Force-Field Functor Theory: Classical Force-Fields which Reproduce Equilibrium Quantum Distributions

TL;DR

This paper introduces a unique and feasible method to derive classical potentials that accurately reproduce quantum distributions, improving classical simulations of quantum systems like liquid hydrogen.

Contribution

It demonstrates the existence, uniqueness, and practical construction of a map between local potentials and effective classical potentials for quantum systems.

Findings

The map is unique and must exist.

The method accurately reproduces quantum distributions in simulations.

Radial distribution functions match path integral Monte Carlo results.

Abstract

Feynman and Hibbs were the first to variationally determine an effective potential whose associated classical canonical ensemble approximates the exact quantum partition function. We examine the existence of a map between the local potential and an effective classical potential which matches the exact quantum equilibrium density and partition function. The usefulness of such a mapping rests in its ability to readily improve Born-Oppenheimer potentials for use with classical sampling. We show that such a map is unique and must exist. To explore the feasibility of using this result to improve classical molecular mechanics, we numerically produce a map from a library of randomly generated one-dimensional potential/effective potential pairs then evaluate its performance on independent test problems. We also apply the map to simulate liquid para-hydrogen, finding that the resulting radial pair distribution functions agree well with path integral Monte Carlo simulations. The surprising accessibility and transferability of the technique suggest a quantitative route to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to the powerful ideas and approximations of density functional theory.