Fast Evaluation of Unbiased Atomic Forces in ab initio Variational Monte Carlo via the Lagrangian Technique

Abstract

Ab initio quantum Monte Carlo (QMC) methods are state-of-the-art electronic structure calculations based on highly parallelizable stochastic frameworks for accurate solutions of the many-body Schr{\"o}dinger equation, suitable for modern many-core supercomputer architectures. Despite its potential, one of the major drawbacks that still hinders QMC applications, especially when targeting dynamical properties of large systems or extensive datasets, is the lack of an affordable method to compute atomic forces that are consistent with the corresponding potential energy surfaces (PESs), also known as unbiased atomic forces. Recently, one of the authors in the present paper proposed a way to obtain unbiased forces with the Jastrow-correlated Slater determinant ansatz, where the determinant part is frozen to the values obtained by a mean-field method, such as Density Functional Theory. However, the proposed method has a significant drawback for its applications: for a system with $N$ nuclei, one requires 6$N$ additional DFT calculations to get unbiased forces. This paper presents a way to replace the 6$N$ DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. We also demonstrate that the developed unbiased VMC force calculation improves not only the consistency with PESs, but also its accuracy, by investigating three molecules from the rMD17 benchmark set, and comparing the unbiased VMC forces with those obtained by CCSD(T) calculations. We found that the bare VMC forces are biased from the CCSD(T) ones, while the unbiased ones give values closer to those of the CCSD(T) ones. Our benchmark test also reveals that the unbiased VMC forces yield very consistent values with hybrid and meta GGAs, but do not necessarily yield values that are very close to those of CCSD(T).