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Physical Review B

Efficient calculation of unbiased expectation values in diffusion quantum Monte Carlo

Manolo C. Per, Ian K. Snook, Salvy P. Russo

Published Nov 29, 2012• 12 citations
Publisher

Abstract

Despite the proven utility of quantum Monte Carlo methods in addressing the quantum many-body problem, many important observables are difficult to calculate due to the presence of large, and sometimes divergent, statistical errors. The present state of the art allows the construction of renormalized estimators which result in finite variances, but which invariably include some systematic bias. We present a simple method for calculating unbiased expectation values of local operators in the diffusion quantum Monte Carlo method which is applicable to both bare and renormalized estimators, allowing the accurate calculation of important properties such as forces.

DOI
10.1103/physrevb.86.201107

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Despite the proven utility of quantum Monte Carlo methods in addressing the quantum many-body problem, many important observables are difficult to calculate due to the presence of large, and sometimes divergent, statistical errors. The present state of the art allows the construction of renormalized estimators which result in finite variances, but which invariably include some systematic bias. We present a simple method for calculating unbiased expectation values of local operators in the diffusion quantum Monte Carlo method which is applicable to both bare and renormalized estimators, allowing the accurate calculation of important properties such as forces.
A Monte Carlo algorithm for computing quantum-mechanical expectation values of coordinate operators in many-body problems is presented. The algorithm, which relies on the forward walking method, fits naturally in a Green's function Monte Carlo calculation, i.e., it does not require side walks or a bilinear sampling method. Our method evidences stability regions large enough to accurately sample unbiased pure expectation values. The proposed algorithm yields accurate results when it is applied to test problems such as the hydrogen atom and the hydrogen molecule. An excellent description of several properties of a fully many-body problem such as liquid 4He^{4}\mathrm{He}4He at zero temperature is achieved.
An alternative approach to conventional quantum chemistry techniques for molecular studies is described.This approach uses the quantum Monte Carlo method, which was developed and used primarily in nuclear and condensed-matter physics.In this approach the many-body Schrodinger equation is re-interpreted as a diffusion equation.Simulation of the appropriate random walk process allows one to calculate expectation values of molecular properties.In principle these expectation values can be calculated exactly, subject only to statistical errors (which may be made arbitrarily small).In preliminary work on small molecules (H 2 , LiH, Li 2 , and H 2 0), we use a simple but accurate approximation to ease the treatment of Fermi statistics.In that approximation, the calculated total energy remains an upper bound to the true energy.How good the bound is, as well as the magnitude of the statistical error, depends on the quality of an "importance function" which guides the diffusion into more probable regions of phase space.With relatively simple importance functions, we obtain from 75-100% of the correlation energy of the above-mentioned molecules.
We introduce a new stochastic method for calculating ground-state properties of quantum systems. Segments of a Langevin random walk guided by a trial wave function are subject to a Metropolis rejection test performed on the time integral of the local energy. The algorithm---which is as simple as variational Monte Carlo---for bosons provides exact expectation values of local observables, as well as their static and dynamic (in imaginary time) response functions, without mixed-estimate nor population-control biases. Our method is demonstrated with a few case applications to 4He{}^{4}\mathrm{He}4He.
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
Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within many-body quantum mechanics, and these multidimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the central limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative generalized central limit theorem and non-normal random error. In both cases the random error is not controlled. Here we consider a new "residual sampling strategy" that reintroduces the central limit theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within variational Monte Carlo are considered in detail, and the approach presented may be generalized to expectation values of other operators, and to other variants of the quantum Monte Carlo method.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).