Refined approach to cellularization: going from Heller's thawed Gaussian approximation to the Herman--Kluk propagator

TL;DR

This work addresses the difficulty of applying the semiclassical Herman–Kluk propagator to large systems due to oscillatory integrands. It introduces a refined cellularization, combining Filinov filtering with an inverse Weierstrass framework to scale cell size with the number of trajectories, ensuring convergence to $C^{HK}(t)$ as $N\to\infty$ and to the thawed Gaussian autocorrelation $C^{TGWP}(t)$ for $N=1$, while keeping the sampling density tied to the initial state. The method is validated on integrable and chaotic model systems, showing improved convergence and damped long-time oscillations in chaotic dynamics, with comparable computational cost to standard HK and FF methods. These results extend the practical reach of semiclassical dynamics by providing a controllable interpolation between HK and TGWP, enabling accurate short-time dynamics and scalable treatment of larger systems. The approach has significant implications for calculating autocorrelation functions and spectra in complex molecular systems where quantum coherence and zero-point effects are important.

Abstract

We present a refined cellularization (Filinov filtering) scheme for the semiclassical Herman--Kluk propagator, which employs the inverse Weierstrass transform and optimal scaling of the cell's size with the number of cells, and was previously used only in the context of the dephasing representation. In the new methodology, the sampling density for the cell centers correlates with the cell size, allowing for an effective sampling of the phase space covered by the initial state of the system. The main advantage of the presented approach is that, unlike the standard cellularization, it converges to the original Herman--Kluk result in the limit of an infinite number of trajectories and to the thawed Gaussian approximation when a single trajectory is used. We illustrate the performance of the refined cellularization scheme by calculating autocorrelation functions and spectra of both integrable and chaotic model systems.

Figures (3)

• Figure 1: Comparison of standard and refined cellularization schemes for $D=1$. Standard sampling: (a) Centers (dots) of $N=4$ Gaussian functions are sampled from the Husimi function of the initial state (dashed circle). (b) Corresponding basis functions (solid circles) have widths independent of $N$. Proposed sampling: (c) For $N=1$, there is no freedom in choosing the center and the method agrees with the thawed Gaussian approximation. (d) For $N=4$, Gaussian centers are sampled from the grey disk (of radius $\sqrt{3}/2$ times the radius of the initial state) and their width is $1/2$ of the initial state's width.
• Figure 2: The collinear NCO model: Comparison of the refined Filinov filtering (RFF), Herman--Kluk (HK) and standard Filinov filtering (FF) results calculated for different values of the filtering parameter $\sigma$. (a) Time correlation functions calculated with $N=2^{17}\approx1.3 \times 10^{5}$ trajectories multiplied by the damping function (indicated by a dashed line). (b) Corresponding Franck--Condon spectra. All spectra were scaled so that the highest intensity is $1$. (c) Convergence error $\eta_{\rm vs.\,HK}$ of the damped correlation functions to the fully converged Herman--Kluk result (obtained with $N_{\mathrm{ref}}=2^{24}\approx1.68\times10^{7}$ trajectories) as a function of the number of trajectories $N$. Each line was produced by averaging the error over $10$ independent simulations. For clarity, the standard Filinov result for $\sigma=10^3$ is only shown in panel (c). (d) Statistical convergence $\eta_{\rm stat}$ of the damped correlation functions to $C_{N_{\rm ref}}$ ($N_{\rm ref}=2^{17}$) within each method. Each line was produced by averaging the error over $10$ independent simulations.
• Figure 3: The chaotic quartic oscillator: Comparison of the refined Filinov filtering (RFF), standard Filinov filtering (FF) and Herman--Kluk (HK) results. (a) Undamped time correlation function calculated with $N= 2^{17}\approx1.3\times10^{5}$ trajectories. (b) Short-time behavior of correlation functions from panel (a) multiplied by a damping function (shown by a dashed grey curve). (c) Convergence error $\eta_{\rm stat}$ of the damped correlation function to a reference solution $C_{N_{\rm ref}}$ ($N_{\rm ref}=2^{17}$) within each method. Each line was produced by averaging the error over $10$ independent simulations.