Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions

Abstract

We propose a method for fitting potential energy surfaces with a sum of component functions of lower dimensionality. This form facilitates quantum dynamics calculations. We show that it is possible to reduce the dimensionality of the component functions by introducing new and redundant coordinates obtained with linear transformations. The transformations are obtained from a neural network. Different coordinates are used for different component functions and the new coordinates are determined as the potential is fitted. The quality of the fits and the generality of the method are illustrated by fitting reference potential surfaces of hydrogen peroxide and of the reaction OH+H(2)-->H(2)O+H.

Neural network‐based approaches for building high dimensional and quantum dynamics‐friendly potential energy surfaces

“…(6) and (7) grows combinatorially with d and k. For example, even if k 5 4 is enough, there are 220 component functions for a sixatom molecule. We have shown [25] that it is possible to reduce substantially the number of terms using an expression similar to Eq. (7) in transformed coordinates.…”

“…Finding the coordinates q can be thought of as adding a layer, with linear neurons, to a classic NN. [25,28] When L k > 1, usually the total number of new coordinates kL k > d, which is why the method was called Redundant Coordinate HMDR neural network (RC-HDMR-NN). [25] We have shown that this approach does result in a substantially smaller number of terms and/or required k compared to RS-HDMR [Eqs.…”

“…An accuracy of 10 cm 21 is achieved with only 12 two-dimensional functions. [25] The advantage of using transformed coordinates becomes more important as the dimensionality is increased. For example, to fit a 12-dimensional PES of vinyl bromide, [27] one would require 66 twodimensional, 220 three-dimensional, 495 four-dimensional, and 792 five-dimensional terms using the standard multimode/HMDR approach.…”

“…The error of the expnn PES is comparable to the cumulative error of the original 1 potfit fits. [25,26,108] Brown and coworkers have used expnn to fit a PES of CS 2 to highly accurate ab initio data and to compute its vibrational levels. [117] They also showed that expnn can be interfaced with the multiconfiguration time dependent Hartree algorithm.…”

“…[22] It has been demonstrated that NN PESs are very accurate for small molecules and as good as other methods for larger molecules. [23][24][25][26][27][28][29][30][31][32] If used with exponential neurons [108] or with product neurons, [33] they yield PESs in sum-of-products (SOP) form. The SOP form facilitates quantum dynamics calculations.…”