Real-time dynamics with bead-Fourier path integrals I: Bead-Fourier CMD

TL;DR

This work introduces bead-Fourier PIs (BF-PIs) as an efficient hybrid representation between discretized and Fourier imaginary-time path integrals and embeds them into centroid molecular dynamics (BF-CMD) to compute an effective potential for real-time simulations. By evaluating two estimators (pure-bead and continuous) for the BF-based free energy, the authors demonstrate that at low temperatures a single Fourier component can reduce the required bead count by up to a factor of four for mildly anharmonic and quartic one-dimensional models, while preserving accuracy in key dynamical observables. The approach achieves exact results for the harmonic oscillator with the bead estimator across all beads and temperatures, and shows favorable convergence behavior for anharmonic systems, suggesting BF-CMD as a promising route to more efficient quantum dynamics in larger systems. Future work aims to extend BF-CMD to adiabatic/partially adiabatic formulations and to assess its impact on the curvature problem in vibrational spectra, with potential extensions to RPMD and non-adiabatic PI methods.

Abstract

Developing new methods for the accurate and efficient calculations of real-time quantum correlation functions is deemed one of the most challenging problems of modern condensed matter theory. Many popular methods, such as centroid molecular dynamics (CMD), make use of Feynman path integrals (PIs) to efficiently introduce nuclear quantum effects into classical dynamical simulations. Conventional CMD methods use the discretized form of the PI formalism to represent a quantum particle using a series of replicas, or "beads", connected with harmonic springs to create an imaginary time ring polymer. The alternative Fourier PI methodology, instead, represents the imaginary time path using a Fourier sine series. Presented as an intermediary between the two formalisms, bead-Fourier PIs (BF-PIs) have been shown to reduce the number of beads needed to converge equilibrium properties by including a few terms of the Fourier series. Here, a new CMD method is presented where the effective potential is calculated using BF-PIs as opposed to the typical discretized PIs. We demonstrate the accuracy and efficiency of this new BF-CMD method for a series of 1D model systems and show that at low temperatures, one can achieve a 4-fold reduction in the number of beads with the addition of a single Fourier component. The developed methodology is general and can be extended to other closely related methods, such as ring polymer molecular dynamics (RPMD), as well as non-adiabatic PI methods.

Figures (5)

• Figure 1: Kubo-transformed position autocorrelation function for the harmonic oscillator with $\beta=1$ and free energy calculated using the (a) continuous estimator with $k_{\mathrm{max}}=1$ and (b) bead estimator with $k_{\mathrm{max}}=0$ for all numbers of beads. Exact results are given in black dots.
• Figure 2: Free energy for the harmonic oscillator at $\beta=8$ using the continuous estimator with varying number of beads and Fourier components. Standard classical potential shown in black dots is given as reference.
• Figure 3: Kubo-transformed position correlation function for the harmonic oscillator with $\beta=8$. Free energy estimated with (a) continuous estimator. Number of Fourier components used for $n=1:5$; $n=2$, $n=4$, and $n=8:3$; and $n=16$ and $n=32:1$. (b) Free energy estimated with bead estimator and all beads using $k_{\mathrm{max}}=0$. Exact results given as black dots.
• Figure 4: Kubo-transformed position autocorrelation function for the mildly anharmonic oscillator with the free energy found using the bead estimator at (top) $\beta=1$ and (bottom) $\beta=8$. All beads have $k_{\mathrm{max}}=0$ for $\beta=1$. For $\beta=8$, $n=2$, $n=4$, and $n=8$ use one Fourier component while $n=16$ and $n=32$ use only the linear path. $n=1$ corresponds to classical results. Exact results are given in black dots, and CMD results are given in black pentagons.
• Figure 5: Kubo-transformed position autocorrelation function for the quartic oscillator with the free energy found using the bead estimator at (top) $\beta=1$ and (bottom) $\beta=8$. All beads have $k_{\mathrm{max}}=0$ for $\beta=1$. For $\beta=8$, $n=2$ and $n=4$ use 5 Fourier components, $n=8$ and $n=16$ use 1 Fourier component, and $n=32$ uses only the linear path. $n=1$ corresponds to classical results. Exact results are given in black dots and CMD results in black pentagons.