Discretising the Herman--Kluk Propagator

TL;DR

This paper develops a discretised version of the Herman--Kluk propagator using Monte Carlo methods, analyzes its accuracy with symplectic discretisation, and validates findings through numerical experiments in quantum molecular dynamics.

Contribution

It introduces a phase space integral formulation of the Herman--Kluk propagator and assesses the accuracy of symplectic time discretisation combining backward error analysis with Fourier integral operator calculus.

Findings

Numerical experiments confirm theoretical accuracy predictions.

Discretisation methods effectively approximate the propagator.

Symplectic discretisation maintains structure and improves stability.

Abstract

The Herman--Kluk propagator is a popular semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman--Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two- and six-dimensional model systems support our theoretical results.