How to obtain thermostatted ring polymer molecular dynamics from exact quantum dynamics and when to use it

TL;DR

This paper derives thermostatted ring polymer molecular dynamics (TRPMD) from exact quantum dynamics using Matsubara dynamics, providing insights into its accuracy, limitations, and suitable applications, especially for spectral calculations.

Contribution

It introduces a derivation of TRPMD from Matsubara dynamics, clarifies error terms, and predicts when TRPMD accurately reproduces quantum dynamics.

Findings

Careful addition of friction improves oscillation frequencies in harmonic systems.

TRPMD accurately predicts spectra but fails for fluctuation dynamics at barriers.

The derivation links TRPMD to exact quantum dynamics via a complex contour integral.

Abstract

We obtain thermostatted ring polymer molecular dynamics (TRPMD) from exact quantum dynamics via Matsubara dynamics, a recently-derived form of linearization which conserves the quantum Boltzmann distribution. Performing a contour integral in the complex quantum Boltzmann distribution of Matsubara dynamics, replacement of the imaginary Liouvillian which results with a Fokker-Planck term gives TRPMD. We thereby provide error terms between TRPMD and quantum dynamics and predict the systems in which they are likely to be small. Using a harmonic analysis we show that careful addition of friction causes the correct oscillation frequency of the higher ring-polymer normal modes in a harmonic well, which we illustrate with calculation of the position-squared autocorrelation function. However, no physical friction parameter will produce the correct fluctuation dynamics for a parabolic barrier. The results in this paper are consistent with previous numerical studies and advise the use of TRPMD for the computation of spectra.