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

Abstract

Reaction path optimization is being used more frequently as an alternative to the standard practice of locating a transition state and following the path downhill. The Variational Reaction Coordinate (VRC) method was proposed as an alternative to chain-of-states methods like nudged elastic band and string method. The VRC method represents the path using a linear expansion of continuous basis functions, allowing the path to be optimized variationally by updating the expansion coefficients to minimize the line integral of the potential energy gradient norm, referred to as the Variational Reaction Energy (VRE) of the path. When constraints are used to control the spacing of basis functions and to couple the minimization of the VRE with the optimization of one or more individual points along the path (representing transition states and intermediates), an approximate path as well as the converged geometries of transition states and intermediates along the path are determined in only a few iterations. This algorithmic efficiency comes at a high per-iteration cost due to numerical integration of the VRE derivatives. In the present work, methods for incorporating redundant internal coordinates and potential energy surface interpolation into the VRC method are described. With these methods, the per-iteration cost, in terms of the number of potential energy surface evaluations, of the VRC method is reduced while the high algorithmic efficiency is maintained.

Dynamics Sampling in Transition Pathway Space

“…Following the same line of thinking, nudged elastic band (NEB) methods were developed by projecting out perpendicular components of elastic forces and parallel components of the true force with respect to the path under minimization. , In zero temperature string (ZTS) methods, the states along the transition pathway are evenly distributed and re-distributed along the fitted path after each step of minimization or simulation. − It should be noted that the ZTS method could automatically optimize end structures to local minima, regardless of whether the starting pathway connects two minima or not. Many variations of chain-of-states methods with improved efficiency and accuracy have been developed subsequently, but will not be covered in detail. − …”

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

“…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.…”

Locating Transition States by Variational Reaction Path Optimization with an Energy-Derivative-Free Objective Function

“…Another notable variational approach is a method that minimizes an objective function defined as the line integral of the Euclidean norm of the energy gradient. − In the present paper, we refer to this approach as the gradient norm method. Unlike the elastic band method, the gradient norm method provides MEPs as optimal solutions.…”

“…Many double-ended methods are based on or relevant to variational formalism, where an objective functional of a reaction path is minimized. − In the following, we pick up some relevant methods for this study. For a more comprehensive review, see refs and , for example.…”