Sampling strategies for the Herman-Kluk propagator of the wavefunction

TL;DR

This work analyzes how initial phase-space sampling affects the accuracy of the Herman–Kluk propagator when evolving the wavefunction. It compares sampling from the Husimi density $\rho_{\rm H}(z)$ with its square root $\tilde{\rho}(z)$, derives analytical convergence and variance expressions, and validates them through numerical experiments on a harmonic oscillator and Morse potentials. The results show that sqrt-Husimi sampling yields finite second moments and significantly faster Monte Carlo convergence, across dimensionality and anharmonicity, providing practical guidance for high-dimensional semiclassical wavefunction propagation. The findings have direct implications for efficiently applying HK-based semiclassical methods to time-dependent quantum dynamics and can inform sampling strategies for related observables.

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.

Figures (8)

• Figure 1: Upper panel: Spectra of a Morse potential evaluated using the exact quantum dynamics, Herman--Kluk (HK) propagator, and thawed Gaussian approximation (TGA). Both approximations yield accurate results. Lower panel: Position density at time $t\approx 392$ fs propagated in the same Morse potential. In contrast to the Herman--Kluk propagator, the thawed Gaussian approximation does not capture interference between faster and slower components of the wavepacket. For more details, see the last paragraph of Sec. IV.
• Figure 2: Sampling error of the initial wavefunction in one (upper panel) and four (lower panel) dimensions as a function of the number $N$ of Monte Carlo points. The plot displays the error for sampling from the Husimi density \ref{['Husimi']} and its approximated convergence (marked lines) as well as the error for sampling from the square root of the Husimi density \ref{['sqrt-Husimi']} and its analytical error estimation (dotted line).
• Figure 3: Dependence of the sampling error of the initial wavefunction on dimension $D$ for $N=100\cdot 2^{13} \approx 8\cdot 10^5$ points sampled from either the Husimi density \ref{['Husimi']} or its square root \ref{['sqrt-Husimi']}. The analytical error estimate for the latter sampling is shown by the dotted line.
• Figure 4: Time dependence of the sampling error of the Herman--Kluk wavefunction propagated in a harmonic oscillator. The upper panel is produced by one independent run with $N=2^{16}=65536$ trajectories, whereas the lower panel is produced by $K=100$ independent runs, each with $N=2^{16}=65536$ trajectories, and averaging the square of the error over the $K$ runs. The analytical error estimate for the sampling from the square root of the Husimi density \ref{['sqrt-Husimi']} is shown with the dotted line.
• Figure 5: Sampling error between the Herman--Kluk wavefunctions obtained with $N$ and $2N$ Monte Carlo quadrature points as a function of $N$. The wavefunctions are calculated in a Morse potential with anharmonicity parameter $\chi=0.005$ after approximately one (solid line) or ten oscillations (dashed line). The upper panel shows both errors and their approximated convergence rates for \ref{['Husimi']}. Similarly, \ref{['sqrt-Husimi']} is displayed in the lower panel.