Heat capacity estimators for random series path-integral methods by finite-difference schemes

TL;DR

This paper introduces finite-difference schemes for heat capacity estimators in path integral simulations, reducing variance issues and avoiding derivatives of the potential, demonstrated through quantum Monte Carlo simulations.

Contribution

It presents a stable finite-difference approach for heat capacity estimators that require only potential evaluations, improving upon previous methods with high variance or derivative requirements.

Findings

Finite-difference schemes yield finite variance estimators.

Second order central-difference scheme is sufficient for most cases.

Method demonstrated on Ne_13 cluster with effective results.

Abstract

Previous heat capacity estimators used in path integral simulations either have large variances that grow to infinity with the number of path variables or require the evaluation of first and second order derivatives of the potential. In the present paper, we show that the evaluation of the total energy by the T-method estimator and of the heat capacity by the TT-method estimator can be implemented by a finite difference scheme in a stable fashion. As such, the variances of the resulting estimators are finite and the evaluation of the estimators requires the potential function only. By comparison with the task of computing the partition function, the evaluation of the estimators requires k + 1 times more calls to the potential, where k is the order of the difference scheme employed. Quantum Monte Carlo simulations for the Ne_13 cluster demonstrate that a second order central-difference scheme should suffice for most applications.