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California Univ., Berkeley Report

Quantum Monte Carlo for molecules

William A. Lester, PETER J. REYNOLDS

Published May 1, 1983• 3 citations
OpenAlex

Abstract

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.

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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.
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).
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
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.
Most methods of quantum chemistry calculate total energies rather than directly the energy differences that are of interest to chemists. In the case of statistical methods like quantum Monte Carlo the statistical errors in the absolute values need to be considerably smaller than their difference. The calculation of small energy differences is therefore particularly time consuming. Correlated sampling techniques provide the possibility to compute directly energy differences by simulating the underlying systems with the same stochastic process. The smaller the energy difference the smaller its statistical error. Correlated sampling is well established in variational quantum Monte Carlo, but it is much more difficult to implement in diffusion quantum Monte Carlo because of the fixed node approximation. A correlated sampling formalism and a corresponding algorithm based on a transformed Schrödinger equation having the form of a Kolmogorov’s backward equation is derived. The numerical verification of the presented algorithm is given for the harmonic oscillator. The extension of the algorithm to electron structure calculations is discussed.
Two aspects of quantum Monte Carlo are discussed. First, we review a procedure for obtaining trial wavefunctions for use in quantum Monte Carlo simulations that have both smaller statistical errors and improved expectation values than commonly used functions. Second, we present a correlated sampling approach for calculating energy differences in variational Monte Carlo much more accurately than the values of the energies. This method is used to calculate the potential energy surfaces of H2 and BH. In the case of BH, at bond lengths 10% different from the reference geometry, the reduced statistical error in the relative energy results in a factor of 3000 saving in computer time.
The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.