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

Kousuke Nakano, Stefano Battaglia, Jürg Hutter

TL;DR

This work extends the Lagrangian framework to variational Monte Carlo to compute atomic forces and pressures that are unbiased with respect to the potential energy surface. By replacing the previous requirement of $3N$ DFT calculations with a single coupled-perturbed KS/HF step, it achieves scalable, PES-consistent force evaluations within the Jastrow-Slater determinant context. The approach yields forces that are not only self-consistent but also closer to CCSD(T) references than biased VMC, as demonstrated on rMD17 molecules. This has practical significance for large-scale QMC simulations and for generating high-quality training data for machine-learning potentials, with future extensions to DMC and broader benchmarks anticipated.

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 Schrödinger 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 large amounts of configurations, 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 DFT. However, the proposed method has a significant drawback for its applications: for a system with $N$ nuclei, one requires $3N$ additional DFT calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the $3N$ DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. This improves the computational cost and scalability of the method. 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 corrected VMC forces with those obtained by the Coupled-Cluster Singles and Doubles with perturbative Triples [CCSD(T)] calculations. We found that the bare VMC forces are significantly biased from the CCSD(T) ones, while the unbiased ones give values much closer to those of the CCSD(T) ones.

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

TL;DR

This work extends the Lagrangian framework to variational Monte Carlo to compute atomic forces and pressures that are unbiased with respect to the potential energy surface. By replacing the previous requirement of DFT calculations with a single coupled-perturbed KS/HF step, it achieves scalable, PES-consistent force evaluations within the Jastrow-Slater determinant context. The approach yields forces that are not only self-consistent but also closer to CCSD(T) references than biased VMC, as demonstrated on rMD17 molecules. This has practical significance for large-scale QMC simulations and for generating high-quality training data for machine-learning potentials, with future extensions to DMC and broader benchmarks anticipated.

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 Schrödinger 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 large amounts of configurations, 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 DFT. However, the proposed method has a significant drawback for its applications: for a system with nuclei, one requires additional DFT calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the DFT calculations with a single coupled-perturbed Kohn-Sham calculation, following the so-called Lagrangian technique established in quantum chemistry. This improves the computational cost and scalability of the method. 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 corrected VMC forces with those obtained by the Coupled-Cluster Singles and Doubles with perturbative Triples [CCSD(T)] calculations. We found that the bare VMC forces are significantly biased from the CCSD(T) ones, while the unbiased ones give values much closer to those of the CCSD(T) ones.

Paper Structure

This paper contains 12 sections, 68 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods: Lagrangian technique extended to VMC
  3. Validation of the method
  4. Application to organic molecules in rMD17 dataset
  5. Concluding remarks and future perspectives
  6. Details of the Variational Monte Carlo calculations
  7. Variational Monte Carlo Methods
  8. Wavefunction Ansatz
  9. Computing derivatives
  10. Error propagation
  11. Computational details
  12. Derivation of Lagrange multipliers equations

Figures (3)

  • Figure 1: Validation of the Lagrangian technique for Cl$_2$. (a) Potential energy surface (PES) as a function of bond length (green circles) and its fit (green dotted line). The reference force on the right-hand Cl atom (Cl$_2$ oriented along the $x$ axis), obtained from the derivative of the fitted PES, is shown as a blue dashed line. Red diamonds, purple squares, and cyan triangles denote forces evaluated with the HF + Pulay terms, HF + Pulay terms + NV correction (via FDM), and HF + Pulay terms + NV correction (via LR), respectively. (b) Comparison between the PES-derived (reference) forces and those obtained from the estimators. Inset: direct comparison between HF + Pulay terms + NV correction (via FDM) and HF + Pulay terms + NV correction (via LR).
  • Figure 2: Validation of the Lagrangian technique for cBN. (a) Equation of state (EOS) as a function of volume (green circles) and its fit (green dotted line). The reference pressure obtained from the derivative of the fitted EOS, is shown as a blue dashed line. Red diamonds, purple squares, and cyan triangles denote pressures evaluated with the HF + Pulay terms, HF + Pulay terms + NV correction (via FDM), and HF + Pulay terms + NV correction (via LR), respectively. (b) Comparison between the EOS-derived (reference) pressures and those obtained from the estimators. Inset: direct comparison between HF + Pulay terms + NV correction (via FDM) and HF + Pulay terms + NV correction (via LR). (c) PES as a function of a displacement of the B atom located at the origin (0,0,0) of the unit cell in the $x$ direction (green circles) and its fit (green dotted line). The reference force on the B atom, obtained from the derivative of the fitted PES, is shown as a blue dashed line. Red diamonds, purple squares, and cyan triangles denote forces evaluated with the HF + Pulay terms, HF + Pulay terms + NV correction (via FDM), and HF + Pulay terms + NV correction (via LR), respectively. (d) Comparison between the PES-derived (reference) forces and those obtained from the estimators. Inset: direct comparison between HF + Pulay terms + NV correction (via FDM) and HF + Pulay terms + NV correction (via LR).
  • Figure 3: Violin plots showing, for each configuration of (a) Ethanol, (b) Benzene, and (c) Malonaldehyde, the RMSE of atomic forces for the selected methods relative to CCSD(T). The unit is kcal/mol/Å.