Variational Monte Carlo Achieves Unbiased Atomic Forces for Accurate Electronic Structure Calculations on Modern-core Supercomputers

Accurate calculation of atomic forces remains a significant challenge in materials science and quantum chemistry, hindering the ability to model dynamic processes in complex systems. Kousuke Nakano from the National Institute for Materials Science, alongside Stefano Battaglia and Jürg Hutter from the University of Zurich, now present a method to efficiently and accurately determine these forces within variational Monte Carlo calculations. Their approach overcomes a key limitation of previous techniques by replacing computationally expensive density functional theory calculations with a single, coupled-perturbed Kohn-Sham calculation using the Lagrangian technique. This advancement not only improves the computational cost and scalability of force calculations, but also demonstrably increases their accuracy, bringing results much closer to the gold standard of coupled-cluster calculations and ensuring greater consistency with potential energy surfaces.

State-of-the-art electronic structure calculations utilize highly parallelizable stochastic frameworks to accurately solve the many-body Schrödinger equation, making them suitable for modern many-core supercomputer architectures. Despite this potential, a major drawback hindering Quantum Monte Carlo applications, particularly when targeting dynamical properties of large systems or large amounts of configurations, remains the lack of an affordable method to compute atomic forces consistent with the corresponding potential energy surfaces, also known as unbiased atomic forces. Recently, one of the authors of this work proposed a method to obtain these unbiased forces using the Jacobian.

Quantum Monte Carlo Methods and Applications

This extensive collection of references details research in computational chemistry, quantum Monte Carlo methods, and electronic structure calculations. The bibliography covers core methods like Variational Monte Carlo, Diffusion Monte Carlo, and Auxiliary-Field Quantum Monte Carlo, alongside electronic structure techniques such as Hartree-Fock, Møller-Plesset Perturbation Theory, and Coupled Cluster methods. It also includes references to Density Functional Theory and various functionals used in these calculations. The collection highlights wavefunction-based methods and many-body perturbation theory, alongside essential software packages like QMCpack, Psi4, OpenMolcas, and Gaussian. Other programs mentioned include PySCF, Molpro, ACESII, NWChem, and CP2K, alongside libraries such as Libint. Key concepts covered include electron correlation, basis sets, pseudopotentials, dispersion corrections, and symmetry-adapted orbitals.

Lagrangian Technique Streamlines Atomic Force Calculations

Scientists have developed a new method for calculating atomic forces within highly accurate quantum Monte Carlo calculations, addressing a longstanding limitation in modeling complex systems. This breakthrough improves the computational efficiency and scalability of these calculations, enabling more precise simulations of material properties and chemical reactions. The work centers on calculating “unbiased” atomic forces, which are consistent with the underlying energy landscape of the system, a crucial aspect for reliable simulations of dynamic processes. The team replaced computationally expensive Density Functional Theory calculations, previously required to obtain these unbiased forces, with a streamlined approach based on a Lagrangian technique.

This technique involves solving a set of equations to determine Lagrange multipliers, which effectively correct the initial force calculations. The method accurately determines these multipliers by projecting stationary conditions onto occupied and virtual orbital spaces, allowing for efficient calculation of the necessary corrections. This advancement significantly reduces the computational cost, particularly for larger systems. Experiments demonstrate that the new unbiased force calculations substantially improve accuracy when compared to standard Variational Monte Carlo forces, as validated against Coupled-Cluster Singles and Doubles with perturbative Triples calculations.

Specifically, the team found that the initial Variational Monte Carlo forces exhibited significant bias, while the corrected, unbiased forces closely matched the highly accurate Coupled-Cluster results. The method was tested on three molecules from the rMD17 benchmark set, confirming its ability to deliver accurate force calculations. Furthermore, the researchers developed a method for estimating the statistical errors that propagate through the calculations, employing a resampling technique known as the Jackknife method to assess the reliability of the results. This error analysis is crucial for ensuring the accuracy and trustworthiness of the simulations. The resulting method delivers a substantial improvement in both the efficiency and accuracy of atomic force calculations within quantum Monte Carlo simulations, paving the way for more detailed and reliable modeling of complex materials and chemical processes.

Unbiased Forces Enable Scalable VMC Simulations

This work presents a significant advancement in the accuracy and efficiency of calculating atomic forces within Variational Monte Carlo (VMC) calculations, a cornerstone of electronic structure modelling. Researchers developed a method to obtain unbiased atomic forces, crucial for simulating molecular dynamics and generating accurate training data for machine learning potentials. The team addressed a key limitation of previous approaches, which required computationally expensive Density Functional Theory (DFT) calculations for each atomic nucleus, by replacing these with a single coupled-perturbed calculation. This innovation substantially improves the scalability of VMC force calculations, particularly for larger systems.

Investigations using a benchmark set of molecules demonstrate that the newly developed unbiased force evaluation not only ensures consistency with potential energy surfaces but also significantly improves the accuracy of VMC forces, bringing them closer to the highly accurate Coupled-Cluster Singles and Doubles with perturbative Triples calculations. While VMC offers a potentially favourable scaling with system size and is well-suited for parallel computing, the researchers acknowledge that practical implementation details currently complicate direct comparisons of computational efficiency with other methods. Future work will focus on expanding the benchmark to include a wider range of chemical systems and employing Diffusion Monte Carlo calculations to further validate the approach and comprehensively assess its performance.