Path optimization by a variational reaction coordinate method. I. Development of formalism and algorithms

Abstract

The development of algorithms to optimize reaction pathways between reactants and products is an active area of study. Existing algorithms typically describe the path as a discrete series of images (chain of states) which are moved downhill toward the path, using various reparameterization schemes, constraints, or fictitious forces to maintain a uniform description of the reaction path. The Variational Reaction Coordinate (VRC) method is a novel approach that finds the reaction path by minimizing the variational reaction energy (VRE) of Quapp and Bofill. The VRE is the line integral of the gradient norm along a path between reactants and products and minimization of VRE has been shown to yield the steepest descent reaction path. In the VRC method, we represent the reaction path by a linear expansion in a set of continuous basis functions and find the optimized path by minimizing the VRE with respect to the linear expansion coefficients. Improved convergence is obtained by applying constraints to the spacing of the basis functions and coupling the minimization of the VRE to the minimization of one or more points along the path that correspond to intermediates and transition states. The VRC method is demonstrated by optimizing the reaction path for the Müller-Brown surface and by finding a reaction path passing through 5 transition states and 4 intermediates for a 10 atom Lennard-Jones cluster.

Reliable and efficient reaction path and transition state finding for surface reactions with the growing string method

“…The algorithms designed to locate TSs and RPs are usually classified as single‐ended or double‐ended . Single‐ended methods start from a single initial state and refine it systematically to locate a TS.…”

Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces

“…A solution of Eq. (12) or that is the same Q(ϕ N +1 ) = 0 is obtained by taking ϕ N +1 (q, ν) = ϕ(q, ν) = V (q) − ν = 0 recovering the eikonal equation (6). The wavefront of Eq.…”

“…The difference between them is the way to parametrize and to modify the guess of the current curve to make the residuum vector to the zero vector within some threshold. Here can be cited all the chain-of-states methods [7] like the widely used Nudged Elastic Band (NEB) [8,9], and the minimization of the Weierstrass E-function [3,10,11] where the error is minimized using a Runge-Kutta technique, the path optimization by a variational reaction coordinate due to Birkholz and Schlegel [12,13] and more recently the geodesic interpolation method [6]. It is worth to note that all these methods locate the IRC path which is a curve representation of a static reaction path, however, the category of an MEP is not guaranteed except if the whole IRC path is located on a deep valley floor.…”

Path optimization by a variational reaction coordinate method. II. Improved computational efficiency through internal coordinates and surface interpolation

“…In the algorithm outlined in Section V D, the energy, the n RIC elements of the RIC gradient, and the 1 2 n RIC × (n RIC + 1) unique elements of the RIC Hessian are fit as functions of t using polyharmonic splines at the start of each macroiteration. The PES data used in the fit are evaluated at the reactant, product, and an additional number of geometries that are equally spaced along the current path.…”

“…This algorithmic efficiency comes at the expense of a high periteration cost, due to the necessity of evaluating the VRE and its derivatives by numerical quadrature methods, and our previous tests were limited to demonstrations on analytical test surfaces. 1 This paper focuses on the development of additional methods to improve the applicability of the FVRC method to the study of chemical reactions, as well as to reduce the per-iteration cost to something comparable to existing CoS path optimization approaches.…”