Interpolating moving least-squares methods for fitting potential energy surfaces: A strategy for efficient automatic data point placement in high dimensions

Abstract

An accurate and efficient method for automated molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating moving least-squares (IMLS) method is developed with the flexibility to fit various ab initio data: (1) energies, (2) energies and gradients, or (3) energies, gradients, and Hessian data. The method is automated and flexible so that a PES can be optimally generated for trajectories, spectroscopy, or other applications. High efficiency is achieved by employing local IMLS in which fitting coefficients are stored at a limited number of expansion points, thus eliminating the need to perform weighted least-squares fits each time the potential is evaluated. An automatic point selection scheme based on the difference in two successive orders of IMLS fits is used to determine where new ab initio data need to be calculated for the most efficient fitting of the PES. A simple scan of the coordinate is shown to work well to identify these maxima in one dimension, but this search strategy scales poorly with dimension. We demonstrate the efficacy of using conjugate gradient minimizations on the difference surface to locate optimal data point placement in high dimensions. Results that are indicative of the accuracy, efficiency, and scalability are presented for a one-dimensional model potential (Morse) as well as for three-dimensional (HCN), six-dimensional (HOOH), and nine-dimensional (CH4) molecular PESs.

Generation of Potential Energy Surfaces in High Dimensions and Their Haptic Exploration

“…IMLS [19,20,21,22] is our method of choice. [4] It provides a very general and therefore adaptable interpolation scheme upon which one can improve in future work.…”

Predicting accurate vibrational frequencies for highly anharmonic systems

“…Beautiful work by Bowman et al on water trimer vibrational energy calculations 4 employed a weighted least-squares fit of the ab initio energies calculated as a function of the complete set of internuclear distances. Interpolating moving least-squares methods for fitting potential energy surfaces were employed by Thompson et al, 5 while Ohno et al 6 applied a hypersphere search method combined with least squares fitting. Manzhos and Carrington 7 combined high dimensional model representation and neural networks approaches into a novel method for fitting potential energy surfaces.…”

Communication: An accurate global potential energy surface for the ground electronic state of ozone

“…32 Our automated PES generation method described previously [34][35][36] was used to fit the surface out to R(O-OO) = 10 bohrs in Jacobi coordinates including 2663 ab initio data (some obtained by permutation). No energy-dependent bias was used for the point selection, so the four wells (including the high-energy ring-minimum) and the asymptotic region were treated equally.…”