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9.1 Equilibrium Geometries and Transition-State Structures

9.1.1 Overview

(February 4, 2022)

Molecular potential energy surfaces rely on the Born-Oppenheimer separation of nuclear and electronic motion. Minima on such energy surfaces correspond to the classical picture of equilibrium geometries, and transition state structures correspond to first-order saddle points. Both equilibrium and transition-state structures are stationary points for which the energy gradient vanishes. Characterization of such critical points requires consideration of the eigenvalues of the Hessian (second derivative matrix): minimum-energy, equilibrium geometries possess Hessians whose eigenvalues are all positive, whereas transition-state structures are defined by a Hessian with precisely one negative eigenvalue. (The latter is therefore a local maximum along the reaction path between minimum-energy reactant and product structures, but a minimum in all directions perpendicular to this reaction path.

The quality of a geometry optimization algorithm is of major importance; even the fastest integral code in the world will be useless if combined with an inefficient optimization algorithm that requires excessive numbers of steps to converge. Q-Chem incorporates a geometry optimization package (Optimize—see Appendix A) developed by the late Jon Baker over more than ten years.

The key to optimizing a molecular geometry successfully is to proceed from the starting geometry to the final geometry in as few steps as possible. Four factors influence the path and number of steps:

  • starting geometry

  • optimization algorithm

  • quality of the Hessian (and gradient)

  • coordinate system

Q-Chem controls the last three of these, but the starting geometry is solely determined by the user, and the closer it is to the converged geometry, the fewer optimization steps will be required. Decisions regarding the optimization algorithm and the coordinate system are generally made by the Optimize package (i.e., internally, within Q-Chem) to maximize the rate of convergence. Although users may override these choices in many cases, this is not generally recommended.

Level of Theory Analytical Maximum Angular Analytical Maximum Angular
(Algorithm) Gradients Momentum Type Hessian Momentum Type
HF/DFT k k
RI-MP2 h
CIS/TDDFT (except RO) h g
Table 9.1: Gradients and Hessians currently available for geometry optimizations with maximum angular momentum types for analytical derivative calculations (higher angular momentum are supported when GEN_SCFMAN = TRUE).

Another consideration when trying to minimize the optimization time concerns the quality of the gradient and Hessian. A higher-quality Hessian (i.e., analytical versus approximate) will in many cases lead to faster convergence, in the sense of requiring fewer optimization steps. However, the construction of an analytical Hessian requires significant computational effort and may outweigh the advantage of fewer optimization cycles. Currently available analytical gradients and Hessians are summarized in Table 9.1.

Features of Q-Chem’s geometry and transition-state optimization capabilities include:

  • Cartesian, Z-matrix or internal coordinate systems

  • Eigenvector Following (EF) or GDIIS algorithms

  • Constrained optimizations

  • Equilibrium structure searches

  • Transition structure searches

  • Hessian-free characterization of stationary points

  • Initial Hessian and Hessian update options

  • Reaction pathways using intrinsic reaction coordinates (IRC)

  • Optimization of minimum-energy crossing points (MECPs) along conical seams