Multi-level Monte Carlo path integral molecular dynamics for thermal average calculation in the nonadiabatic regime

TL;DR

This paper introduces a multi-level Monte Carlo path integral molecular dynamics method to efficiently compute thermal averages in nonadiabatic quantum systems, reducing computational cost while maintaining accuracy.

Contribution

It reformulates the ring polymer approach for multi-electronic states and develops an MLMC method to optimize sampling efficiency and variance reduction.

Findings

Reduced variance in thermal average estimates

Improved computational efficiency over traditional methods

Validated approach through numerical experiments

Abstract

With the path integral approach, the thermal average in a multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.