Variance reduction for forces and pressure in variational Monte Carlo

TL;DR

This paper introduces practical variance reduction techniques for Monte Carlo estimators in electronic structure calculations, improving the accuracy of forces and pressure estimates in variational Monte Carlo methods, with broad applicability.

Contribution

It proposes new variance reduction strategies for Pulay and Hellmann-Feynman forces, including modifications for divergence control and estimators for periodic systems, applicable to molecular dynamics and other observables.

Findings

Variance divergence softened from power-law to logarithmic.

Regularizations effectively suppress outliers with minimal bias.

Enhanced force estimators enable accurate molecular dynamics simulations.

Abstract

We present simple and practical strategies to reduce the variance of Monte Carlo estimators. Our focus is on variational Monte Carlo calculations of atomic forces and pressure in electronic systems, although we show that the underlying ideas apply more broadly to other observables, like pair-correlation and angular-distribution functions, and other methods, including molecular dynamics. For Pulay-type contributions, we show that a minor modification based on the Metropolis acceptance ratio softens the power-law divergence of the variance to a logarithmic one, and that inexpensive regularizations can further suppress outliers at the price of a controlled small bias. For Hellmann-Feynman forces, we derive compact variance-reduced estimators for periodic systems that are straightforward to implement in standard Monte Carlo codes. The approach is illustrated for high-pressure metallic hydrogen with more than a hundred atoms described by neural quantum states, including an application to molecular dynamics driven by the improved forces.