Sampling strategies for the Herman-Kluk propagator of the wavefunction

TL;DR

This paper explores two grid-free sampling strategies for the Herman-Kluk propagator to evolve wavefunctions, demonstrating that sampling from the square root of the Husimi density improves convergence in harmonic and anharmonic potentials.

Contribution

It introduces and validates two novel sampling strategies for the Herman-Kluk propagator applied to wavefunctions, with analytical convergence estimates and numerical validation.

Findings

Sampling from the square root of Husimi density yields faster convergence.

Validated convergence estimates through numerical experiments.

Effective in both harmonic and anharmonic potentials.

Abstract

When the semiclassical Herman-Kluk propagator is used for evaluating quantum-mechanical observables or time-correlation functions, the initial conditions for the guiding trajectories are typically sampled from the Husimi density. Here, we employ this propagator to evolve the wavefunction itself. We investigate two grid-free strategies for the initial sampling of the Herman-Kluk propagator applied to the wavefunction and validate the resulting time-dependent wavefunctions evolved in harmonic and anharmonic potentials. In particular, we consider Monte Carlo quadratures based either on the initial Husimi density or on its square root as possible and most natural sampling densities. We prove analytical convergence error estimates and validate them with numerical experiments on the harmonic oscillator and on a series of Morse potentials with increasing anharmonicity. In all cases, sampling from the square root of Husimi density leads to faster convergence of the wavefunction.