9.1.1 Introduction

Molecular potential energy surfaces rely on the Born-Oppenheimer separation of nuclear and electronic motion. Of particular interest are the critical points on these surfaces, i.e. where the gradient of the energy vanishes. Characterization of a critical point requires consideration of the eigenvalues of the Hessian (second derivative matrix) calculated at that point. An equilibrium geometry corresponds to a critical point where the eigenvalues of the Hessian are all positive, whereas a transition-state structure is defined as a first-order saddle point, and therefore has a Hessian with precisely one negative eigenvalue. The latter is a local maximum along the reaction path between the minima corresponding to the reactants and products, and 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 is currently transitioning to a new geometry optimization driver, Libopt3, which improves on the older Optimize driver. Details on the difference in capabilities between the two drivers are provided in Table 9.1.

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

• •

  coordinate system

• •

  optimization algorithm

• •

  quality of the Hessian (and gradient)

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 Libopt3 and Optimize drivers (i.e., internally, within Q-Chem) to maximize the rate of convergence. Although users may override these choices, caution should be exercised when doing so as changes may significantly impact the computational cost.

Q-Chem provides the capability to optimize a molecule using Cartesian, Z-matrix, redundant internal, or delocalized internal coordinates. The last two of these are generated automatically from the Cartesian coordinates, and delocalized internal coordinates are usually the best choice. These coordinates were developed by Baker et al., ⁵⁸ Baker J., Kessi A., Delley B.
J. Chem. Phys.
(1996), 105, pp. 192. Link and can be considered as an extension of the natural internal coordinates developed by Pulay et al. ¹⁰⁵⁸ Pulay P. et al.
J. Am. Chem. Soc.
(1979), 101, pp. 2550. Link ^{, 368} Fogarasi G. et al.
J. Am. Chem. Soc.
(1992), 114, pp. 8191. Link and the redundant optimization method of Pulay and Fogarasi. ¹⁰⁵⁹ Pulay P., Fogarasi G.
J. Chem. Phys.
(1992), 96, pp. 2856. Link

The heart of the geometry optimization in Q-Chem (for both minima and transition states) is Baker’s eigenvector-following (EF) algorithm. ⁵⁹ Baker J.
J. Comput. Chem.
(1986), 7, pp. 385. Link This was developed following the work of Cerjan and Miller, ²⁰⁴ Cerjan C. J., Miller W. H.
J. Chem. Phys.
(1981), 75, pp. 2800. Link and of Simons and co-workers. ¹¹⁷⁰ Simons J. et al.
J. Phys. Chem.
(1983), 87, pp. 2745. Link ^{, 66} Banerjee A. et al.
J. Phys. Chem.
(1985), 89, pp. 52. Link The Hessian mode-following option incorporated into this algorithm is capable of locating a transition state by walking uphill from the associated minimum. By following the lowest Hessian mode, the EF algorithm can locate a transition state starting from any reasonable input geometry and Hessian.

An additional option available for minimization is Pulay’s GDIIS algorithm, ²⁶⁸ Csaszar P., Pulay P.
J. Mol. Struct. (Theochem)
(1984), 114, pp. 31. Link which is based on the well known DIIS technique for accelerating SCF convergence. ¹⁰⁶² Pulay P.
J. Comput. Chem.
(1982), 3, pp. 556. Link GDIIS must be specifically requested, as the EF algorithm is the default.

Q-Chem incorporates a very accurate and efficient Lagrange multiplier algorithm for constrained optimization. This was originally developed for use with Cartesian coordinates ⁶⁰ Baker J.
J. Comput. Chem.
(1992), 13, pp. 240. Link ^{, 56} Baker J., Bergeron D.
J. Comput. Chem.
(1993), 14, pp. 1339. Link and can handle constraints that are not necessarily satisfied by the starting geometry. The Lagrange multiplier approach has been modified for use with delocalized internal coordinates ⁶¹ Baker J.
J. Comput. Chem.
(1997), 18, pp. 1079. Link which is much more efficient and is now the default within Q-Chem. The Lagrange multiplier code can locate constrained transition states as well as minima.

Another consideration when trying to minimize the total 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. This is why by default for transition-state optimization with Libopt3 the exact Hessian will be calculated. 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 9.2.

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

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  Cartesian, Z-matrix or internal coordinate systems

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  Eigenvector Following (EF) or GDIIS algorithms

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  Constrained optimizations

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  Equilibrium structure searches

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  Transition structure searches

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  Hessian-free characterization of stationary points

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  Initial Hessian and Hessian update options

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  Reaction pathways using intrinsic reaction coordinates (IRC)

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