Sparse Low Rank Approximation of Potential Energy Surfaces with Applications in Estimation of Anharmonic Zero Point Energies and Frequencies

TL;DR

This paper introduces a sparse, low-rank tensor approximation method for potential energy surfaces that enhances efficiency in calculating molecular zero point energies and frequencies, especially for larger molecules.

Contribution

It develops a novel approach combining compressed sensing, tensor rank reduction, and quadrature for efficient PES representation and vibrational energy estimation.

Findings

Improved computational scaling with molecular size.

Effective sparse tensor representation of PES.

Accurate zero point energies and frequencies obtained.

Abstract

We propose a method that exploits sparse representation of potential energy surfaces (PES) on a polynomial basis set selected by compressed sensing. The method is useful for studies involving large numbers of PES evaluations, such as the search for local minima, transition states, or integration. We apply this method for estimating zero point energies and frequencies of molecules using a three step approach. In the first step, we interpret the PES as a sparse tensor on polynomial basis and determine its entries by a compressed sensing based algorithm using only a few PES evaluations. Then, we implement a rank reduction strategy to compress this tensor in a suitable low-rank canonical tensor format using standard tensor compression tools. This allows representing a high dimensional PES as a small sum of products of one dimensional functions. Finally, a low dimensional Gauss-Hermite quadrature rule is used to integrate the product of sparse canonical low-rank representation of PES and Green's function in the second-order diagrammatic vibrational many-body Green's function theory (XVH2) for estimation of zero-point energies and frequencies. Numerical tests on molecules considered in this work suggest a more efficient scaling of computational cost with molecular size as compared to other methods.