Relation of centroid molecular dynamics and ring-polymer molecular dynamics to exact quantum dynamics

TL;DR

This paper clarifies the theoretical foundations of centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD) by relating them to Matsubara dynamics, showing they are approximations that discard certain quantum fluctuations.

Contribution

It provides a unified theoretical framework connecting CMD and RPMD to Matsubara dynamics, revealing the specific approximations involved.

Findings

CMD is a mean-field approximation to Matsubara dynamics.

RPMD results from discarding a term in the Matsubara Liouvillian.

Explicit formulas for the terms omitted by CMD and RPMD.

Abstract

We recently obtained a quantum-Boltzmann-conserving classical dynamics by making a single change to the derivation of the `Classical Wigner' approximation. Here, we show that the further approximation of this `Matsubara dynamics' gives rise to two popular heuristic methods for treating quantum Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field approximation to Matsubara dynamics, obtained by discarding (classical) fluctuations around the centroid, and that RPMD is the result of discarding a term in the Matsubara Liouvillian which shifts the frequencies of these fluctuations. These findings are consistent with previous numerical results, and give explicit formulae for the terms that CMD and RPMD leave out.