Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

TL;DR

This paper introduces improved quantum Monte Carlo estimators for the electron pair density that significantly reduce variance and bias, enabling more accurate and efficient calculations of pair-correlation functions in atoms and molecules.

Contribution

The authors develop zero-variance zero-bias estimators for the intracule density, enhancing the efficiency and accuracy of quantum Monte Carlo calculations for electron pair distributions.

Findings

Estimators drastically reduce variance in pair density calculations.

Systematic bias is minimized with improved estimators and optimized wave functions.

Converged intracule densities achieved for various atoms and molecules.

Abstract

We construct improved quantum Monte Carlo estimators for the spherically- and system-averaged electron pair density (i.e. the probability density of finding two electrons separated by a relative distance u), also known as the spherically-averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth non-local operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.