quantum Monte Carlo (QMC) is a state-of-the-art numerical approach for evaluating accurate expectation values of many-body wave functions. However, one of the major drawbacks that still hinders widespread QMC applications is the lack of an affordable scheme to compute unbiased atomic forces. In this study, we propose an efficient method to obtain unbiased atomic forces and pressures in the variational Monte Carlo (VMC) framework with the Jastrow-correlated Slater determinant ansatz or the Jastrow antisymmetrized geminal power ansatz, exploiting the gauge-invariant and locality properties of their geminal representation. We demonstrate the effectiveness of our method for H 2 and Cl 2 molecules and for the cubic boron nitride crystal. Our framework has a better algorithmic scaling with the system size than the traditional finite-difference method and, in practical applications, is as efficient as single-point VMC calculations. Thus, it paves the way to study dynamical properties of materials, such as phonons, and is beneficial for pursuing more reliable machine-learning interatomic potentials based on unbiased VMC forces. Published by the American Physical Society 2024
quantum Monte Carlo (QMC) is a state-of-the-art numerical approach for evaluating accurate expectation values of many-body wave functions. However, one of the major drawbacks that still hinders widespread QMC applications is the lack of an affordable scheme to compute unbiased atomic forces. In this study, we propose an efficient method to obtain unbiased atomic forces and pressures in the variational Monte Carlo (VMC) framework with the Jastrow-correlated Slater determinant ansatz or the Jastrow antisymmetrized geminal power ansatz, exploiting the gauge-invariant and locality properties of their geminal representation. We demonstrate the effectiveness of our method for H 2 and Cl 2 molecules and for the cubic boron nitride crystal. Our framework has a better algorithmic scaling with the system size than the traditional finite-difference method and, in practical applications, is as efficient as single-point VMC calculations. Thus, it paves the way to study dynamical properties of materials, such as phonons, and is beneficial for pursuing more reliable machine-learning interatomic potentials based on unbiased VMC forces. Published by the American Physical Society 2024
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 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 [K. Nakano, M. Casula, and G. Tenti, Phys. Rev. B 109, 205151 (2024)]. However, the proposed method has a significant drawback for its applications: for a system with N nuclei, one requires 6N additional density functional theory (DFT) calculations to get unbiased forces, which is not negligible as the system size increases. This paper presents a way to replace the 6N 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 variational Monte Carlo (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 the coupled-cluster singles and doubles with perturbative triples [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 generalized gradient approximations (e.g., ωB97X-D3BJ and ωB97M-D3BJ), but do not necessarily yield values that are very close to those of CCSD(T). Our finding paves the way to generate machine learning interatomic potentials based on VMC forces more efficiently and accurately.
Quantum Monte Carlo methods are accurate and promising many body techniques for electronic structure calculations which, in the last years, are encountering a growing interest thanks to their favorable scaling with the system size and their efficient parallelization, particularly suited for the modern high performance computing facilities. The ansatz of the wave function and its variational flexibility are crucial points for both the accurate description of molecular properties and the capabilities of the method to tackle large systems. In this paper, we extensively analyze, using different variational ansatzes, several properties of the water molecule, namely, the total energy, the dipole and quadrupole momenta, the ionization and atomization energies, the equilibrium configuration, and the harmonic and fundamental frequencies of vibration. The investigation mainly focuses on variational Monte Carlo calculations, although several lattice regularized diffusion Monte Carlo calculations are also reported. Through a systematic study, we provide a useful guide to the choice of the wave function, the pseudopotential, and the basis set for QMC calculations. We also introduce a new method for the computation of forces with finite variance on open systems and a new strategy for the definition of the atomic orbitals involved in the Jastrow-Antisymmetrised Geminal power wave function, in order to drastically reduce the number of variational parameters. This scheme significantly improves the efficiency of QMC energy minimization in case of large basis sets.
Quantum Monte Carlo methods are accurate and promising many body techniques for electronic structure calculations which, in the last years, are encountering a growing interest thanks to their favorable scaling with the system size and their efficient parallelization, particularly suited for the modern high performance computing facilities. The ansatz of the wave function and its variational flexibility are crucial points for both the accurate description of molecular properties and the capabilities of the method to tackle large systems. In this paper, we extensively analyze, using different variational ansatzes, several properties of the water molecule, namely: the total energy, the dipole and quadrupole momenta, the ionization and atomization energies, the equilibrium configuration, and the harmonic and fundamental frequencies of vibration. The investigation mainly focuses on variational Monte Carlo calculations, although several lattice regularized diffusion Monte Carlo calculations are also reported. Through a systematic study, we provide a useful guide to the choice of the wave function, the pseudo potential, and the basis set for QMC calculations. We also introduce a new strategy for the definition of the atomic orbitals involved in the Jastrow - Antisymmetrised Geminal power wave function, in order to drastically reduce the number of variational parameters. This scheme significantly improves the efficiency of QMC energy minimization in case of large basis sets.
Accurate ab initio calculations are of fundamental importance in physics, chemistry, biology, and materials science, which have witnessed rapid development in the last couple of years with the help of machine learning computational techniques such as neural networks. Most of the recent efforts applying neural networks to ab initio calculation have been focusing on the energy of the system. In this study, we take a step forward and look at the interatomic force obtained with neural network wavefunction methods by implementing and testing several commonly used force estimators in variational quantum Monte Carlo (VMC). Our results show that neural network ansatz can improve the calculation of interatomic force upon traditional VMC. The relationship between the force error and the quality of the neural network, the contribution of different force terms, and the computational cost of each term is also discussed to provide guidelines for future applications. Our work demonstrates that it is promising to apply neural network wavefunction methods in simulating structures/dynamics of molecules/materials and provide training data for developing accurate force fields.
Solutions to the Schrodinger equation can be used to predict the electronic structure of molecules and materials and therefore infer their complex physical and chemical properties. Variational Quantum Monte Carlo (VMC) is a technique that can be used to solve the weak form of the Schrodinger equation. Applying VMC to systems with N electrons involves evaluating the determinant of an N by N matrix. The evaluation of this determinant scales as O(N3) and is the main computational cost in the VMC process. In this work, we investigate an alternative VMC technique based on the Vandermonde determinant. The Vandermonde determinant is a product of pairwise differences and so evaluating it scales as O(N2). Therefore, this approach reduces the computational cost by a factor of N. The Vandermonde determinant was implemented in PyTorch and the performance was assessed in approximating the ground state energy of various quantum systems against existing techniques. The performance is evaluated in a variety of systems, starting with the one-dimensional particle in a box, and then considering more complicated atomic systems with multiple particles. The Vandermonde determinant was also implemented in PauliNet, a deep-learning architecture for VMC. The new method is shown to be computationally efficient, and results in a speed-up as large as 5X. In these cases, the new ansatz obtains a reasonable approximation for wavefunctions of atomic systems, but does not reach the accuracy of the Hartree-Fock method that relies on the Slater determinant. It is observed that while the use of neural networks in VMC can result in highly accurate solutions, further work is necessary to determine an appropriate balance between computational time and accuracy.
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ