A Sparse Interpolation Algorithm for Dynamical Simulations in Computational Chemistry

Abstract

In this paper we present a new implementation of Smolyak's sparse grid interpolation algorithm designed for dynamical simulations. The implementation is motivated by an application to quantum chemistry where the goal is to simulate photo-induced molecular transformations. A molecule conforms to a geometry that minimizes its potential energy, and many molecules have multiple potential energy minima. These geometries correspond to local minima of the molecule's potential energy surface, and one can simulate how a molecule transitions from one geometry to another by following the steepest descent path, or reaction path, on potential energy surfaces. Molecular vibrations and thermal fluctuations cause randomness in dynamics, so one must follow several paths simultaneously to more accurately simulate possible reaction paths. Current algorithms for reaction path following are too computationally burdensome for molecules of moderate size, but Smolyak's interpolation algorithm offers a cheap surrogate for potential energy surfaces. While current implementations of Smolyak's algorithm are not designed to simultaneously follow multiple reaction paths efficiently, our implementation of Smolyak's algorithm achieves this efficiency by recursively defining Lagrange basis polynomials and making use of an efficient reformulation of Smolyak's algorithm. In this paper we describe our new implementation of Smolyak's algorithm and compare performance times to MATLAB's Sparse Grid Interpolation Toolbox to demonstrate its computational savings. We also present dynamical simulations for the photoisomerization of 2-butene as an example of our reaction path following method.