Contents

Chapter 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

• 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

• 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

• 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

• 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

• 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

• 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

1.8  Citing Q-Chem

Chapter 2
Installation

2.1   Q-Chem Installation Requirements

2.1.1  Execution Environment

• 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).

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

#
setenv  QC         qchem_root_directory_name
setenv  QCSCRATCH  scratch_directory_name
source  $QC/qcenv.csh
#

# 
export QC=qchem_root_directory_name
export QCSCRATCH=scratch_directory_name
. $QC/qcenv.sh
#

2.6  Further Customization

• User input file
• $HOME/.qchemrc file
• $QC/config/preferences file
• Program defaults

2.6.1  .qchemrc and Preferences File Format

$rem
   rem_variable   option   comment
   rem_variable   option   comment
   ...
$end

2.6.2  Recommendations

2.7  Running Q-Chem

1. 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).
2. 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.
3. 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.

qchem infile outfile
qchem infile outfile savename
qchem --save infile outfile savename

localhost-1> qchem water.in water.out &

localhost-2> qchem water.com water.log water &

2.7.1  Running Q-Chem in parallel

qchem -nt nthreads infile outfile
qchem -nt nthreads infile outfile save
qchem -save -nt nthreads infile outfile save

qchem -np n infile outfile
qchem -np n infile outfile savename
qchem -save -np n infile outfile savename

2.8   IQmol Installation Requirements

1. Select the Calculation→Edit Servers menu option. A dialog will appear with a list of configured servers (which will initially be empty).
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. The $QC environment variable should be entered in the given box.
8. 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.

2.9  Testing and Exploring Q-Chem

Chapter 3
Q-Chem Inputs

3.1   IQmol

3.2  General Form

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.

• User input file (required)
• .qchemrc file in $HOME (optional)
• preferences file in $QC/config (optional)
• Internal program defaults and calculation results (built-in)

3.3  Molecular Coordinate Input ($molecule)

3.3.1  Reading Molecular Coordinates From a Previous Calculation

• 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.

localhost-1> qchem job1.in job1.out job1
localhost-2> qchem job2.in job2.out job1

3.3.2  Reading Molecular Coordinates from Another File

3.4  Cartesian Coordinates

atom  x-coordinate  y-coordinate  z-coordinate

3.4.1  Examples

3.5  Z-matrix Coordinates

1. First atom.
2. Second atom, reference atom, distance.
3. 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.
4. Fourth atom, reference atom A, distance, reference atom B, angle, reference atom C, dihedral angle (A, B, C and the fourth atom).
5. All subsequent atoms follow the same basic form as (4)

• 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.

3.5.1  Dummy Atoms

3.6  Job Specification: The $rem Array Concept

3.7  $rem Array Format in Q-Chem Input

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.

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)

$xc_functional
   X  exchange_symbol  coefficient
   X  exchange_symbol  coefficient
   ...
   C  correlation_symbol  coefficient
   C  correlation_symbol  coefficient
   ...
   K  coefficient
$end

3.26  Multiple Jobs in a Single File: Q-Chem Batch Job Files

1. To use information from a prior job in a later job
2. To keep projects together in a single file

• 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).

3.27   Q-Chem Output File

• 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

3.28   Q-Chem Scratch Files

MAX_SUB_FILE_NUM

Sets the maximum number of sub files allowed.

TYPE:
    

INTEGER

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

H(R,r)Ψ(R,r)=E(R)Ψ(R,r)

(4.1)

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, M_A is the ratio of the mass of nucleus A to the mass of an electron, R_AB = |R_A − R_B| is the distance between the Ath and Bth nucleus, r_ij = |r_i − r_j| is the distance between the ith and jth electrons, r_iA = | r_i − R_A| 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, E_elec, and electronic wave function, Ψ_elec, 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:

Etot = Eelec +Enuc

(4.6)

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],

Ψ = 1 √   n!   ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ χ1 (1) χ 2(1) … χn (1) χ1 (2) χ2 (2) … χn (2) : : : χ1 (n) χ2 (n) … χn (n) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

(4.7)

where χ_i, a spin orbital, is the product of a molecular orbital ψ_i 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

f(i)χ(xi )=εχ(xi )

(4.8)

where the Fock operator f(i) can be written

f(i)=− 1 2 ∇i2 +υeff(i)

(4.9)

Here x_i are spin and spatial coordinates of the ith electron, χ are the spin orbitals and υ^eff 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 υ^eff 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 (ϕ_μ) 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,

ψi = ∑ μ  cμi ϕμ

(4.10)

Eq. (4.8) reduces to the Roothaan-Hall matrix equation,

FC=εSC

(4.11)

where F is the Fock matrix, C is a square matrix of molecular orbital coefficients, S is the overlap matrix with elements

Sμν = ⌠ ⌡ ϕμ (r) ϕν (r)dr

(4.12)

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

Fα Cα = εα SCα Fβ Cβ = εβ SCβ (4.13)

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 υ^eff 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

Fμν = Hμνcore +Jμν −Kμν

(4.17)

where the core Hamiltonian matrix elements

Hμνcore = Tμν +Vμν

(4.18)

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

Jμν = ∑ λσ  Pλσ ( μν|λσ )

(4.21)

and

Kμν = 1 2 ∑ λσ  Pλσ ( μλ|νσ )

(4.22)

respectively, where the density matrix elements are

Pμν = 2 N/2 ∑ a=1  Cμa Cνa

(4.23)

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:

Pμνα = nα ∑ a=1  Cμaα Cνaα Pμνβ = nβ ∑ a=1  Cμaβ Cνaβ (4.25)

The total electron density matrix P^T 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 ψ_i^α = ψ_i^β, 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:      STRING DEFAULT:      SP 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:      Job dependent

EXCHANGE      Specifies the exchange level of theory. TYPE:      STRING DEFAULT:      No default OPTIONS:      HF Exact (Hartree-Fock). RECOMMENDATION:      Use HF for Hartree-Fock calculations.

BASIS      Specifies the basis sets to be used. TYPE:      STRING DEFAULT:      No default basis set 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:      INTEGER/LOGICAL DEFAULT:      FALSE 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:      INTEGER 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:      INTEGER DEFAULT:      5 For single point energy calculations. 7 For geometry optimizations and vibrational analysis. 8 For SSG calculations, see Chapter 5. OPTIONS:      User-defined 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:      LOGICAL 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:      INTEGER DEFAULT:      15 15 Megabytes. OPTIONS:      User defined size. RECOMMENDATION:      Use the default, or consult your systems administrator for hardware limits.

DIRECT_SCF      Controls direct SCF. TYPE:      LOGICAL DEFAULT:      Determined by program. 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:      INTEGER 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:      Use default.

STABILITY_ANALYSIS      Performs stability analysis for a HF or DFT solution. TYPE:      LOGICAL DEFAULT:      FALSE 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:      INTEGER 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:      INTEGER DEFAULT:      0 No extra print out. 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:      LOGICAL 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:      LOGICAL 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:      LOGICAL DEFAULT:      FALSE Do determine the point group (disabled for RIMP2 jobs). OPTIONS:      TRUE/FALSE 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:      INTEGER DEFAULT:      5 corresponding to 10−5. OPTIONS:      User defined. 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,…,r_N) 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

E = ET + EV + EJ + EXC

(4.30)

where E_T is the kinetic energy, E_V is the electron-nuclear interaction energy, E_J is the Coulomb self-interaction of the electron density ρ(r) and E_XC is the exchange-correlation energy. Adopting an unrestricted format, the alpha and beta total electron densities can be written as

ρα(r) = nα ∑ i=1  |ψiα|2 ρβ(r) = nβ ∑ i=1  |ψiβ |2 (4.31)

where n_α and n_β are the number of alpha and beta electron respectively and, ψ_i are the Kohn-Sham orbitals. Thus, the total electron density is

ρ(r) = ρα (r) + ρβ(r)

(4.32)

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

ET = 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) EV = − M ∑ A=1  ZA ρ(r) |r−RA| dr = − ∑ μν  PμνT ∑ A      ϕμ(r) ⎢ ⎢ ZA |r−RA| ⎢ ⎢ ϕν(r)     (4.35) EJ = 1 2     ρ(r1) ⎢ ⎢ 1 |r1 − r2| ⎢ ⎢ ρ(r2)     = 1 2 ∑ μν  ∑ λσ  PμνT PλσT (μν|λσ) (4.36) EXC = ⌠ ⌡ f[ρ(r),∇ρ(r),…] dr (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

Fα Cα = εα SCα (4.38) Fβ Cβ = εβ SCβ (4.39)

where the Fock matrix elements are generalized to

Fμνα = Hμνcore + Jμν − FμνXCα (4.40) Fμνβ = Hμνcore + Jμν − FμνXCβ (4.41)

where F_μν^XCα and F_μν^XCβ 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

FμνXCα = Kμνα

(4.42)

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,[83]
• 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 E_C is the correlation energy, E_X^GGA is the (local) GGA exchange energy, and E_X^HF is the (non-local) Hartree-Fock exchange energy. The constant C_HF denotes the fraction of Hartree-Fock exchange in the functional, therefore C_HF = 0 for GGAs, C_HF = 0.20 for B3LYP, C_HF = 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 E_C 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, E_X^{HF, 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, E_XC = E_C + E_X^HF. 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 E_X^{GGA, 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:      LOGICAL DEFAULT:      FALSE 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:      INTEGER DEFAULT:      No default OPTIONS:      n Corresponding to ω = n/1000, in units of bohr−1 RECOMMENDATION:      None

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 C_HF = 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 (C_HF = 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 C_GGA,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+ϵ_HOMO)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 E_X^GGA to E_X^GGA,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 E_c^ss, and an opposite-spin (os) component E_c^os of PT2 correlation energy. The two scaling parameters, c_ss and c_os, 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 E_c^nl, 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,

Edisp = −s6 ∑ A  ∑ B < A  C6AB RAB6 fdmp(RAB) (4.54) C6AB = √   C6AC6B   , (4.55) fdmp(RAB) = 1 1+e−d(RAB/RAB0 − 1) (4.56)

where s₆ is a global scaling parameter (near unity), f_dmp 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 R_AB⁰ 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:      LOGICAL DEFAULT:      None 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:      NONE

DFT_D_A      Controls the strength of dispersion corrections in the Chai-Head-Gordon DFT-D scheme in Eq.(3) of Ref. . TYPE:      INTEGER DEFAULT:      600 OPTIONS:      n Corresponding to a = n/100. RECOMMENDATION:      Use default, i.e., a=6.0

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 h_σ(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, R_ij is the distance between atoms i and j, and E_ij^C is the sum of the absolute values of the correlation energy of free atoms i and j. The dispersion coefficients C_6,ij is computed as

C6,ij= 〈dX2〉i〈dX2〉jαiαj 〈dX2〉iαj+〈dX2〉jαi

(4.59)

where 〈d_X²〉_i is the exchange hole dipole moment of the atom, and α_i 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 C_6,ij, C_8,ij and C_10,ij are dispersion coefficients computed at higher-order multipole (including dipole, quadrupole and octopole) moments of the exchange hole [127]. In above, R_vdW,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 R_C,ij between atoms i and j:

RvdW,ij=a1RC,ij+a2

(4.61)

The critical distance, R_C,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      Basic vdW job control TYPE:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      1 OPTIONS:      1 use Becke's damping function including C6 term only. 2 use Becke's damping function with higher-order (C8,C10) terms. RECOMMENDATION:      none

DFTVDW_MOL1NATOMS      The number of atoms in the first monomer in dimer calculation TYPE:      INTEGER DEFAULT:      0 OPTIONS:      0-NATOMS default 0 RECOMMENDATION:      none

DFTVDW_KAI      Damping factor K for C6 only damping function TYPE:      INTEGER DEFAULT:      800 OPTIONS:      10-1000 default 800 RECOMMENDATION:      none

DFTVDW_ALPHA1      Parameter in XDM calculation with higher-order terms TYPE:      INTEGER DEFAULT:      83 OPTIONS:      10-1000 RECOMMENDATION:      none

DFTVDW_ALPHA2      Parameter in XDM calculation with higher-order terms. TYPE:      INTEGER DEFAULT:      135 OPTIONS:      10-1000 RECOMMENDATION:      none

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:      LOGICAL DEFAULT:      1 OPTIONS:      1 use density correction when applicable (default). 0 do not use this correction (for debugging purpose) RECOMMENDATION:      none

DFTVDW_PRINT      Printing control for VDW code TYPE:      INTEGER DEFAULT:      1 OPTIONS:      0 no printing. 1 minimum printing (default) 2 debug printing RECOMMENDATION:      none

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 E_KS-DFT is the total energy from KS-DFT and E_disp is the dispersion correction as a sum of two- and three-body energies,

Edisp = E(2)+E(3)

(4.64)

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:      LOGICAL DEFAULT:      None 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:      NONE

Grimme suggested three scaling factors s₆, s_r,6 and s₈ that were optimized for several functionals (see Table IV in Ref. ). For example, s_r,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 s_r,6 and s₈. Otherwise, the default values of s₆, s_r,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:      INTEGER DEFAULT:      1000 OPTIONS:      n Corresponding to s6 = n/1000. RECOMMENDATION:      NONE

DFT_D3_RS6      Controls the strength of dispersion corrections, sr6, in the Grimme's DFT-D3 method (see Table IV in Ref. ). TYPE:      INTEGER DEFAULT:      1000 OPTIONS:      n Corresponding to sr6 = n/1000. RECOMMENDATION:      NONE

DFT_D3_S8      Controls the strength of dispersion corrections, s8, in Grimme's DFT-D3 method (see Table IV in Ref. ). TYPE:      INTEGER DEFAULT:      1000 OPTIONS:      n Corresponding to s8 = n/1000. RECOMMENDATION:      NONE

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:      LOGICAL DEFAULT:      FALSE OPTIONS:      FALSE (or 0) Do not apply the three-body interaction term TRUE Apply the three-body interaction term RECOMMENDATION:      NONE

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, E_c^ss, as well as an opposite-spin (os) component, E_c^os, which are added to the total energy obtained from the KS-DFT calculation. Two scaling parameters, c_ss and c_os, 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:      LOGICAL DEFAULT:      FALSE OPTIONS:      FALSE (or 0) Do not apply the DH-DFT scheme TRUE (or 1) Apply DH-DFT scheme RECOMMENDATION:      NONE

DH_SS      Controls the strength of the same-spin component of PT2 correlation energy. TYPE:      INTEGER DEFAULT:      0 OPTIONS:      n Corresponding to css = n/1000000 in Eq. (4.65). RECOMMENDATION:      NONE

DH_OS      Controls the strength of the opposite-spin component of PT2 correlation energy. TYPE:      INTEGER DEFAULT:      0 OPTIONS:      n Corresponding to cos = n/1000000 in Eq. (4.65). RECOMMENDATION:      NONE

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 U_XC,λ is the exchange correlation potential energy at a coupling strength λ. If one assumes a linear model of the latter:

UXC,λ = a+bλ ,

(4.67)

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|)/(ρ^4/3)] 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]:

vxcLB94 = vxcLDA + vxLB

(4.70)

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:      INTEGER DEFAULT:      500 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 E_XC 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 w_Ai are the quadrature weights, and the grid points r_Ai are given by

rAi = RA +ri

(4.73)

where R_A is the position of nucleus A, with the r_i 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 2N_ϕ ϕ-points, N_θ 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:      STRING 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:      STRING DEFAULT:      None No correlation. 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:      STRING 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:      INTEGER 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:      INTEGER DEFAULT:      No 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:      LOGICAL DEFAULT:      FALSE 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:      INTEGER DEFAULT:      1 SG-1 hybrid 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:      LOGICAL DEFAULT:      FALSE OPTIONS:      TRUE/FALSE 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:      INTEGER DEFAULT:      1 SG-1 grid OPTIONS:      Same as for XC_GRID 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:

1. Form and translate multipoles.
2. Convert multipoles to local Taylor expansions.
3. Translate Taylor information to the lowest level.
4. Evaluate Taylor expansions to obtain the far-field potential.
5. 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:      INTEGER DEFAULT:      15 For single point SCF accuracy 25 For tighter convergence (optimizations) OPTIONS:      n Use multipole expansions of order n RECOMMENDATION:      Use default.

GRAIN      Controls the number of lowest-level boxes in one dimension for CFMM. TYPE:      INTEGER 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:      LOGICAL 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

Fm=Fm−1+∆Jm−1− 1 2 ∆Km−1

(4.75)

where m is the SCF cycle, and ∆J^m and ∆K^m are computed using the difference density

∆Pm=Pm−Pm−1

(4.76)

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, ∆P^m 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:      INTEGER 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:      INTEGER DEFAULT:      0 Turns VARTHRESH off OPTIONS:      n User-defined threshold 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:      LOGICAL DEFAULT:      TRUE 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:      INTEGER DEFAULT:      SCF_CONVERGENCE + 3 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:      INTEGER DEFAULT:      SCF_CONVERGENCE + 3 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:      INTEGER 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:      INTEGER 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      1 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:      INTEGER DEFAULT:      5 Logarithmic part of the FTC classification threshold. Corresponds to 10−5 OPTIONS:      n User specified RECOMMENDATION:      Use the default.

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:      INTEGER 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:      n User specified. 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      1 OPTIONS:      im, an integer RECOMMENDATION:      a prefactor in the threshold for mrxc error control: im*10.0−io

MRXC_CLASS_THRESH_ORDER      Controls the of smoothness precision TYPE:      INTEGER DEFAULT:      6 OPTIONS:      io, an integer 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:      INTEGER DEFAULT:      6 OPTIONS:      n, an integer 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) P_λσ 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 P_K must be determined to accurately represent the potential. This is done by performing a least-squared minimization of the difference between P_λσ λσ(r₁) and P_K 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:      LOGICAL DEFAULT:      FALSE RI will not be used to compute J. OPTIONS:      TRUE Turn on RI for J. 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:      LOGICAL DEFAULT:      FALSE RI will not be used to compute K. OPTIONS:      TRUE Turn on RI for K. 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:      LOGICAL DEFAULT:      FALSE ARI will not be used by default for an RI-JK calculation. OPTIONS:      TRUE Turn on ARI. 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:      INTEGER DEFAULT:      4 A value of 4 Å will be added to the atomic van der Waals radius. OPTIONS:      n User defined radius. 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:      INTEGER DEFAULT:      5 A value of 5 Å will be added to the atomic van der Waals radius. OPTIONS:      n User defined radius. 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.

1. 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:
   1. 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,
   2. The SAD guess is not available for general (read-in) basis sets. All internal basis sets support the SAD guess.
   3. The SAD guess is not idempotent and thus requires at least two SCF iterations to ensure proper SCF convergence (idempotency of the density).
2. 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 c_x is a constant.
   
   

   Hμυ = cx Sμυ (Hμμ +Hυυ ) / (Hμμ +Hυυ ) 2 2

   (4.77)
3. 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:      STRING 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:      LOGICAL DEFAULT:      False 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:

1. 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.
2. 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 N_α (or N_β) 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:      INTEGER DEFAULT:      0 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:      None

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:      INTEGER 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:      STRING 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:      STRING 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:      None

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:      INTEGER DEFAULT:      50 OPTIONS:      User-defined. RECOMMENDATION:      Increase for slowly converging systems such as those containing transition metals.

SCF_ALGORITHM      Algorithm used for converging the SCF. TYPE:      STRING DEFAULT:      DIIS Pulay DIIS. 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:      INTEGER DEFAULT:      5 For single point energy calculations. 7 For geometry optimizations and vibrational analysis. 8 For SSG calculations, see Chapter 5. OPTIONS:      n Corresponding to 10−n 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:

SPF−FPS=0

(4.78)

During the SCF cycles, prior to achieving self-consistency, it is therefore possible to define an error vector e_i, which is non-zero except at convergence:

SPi Fi −Fi Pi S=ei

(4.79)

Here, P_i is obtained from diagonalization of ∧F_i , and

^ F   k  = k−1 ∑ j=1  cj Fj

(4.80)

The DIIS coefficients c_k, 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

∑ k  ck = 1

(4.82)

is imposed to yield a set of linear equations, of dimension N+1:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ e1 ·e1 … e1 ·eN 1 : ··· : : eN ·e1 … eN ·eN 1 1 … 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c1 : cN λ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 : 0 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:      INTEGER DEFAULT:      15 OPTIONS:      User-defined RECOMMENDATION:      None

DIIS_PRINT      Controls the output from DIIS SCF optimization. TYPE:      INTEGER DEFAULT:      0 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:      Use default

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:      LOGICAL DEFAULT:      FALSE OPTIONS:      TRUE, FALSE 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:      INTEGER DEFAULT:      50 OPTIONS:      1 Only a single Roothaan step before switching to (G)DM n n DIIS iterations before switching to (G)DM. RECOMMENDATION:      None

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:      INTEGER DEFAULT:      2 OPTIONS:      User-defined. RECOMMENDATION:      None

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:      LOGICAL DEFAULT:      FALSE 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:      LOGICAL DEFAULT:      FALSE OPTIONS:      FALSE Printing is turned off TRUE Printing is turned on. RECOMMENDATION:      None

MOM_START      Determines when MOM is switched on to stabilize DIIS iterations. TYPE:      INTEGER DEFAULT:      0 (FALSE) 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:

P(x) = ∑ i  xi Pi

(4.85)

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 x_i 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 δF_xc(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:      INTEGER DEFAULT:      0 OPTIONS:      0 No print out 1 RCA summary information 2 Level 1 plus RCA coefficients 3 Level 2 plus RCA iteration details RECOMMENDATION:      None

MAX_RCA_CYCLES      The maximum number of RCA iterations before switching to DIIS when SCF_ALGORITHM is RCA_DIIS. TYPE:      INTEGER DEFAULT:      50 OPTIONS:      N N RCA iterations before switching to DIIS RECOMMENDATION:      None

THRESH_RCA_SWITCH      The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS. TYPE:      INTEGER DEFAULT:      3 OPTIONS:      N Algorithm changes from RCA to DIIS when Error is less than 10−N. RECOMMENDATION:      None

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

H−He: s p s d p s s p s − − s Li−Ne: s p s d p f s d p s s p s d p − s d p s cc−pVTZ rcc−pVTZ cc−pVTZ rcc−pVTZ

H−He: s p s d p f s d p s s − s − p − s − − s Li−Ne: s p s d p f s d g p f s d p s s p s d p − s d − p − s d p s cc−pVQZ rcc−pVQZ cc−pVQZ rcc−pVQZ

H−He: s p s p s′ s p s − s′ Li−Ne: s p s d p s d′ p′ s′ s p s d p s − p′ s′ aug−cc−pVDZ racc−pVDZ aug−cc−pVDZ racc−pVDZ

H−He: s p s d p s d′ p′ s′ s p s − p s − − s′ Li−Ne: s p s d p f s d p f′ s d′ p′ s′ s p s d p − s d p − s − p′ s′ aug−cc−pVTZ racc−pVTZ aug−cc−pVTZ racc−pVTZ

H−He: s p s d p f s d p f′ s d′ p′ s′ s p s − p − s − p − s − − s′ Li−Ne: s p s d p f s d g p f s d g′ p f′ s d′ p′ s′ s p s d p − s d − p − s d − p − s − p′ s′ aug−cc−pVQZ racc−pVQZ aug−cc−pVQZ racc−pVQZ

#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:      LOGICAL DEFAULT:      FALSE 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:      LOGICAL DEFAULT:      FALSE OPTIONS:      Single-point energy only 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:      STRING DEFAULT:      None OPTIONS:      None RECOMMENDATION:      See reference for recommended basis set, functional, and grid pairings.

DFPT_XC_GRID      Specifies the secondary grid in a HFPC/DFPC calculation. TYPE:      STRING DEFAULT:      None OPTIONS:      None RECOMMENDATION:      See reference for recommended basis set, functional, and grid pairings.

DFPT_EXCHANGE      Specifies the secondary functional in a HFPC/DFPC calculation. TYPE:      STRING DEFAULT:      None OPTIONS:      None 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 M_S 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:

⌠ ⌡ ρ(r) O(r) d3r = N

(4.92)

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:

O(r) = ∑ A,σ  CσA wA(r)

(4.94)

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 C_A^σ 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 (C_α) 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 (C_A^α=C_A^β = C_A) 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 (C_A^α−C_A^β = C_A) 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:      LOGICAL DEFAULT:      FALSE 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:      LOGICAL DEFAULT:      TRUE 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:      LOGICAL DEFAULT:      FALSE 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:      INTEGER DEFAULT:      5 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:      LOGICAL DEFAULT:      TRUE OPTIONS:      TRUE Crash if constraint iterations do not converge. FALSE Do not crash. RECOMMENDATION:      Use default.

CDFT_BECKE_POP      Whether the calculation should print the Becke atomic charges at convergence TYPE:      LOGICAL DEFAULT:      TRUE 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]:

H12 = 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+V_Ciω_Ci with eigenvalue E_i+V_CiN_Ci). 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|ω_Ci|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:      LOGICAL DEFAULT:      FALSE 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:      INTEGER DEFAULT:      0 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:      BOOLEAN DEFAULT:      FALSE 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:      BOOLEAN DEFAULT:      FALSE 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:      INTEGER DEFAULT:      4 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      No default OPTIONS:      n Corresponding to ω = n/1000, in units of bohr−1 RECOMMENDATION:      None

INTEGRAL_2E_OPR      Determines the two-electron operator. TYPE:      INTEGER DEFAULT:      -2 Coulomb Operator. 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:      INTEGER DEFAULT:      0 OPTIONS:      0 Use efficient (and riskier) strategy to converge PAOs. 1 Use conservative (and slower) strategy to converge PAOs. RECOMMENDATION:      None

PAO_METHOD      Controls evaluation of polarized atomic orbitals (PAOs). TYPE:      STRING 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:      None

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 d_wx²=〈^wΨ| ^w∧ρ− ^x∧ρ | ^wΨ〉, where ^wΨ is a Slater determinant formed from the orthonormal orbitals, ^wϕ_i, of solution w, and ^w∧ρ is the one-particle density operator for ^wΨ. This definition is equivalent to d_wx²=N−^wP^μνS_νσ·^xP^στS_τμ. and is easily calculated.

d_wx² 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,

~ E   =E+ ∑ x  Nx e−λx d0x2 (4.95)

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, N_x and λ_x 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      5 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:      INTEGER DEFAULT:      300 OPTIONS:      n Restart after n steps. 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:      INTEGER DEFAULT:      10100 meaning 1.01 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:      INTEGER DEFAULT:      02000 meaning 2.000 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:      INTEGER DEFAULT:      01000 meaning 1.000 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:      INTEGER 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:      INTEGER DEFAULT:      10 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:      INTEGER DEFAULT:      00100 meaning 0.1 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      00200 meaning 00.200 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:      INTEGER DEFAULT:      0 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:

1. Perturbative treatment of pair correlations between electrons, typically capable of recovering 80% or so of the correlation energy in stable molecules.
2. 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.
3. 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:      STRING DEFAULT:      None No Correlation 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

H=H0 +λV

(5.1)

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 Ψ0 = E(0)Ψ0

(5.4)

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:

E(0)=〈Ψ0 |H0 | Ψ0 〉

(5.7)

E(1)=〈Ψ0 |V| Ψ0 〉

(5.8)

E(2)=〈 Ψ0 |V| Ψ(1) 〉

(5.9)

Thus, the Hartree-Fock energy

E0 = 〈 Ψ0 |H0 +V| Ψ0 〉

(5.10)

is simply the sum of the zeroth- and first- order energies

E0 = E(0)+E(1)

(5.11)

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:      INTEGER 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:      INTEGER DEFAULT:      2000 (2 Gb) 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:      INTEGER 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:      STRING DEFAULT:      Program determined. 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 OPTIONS:      n Freeze n virtual orbitals. RECOMMENDATION:      None

CORE_CHARACTER      Selects how the core orbitals are determined in the frozen-core approximation. TYPE:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 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   | jb) ( - 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:      STRING 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:      None

EPAO_ITERATE      Controls iterations for EPAO calculations (see PAO_METHOD). TYPE:      INTEGER 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:      INTEGER 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:      INTEGER 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:      INTEGER DEFAULT:      2000 (2 Gb) 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:      LOGICAL DEFAULT:      FALSE 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:      INTEGER DEFAULT:      250 OPTIONS:      n user-defined threshold 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      10000 (corresponds to 1 OPTIONS:      n user-defined threshold 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:      STRING DEFAULT:      None No Correlation 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:

ECCSD = 〈Φ0 | ^ H   |ΨCCSD 〉 =     Φ0 ⎢ ⎢ ^ H   ⎢ ⎢ ⎛ ⎝ 1+ ^ T   1  + 1 2 ^ T   2 1  + ^ T   2  ⎞ ⎠ Φ0     C  (5.27) 0 =     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) 0 =     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:

ECCD =     Φ0 ⎢ ⎢ ^ H   ⎢ ⎢ ⎛ ⎝ 1+ ^ T   2  ⎞ ⎠ Φ0     C  (5.33) 0 =     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

∂ECCD ∂θia =0

(5.35)

where the rotation angle θ_i^a 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:      INTEGER DEFAULT:      6 Energies. 7 Gradients. OPTIONS:      n Corresponding to 10−n convergence criterion. Amplitude convergence is set automatically to match energy convergence. RECOMMENDATION:      Use default

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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 no scaling 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:      INTEGER DEFAULT:      200 OPTIONS:      n up to n iterations to achieve convergence. RECOMMENDATION:      None

CC_PRINT      Controls the output from post-MP2 coupled-cluster module of Q-Chem TYPE:      INTEGER DEFAULT:      1 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:      INTEGER DEFAULT:      3 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:      LOGICAL DEFAULT:      FALSE 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:      LOGICAL DEFAULT:      TRUE OPTIONS:      TRUE/FALSE 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

∂ECCD ∂θij′ =0

(5.39)

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:

∂ECCD ∂θab′ =0

(5.40)

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:      INTEGER 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:      INTEGER DEFAULT:      none 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:      INTEGER DEFAULT:      none 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:      BOOL DEFAULT:      FALSE 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:      BOOL DEFAULT:      FALSE 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:      INTEGER 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      3 OPTIONS:      n User-defined 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:      INTEGER 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:      INTEGER DEFAULT:      100 Corresponding to 1.0 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:      INTEGER DEFAULT:      0 (FALSE) 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:      INTEGER DEFAULT:      None OPTIONS:      range 0000-10000 abcd Corresponding to ab.cd% RECOMMENDATION:      None

CC_FNO_USEPOP      Selection of the truncation scheme TYPE:      INTEGER DEFAULT:      1 OCCT OPTIONS:      0 POVO RECOMMENDATION:      None

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., 3p_x 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:      LOGICAL DEFAULT:      FALSE (no one-particle properties will be calculated) OPTIONS:      FALSE, TRUE 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:      LOGICAL DEFAULT:      FALSE (no two-particle properties will be calculated) OPTIONS:      FALSE, TRUE 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:      LOGICAL DEFAULT:      FALSE (no orbital response will be calculated) OPTIONS:      FALSE, TRUE 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:      INTEGER 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:      INTEGER DEFAULT:      50% of MEM_TOTAL. If MEM_TOTAL is not set, use 1.5 Gb. A minimum of 192 Mb is hard-coded. OPTIONS:      n Integer number of Mb 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:      INTEGER 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:      STRING DEFAULT:      None No Correlation 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 n_active is the number of active electrons, and the t_i 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 t_i, 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:      LOGICAL DEFAULT:      same value as UNRESTRICTED OPTIONS:      TRUE/FALSE 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:      INTEGER DEFAULT:      0 OPTIONS:      n User-Defined 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:      INTEGER DEFAULT:      0 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:      INTEGER DEFAULT:      0 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 activ