9.3.2 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 method ⁸⁸⁸ Mills G., Jónsson H. H.
Phys. Rev. Lett.
(1994), 72, pp. 1124. Link ^{, 517} Henkelman G., Jónsson H.
J. Chem. Phys.
(2000), 113, pp. 9978. Link and the string method ³³⁸ E. W., Ren W., Vanden-Eijnden E.
Phys. Rev. B
(2002), 66, pp. 052301. Link 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 method ¹⁰⁰⁵ Peters B. et al.
J. Chem. Phys.
(2004), 120, pp. 7877. Link and freezing-string method ⁹⁷ Behn A. et al.
J. Chem. Phys.
(2011), 135, pp. 224108. Link ^{, 1162} Sharada S. M. et al.
J. Chem. Theory Comput.
(2012), 8, pp. 5166. Link 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.  97 Behn A. et al.
J. Chem. Phys.
(2011), 135, pp. 224108. Link and 1162 Sharada S. M. et al.
J. Chem. Theory Comput.
(2012), 8, pp. 5166. Link for a guide to a typical use of this method.

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 $n$th atom in the reactant moves toward the $n$th 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:

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 9.1.1. If the string returns a pathway that is unreasonable, check whether the atoms in the two input geometries are in the correct order.