the $reorder_mo feature to adjust
the SCF orbitals appropriately. |
|
| GVB_OLD_UPP
Which unrestricted algorithm to use for GVB. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use Unrestricted-in-Active Pairs |
1 | Use Unrestricted Implementation described in Ref. |
RECOMMENDATION:
Only works for Unrestricted PP and no other GVB model. |
|
|
|
GVB_ORB_CONV
The GVB-CC wavefunction is considered converged when the root-mean-square
orbital gradient and orbital step sizes are less than
10−GVB_ORB_CONV. Adjust THRESH simultaneously. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use 6 for PP(2) jobs or geometry optimizations.
Tighter convergence (i.e. 7 or higher) cannot always be reliably achieved. |
|
| GVB_ORB_MAX_ITER
Controls the number of orbital iterations allowed in GVB-CC calculations.
Some jobs, particularly unrestricted PP jobs can require 500-1000 iterations. |
TYPE:
DEFAULT:
OPTIONS:
User-defined number of iterations. |
RECOMMENDATION:
Default is typically adequate, but some jobs, particularly UPP jobs, can
require 500-1000 iterations if converged tightly. |
|
|
|
GVB_ORB_SCALE
Scales the default orbital step size by n/1000. |
TYPE:
DEFAULT:
1000 | Corresponding to 100% |
OPTIONS:
RECOMMENDATION:
Default is usually fine, but for some stretched geometries it
can help with convergence to use smaller values. |
|
| GVB_POWER
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spin-unrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term included in
the energy denominator: -(c/10000)(etijp−1)/(e1−1) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should be decreased if unrestricted amplitudes do not converge or
converge slowly at dissociation, and should be kept even valued. |
|
|
|
GVB_PRINT
Controls the amount of information printed during a GVB-CC job. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Should never need to go above 0 or 1. |
|
| GVB_REGULARIZE
Coefficient for GVB_IP exchange type amplitude regularization
to improve the convergence of the amplitude equations especially
for spin-unrestricted amplitudes near dissociation. This is the
leading coefficient for an amplitude dampening term -(c/10000)(etijp−1)/(e1−1) |
TYPE:
DEFAULT:
0 for restricted | 1 for unrestricted |
OPTIONS:
RECOMMENDATION:
Should be increased if unrestricted amplitudes do not converge or
converge slowly at dissociation. Set this to zero to remove
all dynamically-valued amplitude regularization. |
|
|
|
GVB_REORDER_1
Tells the code which two pairs to swap first |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified
in GVB_REORDER_PAIRS ≥ 1. |
|
| GVB_REORDER_2
Tells the code which two pairs to swap second |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example swapping
pair 1 and 2 would get the input 001002. Must be specified in GVB_REORDER_PAIRS ≥ 2. |
|
|
|
GVB_REORDER_3
Tells the code which two pairs to swap third |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002.
Must be specified in GVB_REORDER_PAIRS ≥ 3. |
|
| GVB_REORDER_4
Tells the code which two pairs to swap fourth |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must
be specified in GVB_REORDER_PAIRS ≥ 4. |
|
|
|
GVB_REORDER_5
Tells the code which two pairs to swap fifth |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This is in the format of two 3-digit pair indices that
tell the code to swap pair XXX with YYY, for example
swapping pair 1 and 2 would get the input 001002. Must be
specified in GVB_REORDER_PAIRS ≥ 5. |
|
| GVB_REORDER_PAIRS
Tells the code how many GVB pairs to switch around |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for the user to change the order the active
pairs are placed in after the orbitals are read in or are
guessed using localization and the Sano procedure. Up to 5
sequential pair swaps can be made, but it is best to leave this alone. |
|
|
|
GVB_RESTART
Restart a job from previously-converged GVB-CC orbitals. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Useful when trying to converge to the same GVB
solution at slightly different geometries, for example. |
|
| GVB_SHIFT
Value for a statically valued energy shift in the energy
denominator used to solve the coupled cluster amplitude equations, n/10000. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Default is fine, can be used in lieu of the dynamically
valued amplitude regularization if it does not aid convergence. |
|
|
|
GVB_SYMFIX
Should GVB use a symmetry breaking fix |
TYPE:
DEFAULT:
OPTIONS:
0 | no symmetry breaking fix |
1 | symmetry breaking fix with virtual orbitals spanning the active space |
2 | symmetry breaking fix with virtual orbitals spanning the whole virtual space |
RECOMMENDATION:
It is best to stick with type 1 to get a symmetry breaking correction
with the best results coming from CORRELATION=NP and GVB_SYMFIX=1. |
|
| GVB_SYMPEN
Sets the pre-factor for the amplitude regularization term for the SB amplitudes |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the pre-factor for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e(c*100)*t2−1). |
|
|
|
GVB_SYMSCA
Sets the weight for the amplitude regularization term for the SB amplitudes |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Sets the weight for the amplitude regularization term for
the SB amplitudes: −(γ/1000)(e(c*100)*t2−1). |
|
| GVB_TRUNC_OCC
Controls how many pairs' occupied orbitals are truncated from the GVB active space |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing the n
lowest energy occupied orbitals from the GVB active space
while leaving their paired virtual orbitals in the active space.
Only the models including the SIP and DIP amplitudes (ie NP and 2P)
benefit from this all other models this equivalent to just
reducing the total number of pairs. |
|
|
|
GVB_TRUNC_VIR
Controls how many pairs' virtual orbitals are truncated from the GVB active space |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
This allows for asymmetric GVB active spaces removing
the n highest energy occupied orbitals from the GVB
active space while leaving their paired virtual orbitals
in the active space. Only the models including the SIP
and DIP amplitudes (ie NP and 2P) benefit from this all
other models this equivalent to just reducing the total number of pairs. |
|
| GVB_UNRESTRICTED
Controls restricted versus unrestricted PP jobs. Usually handled
automatically. |
TYPE:
DEFAULT:
same value as UNRESTRICTED |
OPTIONS:
RECOMMENDATION:
Set this variable explicitly only to do a UPP job from an RHF
or ROHF initial guess. Leave this variable alone and specify
UNRESTRICTED=TRUE to access the new Unrestricted-in-Active-Pairs
GVB code which can return an RHF or ROHF solution if used with GVB_DO_ROHF |
|
|
|
HESS_AND_GRAD
Enables the evaluation of both analytical gradient and Hessian in a single job |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluates both gradient and Hessian. |
FALSE | Evaluates Hessian only. |
RECOMMENDATION:
Use only in a frequency (and thus Hessian) evaluation. |
|
| HFPT_BASIS
Specifies the secondary basis in a HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings. |
|
|
|
HFPT
Activates HFPC/DFPC calculation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use Dual-Basis to capture large-basis effects at smaller basis cost. See reference for recommended basis set, functional, and grid pairings. |
|
| HF_LR
Sets the fraction of Hartree-Fock exchange at r12=∞. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to HF_LR = n/1000 |
RECOMMENDATION:
|
|
|
HF_SR
Sets the fraction of Hartree-Fock exchange at r12=0. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to HF_SR = n/1000 |
RECOMMENDATION:
|
| HIRSHFELD_READ
Switch to force reading in of isolated atomic densities. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Read in isolated atomic densities from previous Hirshfeld calculation from disk. |
FALSE | Generate new isolated atomic densities. |
RECOMMENDATION:
Use default unless system is large. Note, atoms should be in the same order
with same basis set used as in the previous Hirshfeld
calculation (although coordinates can change). The previous calculation
should be run with the -save switch. |
|
|
|
HIRSHFELD_SPHAVG
Controls whether atomic densities should be spherically averaged in pro-molecule. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Spherically average atomic densities. |
FALSE | Do not spherically average.
|
RECOMMENDATION:
|
| HIRSHFELD
Controls running of Hirshfeld population analysis. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculate Hirshfeld populations. |
FALSE | Do not calculate Hirshfeld populations.
|
RECOMMENDATION:
|
|
|
ICVICK
Specifies whether to perform cavity check |
TYPE:
DEFAULT:
OPTIONS:
0 | no cavity check, use only the outer cavity |
1 | cavity check, generating both the inner and outer cavities and compare. |
RECOMMENDATION:
Consider turning off cavity check only if the molecule has a hole and if a star
(outer) surface is expected. |
|
| IDERIV
Controls the order of derivatives that are evaluated analytically. The user
is not normally required to specify a value, unless numerical derivatives
are desired. The derivatives will be evaluated numerically if IDERIV
is set lower than JOBTYPE requires. |
TYPE:
DEFAULT:
| Set to the order of derivative that JOBTYPE requires |
OPTIONS:
2 | Analytic second derivatives of the energy (Hessian) |
1 | Analytic first derivatives of the energy. |
0 | Analytic energies only. |
RECOMMENDATION:
Usually set to the maximum possible for efficiency. Note that IDERIV
will be set lower if analytic derivatives of the requested order are not
available. |
|
|
|
IGDEFIELD
Triggers the calculation of the electrostatic potential and/or the electric field
at the positions of the MM charges. |
TYPE:
DEFAULT:
OPTIONS:
O | Computes ESP. |
1 | Computes ESP and EFIELD. |
2 | Computes EFIELD. |
RECOMMENDATION:
Must use this $rem when IGDESP is specified. |
|
| IGDESP
Controls evaluation of the electrostatic potential on a grid of points. If
enabled, the output is in an ACSII file, plot.esp, in the format x, y, z, esp
for each point. |
TYPE:
DEFAULT:
none no electrostatic potential evaluation |
OPTIONS:
−1 | read grid input via the $plots section of the input deck |
0 | Generate the ESP values at all nuclear positions. |
+n | read n grid points in bohrs (!) from the ACSII file ESPGrid. |
RECOMMENDATION:
|
|
|
IGNORE_LOW_FREQ
Low frequencies that should be treated as rotation can be ignored during |
anharmonic correction calculation. |
TYPE:
DEFAULT:
300 | Corresponding to 300 cm−1. |
OPTIONS:
n | Any mode with harmonic frequency less than n will be ignored. |
RECOMMENDATION:
|
| INCDFT_DENDIFF_THRESH
Sets the threshold for screening density matrix values in the IncDFT procedure. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. |
|
|
|
INCDFT_DENDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening density matrix
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_DENDIFF_THRESH is reached. This means this
value must be set lower than INCDFT_DENDIFF_THRESH. |
TYPE:
DEFAULT:
0 | Variable threshold is not used. |
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. |
|
| INCDFT_GRIDDIFF_THRESH
Sets the threshold for screening functional values in the IncDFT procedure |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to
tighten the threshold. |
|
|
|
INCDFT_GRIDDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening the functional
values in the IncDFT procedure. The threshold will begin at this value and
then vary depending on the error in the current SCF iteration until the value
specified by INCDFT_GRIDDIFF_THRESH is reached. This means that
this value must be set lower than INCDFT_GRIDDIFF_THRESH. |
TYPE:
DEFAULT:
0 | Variable threshold is not used. |
OPTIONS:
n | Corresponding to a threshold of 10−n. |
RECOMMENDATION:
If the default value causes convergence problems, set this value higher
to tighten accuracy. If this fails, set to 0 and use a static threshold. |
|
| INCDFT
Toggles the use of the IncDFT procedure for DFT energy calculations. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use IncDFT |
TRUE | Use IncDFT |
RECOMMENDATION:
Turning this option on can lead to faster SCF calculations, particularly
towards the end of the SCF. Please note that for some systems use of this
option may lead to convergence problems. |
|
|
|
INCFOCK
Iteration number after which the incremental Fock matrix algorithm is
initiated |
TYPE:
DEFAULT:
1 | Start INCFOCK after iteration number 1 |
OPTIONS:
User-defined (0 switches INCFOCK off) |
RECOMMENDATION:
May be necessary to allow several iterations before switching on INCFOCK. |
|
| INTCAV
A flag to select the surface integration method. |
TYPE:
DEFAULT:
OPTIONS:
0 | Single center Lebedev integration. |
1 | Single center spherical polar integration. |
RECOMMENDATION:
The Lebedev integration is by far the more efficient. |
|
|
|
INTEGRALS_BUFFER
Controls the size of in-core integral storage buffer. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default, or consult your systems administrator for hardware limits. |
|
| INTEGRAL_2E_OPR
Determines the two-electron operator. |
TYPE:
DEFAULT:
OPTIONS:
-1 | Apply the CASE approximation. |
-2 | Coulomb Operator. |
RECOMMENDATION:
Use default unless the CASE operator is desired. |
|
|
|
INTRACULE
Controls whether intracule properties are calculated (see also the
$intracule section). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | No intracule properties. |
TRUE | Evaluate intracule properties. |
RECOMMENDATION:
|
| IOPPRD
Specifies the choice of system operator form. |
TYPE:
DEFAULT:
OPTIONS:
0 | Symmetric form. |
1 | Non-symmetric form. |
RECOMMENDATION:
The default uses more memory but is generally more efficient, we recommend its
use unless there is shortage of memory available. |
|
|
|
IROTGR
Rotation of the cavity surface integration grid. |
TYPE:
DEFAULT:
OPTIONS:
0 | No rotation. |
1 | Rotate initial xyz axes of the integration grid to coincide |
| with principal moments of nuclear inertia (relevant if ITRNGR=1) |
2 | Rotate initial xyz axes of integration grid to coincide with |
| principal moments of nuclear charge (relevant if ITRNGR=2) |
3 | Rotate initial xyz axes of the integration grid through user-specified |
| Euler angles as defined by Wilson, Decius, and Cross. |
RECOMMENDATION:
The default is recommended unless the knowledgeable user has good reason otherwise. |
|
| ISHAPE
A flag to set the shape of the cavity surface. |
TYPE:
DEFAULT:
OPTIONS:
0 | use the electronic iso-density surface. |
1 | use a spherical cavity surface. |
RECOMMENDATION:
|
|
|
ISOTOPES
Specifies if non-default masses are to be used in the frequency calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use default masses only. |
TRUE | Read isotope masses from $isotopes section. |
RECOMMENDATION:
|
| ITRNGR
Translation of the cavity surface integration grid. |
TYPE:
DEFAULT:
OPTIONS:
0 | No translation (i.e., center of the cavity at the origin |
| of the atomic coordinate system) |
1 | Translate to the center of nuclear mass. |
2 | Translate to the center of nuclear charge. |
3 | Translate to the midpoint of the outermost atoms. |
4 | Translate to midpoint of the outermost non-hydrogen atoms. |
5 | Translate to user-specified coordinates in Bohr. |
6 | Translate to user-specified coordinates in Angstroms. |
RECOMMENDATION:
The default value is recommended unless the single-center integrations
procedure fails. |
|
|
|
JOBTYPE
Specifies the type of calculation. |
TYPE:
DEFAULT:
OPTIONS:
SP | Single point energy. |
OPT | Geometry Minimization. |
TS | Transition Structure Search. |
FREQ | Frequency Calculation. |
FORCE | Analytical Force calculation. |
RPATH | Intrinsic Reaction Coordinate calculation. |
NMR | NMR chemical shift calculation. |
BSSE | BSSE calculation. |
EDA | Energy decomposition analysis. |
RECOMMENDATION:
|
| LB94_BETA
Set the β parameter of LB94 xc potential |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to β = n/10000. |
RECOMMENDATION:
Use default, i.e., β = 0.05 |
|
|
|
LINEQ
Flag to select the method for solving the linear equations that determine the
apparent point charges on the cavity surface. |
TYPE:
DEFAULT:
OPTIONS:
0 | use LU decomposition in memory if space permits, else switch to LINEQ=2 |
1 | use conjugate gradient iterations in memory if space permits, else use LINEQ=2 |
2 | use conjugate gradient iterations with the system matrix stored externally on disk. |
RECOMMENDATION:
The default should be sufficient in most cases. |
|
| LINK_ATOM_PROJECTION
Controls whether to perform a link-atom projection |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Performs the projection |
FALSE | No projection |
RECOMMENDATION:
Necessary in a full QM/MM Hessian evaluation on a system with link atoms |
|
|
|
LIN_K
Controls whether linear scaling evaluation of exact exchange (LinK) is used. |
TYPE:
DEFAULT:
Program chooses, switching on LinK whenever CFMM is used. |
OPTIONS:
TRUE | Use LinK |
FALSE | Do not use LinK |
RECOMMENDATION:
Use for HF and hybrid DFT calculations with large numbers of atoms. |
|
| LOBA_THRESH
Specifies the thresholds to use for LOBA |
TYPE:
DEFAULT:
OPTIONS:
aabb | |
aa | specifies the threshold to use for localization |
bb | specifies the threshold to use for occupation |
Both are measured in %
|
RECOMMENDATION:
Decrease bb to see the smaller contributions to orbitals. Values of
aa between 40 and 75 have been shown to given meaningful results. |
|
|
|
LOBA
Specifies the methods to use for LOBA |
TYPE:
DEFAULT:
OPTIONS:
ab | |
a | specifies the localization method |
| 0 Perform Boys localization. |
| 1 Perform PM localization. |
| 2 Perform ER localization. |
b | specifies the population analysis method |
| 0 Do not perform LOBA. This is the default. |
| 1 Use Mulliken population analysis. |
| 2 Use Löwdin population analysis. |
RECOMMENDATION:
Boys Localization is the fastest. ER will require an auxiliary basis set. |
LOBA 12 provides a reasonable speed/accuracy compromise. |
|
| LOCAL_INTERP_ORDER
Controls the order of the B-spline |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The default value is sufficiently accurate |
|
|
|
LOC_CIS_OV_SEPARATE
Decide whether or not to localized the "occupied" and "virtual" components of the localized diabatization
function, i.e., whether to localize the electron attachments and detachments separately. |
TYPE:
DEFAULT:
FALSE | Do not separately localize electron attachments and detachments. |
OPTIONS:
RECOMMENDATION:
If one wants to use Boys localized diabatization for energy transfer (as opposed to electron transfer)
, this is a necessary option. ER is more rigorous technique, and does not require this OV feature, but will be somewhat slower. |
|
| LOWDIN_POPULATION
Run a Löwdin population analysis instead of a Mulliken. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate Löwdin Populations. |
TRUE | Run Löwdin Population analyses instead of Mulliken. |
RECOMMENDATION:
|
|
|
LRC_DFT
Controls the application of long-range-corrected DFT |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not apply long-range correction. |
TRUE | (or 1) Use the long-range-corrected version of the requested functional.
|
RECOMMENDATION:
Long-range correction is available for any combination of Hartree-Fock,
B88, and PBE exchange (along with any stand-alone correlation functional).
|
|
| MAKE_CUBE_FILES
Requests generation of cube files for MOs, NTOs, or NBOs. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not generate cube files. |
TRUE | Generate cube files for MOs and densities. |
NTOS | Generate cube files for NTOs. |
NBOS | Generate cube files for NBOs.
|
RECOMMENDATION:
|
|
|
MAX_CIS_CYCLES
Maximum number of CIS iterative cycles allowed |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of cycles |
RECOMMENDATION:
Default is usually sufficient. |
|
| MAX_CIS_SUBSPACE
Maximum number of subspace vectors allowed in the CIS iterations |
TYPE:
DEFAULT:
As many as required to converge all roots |
OPTIONS:
n | User-defined number of subspace vectors |
RECOMMENDATION:
The default is usually appropriate, unless a large number of states are
requested for a large molecule.
The total memory required to store the subspace vectors is bounded
above by 2nOV, where O and V represent the number of occupied
and virtual orbitals,
respectively. n can be reduced to save memory, at the cost of a
larger number of CIS iterations. Convergence may be impaired if n
is not much larger than CIS_N_ROOTS. |
|
|
|
MAX_DIIS_CYCLES
The maximum number of DIIS iterations before switching to (geometric) direct
minimization when SCF_ALGORITHM is DIIS_GDM or
DIIS_DM. See also THRESH_DIIS_SWITCH. |
TYPE:
DEFAULT:
OPTIONS:
1 | Only a single Roothaan step before switching to (G)DM |
n | n DIIS iterations before switching to (G)DM. |
RECOMMENDATION:
|
| MAX_RCA_CYCLES
The maximum number of RCA iterations before switching to DIIS when SCF_ALGORITHM is RCA_DIIS. |
TYPE:
DEFAULT:
OPTIONS:
N | N RCA iterations before switching to DIIS |
RECOMMENDATION:
|
|
|
MAX_SCF_CYCLES
Controls the maximum number of SCF iterations permitted. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase for slowly converging systems such as those containing transition
metals. |
|
| MAX_SUB_FILE_NUM
Sets the maximum number of sub files allowed. |
TYPE:
DEFAULT:
16 Corresponding to a total of 32Gb for a given file. |
OPTIONS:
n | User-defined number of gigabytes. |
RECOMMENDATION:
Leave as default, or adjust according to your system limits. |
|
|
|
MAX_SUB_FILE_NUM
Sets the maximum number of sub files allowed. |
TYPE:
DEFAULT:
16 Corresponding to a total of 32Gb for a given file. |
OPTIONS:
n | User-defined number of gigabytes. |
RECOMMENDATION:
Leave as default, or adjust according to your system limits. |
|
| MEM_STATIC
Sets the memory for individual program modules |
TYPE:
DEFAULT:
64 | corresponding to 64 Mb |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
At least 150(N2 + N)D of MEM_STATIC is required
(N: number of basis functions, D: size of a double precision storage, usually 8).
Because a number of matrices with N2 size also need to be
stored, 32-160 Mb of additional MEM_STATIC is needed. |
|
|
|
MEM_STATIC
Sets the memory for Fortran AO integral calculation and transformation modules. |
TYPE:
DEFAULT:
64 | corresponding to 64 Mb. |
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
For direct and semi-direct MP2 calculations, this must exceed OVN +
requirements for AO integral evaluation (32-160 Mb), as discussed above. |
|
| MEM_TOTAL
Sets the total memory available to Q-Chem |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes |
RECOMMENDATION:
The minimum memory requirement of RI-CIS(D) is approximately
MEM_STATIC + max(3SVXD, 3X2D)
(S: number of excited states, X: number of auxiliary basis
functions, D: size of a double precision storage, usually 8). However, because
RI-CIS(D) uses a batching scheme for efficient evaluations of electron
repulsion integrals, specifying more memory will significantly speed up the
calculation. Put as much memory as possible if you are not sure
what to use, but never put any more than what is available.
The minimum memory requirement of SOS-CIS(D) and SOS-CIS(D0) is approximately
MEM_STATIC + 20 X2 D. SOS-CIS(D0) gradient calculation
becomes more efficient when 30 X2 D more memory space is given.
Like in RI-CIS(D), put as much memory as possible if you are not sure what to use.
The actual memory size used in these calculations will be printed out in the output file
to give a guide about the required memory. |
|
|
|
MEM_TOTAL
Sets the total memory available to Q-Chem, in megabytes. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of megabytes. |
RECOMMENDATION:
Use default, or set to the physical memory of your machine. Note that if more
than 1GB is specified for a CCMAN job, the memory is allocated as
follows |
12% | MEM_STATIC |
50% | CC_MEMORY |
35% | Other memory requirements:
|
|
|
| METECO
Sets the threshold criteria for discarding shell-pairs. |
TYPE:
DEFAULT:
2 | Discard shell-pairs below 10−THRESH. |
OPTIONS:
1 | Discard shell-pairs four orders of magnitude below machine precision. |
2 | Discard shell-pairs below 10−THRESH. |
RECOMMENDATION:
|
|
|
MGC_AMODEL
Choice of approximate cluster model. |
TYPE:
DEFAULT:
Determines | how the CC equations are approximated: |
OPTIONS:
0% | Local Active-Space Amplitude iterations. |
| (pre-calculate GVB orbitals with |
| your method of choice (RPP is good)). |
| |
7% | Optimize-Orbitals using the VOD 2-step solver. |
| (Experimental only use with MGC_AMPS = 2, 24 ,246) |
| |
8% | Traditional Coupled Cluster up to CCSDTQPH. |
9% | MR-CC version of the Pair-Models. (Experimental) |
RECOMMENDATION:
|
| MGC_AMPS
Choice of Amplitude Truncation |
TYPE:
DEFAULT:
OPTIONS:
2 ≤ n ≤ 123456, a sorted list of integers for every amplitude |
which will be iterated. Choose 1234 for PQ and 123456 for PH |
RECOMMENDATION:
|
|
|
MGC_LOCALINTER
Pair filter on an intermediate. |
TYPE:
DEFAULT:
OPTIONS:
Any nonzero value enforces the pair constraint on intermediates, |
significantly reducing computational cost. Not recommended for ≤ 2 pair locality |
RECOMMENDATION:
|
| MGC_LOCALINTS
Pair filter on an integrals. |
TYPE:
DEFAULT:
OPTIONS:
Enforces a pair filter on the 2-electron integrals, significantly |
reducing computational cost. Generally useful. for more than 1 pair locality. |
RECOMMENDATION:
|
|
|
MGC_NLPAIRS
Number of local pairs on an amplitude. |
TYPE:
DEFAULT:
OPTIONS:
Must be greater than 1, which corresponds to the PP model.
2 for PQ, and 3 for PH. |
RECOMMENDATION:
|
| MGEMM_THRESH
Sets MGEMM threshold to determine the separation
between "large" and "small" matrix elements.
A larger threshold value will result in a value closer
to the single-precision result. Note that the desired factor
should be multiplied by 10000 to ensure an integer value. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
For small molecules and basis sets up to triple-ζ, the
default value suffices to not deviate too much from the
double-precision values. Care should be taken to reduce
this number for larger molecules and also larger basis-sets. |
|
|
|
MM_CHARGES
Requests the calculation of multipole-derived charges (MDCs). |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Calculates the MDCs and also the traceless form of the multipole moments |
RECOMMENDATION:
Set to TRUE if MDCs or the traceless form of the multipole
moments are desired. The calculation does not take long. |
|
| MODEL_SYSTEM_CHARGE
Specifies the QM subsystem charge if different from the $molecule section. |
TYPE:
DEFAULT:
OPTIONS:
n | The charge of the QM subsystem. |
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system)
has a charge that is different from the total system charge. |
|
|
|
MODEL_SYSTEM_MULT
Specifies the QM subsystem multiplicity if different from the $molecule section. |
TYPE:
DEFAULT:
OPTIONS:
n | The multiplicity of the QM subsystem. |
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system)
has a multiplicity that is different from the total system multiplicity.
ONIOM calculations must be closed shell. |
|
| MODE_COUPLING
Number of modes coupling in the third and fourth derivatives calculation. |
TYPE:
DEFAULT:
2 | for two modes coupling. |
OPTIONS:
n | for n modes coupling, Maximum value is 4. |
RECOMMENDATION:
|
|
|
MOLDEN_FORMAT
Requests a MolDen-formatted input file containing information from a Q-Chem
job. |
TYPE:
DEFAULT:
OPTIONS:
True | Append MolDen input file at the end of the Q-Chem output file. |
RECOMMENDATION:
|
| MOM_PRINT
Switches printing on within the MOM procedure. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Printing is turned off |
TRUE | Printing is turned on. |
RECOMMENDATION:
|
|
|
MOM_START
Determines when MOM is switched on to stabilize DIIS iterations. |
TYPE:
DEFAULT:
OPTIONS:
0 (FALSE) | MOM is not used |
n | MOM begins on cycle n. |
RECOMMENDATION:
Set to 1 if preservation of initial orbitals is desired. If MOM is to be
used to aid convergence, an SCF without MOM should be run to determine when
the SCF starts oscillating. MOM should be set to start just before the
oscillations. |
|
| MOPROP_CONV_1ST
Sets the convergence criteria for CPSCF and 1st order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n < 10 | Convergence threshold set to 10−n. |
RECOMMENDATION:
|
|
|
MOPROP_CONV_2ND
Sets the convergence criterion for second-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n < 10 | Convergence threshold set to 10−n. |
RECOMMENDATION:
|
| MOPROP_DIIS_DIM_SS
Specified the DIIS subspace dimension. |
TYPE:
DEFAULT:
OPTIONS:
0 | No DIIS. |
n | Use a subspace of dimension n. |
RECOMMENDATION:
|
|
|
MOPROP_DIIS
Controls the use of Pulays DIIS. |
TYPE:
DEFAULT:
OPTIONS:
0 | Turn off DIIS. |
5 | Turn on DIIS. |
RECOMMENDATION:
|
| MOPROP_MAXITER_2ND
The maximal number of iterations for second-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | Set maximum number of iterations to n. |
RECOMMENDATION:
|
|
|
MOPROP_PERTNUM
Set the number of perturbed densities that will to be treated together. |
TYPE:
DEFAULT:
OPTIONS:
0 | All at once. |
n | Treat the perturbed densities batch-wise. |
RECOMMENDATION:
|
| MOPROP_criteria_1ST
The maximal number of iterations for CPSCF and first-order TDSCF. |
TYPE:
DEFAULT:
OPTIONS:
n | Set maximum number of iterations to n. |
RECOMMENDATION:
|
|
|
MOPROP
Specifies the job for mopropman. |
TYPE:
DEFAULT:
OPTIONS:
1 | NMR chemical shielding tensors. |
2 | Static polarizability. |
100 | Dynamic polarizability. |
101 | First hyperpolarizability. |
102 | First hyperpolarizability, reading First order results from disk. |
103 | First hyperpolarizability using Wigner's (2n+1) rule. |
104 | First hyperpolarizability using Wigner's (2n+1) rule, reading |
| first order results from disk. |
RECOMMENDATION:
|
| MRXC_CLASS_THRESH_MULT
Controls the of smoothness precision |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
a prefactor in the threshold for mrxc error control:
im*10.0−io |
|
|
|
MRXC_CLASS_THRESH_ORDER
Controls the of smoothness precision |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The exponent in the threshold of the mrxc error control:
im*10.0−io |
|
| MRXC
Controls the use of MRXC. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use MRXC |
1 | Use MRXC in the evaluation of the XC part |
RECOMMENDATION:
MRXC is very efficient for medium and large molecules,
especially when medium and large basis sets are used. |
|
|
|
MULTIPOLE_ORDER
Determines highest order of multipole moments to print if wavefunction
analysis requested. |
TYPE:
DEFAULT:
OPTIONS:
n | Calculate moments to nth order. |
RECOMMENDATION:
Use default unless higher multipoles are required. |
|
| NBO
Controls the use of the NBO package. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not invoke the NBO package. |
1 | Do invoke the NBO package, for the ground state. |
2 | Invoke the NBO package for the ground state, and also each |
| CIS, RPA, or TDDFT excited state.
|
RECOMMENDATION:
|
|
|
NL_CORRELATION
Specifies a non-local correlation functional that includes non-empirical dispersion. |
TYPE:
DEFAULT:
None | No non-local correlation. |
OPTIONS:
None | No non-local correlation |
vdW-DF-04 | the non-local part of vdW-DF-04 |
vdW-DF-10 | the nonlocal part of vdW-DF-10 (aka vdW-DF2) |
VV09 | the nonlocal part of VV09 |
VV10 | the nonlocal part of VV10 |
RECOMMENDATION:
Do not forget to add the LSDA correlation (PW92 is recommended) when
using vdW-DF-04, vdW-DF-10, or VV09. VV10 should be used with PBE
correlation. Choose exchange functionals carefully: HF, rPW86, revPBE, and
some of the LRC exchange functionals are among the recommended choices. |
|
| NL_GRID
Specifies the grid to use for non-local correlation. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default unless computational cost becomes prohibitive, in which case SG-0 may be used.
XC_GRID should generally be finer than NL_GRID. |
|
|
|
NL_VV_B
Sets the parameter b in VV10. This parameter controls the short range behavior
of the nonlocal correlation energy. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to b = n/100 |
RECOMMENDATION:
The optimal value depends strongly on the exchange functional used.
b = 5.9 is recommended for rPW86. See further details in Ref. [124]. |
|
| NL_VV_C
Sets the parameter C in VV09 and VV10. This parameter is fitted to asymptotic
van der Waals C6 coefficients. |
TYPE:
DEFAULT:
89 | for VV09 |
No default | for VV10 |
OPTIONS:
n | Corresponding to C = n/10000 |
RECOMMENDATION:
C = 0.0093 is recommended when a semilocal exchange functional is used.
C = 0.0089 is recommended when a long-range corrected (LRC) hybrid functional is used.
See further details in Ref. [124]. |
|
|
|
NOCI_PRINT
Specify the debug print level of NOCI |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase this for more debug information |
|
| NPTLEB
The number of points used in the Lebedev grid for the single-center surface
integration. (Only relevant if INTCAV=0). |
TYPE:
DEFAULT:
OPTIONS:
Valid choices are:
| 6, 18, 26, 38, 50, 86, 110, 146, 170, 194, 302, 350, 434, 590, 770, |
| 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, |
| 4802, or 5294. |
RECOMMENDATION:
The default value has been found adequate to obtain the energy to within 0.1
kcal/mol for solutes the size of mono-substituted benzenes. |
|
|
|
NPTTHE, NPTPHI
The number of (θ,ϕ) points used for single-centered surface
integration (relevant only if INTCAV=1). |
TYPE:
DEFAULT:
OPTIONS:
θ,ϕ specifying the number of points. |
RECOMMENDATION:
These should be multiples of 2 and 4 respectively, to provide symmetry
sufficient for all Abelian point groups. Defaults are too small for all but
the tiniest and simplest solutes. |
|
| NTO_PAIRS
Controls the writing of hole/particle NTO pairs for excited state. |
TYPE:
DEFAULT:
OPTIONS:
N | Write N NTO pairs per excited state. |
RECOMMENDATION:
If activated (N > 0), a minimum of two NTO pairs will be printed for each state.
Increase the value of N if additional NTOs are desired.
|
|
|
|
NVO_LIN_CONVERGENCE
Target error factor in the preconditioned conjugate gradient solver of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Solution of the single-excitation amplitude equations is considered converged if the maximum residual is less than 10−n multiplied by the current DIIS error. For the ARS correction, n is automatically set to 1 since the locally-projected DIIS error is normally several orders of magnitude smaller than the full DIIS error. |
|
| NVO_LIN_MAX_ITE
Maximum number of iterations in the preconditioned conjugate gradient solver of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of iterations. |
RECOMMENDATION:
|
|
|
NVO_METHOD
Sets method to be used to converge solution of the single-excitation amplitude equations. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Experimental option. Use default. |
|
| NVO_TRUNCATE_DIST
Specifies which atomic blocks of the Fock matrix are used to construct the preconditioner. |
TYPE:
DEFAULT:
OPTIONS:
n > 0 | If distance between a pair of atoms is more than n angstroms |
| do not include the atomic block. |
-2 | Do not use distance threshold, use NVO_TRUNCATE_PRECOND instead. |
-1 | Include all blocks. |
0 | Include diagonal blocks only. |
RECOMMENDATION:
This option does not affect the final result. However, it affects the rate of the PCG algorithm convergence. For small systems use default. |
|
|
|
NVO_TRUNCATE_PRECOND
Specifies which atomic blocks of the Fock matrix are used to construct the preconditioner. This variable is used only if NVO_TRUNCATE_DIST is set to −2. |
TYPE:
DEFAULT:
OPTIONS:
n | If the maximum element in an atomic block is less than 10−n do not include |
| the block. |
RECOMMENDATION:
Use default. Increasing n improves convergence of the PCG algorithm but overall may slow down calculations. |
|
| NVO_UVV_MAXPWR
Controls convergence of the Taylor series when calculating the Uvv block from the single-excitation amplitudes. If the series is not converged at the nth term, more expensive direct inversion is used to calculate the Uvv block. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
NVO_UVV_PRECISION
Controls convergence of the Taylor series when calculating the Uvv block from the single-excitation amplitudes. Series is considered converged when the maximum element of the term is less than 10−n. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
NVO_UVV_PRECISION must be the same as or larger than THRESH. |
|
| N_FROZEN_CORE
Controls the number of frozen core orbitals |
TYPE:
DEFAULT:
0 | No frozen core orbitals |
OPTIONS:
FC | Frozen core approximation |
n | Freeze n core orbitals |
RECOMMENDATION:
There is no computational advantage to using frozen core for CIS, and
analytical derivatives are only available when no orbitals are frozen. It is
helpful when calculating CIS(D) corrections (see Sec. 6.4). |
|
|
|
N_FROZEN_CORE
Sets the number of frozen core orbitals in a post-Hartree-Fock calculation. |
TYPE:
DEFAULT:
OPTIONS:
FC | Frozen Core approximation (all core orbitals frozen). |
n | Freeze n core orbitals. |
RECOMMENDATION:
While the default is not to freeze orbitals, MP2 calculations are more
efficient with frozen core orbitals. Use FC if possible. |
|
| N_FROZEN_VIRTUAL
Controls the number of frozen virtual orbitals. |
TYPE:
DEFAULT:
0 | No frozen virtual orbitals |
OPTIONS:
n | Freeze n virtual orbitals |
RECOMMENDATION:
There is no computational advantage to using frozen virtuals for CIS, and
analytical derivatives are only available when no orbitals are frozen. |
|
|
|
N_FROZEN_VIRTUAL
Sets the number of frozen virtual orbitals in a post-Hartree-Fock
calculation. |
TYPE:
DEFAULT:
OPTIONS:
n | Freeze n virtual orbitals. |
RECOMMENDATION:
|
| N_I_SERIES
Sets summation limit for series expansion evaluation of in(x). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy. |
|
|
|
N_J_SERIES
Sets summation limit for series expansion evaluation of jn(x). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy. |
|
| N_SOL
Specifies number of atoms included in the Hessian |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
N_WIG_SERIES
Sets summation limit for Wigner integrals. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Increase n for greater accuracy. |
|
| OMEGA2
Sets the Coulomb attenuation parameter for the long-range component. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω2 = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
|
|
OMEGA
Sets the Coulomb attenuation parameter ω. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
| OMEGA
Sets the Coulomb attenuation parameter for the short-range component. |
TYPE:
DEFAULT:
OPTIONS:
n | Corresponding to ω = n/1000, in units of bohr−1 |
RECOMMENDATION:
|
|
|
PAO_ALGORITHM
Algorithm used to optimize polarized atomic orbitals (see PAO_METHOD) |
TYPE:
DEFAULT:
OPTIONS:
0 | Use efficient (and riskier) strategy to converge PAOs. |
1 | Use conservative (and slower) strategy to converge PAOs. |
RECOMMENDATION:
|
| PAO_METHOD
Controls evaluation of polarized atomic orbitals (PAOs). |
TYPE:
DEFAULT:
EPAO | For local MP2 calculations Otherwise no default. |
OPTIONS:
PAO | Perform PAO-SCF instead of conventional SCF. |
EPAO | Obtain EPAO's after a conventional SCF. |
RECOMMENDATION:
|
|
|
PAO_METHOD
Controls the type of PAO calculations requested. |
TYPE:
DEFAULT:
EPAO | For local MP2, EPAOs are chosen by default. |
OPTIONS:
EPAO | Find EPAOs by minimizing delocalization function. |
PAO | Do SCF in a molecule-optimized minimal basis. |
RECOMMENDATION:
|
| PBHT_ANALYSIS
Controls whether overlap analysis of electronic excitations is performed. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform overlap analysis |
TRUE | Perform overlap analysis |
RECOMMENDATION:
|
|
|
PBHT_FINE
Increases accuracy of overlap analysis |
TYPE:
DEFAULT:
OPTIONS:
FALSE | |
TRUE | Increase accuracy of overlap analysis |
RECOMMENDATION:
|
| PCM_PRINT
Controls the printing level during PCM calculations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Prints PCM energy and basic surface grid information. Minimal additional printing. |
1 | Level 0 plus PCM solute-solvent interaction energy components and Gauss Law error. |
2 | Level 1 plus surface grid switching parameters and a .PQR file for visualization of |
| the cavity surface apparent surface charges. |
3 | Level 2 plus a .PQR file for visualization of the electrostatic potential at the surface |
| grid created by the converged solute. |
4 | Level 3 plus additional surface grid information, electrostatic potential and apparent |
| surface charges on each SCF cycle. |
5 | Level 4 plus extensive debugging information. |
RECOMMENDATION:
Use the default unless further information is desired. |
|
|
|
PHESS
Controls whether partial Hessian calculations are performed. |
TYPE:
DEFAULT:
0 | Full Hessian calculation |
OPTIONS:
0 | Full Hessian calculation |
1 | Partial Hessian calculation |
2 | Vibrational subsystem analysis (massless) |
3 | Vibrational subsystem analysis (weighted) |
RECOMMENDATION:
|
| PH_FAST
Lowers integral cutoff in partial Hessian calculation is performed. |
TYPE:
DEFAULT:
FALSE | Use default cutoffs |
OPTIONS:
TRUE | Lower integral cutoffs |
RECOMMENDATION:
|
|
|
PIMC_ACCEPT_RATE
Acceptance rate for MC/PIMC simulations when Cartesian or normal-mode displacements are utilized. |
TYPE:
DEFAULT:
OPTIONS:
0 < n < 100 | User-specified rate, given as a whole-number percentage. |
RECOMMENDATION:
Choose acceptance rate to maximize sampling efficiency, which is typically signified by the mean-square displacement (printed in the job output). Note that the maximum displacement is adjusted during the warmup run to achieve roughly this acceptance rate. |
|
| PIMC_MCMAX
Number of Monte Carlo steps to sample. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of steps to sample. |
RECOMMENDATION:
This variable dictates the statistical convergence of MC/PIMC simulations. Recommend setting to at least 100000 for converged simulations. |
|
|
|
PIMC_MOVETYPE
Selects the type of displacements used in MC/PIMC simulations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Cartesian displacements of all beads, with occasional (1%) center-of-mass moves. |
1 | Normal-mode displacements of all modes, with occasional (1%) center-of-mass moves. |
2 | Levy flights without center-of-mass moves. |
RECOMMENDATION:
Except for classical sampling (MC) or small bead-number quantum sampling (PIMC), Levy flights should be utilized. For Cartesian and normal-mode moves, the maximum displacement is adjusted
during the warmup run to the desired acceptance rate (controlled by PIMC_ACCEPT_RATE). For Levy flights, the acceptance is solely controlled by PIMC_SNIP_LENGTH. |
|
| PIMC_NBEADSPERATOM
Number of path integral time slices ("beads") used on each atom of a PIMC simulation. |
TYPE:
DEFAULT:
OPTIONS:
1 | Perform classical Boltzmann sampling. |
> 1 | Perform quantum-mechanical path integral sampling. |
RECOMMENDATION:
This variable controls the inherent convergence of the path integral simulation. The 1-bead limit is purely classical sampling; the infinite-bead limit is exact quantum mechanical sampling. Using 32 beads is reasonably converged for room-temperature simulations of molecular systems. |
|
|
|
PIMC_SNIP_LENGTH
Number of "beads" to use in the Levy flight movement of the ring polymer. |
TYPE:
DEFAULT:
OPTIONS:
3 ≤ n ≤ PIMC_NBEADSPERATOM | User-specified length of snippet. |
RECOMMENDATION:
Choose the snip length to maximize sampling efficiency. The efficiency can be estimated by the mean-square displacement between configurations, printed at the end of the output file. This efficiency will typically, however, be a trade-off between the mean-square displacement (length of statistical correlations) and the number of beads moved. Only the moved beads require recomputing the potential, i.e., a call to Q-Chem for the electronic energy.
(Note that the endpoints of the snippet remain fixed during a single move, so n−2 beads are actually moved for a snip length of n. For 1 or 2 beads in the simulation, Cartesian moves should be used instead.) |
|
| PIMC_TEMP
Temperature, in Kelvin (K), of path integral simulations. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of Kelvin for PIMC or classical MC simulations. |
RECOMMENDATION:
|
|
|
PIMC_WARMUP_MCMAX
Number of Monte Carlo steps to sample during an equilibration period of MC/PIMC simulations. |
TYPE:
DEFAULT:
OPTIONS:
| User-specified number of steps to sample. |
RECOMMENDATION:
Use this variable to equilibrate the molecule/ring polymer before collecting production statistics. Usually a short run of roughly 10% of PIMC_MCMAX is sufficient. |
|
| POP_MULLIKEN
Controls running of Mulliken population analysis. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | (or 0) Do not calculate Mulliken Population. |
TRUE | (or 1) Calculate Mulliken population |
2 | Also calculate shell populations for each occupied orbital. |
−1 | Calculate Mulliken charges for both the ground state and any CIS, |
| RPA, or TDDFT excited states.
|
RECOMMENDATION:
Leave as TRUE, unless excited-state charges are desired.
Mulliken analysis is a trivial additional calculation, for ground or
excited states. |
|
|
|
PRINT_CORE_CHARACTER
Determines the print level for the CORE_CHARACTER option. |
TYPE:
DEFAULT:
OPTIONS:
0 | No additional output is printed. |
1 | Prints core characters of occupied MOs. |
2 | Print level 1, plus prints the core character of AOs. |
RECOMMENDATION:
Use default, unless you are uncertain about what the core character is. |
|
| PRINT_DIST_MATRIX
Controls the printing of the inter-atomic distance matrix |
TYPE:
DEFAULT:
OPTIONS:
0 | Turns off the printing of the distance matrix |
n | Prints the distance matrix if the number of atoms in the molecule |
| is less than or equal to n. |
RECOMMENDATION:
Use default unless distances are required for large systems |
|
|
|
PRINT_GENERAL_BASIS
Controls print out of built in basis sets in input format |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Print out standard basis set information |
FALSE | Do not print out standard basis set information |
RECOMMENDATION:
Useful for modification of standard basis sets. |
|
| PRINT_ORBITALS
Prints orbital coefficients with atom labels in analysis part of output. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not print any orbitals. |
TRUE | Prints occupied orbitals plus 5 virtuals. |
NVIRT | Number of virtuals to print. |
RECOMMENDATION:
Use TRUE unless more virtuals are desired. |
|
|
|
PRINT_RADII_GYRE
Controls printing of MO centroids and radii of gyration. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | (or 1) Calculate the centroid and radius of gyration for each MO and density. |
FALSE | (or 0) Do not calculate these quantities.
|
RECOMMENDATION:
|
| PROJ_TRANSROT
Removes translational and rotational drift during AIMD trajectories. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not apply translation/rotation corrections. |
TRUE | Apply translation/rotation corrections. |
RECOMMENDATION:
When computing spectra (see AIMD_NUCL_DACF_POINTS, for example), this option can be utilized to remove artificial, contaminating peaks stemming from translational and/or rotational motion. Recommend setting to TRUE for all dynamics-based spectral simulations. |
|
|
|
PSEUDO_CANONICAL
When SCF_ALGORITHM = DM, this controls the way the initial
step, and steps after subspace resets are taken. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use Roothaan steps when (re)initializing |
TRUE | Use a steepest descent step when (re)initializing |
RECOMMENDATION:
The default is usually more efficient, but choosing TRUE sometimes
avoids problems with orbital reordering. |
|
| PURECART
TYPE:
Controls the use of pure (spherical harmonic) or Cartesian angular forms |
DEFAULT:
2111 | Cartesian h-functions and pure g,f,d functions |
OPTIONS:
hgfd | Use 1 for pure and 2 for Cartesian. |
RECOMMENDATION:
This is pre-defined for all standard basis sets |
|
|
|
QMMM_CHARGES
Controls the printing of QM charges to file. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Writes a charges.dat file with the Mulliken charges from the QM
region. |
FALSE | No file written. |
RECOMMENDATION:
Use default unless running calculations with CHARMM where charges on the QM
region need to be saved. |
|
| QMMM_FULL_HESSIAN
Trigger the evaluation of the full QM/MM Hessian. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Evaluates full Hessian. |
FALSE | Hessian for QM-QM block only. |
RECOMMENDATION:
|
|
|
QMMM_PRINT
Controls the amount of output printed from a QM/MM job. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Limit molecule, point charge, and analysis printing. |
FALSE | Normal printing. |
RECOMMENDATION:
Use default unless running calculations with CHARMM. |
|
| QM_MM_INTERFACE
Enables internal QM/MM calculations. |
TYPE:
DEFAULT:
OPTIONS:
MM | Molecular mechanics calculation (i.e., no QM region) |
ONIOM | QM/MM calculation using two-layer mechanical embedding |
JANUS | QM/MM calculation using electronic embedding
|
RECOMMENDATION:
The ONIOM model and Janus models are described above. Choosing MM
leads to no electronic structure calculation. However, when using MM, one
still needs to define the $rem variables BASIS and EXCHANGE
in order for Q-Chem to proceed smoothly.
|
|
|
|
QM_MM
Turns on the Q-Chem/ CHARMM interface. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Do QM/MM calculation through the Q-Chem/ CHARMM interface. |
FALSE | Turn this feature off. |
RECOMMENDATION:
Use default unless running calculations with CHARMM. |
|
| RADSPH
Sphere radius used to specify the cavity surface (Only relevant for ISHAPE=1). |
TYPE:
DEFAULT:
Half the distance between the outermost atoms plus 1.4 Angstroms. |
OPTIONS:
Real number specifying the radius in bohr (if positive) or in Angstroms (if negative). |
RECOMMENDATION:
Make sure that the cavity radius is larger than the length of the molecule. |
|
|
|
RAS_ACT_DIFF
Sets the number of alpha vs. beta electrons |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to 1 for an odd number of electrons or a cation, -1 for an anion.
Only works with RASCI2. |
|
| RAS_ACT_OCC
Sets the number of occupied orbitals to enter the RAS active space. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
None. Only works with RASCI2 |
|
|
|
RAS_ACT_ORB
Sets the user-selected active orbitals (RAS2 orbitals). |
TYPE:
DEFAULT:
From RAS_OCC+1 to RAS_OCC+RAS_ACT |
OPTIONS:
[i,j,k...] | The number of orbitals must be equal to the RAS_ACT variable |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_ACT_VIR
Sets the number of virtual orbitals to enter the RAS active space. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
None. Only works with RASCI2. |
|
|
|
RAS_ACT
Sets the number of orbitals in RAS2 (active orbitals). |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_AMPL_PRINT
Defines the absolute threshold (×102) for the CI amplitudes to be printed. |
TYPE:
DEFAULT:
10 | 0.1 minimum absolute amplitude |
OPTIONS:
n | User-defined integer, n ≥ 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_DO_HOLE
Controls the presence of hole excitations in the RAS-CI wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Include hole configurations (RAS1 to RAS2 excitations) |
FALSE | Do not include hole configurations |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_DO_PART
Controls the presence of particle excitations in the RAS-CI wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Include particle configurations (RAS2 to RAS3 excitations) |
FALSE | Do not include particle configurations |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_ELEC
Sets the number of electrons in RAS2 (active electrons). |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_GUESS_CS
Controls the number of closed shell guess configurations in RAS-CI. |
TYPE:
DEFAULT:
OPTIONS:
n | Imposes to start with n closed shell guesses |
RECOMMENDATION:
Only relevant for the computation of singlet states. Only works with RASCI. |
|
|
|
RAS_NATORB_STATE
Allows to save the natural orbitals of a RAS-CI computed state. |
TYPE:
DEFAULT:
OPTIONS:
i | Saves the natural orbitals for the i-th state |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_NATORB
Controls the computation of the natural orbital occupancies. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Compute natural orbital occupancies for all states |
FALSE | Do not compute natural orbital occupancies |
RECOMMENDATION:
None. Only works with RASCI. |
|
|
|
RAS_N_ROOTS
Sets the number of RAS-CI roots to be computed. |
TYPE:
DEFAULT:
OPTIONS:
n | n > 0 Compute n RAS-CI states |
RECOMMENDATION:
None. Only works with RASCI2 |
|
| RAS_OCC
Sets the number of orbitals in RAS1 |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined integer, n > 0 |
RECOMMENDATION:
These are the initial doubly occupied orbitals (RAS1) before
including hole type of excitations.
The RAS1 space starts from the lowest orbital up to RAS_OCC,
i.e. no frozen orbitals option available yet. Only works with RASCI. |
|
|
|
RAS_ROOTS
Sets the number of RAS-CI roots to be computed. |
TYPE:
DEFAULT:
OPTIONS:
n | n > 0 Compute n RAS-CI states |
RECOMMENDATION:
None. Only works with RASCI. |
|
| RAS_SPIN_MULT
Specifies the spin multiplicity of the roots to be computed |
TYPE:
DEFAULT:
OPTIONS:
0 | Compute any spin multiplicity |
2n+1 | User-defined integer, n ≥ 0 |
RECOMMENDATION:
Only for RASCI, which at present only allows for the computation of systems with an even
number of electrons. Thus, RAS_SPIN_MULT only can take odd values. |
|
|
|
RCA_PRINT
Controls the output from RCA SCF optimizations. |
TYPE:
DEFAULT:
OPTIONS:
0 | No print out |
1 | RCA summary information |
2 | Level 1 plus RCA coefficients |
3 | Level 2 plus RCA iteration details |
RECOMMENDATION:
|
| RC_R0
Determines the parameter in the Gaussian weight function used to smooth the
density at the nuclei. |
TYPE:
DEFAULT:
OPTIONS:
0 | Corresponds the traditional delta function spin and charge densities |
n | corresponding to n×10−3 a.u. |
RECOMMENDATION:
We recommend value of 250 for a typical spit valence basis. For basis sets
with increased flexibility in the nuclear vicinity the smaller values of r0
also yield adequate spin density. |
|
|
|
READ_VDW
Controls the input of user-defined atomic radii for ChemSol calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Use default ChemSol parameters. |
TRUE | Read from the $van_der_waals section of the input file. |
RECOMMENDATION:
|
| RHOISO
Value of the electronic iso-density contour used to specify the cavity surface.
(Only relevant for ISHAPE = 0). |
TYPE:
DEFAULT:
OPTIONS:
Real number specifying the density in electrons/bohr3. |
RECOMMENDATION:
The default value is optimal for most situations. Increasing the value
produces a smaller cavity which ordinarily increases the magnitude of the
solvation energy. |
|
|
|
RI_J
Toggles the use of the RI algorithm to compute J. |
TYPE:
DEFAULT:
FALSE | RI will not be used to compute J. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI. |
|
| RI_K
Toggles the use of the RI algorithm to compute K. |
TYPE:
DEFAULT:
FALSE | RI will not be used to compute K. |
OPTIONS:
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI. |
|
|
|
ROTTHE ROTPHI ROTCHI
Euler angles (θ, ϕ, χ) in degrees for user-specified
rotation of the cavity surface. (relevant if IROTGR=3) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| RPATH_COORDS
Determines which coordinate system to use in the IRC search. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use mass-weighted coordinates. |
1 | Use Cartesian coordinates. |
2 | Use Z-matrix coordinates. |
RECOMMENDATION:
|
|
|
RPATH_DIRECTION
Determines the direction of the eigen mode to follow. This will not usually be
known prior to the Hessian diagonalization. |
TYPE:
DEFAULT:
OPTIONS:
1 | Descend in the positive direction of the eigen mode. |
-1 | Descend in the negative direction of the eigen mode. |
RECOMMENDATION:
It is usually not possible to determine in which direction to
go a priori, and therefore both directions will need to be
considered. |
|
| RPATH_MAX_CYCLES
Specifies the maximum number of points to find on the reaction path. |
TYPE:
DEFAULT:
OPTIONS:
n | User-defined number of cycles. |
RECOMMENDATION:
Use more points if the minimum is desired, but not reached using the default. |
|
|
|
RPATH_MAX_STEPSIZE
Specifies the maximum step size to be taken (in thousandths of a.u.). |
TYPE:
DEFAULT:
150 | corresponding to a step size of 0.15 a.u.. |
OPTIONS:
RECOMMENDATION:
|
| RPATH_PRINT
Specifies the print output level. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use default, little additional information is printed at higher levels. Most
of the output arises from the multiple single point calculations that are
performed along the reaction pathway. |
|
|
|
RPATH_TOL_DISPLACEMENT
Specifies the convergence threshold for the step. If a step size is chosen by
the algorithm that is smaller than this, the path is deemed to have reached the
minimum. |
TYPE:
DEFAULT:
5000 | Corresponding to 0.005 a.u. |
OPTIONS:
n | User-defined. Tolerance = n/1000000. |
RECOMMENDATION:
Use default. Note that this option only controls the
threshold for ending the RPATH job and does nothing to
the intermediate steps of the calculation. A smaller value will
provide reaction paths that end closer to the true minimum.
Use of smaller values without adjusting RPATH_MAX_STEPSIZE,
however, can lead to oscillations about the minimum. |
|
| RPA
Do an RPA calculation in addition to a CIS calculation |
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an RPA calculation |
True | Do an RPA calculation. |
RECOMMENDATION:
|
|
|
SAVE_LAST_GPX
Save last G[Px] when calculating dynamic
polarizabilities in order to call mopropman in a second run with MOPROP = 102. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| SCALE_NUCLEAR_CHARGE
Scales charge of each nuclei by a certain value. The nuclear repulsion energy
is calculated for the unscaled nuclear charges. |
TYPE:
DEFAULT:
OPTIONS:
n a total positive charge of (1+n / 100)e is added to the molecule. |
RECOMMENDATION:
|
|
|
SCF_ALGORITHM
Algorithm used for converging the SCF. |
TYPE:
DEFAULT:
OPTIONS:
DIIS | Pulay DIIS. |
DM | Direct minimizer. |
DIIS_DM | Uses DIIS initially, switching to direct minimizer for later iterations |
| (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). |
DIIS_GDM | Use DIIS and then later switch to geometric direct minimization |
| (See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES). |
GDM | Geometric Direct Minimization. |
RCA | Relaxed constraint algorithm |
RCA_DIIS | Use RCA initially, switching to DIIS for later iterations (see |
| THRESH_RCA_SWITCH and MAX_RCA_CYCLES described |
| later in this chapter) |
ROOTHAAN | Roothaan repeated diagonalization. |
RECOMMENDATION:
Use DIIS unless performing a restricted open-shell calculation, in which case GDM is recommended.
If DIIS fails to find a reasonable approximate solution in the initial iterations,
RCA_DIIS is the recommended fallback option.
If DIIS approaches the correct solution but fails to finally converge,
DIIS_GDM is the recommended fallback. |
|
| SCF_CONVERGENCE
SCF is considered converged when the wavefunction error is less that
10−SCF_CONVERGENCE. Adjust the value of THRESH at the same
time. Note that in Q-Chem 3.0 the DIIS error is measured by the maximum error
rather than the RMS error as in previous versions. |
TYPE:
DEFAULT:
5 | For single point energy calculations. |
7 | For geometry optimizations and vibrational analysis. |
8 | For SSG calculations, see Chapter 5. |
OPTIONS:
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger
values provide more significant figures, at greater computational cost. |
|
|
|
SCF_FINAL_PRINT
Controls level of output from SCF procedure to Q-Chem output file at the
end of the SCF. |
TYPE:
DEFAULT:
OPTIONS:
0 | No extra print out. |
1 | Orbital energies and break-down of SCF energy. |
2 | Level 1 plus MOs and density matrices. |
3 | Level 2 plus Fock and density matrices. |
RECOMMENDATION:
The break-down of energies is often useful (level 1). |
|
| SCF_GUESS_ALWAYS
Switch to force the regeneration of a new initial guess for each series of
SCF iterations (for use in geometry optimization). |
TYPE:
DEFAULT:
OPTIONS:
False | Do not generate a new guess for each series of SCF iterations in an |
| optimization; use MOs from the previous SCF calculation for the guess, |
| if available. |
True | Generate a new guess for each series of SCF iterations in a geometry |
| optimization. |
RECOMMENDATION:
Use default unless SCF convergence issues arise |
|
|
|
SCF_GUESS_MIX
Controls mixing of LUMO and HOMO to break symmetry in the initial guess. For
unrestricted jobs, the mixing is performed only for the alpha orbitals. |
TYPE:
DEFAULT:
0 (FALSE) | Do not mix HOMO and LUMO in SCF guess. |
OPTIONS:
0 (FALSE) | Do not mix HOMO and LUMO in SCF guess. |
1 (TRUE) | Add 10% of LUMO to HOMO to break symmetry. |
n | Add n×10% of LUMO to HOMO (0 < n < 10). |
RECOMMENDATION:
When performing unrestricted calculations on molecules with an even number of
electrons, it is often necessary to break alpha / beta symmetry in the initial
guess with this option, or by specifying input for $occupied. |
|
| SCF_GUESS_PRINT
Controls printing of guess MOs, Fock and density matrices. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not print guesses. |
SAD | |
1 | Atomic density matrices and molecular matrix. |
2 | Level 1 plus density matrices. |
CORE and GWH | |
1 | No extra output. |
2 | Level 1 plus Fock and density matrices and, MO coefficients and |
| eigenvalues. |
READ | |
1 | No extra output |
2 | Level 1 plus density matrices, MO coefficients and eigenvalues. |
RECOMMENDATION:
|
|
|
SCF_GUESS
Specifies the initial guess procedure to use for the SCF. |
TYPE:
DEFAULT:
SAD | Superposition of atomic density (available only with standard basis
sets) |
GWH | For ROHF where a set of orbitals are required. |
FRAGMO | For a fragment MO calculation |
OPTIONS:
CORE | Diagonalize core Hamiltonian |
SAD | Superposition of atomic density |
GWH | Apply generalized Wolfsberg-Helmholtz approximation |
READ | Read previous MOs from disk |
FRAGMO | Superimposing converged fragment MOs |
RECOMMENDATION:
SAD guess for standard basis sets. For general basis sets, it is best to use
the BASIS2 $rem. Alternatively, try the GWH or core Hamiltonian
guess. For ROHF it can be useful to READ guesses from an SCF calculation on the
corresponding cation or anion. Note that because the density is made spherical,
this may favor an undesired state for atomic systems, especially transition
metals. Use FRAGMO in a fragment MO calculation. |
|
| SCF_MINFIND_INCREASEFACTOR
Controls how the height of the penalty function
changes when repeatedly trapped at the same solution |
TYPE:
DEFAULT:
OPTIONS:
abcde | corresponding to a.bcde |
RECOMMENDATION:
If the algorithm converges to a solution which corresponds
to a previously located solution, increase both the
normalization N and the width lambda of the penalty function there. Then do a restart. |
|
|
|
SCF_MINFIND_INITLAMBDA
Control the initial width of the penalty function. |
TYPE:
DEFAULT:
OPTIONS:
abcde | corresponding to ab.cde |
RECOMMENDATION:
The initial inverse-width (i.e., the inverse-variance) of the
Gaussian to place to fill solution's well. Measured in electrons(−1).
Increasing this will repeatedly converging on the same solution. |
|
| SCF_MINFIND_INITNORM
Control the initial height of the penalty function. |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The initial normalization of the Gaussian to place to fill a well. Measured in Hartrees. |
|
|
|
SCF_MINFIND_MIXENERGY
Specify the active energy range when doing Active mixing |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The standard deviation of the Gaussian distribution used
to select the orbitals for mixing (centered on the Fermi level). Measured in Hartree.
To find less-excited solutions, decrease this value |
|
| SCF_MINFIND_MIXMETHOD
Specify how to select orbitals for random mixing |
TYPE:
DEFAULT:
OPTIONS:
0 | Random mixing: select from any orbital to any orbital. |
1 | Active mixing: select based on energy, decaying with distance from the Fermi level. |
2 | Active Alpha space mixing: select based on energy, decaying with distance from the |
| Fermi level only in the alpha space. |
RECOMMENDATION:
Random mixing will often find very high energy solutions.
If lower energy solutions are desired, use 1 or 2. |
|
|
|
SCF_MINFIND_NRANDOMMIXES
Control how many random mixes to do to generate new orbitals |
TYPE:
DEFAULT:
OPTIONS:
n | Perform n random mixes. |
RECOMMENDATION:
This is the number of occupied/virtual pairs to attempt to mix,
per separate density (i.e., for unrestricted calculations both
alpha and beta space will get this many rotations). If this
is negative then only mix the highest 25% occupied and lowest 25% virtuals. |
|
| SCF_MINFIND_RANDOMMIXING
Control how to choose new orbitals after locating a solution |
TYPE:
DEFAULT:
00200 meaning .02 radians |
OPTIONS:
abcde corresponding to a.bcde radians |
RECOMMENDATION:
After locating an SCF solution, the orbitals are mixed
randomly to move to a new position in orbital space. For each
occupied and virtual orbital pair picked at random and rotate
between them by a random angle between 0 and this. If this
is negative then use exactly this number, e.g., −15708 will
almost exactly swap orbitals. Any number < −15708 will cause the orbitals to be swapped exactly. |
|
|
|
SCF_MINFIND_READDISTTHRESH
The distance threshold at which to consider two solutions the same |
TYPE:
DEFAULT:
OPTIONS:
abcde corresponding to ab.cde |
RECOMMENDATION:
The threshold to regard a minimum as the same as a read in
minimum. Measured in electrons. If two minima are closer
together than this, reduce the threshold to distinguish them. |
|
| SCF_MINFIND_RESTARTSTEPS
Restart with new orbitals if no minima have been found within this many steps |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
If the SCF calculation spends many steps not finding a
solution, lowering this number may speed up solution-finding.
If the system converges to solutions very
slowly, then this number may need to be raised. |
|
|
|
SCF_MINFIND_RUNCORR
Run post-SCF correlated methods on multiple SCF solutions |
TYPE:
DEFAULT:
OPTIONS:
If this is set > 0, then run correlation methods for all found SCF solutions. |
RECOMMENDATION:
Post-HF correlation methods should function correctly with excited
SCF solutions, but their convergence is often much more difficult owing to intruder states. |
|
| SCF_MINFIND_WELLTHRESH
Specify what SCF_MINFIND believes is the basin of a solution |
TYPE:
DEFAULT:
OPTIONS:
n for a threshold of 10−n |
RECOMMENDATION:
When the DIIS error is less than 10−n, penalties are switched
off to see whether it has converged to a new solution. |
|
|
|
SCF_PRINT_FRGM
Controls the output of Q-Chem jobs on isolated fragments. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | The output is printed to the parent job output file. |
FALSE | The output is not printed. |
RECOMMENDATION:
Use TRUE if details about isolated fragments are important. |
|
| SCF_PRINT
Controls level of output from SCF procedure to Q-Chem output file. |
TYPE:
DEFAULT:
0 | Minimal, concise, useful and necessary output. |
OPTIONS:
0 | Minimal, concise, useful and necessary output. |
1 | Level 0 plus component breakdown of SCF electronic energy. |
2 | Level 1 plus density, Fock and MO matrices on each cycle. |
3 | Level 2 plus two-electron Fock matrix components (Coulomb, HF exchange |
| and DFT exchange-correlation matrices) on each cycle. |
RECOMMENDATION:
Proceed with care; can result in extremely large output files at level 2 or higher.
These levels are primarily for program debugging. |
|
|
|
SCF_READMINIMA
Read in solutions from a previous SCF Metadynamics calculation |
TYPE:
DEFAULT:
OPTIONS:
n | Read in n previous solutions and attempt to locate them all. |
−n | Read in n previous solutions, but only attempt to locate solution n. |
RECOMMENDATION:
This may not actually locate all solutions required and will probably
locate others too. The SCF will also stop when the number of
solutions specified in SCF_SAVEMINIMA are found.
Solutions from other geometries may also be read in and used as starting orbitals.
If a solution is found and matches one that is read in within
SCF_MINFIND_READDISTTHRESH, its orbitals are saved in
that position for any future calculations.
The algorithm works by restarting from the orbitals and density
of a the minimum it is attempting to find. After 10 failed
restarts (defined by SCF_MINFIND_RESTARTSTEPS), it moves
to another previous minimum and attempts to locate that instead.
If there are no minima to find, the restart does random mixing
(with 10 times the normal random mixing parameter).
|
|
| SCF_SAVEMINIMA
Turn on SCF Metadynamics and specify how many solutions to locate. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not use SCF Metadynamics |
n | Attempt to find n distinct SCF solutions. |
RECOMMENDATION:
Perform SCF Orbital metadynamics and attempt to locate
n different SCF solutions. Note that these may not all be minima. Many saddle points are often located.
The last one located will be the one used in any post-SCF treatments.
In systems where there are infinite point groups, this procedure
cannot currently distinguish between spatial rotations of different
densities, so will likely converge on these multiply. |
|
|
|
SET_STATE_DERIV
Sets the excited state index for analytical gradient calculation
for geometry optimizations and vibrational analysis with SOS-CIS(D0) |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Check to see that the states do no change order during an optimization.
For closed-shell systems, either CIS_SINGLETS or CIS_TRIPLETS
must be set to false. |
|
| SFX_AMP_OCC_A
Defines a customer amplitude guess vector in SF-XCIS method |
TYPE:
DEFAULT:
OPTIONS:
n | builds a guess amplitude with an α-hole in the nth orbital
(requires SFX_AMP_VIR_B). |
RECOMMENDATION:
Only use when default guess is not satisfactory |
|
|
|
SFX_AMP_VIR_B
Defines a customer amplitude guess vector in SF-XCIS method |
TYPE:
DEFAULT:
OPTIONS:
n | builds a guess amplitude with a β-particle in the nth orbital
(requires SFX_AMP_OCC_A). |
RECOMMENDATION:
Only use when default guess is not satisfactory |
|
| SKIP_CIS_RPA
Skips the solution of the CIS, RPA, TDA or TDDFT equations for wavefunction
analysis. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Set to true to speed up the generation of plot data if the same calculation
has been run previously with the scratch files saved. |
|
|
|
SMX_SOLVATION
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform the SM8 solvation procedure |
TRUE | Perform the SM8 solvation procedure |
RECOMMENDATION:
|
| SMX_SOLVENT
TYPE:
DEFAULT:
OPTIONS:
any name from the list of solvents |
RECOMMENDATION:
|
|
|
SOLUTE_RADIUS
Sets the solvent model cavity radius. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| SOLVENT_DIELECTRIC
Sets the dielectric constant of the solvent continuum. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
SOLVENT_METHOD
Sets the preferred solvent method. |
TYPE:
DEFAULT:
SCRF if SOLVENT_DIELECTRIC > 0 |
OPTIONS:
SCRF | Use the Kirkwood-Onsager SCRF model |
PCM | Use an apparent surface charge polarizable continuum model |
COSMO | USE the COSMO model |
RECOMMENDATION:
None. The PCMs are more sophisticated and may require additional input options. These models
are discussed in Section 10.2.2. |
|
| SOL_ORDER
Determines the order to which the multipole expansion of the solute charge density is carried out. |
TYPE:
DEFAULT:
OPTIONS:
L | Include up to L-th order multipoles. |
RECOMMENDATION:
The multipole expansion is usually converged at order L = 15 |
|
|
|
SOS_FACTOR
Sets the scaling parameter cT |
TYPE:
DEFAULT:
130 | corresponding to 1.30 |
OPTIONS:
RECOMMENDATION:
|
| SOS_UFACTOR
Sets the scaling parameter cU |
TYPE:
DEFAULT:
151 | For SOS-CIS(D), corresponding to 1.51 |
140 | For SOS-CIS(D0), corresponding to 1.40
|
OPTIONS:
RECOMMENDATION:
|
|
|
SPIN_FLIP_XCIS
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an SF-XCIS calculation |
True | Do an SF-XCIS calculation (requires ROHF triplet ground state). |
RECOMMENDATION:
|
| SPIN_FLIP
Selects whether to perform a standard excited state calculation, or a
spin-flip calculation. Spin multiplicity should be set to 3 for systems with
an even number of electrons, and 4 for systems with an odd number of electrons. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
|
|
SRC_DFT
Selects form of the short-range corrected functional |
TYPE:
DEFAULT:
OPTIONS:
1 | SRC1 functional |
2 | SRC2 functional |
RECOMMENDATION:
|
| SSG
Controls the calculation of the SSG wavefunction. |
TYPE:
DEFAULT:
OPTIONS:
0 | Do not compute the SSG wavefunction |
1 | Do compute the SSG wavefunction |
RECOMMENDATION:
See also the UNRESTRICTED and DIIS_SUBSPACE_SIZE $rem
variables. |
|
|
|
STABILITY_ANALYSIS
Performs stability analysis for a HF or DFT solution. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform stability analysis. |
FALSE | Do not perform stability analysis. |
RECOMMENDATION:
Set to TRUE when a HF or DFT solution is suspected to be unstable. |
|
| STS_ACCEPTOR
Define the acceptor molecular fragment. |
TYPE:
DEFAULT:
0 | No acceptor fragment is defined. |
OPTIONS:
i-j | Acceptor fragment is in the ith atom to the jth atom. |
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j. |
|
|
|
STS_DONOR
Define the donor fragment. |
TYPE:
DEFAULT:
0 | No donor fragment is defined. |
OPTIONS:
i-j | Donor fragment is in the ith atom to the jth atom. |
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j. |
|
| STS_FCD
Control the calculation of FCD for ET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform an FCD calculation. |
TRUE | Include an FCD calculation. |
RECOMMENDATION:
|
|
|
STS_FED
Control the calculation of FED for EET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a FED calculation. |
TRUE | Include a FED calculation. |
RECOMMENDATION:
|
| STS_FSD
Control the calculation of FSD for EET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a FSD calculation. |
TRUE | Include a FSD calculation. |
RECOMMENDATION:
For RCIS triplets, FSD and FED are equivalent. FSD will be automatically switched off and perform a FED calculation. |
|
|
|
STS_GMH
Control the calculation of GMH for ET couplings. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform a GMH calculation. |
TRUE | Include a GMH calculation. |
RECOMMENDATION:
When set to true computes Mulliken-Hush electronic couplings. It yields
the generalized Mulliken-Hush couplings as well as the transition dipole
moments for each pair of excited states and for each excited state with
the ground state. |
|
| STS_MOM
Control calculation of the transition moments between excited states in
the CIS and TDDFT calculations (including SF variants). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not calculate state-to-state transition moments. |
TRUE | Do calculate state-to-state transition moments. |
RECOMMENDATION:
When set to true requests the state-to-state dipole transition moments for
all pairs of excited states and for each excited state with the ground
state. |
|
|
|
SVP_CAVITY_CONV
Determines the convergence value of the iterative iso-density cavity procedure. |
TYPE:
DEFAULT:
OPTIONS:
n Convergence threshold set to 10−n. |
RECOMMENDATION:
The default value unless convergence problems arise. |
|
| SVP_CHARGE_CONV
Determines the convergence value for the charges on the cavity. When the
change in charges fall below this value, if the electron density is converged,
then the calculation is considered converged. |
TYPE:
DEFAULT:
OPTIONS:
n Convergence threshold set to 10−n. |
RECOMMENDATION:
The default value unless convergence problems arise. |
|
|
|
SVP_GUESS
Specifies how and if the solvation module will use a given guess for the
charges and cavity points. |
TYPE:
DEFAULT:
OPTIONS:
0 | No guessing. |
1 | Read a guess from a previous Q-Chem solvation computation. |
2 | Use a guess specified by the $svpirf section from the input |
RECOMMENDATION:
It is helpful to also set SCF_GUESS to READ when using a guess
from a previous Q-Chem run. |
|
| SVP_MEMORY
Specifies the amount of memory for use by the solvation module. |
TYPE:
DEFAULT:
OPTIONS:
n corresponds to the amount of memory in MB. |
RECOMMENDATION:
The default should be fine for medium size molecules
with the default Lebedev grid, only increase if needed. |
|
|
|
SVP_PATH
Specifies whether to run a gas phase computation prior to performing the
solvation procedure. |
TYPE:
DEFAULT:
OPTIONS:
0 | runs a gas-phase calculation and after |
| convergence runs the SS(V)PE computation. |
1 | does not run a gas-phase calculation. |
RECOMMENDATION:
Running the gas-phase calculation provides a good guess to start the solvation
stage and provides a more complete set of solvated properties. |
|
| SVP
Sets whether to perform the SS(V)PE iso-density solvation procedure. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not perform the SS(V)PE iso-density solvation procedure. |
TRUE | Perform the SS(V)PE iso-density solvation procedure. |
RECOMMENDATION:
|
|
|
SYMMETRY_DECOMPOSITION
Determines symmetry decompositions to calculate. |
TYPE:
DEFAULT:
OPTIONS:
0 | No symmetry decomposition. |
1 | Calculate MO eigenvalues and symmetry (if available). |
2 | Perform symmetry decomposition of kinetic energy and nuclear attraction |
| matrices. |
RECOMMENDATION:
|
| SYMMETRY
Controls the efficiency through the use of point group symmetry for
calculating integrals. |
TYPE:
DEFAULT:
TRUE | Use symmetry for computing integrals. |
OPTIONS:
TRUE | Use symmetry when available. |
FALSE | Do not use symmetry. This is
always the case for RIMP2 jobs |
RECOMMENDATION:
Use default unless benchmarking.
Note that symmetry usage is disabled for RIMP2, FFT, and QM/MM jobs. |
|
|
|
SYM_IGNORE
Controls whether or not Q-Chem determines the point group of the molecule and reorients the molecule to the standard orientation. |
TYPE:
DEFAULT:
FALSE | Do determine the point group (disabled for RIMP2 jobs). |
OPTIONS:
RECOMMENDATION:
Use default unless you do not want the molecule to be reoriented.
Note that symmetry usage is disabled for RIMP2 jobs. |
|
| SYM_TOL
Controls the tolerance for determining point group symmetry. Differences in
atom locations less than 10−SYM_TOL are treated as zero. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default unless the molecule has high symmetry which is not being
correctly identified. Note that relaxing this tolerance too much may introduce
errors into the calculation. |
|
|
|
THRESH_DIIS_SWITCH
The threshold for switching between DIIS extrapolation and direct minimization
of the SCF energy is 10−THRESH_DIIS_SWITCH when
SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See
also MAX_DIIS_CYCLES |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
|
| THRESH_RCA_SWITCH
The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS. |
TYPE:
DEFAULT:
OPTIONS:
N | Algorithm changes from RCA to DIIS when Error is less than 10−N. |
RECOMMENDATION:
|
|
|
THRESH
Cutoff for neglect of two electron integrals. 10−THRESH (THRESH
≤ 14). |
TYPE:
DEFAULT:
8 | For single point energies. |
10 | For optimizations and frequency calculations. |
14 | For coupled-cluster calculations. |
OPTIONS:
n | for a threshold of 10−n. |
RECOMMENDATION:
Should be at least three greater than SCF_CONVERGENCE. Increase for
more significant figures, at greater computational cost. |
|
| TIME_STEP
Specifies the molecular dynamics time step, in atomic units (1 a.u. = 0.0242
fs). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Smaller time steps lead to better energy conservation; too large a time step
may cause the job to fail entirely. Make the time step as large as possible,
consistent with tolerable energy conservation. |
|
|
|
TRANS_ENABLE
Decide whether or not to enable the molecular transport code. |
TYPE:
DEFAULT:
0 | Do not perform transport calculations. |
OPTIONS:
1 | Perform transport calculations in the Landauer approximation. |
−1 | Print matrices for subsquent calls for tranchem.exe as a stand-alone
post-processing |
| utility, or for generating bulk model files. |
RECOMMENDATION:
|
| TRANX, TRANY, TRANZ
x, y, and z value of user-specified translation (only relevant if
ITRNGR is set to 5 or 6 |
TYPE:
DEFAULT:
OPTIONS:
x, y, and z relative to the origin in the appropriate units. |
RECOMMENDATION:
|
|
|
TRNSS
Controls whether reduced single excitation space is used |
TYPE:
DEFAULT:
FALSE | Use full excitation space |
OPTIONS:
TRUE | Use reduced excitation space |
RECOMMENDATION:
|
| TRTYPE
Controls how reduced subspace is specified |
TYPE:
DEFAULT:
OPTIONS:
1 | Select orbitals localized on a set of atoms |
2 | Specify a set of orbitals |
3 | Specify a set of occupied orbitals, include excitations to all virtual orbitals |
RECOMMENDATION:
|
|
|
UNRESTRICTED
Controls the use of restricted or unrestricted orbitals. |
TYPE:
DEFAULT:
FALSE | (Restricted) Closed-shell systems. |
TRUE | (Unrestricted) Open-shell systems. |
OPTIONS:
TRUE | (Unrestricted) Open-shell systems. |
FALSE | Restricted open-shell HF (ROHF). |
RECOMMENDATION:
Use default unless ROHF is desired. Note that for unrestricted calculations on
systems with an even number of electrons it is usually necessary to break
alpha / beta symmetry in the initial guess, by using SCF_GUESS_MIX or
providing $occupied information (see Section 4.5 on
initial guesses). |
|
| USECUBLAS_THRESH
Sets threshold of matrix size sent to GPU
(smaller size not worth sending to GPU). |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
Use the default value. Anything less can
seriously hinder the GPU acceleration |
|
|
|
USER_CONNECT
Enables explicitly defined bonds. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Bond connectivity is read from the $molecule section |
FALSE | Bond connectivity is determined by atom proximity |
RECOMMENDATION:
Set to TRUE if bond connectivity is known, in which case this connectivity must be
specified in the $molecule section. This greatly accelerates MM calculations.
|
|
| USE_MGEMM
Use the mixed-precision matrix scheme (MGEMM)
if you want to make calculations in your card in single-precision
(or if you have a single-precision-only GPU), but leave some parts
of the RI-MP2 calculation in double precision) |
TYPE:
DEFAULT:
OPTIONS:
0 | MGEMM disabled |
1 | MGEMM enabled |
RECOMMENDATION:
Use when having single-precision cards |
|
|
|
VARTHRESH
Controls the temporary integral cut-off threshold. tmp_thresh = 10−VARTHRESH×DIIS_error |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
3 has been found to be a practical level, and can slightly speed up SCF
evaluation. |
|
| VCI
Specifies the number of quanta involved in the VCI calculation. |
TYPE:
DEFAULT:
OPTIONS:
User-defined. Maximum value is 10. |
RECOMMENDATION:
The availability depends on the memory of the machine.
Memory allocation for VCI calculation is the square of
2*(NVib+NVCI)/NVibNVCI with double precision.
For example, a machine with 1.5 GB memory and for molecules with fewer than 4
atoms, VCI(10) can be carried out, for molecule containing fewer than 5 atoms,
VCI(6) can be carried out, for molecule containing fewer than 6 atoms, VCI(5)
can be carried out. For molecules containing fewer than 50 atoms, VCI(2) is
available. VCI(1) and VCI(3) usually overestimated the true energy while
VCI(4) usually gives an answer close to the converged energy. |
|
|
|
VIBMAN_PRINT
Controls level of extra print out for vibrational analysis. |
TYPE:
DEFAULT:
OPTIONS:
1 | Standard full information print out. |
| If VCI is TRUE, overtones and combination bands are also printed. |
3 | Level 1 plus vibrational frequencies in atomic units. |
4 | Level 3 plus mass-weighted Hessian matrix, projected mass-weighted Hessian |
| matrix. |
6 | Level 4 plus vectors for translations and rotations projection matrix. |
RECOMMENDATION:
|
| WANG_ZIEGLER_KERNEL
Controls whether to use the Wang-Ziegler
non-collinear exchange-correlation kernel in a SFDFT calculation. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use non-collinear kernel |
TRUE | Use non-collinear kernel |
RECOMMENDATION:
|
|
|
WAVEFUNCTION_ANALYSIS
Controls the running of the default wavefunction analysis tasks. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform default wavefunction analysis. |
FALSE | Do not perform default wavefunction analysis. |
RECOMMENDATION:
|
| WIG_GRID
Specify angular Lebedev grid for Wigner intracule calculations. |
TYPE:
DEFAULT:
OPTIONS:
Lebedev grids up to 5810 points. |
RECOMMENDATION:
Larger grids if high accuracy required. |
|
|
|
WIG_LEB
Use Lebedev quadrature to evaluate Wigner integrals. |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Evaluate Wigner integrals through series summation. |
TRUE | Use quadrature for Wigner integrals. |
RECOMMENDATION:
|
| WIG_MEM
Reduce memory required in the evaluation of W(u,v). |
TYPE:
DEFAULT:
OPTIONS:
FALSE | Do not use low memory option. |
TRUE | Use low memory option. |
RECOMMENDATION:
The low memory option is slower, use default unless memory is limited. |
|
|
|
WRITE_WFN
Specifies whether or not a wfn file is created, which is suitable for use with
AIMPAC. Note that the output to this file is currently limited to f orbitals,
which is the highest angular momentum implemented in AIMPAC. |
TYPE:
DEFAULT:
(NULL) | No output file is created. |
OPTIONS:
filename | Specifies the output file name. The suffix .wfn will |
| be appended to this name. |
RECOMMENDATION:
|
| XCIS
Do an XCIS calculation in addition to a CIS calculation |
TYPE:
DEFAULT:
OPTIONS:
False | Do not do an XCIS calculation |
True | Do an XCIS calculation (requires ROHF ground state). |
RECOMMENDATION:
|
|
|
XC_GRID
Specifies the type of grid to use for DFT calculations. |
TYPE:
DEFAULT:
OPTIONS:
0 | Use SG-0 for H, C, N, and O, SG-1 for all other atoms. |
1 | Use SG-1 for all atoms. |
2 | Low Quality. |
mn | The first six integers correspond to m radial points and the second six |
| integers correspond to n angular points where possible numbers of Lebedev |
| angular points are listed in section 4.3.11. |
−mn | The first six integers correspond to m radial points and the second six |
| integers correspond to n angular points where the number of Gauss-Legendre |
| angular points n = 2N2. |
RECOMMENDATION:
Use default unless numerical integration problems arise. Larger grids may be
required for optimization and frequency calculations. |
|
| XC_SMART_GRID
Uses SG-0 (where available) for early SCF cycles, and switches to the
(larger) grid specified by XC_GRID (which defaults to SG-1, if not
otherwise specified) for final cycles of the SCF. |
TYPE:
DEFAULT:
OPTIONS:
RECOMMENDATION:
The use of the smart grid can save some time on initial SCF cycles. |
|
|
|
XOPT_SEAM_ONLY
Orders an intersection seam search only, no minimization is to perform. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Find a point on the intersection seam and stop. |
FALSE | Perform a minimization of the intersection seam. |
RECOMMENDATION:
In systems with a large number of degrees of freedom it might be useful
to locate the seam first setting this option to TRUE and use that geometry
as a starting point for the minimization. |
|
| XOPT_STATE_1, XOPT_STATE_2
Specify two electronic states the intersection of which will be searched. |
TYPE:
[INTEGER, INTEGER, INTEGER] |
DEFAULT:
No default value (the option must be specified to run this calculation) |
OPTIONS:
[spin, irrep, state] | |
spin = 0 | Addresses states with low spin, |
| see also CC_NLOWSPIN. |
spin = 1 | Addresses states with high spin, |
| see also CC_NHIGHSPIN. |
irrep | Specifies the irreducible representation to which |
| the state belongs, for C2v point group symmetry |
| irrep = 1 for A1, irrep = 2 for A2, |
| irrep = 3 for B1, irrep = 4 for B2. |
state | Specifies the state number within the irreducible |
| representation, state = 1 means the lowest excited |
| state, state = 2 is the second excited state, etc. |
0, 0, -1 | Ground state. |
RECOMMENDATION:
Only intersections of states with different spin
or symmetry can be calculated at this time. |
|
|
|
XPOL
Perform a self-consistent XPol calculation. |
TYPE:
DEFAULT:
OPTIONS:
TRUE | Perform an XPol calculation. |
FALSE | Do not perform an XPol calculation. |
RECOMMENDATION:
|
| Z_EXTRAP_ORDER
Specifies the polynomial order N for Z-vector extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform Z-vector extrapolation. |
OPTIONS:
N | Extrapolate using an Nth-order polynomial (N > 0). |
RECOMMENDATION:
|
|
|
Z_EXTRAP_POINTS
Specifies the number M of old Z-vectors that are retained for use in extrapolation. |
TYPE:
DEFAULT:
0 | Do not perform response equation extrapolation. |
OPTIONS:
M | Save M previous Z-vectors for use in extrapolation (M > N) |
RECOMMENDATION:
Using the default Z-vector convergence settings, a (4,2)=(M,N) extrapolation was shown to provide the greatest speedup. At this setting, a 2-3-fold reduction in iterations was demonstrated. |
|
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This version 4.1 was edited by:
Prof. Anna Krylov and Prof. Martin Head-Gordon
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Version 4.0.1 was edited by:
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Version 4.0 was edited by:
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