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.
|Step Generation Algorithms:|
|Line Search (LS)||✓||–|
|Eigenvector Following (EF)||✓||✓|
|Intrinsic Reaction Coordinates (IRC)||✗||✓|
|Minimum-Energy Crossing Points (MECPs)||✓||✓|
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:
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.,
J. Chem. Phys.
(1996), 105, pp. 192. and can be considered as an extension of the natural internal coordinates developed by Pulay et al. 1020 J. Am. Chem. Soc.
(1979), 101, pp. 2550. , 351 J. Am. Chem. Soc.
(1992), 114, pp. 8191. and the redundant optimization method of Pulay and Fogarasi. 1021 J. Chem. Phys.
(1992), 96, pp. 2856.
The heart of the geometry optimization in Q-Chem (for both minima and
transition states) is Baker’s eigenvector-following (EF)
J. Comput. Chem.
(1986), 7, pp. 385. This was developed following the work of Cerjan and Miller, 192 J. Chem. Phys.
(1981), 75, pp. 2800. and of Simons and co-workers. 1130 J. Phys. Chem.
(1983), 87, pp. 2745. , 64 J. Phys. Chem.
(1985), 89, pp. 52. 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
J. Mol. Struct. (Theochem)
(1984), 114, pp. 31. which is based on the well known DIIS technique for accelerating SCF convergence. 1024 J. Comput. Chem.
(1982), 3, pp. 556. 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
J. Comput. Chem.
(1992), 13, pp. 240. , 54 J. Comput. Chem.
(1993), 14, pp. 1339. 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 59 J. Comput. Chem.
(1997), 18, pp. 1079. 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.
|Level of Theory||Analytical||Maximum Angular||Analytical||Maximum Angular|
|(Algorithm)||Gradients||Momentum Type||Hessian||Momentum Type|
|CIS/TDDFT (except RO)||✓||✓|
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
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