10.2 Improved Algorithms for Transition-Structure Optimization

10.2.1 Freezing String Method

Perhaps the most significant difficulty in locating transition states is to obtain a good initial guess of the geometry to feed into a surface-walking algorithm. This difficulty becomes especially relevant for large systems, for which the dimensionality of the search space is large. Interpolation algorithms are promising for locating good guesses of the minimum-energy pathway connecting reactant and product states as well as approximate saddle-point geometries. For example, the nudged elastic band method636, 366 and the string method244 start from a certain initial reaction pathway connecting the reactant and the product state, and then optimize in discretized path space towards the minimum-energy pathway. The highest-energy point on the approximate minimum-energy pathway becomes a good initial guess for the saddle-point configuration that can subsequently be used with any local surface-walking algorithm.

Inevitably, the performance of any interpolation method heavily relies on the choice of the initial reaction pathway, and a poorly-chosen initial pathway can cause slow convergence, or possibly convergence to an incorrect pathway. The growing-string method714 and freezing-string method74, 834 offer solutions to this problem, in which two string fragments (one representing the reactant state and the other representing the product state) are “grown” (i.e., increasingly-finely defined) until the two fragments join. The freezing-string method offers a choice between Cartesian interpolation and linear synchronous transit (LST) interpolation. It also allows the user to choose between conjugate gradient and quasi-Newton optimization techniques.

Freezing-string calculations are requested by setting JOBTYPE = FSM in the $rem section. Additional job-control keywords are described below, along with examples. Consult Refs. 74 and 834 for a guide to a typical use of this method.

FSM_NNODE
       Specifies the number of nodes along the string
TYPE:
       INTEGER
DEFAULT:
       Undefined
OPTIONS:
       N number of nodes in FSM calculation
RECOMMENDATION:
       N=15. Use 10 to 20 nodes for a typical calculation. Reaction paths that connect multiple elementary steps should be separated into individual elementary steps, and one FSM job run for each pair of intermediates. Use a higher number when the FSM is followed by an approximate-Hessian based transition state search (Section 10.2.2).

FSM_NGRAD
       Specifies the number of perpendicular gradient steps used to optimize each node
TYPE:
       INTEGER
DEFAULT:
       Undefined
OPTIONS:
       N Number of perpendicular gradients per node
RECOMMENDATION:
       Anything between 2 and 6 should work, where increasing the number is only needed for difficult reaction paths.

FSM_MODE
       Specifies the method of interpolation
TYPE:
       INTEGER
DEFAULT:
       2
OPTIONS:
       1 Cartesian 2 LST
RECOMMENDATION:
       In most cases, LST is superior to Cartesian interpolation.

FSM_OPT_MODE
       Specifies the method of optimization
TYPE:
       INTEGER
DEFAULT:
       Undefined
OPTIONS:
       1 Conjugate gradients 2 Quasi-Newton method with BFGS Hessian update
RECOMMENDATION:
       The quasi-Newton method is more efficient when the number of nodes is high.

An example input appears below. Note that the $molecule section includes geometries for two optimized intermediates, separated by ****. The order of the atoms is important, as Q-Chem assumes that the nth atom in the reactant moves toward the nth atom in the product. The FSM string is printed out in the file stringfile.txt, which contains Cartesian coordinates of the structures that connect reactant to product. Each node along the path is labeled in this file, and its energy is provided. The geometries and energies are also printed at the end of the Q-Chem output file, where they are labeled:

----------------------------------------
   STRING
----------------------------------------

Finally, if MOLDEN_FORMAT is set to TRUE, then geometries along the string are printed in a MolDen-readable format at the end of the Q-Chem output file. The highest-energy node can be taken from this file and used to run a transition structure search as described in section 10.1. If the string returns a pathway that is unreasonable, check whether the atoms in the two input geometries are in the correct order.

Example 10.5  Example of the freezing-string method.

$molecule
   0  1
   Si   1.028032  -0.131573  -0.779689
   H    0.923921  -1.301934   0.201724
   H    1.294874   0.900609   0.318888
   H   -1.713989   0.300876  -0.226231
   H   -1.532839   0.232021   0.485307
****
   Si   0.000228  -0.000484  -0.000023
   H    0.644754  -1.336958  -0.064865
   H    1.047648   1.052717   0.062991
   H   -0.837028   0.205648  -1.211126
   H   -0.855603   0.079077   1.213023
$end

$rem
   JOBTYPE        fsm
   FSM_NGRAD      3
   FSM_NNODE      12
   FSM_MODE       2
   FSM_OPT_MODE   2
   METHOD         b3lyp
   BASIS          6-31G
$end