Canonical mean-field molecular dynamics derived from quantum mechanics

TL;DR

This paper introduces a mean-field molecular dynamics approach derived from quantum mechanics that accurately approximates quantum correlation observables, extending applicability to higher temperatures and excited states, with proven error bounds.

Contribution

It develops a rigorous mean-field molecular dynamics method from quantum mechanics, providing error estimates and a path integral representation for the Weyl symbol of the Gibbs density operator.

Findings

Mean-field dynamics approximates quantum correlations with error O(M^{-1}+ tε^2).

Numerical results show comparable or better accuracy than ground state-based methods.

The approach extends to higher temperatures and excited states, excluding conical intersections.

Abstract

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be $\mathcal O(M^{-1})$, provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and $M$ is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain $\mathcal O(M^{-1})$ accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace $h:={\rm Tr}(H e^{-\beta H})/{\rm Tr}(e^{-\beta H})$ with respect to the electron degrees of freedom and $H$ is the Weyl symbol corresponding to a quantum many body Hamiltonian $\widehat{H}$. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy $\mathcal O (M^{-1}+ t\epsilon^2)$, for correlation time $t$ where $\epsilon^2$ is related to the variance of mean value approximation $h$. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.