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.
Highlights
The energy and mass scales of chemical motion lie in a regime between quantum and classical mechanics but for reasons of computational complexity, molecular modeling (MM) is largely performed according to Newton’s laws
The BO surface is incompatible with classical statistical mechanics in the sense that we would not expect a classical simulation on the BO surface to reproduce properties of the real material, except in the limit of infinite temperature
UNIQUENESS AND EXISTENCE Our first step toward developing a theory of force-field functors is to show that the proposed mapping, F V(q) → W(q), exists and is unique. This proof begins in Part A of the current section in which we argue that no two V(q) lead to the same quantum equilibrium density η0(q), which exists by Equations 3 and 4
Summary
The energy and mass scales of chemical motion lie in a regime between quantum and classical mechanics but for reasons of computational complexity, molecular modeling (MM) is largely performed according to Newton’s laws. Many approaches already exist to bridge this gap and study quantum equilibrium properties and dynamics: path integral Monte Carlo (PIMC), ring polymer molecular dynamics (RPMD), centroid molecular dynamics (CMD), variational pathintegral approximations, discretized path-integral approximations, semi-classical approximations, thermal Gaussian molecular dynamics and colored-noise thermostats (Whitlock et al, 1979; Chandler and Wolynes, 1981; Jang et al, 2001; Nakayama and Makri, 2003; Poulsen et al, 2003; Liu and Miller, 2006; Paesani et al, 2006; Fanourgakis et al, 2009; Liu et al, 2009; Ceriotti et al, 2011; Georgescu et al, 2011; Pérez and Tuckerman, 2011) Most of the these methods involve computational overhead significantly beyond classical mechanics and as they approach exactness their cost rapidly increases. We numerically approximate the map in a rudimentary way, and obtain surprisingly good results and transferability for both one dimensional potentials and a model of liquid para-hydrogen
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
