Real-time dynamics with bead-Fourier path integrals. II. Bead-Fourier RPMD

TL;DR

This work extends bead-Fourier path integrals to real-time quantum dynamics by developing BF-RPMD, coupling BF-PIMD paths with RPMD-like dynamics. It demonstrates that, by scaling Fourier amplitudes and thermostating their momenta, BF-RPMD can reproduce accurate quantum time-correlation functions while significantly reducing the number of beads required (up to eightfold in favorable cases) for linear and some nonlinear operators. The method shows comparable performance to standard RPMD for many cases, while outperforming BF-CMD for nonlinear observables, with convergence depending on the system and operator. Overall, BF-RPMD offers a computationally efficient alternative for incorporating nuclear quantum effects into real-time dynamics and motivates further application to more complex, condensed-phase systems.

Abstract

Feynman path integrals (PIs) have found many uses in approximate quantum dynamics methods that are able to efficiently calculate real-time quantum correlation functions. The PIs typically take the form of discrete imaginary time slices over a closed path, where the slices form the ``beads'' of a ring polymer (RP) necklace. Some methods, such as centroid molecular dynamics (CMD), use the RP to generate an effective potential for the dynamics, while others, like RP molecular dynamics (RPMD), directly utilize the RP in real-time dynamics in order to incorporate quantum effects. The standard, discretized bead forms of CMD and RPMD can require a large number of RP beads to provide accurate results for systems where quantum effects are strong, such as at low temperatures. In Paper I, we introduced the bead-Fourier (BF) CMD method, where we utilized the inclusion of a Fourier sine series to reduce the number of beads needed to converge the CMD effective potential up to eightfold. In this work, we extend RPMD to incorporate BF-PIs in the form of BF-RPMD. We study a number of different implementations of the method through the calculation of correlation functions for both linear and non-linear operators. The effectiveness of the BF-RPMD method is sensitive to both the system and form of the operators being studied, but we show that this method is able to produce results on par with standard RPMD, with at worst twofold and up to eightfold reduction in the number of beads by including two to three Fourier components.

Figures (15)

• Figure 1: Quantum energy of the harmonic oscillator of Eq. \ref{['eq:harm']} at $\beta=8$ using BF-PIMD with varying numbers of beads and Fourier components. BF-PIMD with $k_{\mathrm{max}}=0$ includes the linear path between beads, while the first column labeled "PIMD" refers to the standard PIMD. The exact value is given as the black line.
• Figure 2: Position distribution function of the harmonic oscillator (Eq. \ref{['eq:harm']}) from BF-RPMD with $n=4$ and $k_{\mathrm{max}}=1$. The top panel compares the distribution using just the bead positions (blue squares) and integrating over the paths (red diamonds) with the exact quantum distribution (black line). The bottom panel compares the distributions of the bead centroid of Eq. \ref{['eq:bd-cent']} (blue squares) and the BF centroid of Eq. \ref{['eq:bf-cent']} to the classical Boltzmann distribution (black line).
• Figure 3: Kubo-transformed position autocorrelation function for the harmonic oscillator (Eq. \ref{['eq:harm']}) at $\beta=8$ for different BF-RPMD methods using varying number of beads, with the values of $k_{\mathrm{max}}$ given in Table \ref{['tab:ho-corr-1']}. The top row shows the correlation function calculated with the bead estimator, and the bottom row with the continuous estimator. (a-b) Method 1-A, (c-d) Method 1-B, (e-f) Method 2-A, and (g-h) Method 2-B. Exact results are given as black circles.
• Figure 4: Same as Fig. \ref{['fig:ho-corr']} but with $A=x^3$ and the values of $k_{\mathrm{max}}$ given in Table \ref{['tab:ho-corr-3']}
• Figure 5: Kubo transformed autocorrelation functions for the harmonic oscillator (Eq. \ref{['eq:harm']}) at high and low temperatures and for linear and cubic position operators. All BF-RPMD results (shown in blue diamonds) use Method 2-B with the continuous estimator except for $\beta=8$ with $A=x^3$ which uses the bead estimator. $n$;$k_{\mathrm{max}}$ combinations for $\beta=1$; $A=x$: 2; 0, $\beta=1$; $A=x^3$: 2; 1, $\beta=8$; $A=x$: 2; 0, $\beta=8$; $A=x^3$: 16; 1. BF-CMD results (shown in red dash dotted lines) use $n=2$; $k_{\mathrm{max}}=0$ for both temperatures. RPMD results (shown in solid orange lines) use $n=4\beta$. Exact results are shown as black circles.