Towards Quantum Monte Carlo Forces on Heavier Ions: Scaling Properties

TL;DR

This paper benchmarks the accuracy and computational cost of Quantum Monte Carlo forces for heavier ions, revealing how cost scales with effective charge and highlighting the need for improved variance reduction techniques.

Contribution

It provides the first detailed analysis of QMC force scaling with atomic number and introduces a regression technique for heavy-tailed data, informing future applications to complex materials.

Findings

Force cost scales approximately as Z_eff^6.5 in DMC.

System size decreases as Z_eff^{-2}, regardless of variance reduction methods.

Force uncertainties grow rapidly with increasing Z_eff, impacting computational feasibility.

Abstract

Quantum Monte Carlo (QMC) forces have been studied extensively in recent decades because of their importance with spectroscopic observables and geometry optimization. Here we benchmark the accuracy and statistical cost of QMC forces. The zero-variance zero-bias (ZVZB) force estimator is used in standard variational and diffusion Monte Carlo simulations with mean-field based trial wavefunctions and atomic pseudopotentials. Statistical force uncertainties are obtained with a recently developed regression technique for heavy tailed QMC data [P. Lopez Rios and G. J. Conduit, Phys. Rev. E 99, 063312 (2019)]. By considering selected atoms and dimers with elements ranging from H to Zn ($1\leq Z_{\mathrm{eff}} \leq 20$), we assess the accuracy and the computational cost of ZVZB forces as the effective pseudopotential valence charge, $Z_{\mathrm{eff}}$, increases. We find that the cost of QMC energies and forces approximately follow simple power laws in $Z_{\mathrm{eff}}$. The force uncertainty grows more rapidly, leading to a best case cost scaling relationship of approximately $Z_{\mathrm{eff}}^{6.5(3)}$ for DMC. We find the affordable system size decreases as $Z_{\mathrm{eff}}^{-2}$, insensitive to model assumptions or the use of "space warp" variance reduction. Our results predict the practical cost of obtaining forces for a range of materials, such as transition metal oxides where QMC forces have yet to be applied, and underscore the importance of further developing force variance reduction techniques, particularly for atoms with high $\zeff$.