Recent Advances in Quantum Monte Carlo Methods — Part II

The following sections are included:

• PREFACE

• INTRODUCTION

• CONTENTS

The following sections are included:

• Introduction
  • Nodes and the Sign Problem
  • The Plan of Attack
  • The Helium Triplet
  • Nodal Conjectures

• Lithium Atom Ground State

• Beryllium Atom
  • Numerical Arguments and Mathematica® Cross-Sections
  • Proof That Four-Electron ¹S Atomic Ground States Have Only Two Nodal Regions

• Conclusions

• References

One of the main difficulties of quantum Monte Carlo techniques is the lack of an efficient method for computing interatomic forces. To date, most quantum Monte Carlo calculations have been performed on geometries obtained with either density functional theory or conventional quantum chemistry methods. Here, we present a correlated sampling method to efficiently calculate numerical forces and potential energy surfaces in diffusion Monte Carlo. It employs a novel coordinate transformation, earlier used in variational Monte Carlo, to greatly reduce the statistical error. Results are presented from all-electron and pseudopotential calculations of homonuclear diatomic molecules.

The standard diffusion quantum Monte Carlo algorithms for electron structure calculations are based on Slater determinants built from canonical orbitals and long-range correlation functions. It is shown that the scaling of the diffusion quantum Monte Carlo method is improved by an order of magnitude when localized orbitals and short-range correlation functions are employed.

The auxiliary field quantum monte carlo (AFQMC) method is one of the most rigorous numerical methods for strongly correlated electron systems (SCES) in solids. The method does not include any approximations and the possible errors are limited only to the Trotter error and the statistical error, both of which should be reduced arbitrarily small in principle if there were no negative sign problem. The method was originally developed in studies of SCES models such as the Hubbard model, the Kondo model, and the Anderson model but we now know that it can be more generally applied to ab initio problems. Like other quantum monte carlo methods, it also suffers from the negative sign problem. However, it has turned out in these years that a remedy can be devised to reduce the negative sign ratio without much loss of accuracy. Here we will discuss the nature of the negative sign problem in the AFQMC method and how we circumvent the problem.

We examine several trial wavefunction forms for the ground state of the beryllium atom in order to determine which characteristics give the most rapid convergence towards the exact nonrelativistic energy. Our best wavefunction is an exponential Hylleraas expansion in transformed coordinates whose nodal behavior is described by a minimal set of functions. Using this trial wavefunction we compute a number of properties for this system.

We show how to estimate static electrical polarizabilities and hyperpolarizabilities for the H and He atoms up to the sixth degree by using quantum Monte Carlo methods. There are twelve non-zero quantities. Our results are in principle exact, as these are node-less systems and no use is made of the finite field approximation. To test the accuracy and precision of the Monte Carlo estimates, a comparison of our results is made with highly-accurate, ab initio determinations in the literature. Our estimates have the expected accuracy, but their precision range from four digits to only two, deteriorating as the number of perturbations increases.

Although recent advances in quantum-chemical theory and computer technology have greatly facilitated the ab initio calculations of vibrational properties of molecules, systematic discrepancies between the calculated normal-mode frequencies and the observed fundamental frequencies persist and are especially notable in the high-frequency regime because of the nuclear quantum effects which are usually neglected in the calculations. In this paper, a new ab initio computational scheme to treat the molecular quantum dynamics is proposed on the basis of the variational quantum Monte Carlo method to describe the electronic structure combined with the centroid molecular dynamics method for the quantum nuclei in the framework of a fictitious Lagrangian dynamics.

We describe and systematically analyze new implementations of the Projection Operator Imaginary Time Spectral Evolution (POITSE) method for the Monte Carlo evaluation of excited state energies. The POITSE method involves the computation of a correlation function in imaginary time. Decay of this function contains information about excitation energies, which can be extracted by a spectral transform. By incorporating branching processes in the Monte Carlo propagation, we compute these correlation functions with significantly reduced statistical noise. Our approach allows for the stable evaluation of small energy differences in situations where the previous POITSE implementation was limited by this noise.

A method is introduced to optimize excited state trial wavefunctions. The method is applied to ground and vibrationally excited states of bosonic van der Waals clusters of upto seven particles. Employing optimized trial wave functions with three-body correlations, we use correlation function Monte Carlo to estimate the corresponding excited state energies.

We describe the application of the variational and fixed-node diffusion quantum Monte Carlo methods to calculating the energies of excited electronic states. The discussion is illustrated by applications to the low-lying excited states of the sodium dimer and small hydrogenated silicon clusters.

Advances in quantum Monte Carlo (QMC) for electronic structure have made possible studies of systems of practical chemical and physical interest that would have been outside the range of system size for the method a decade ago. In this review we present selected applications of the QMC method to combustion systems that exemplify the significant strides that have been made with the approach.

We have studied the properties of small and medium-sized mercury clusters to gain some insight how bulk properties evolve with cluster size. Mercury clusters are a particularly interesting case, they undergo a transition from van der Waals to co-valent and finally metallic type of bonding with increasing cluster size. Quantum Monte Carlo methods in combination with relativistic pseudopotentials and polarization potentials can be used for these systems with almost invariable accuracy over a large range of cluster sizes and provide an accurate description of electron correlation. Properties like cohesive energies, ionization potentials and electron affinities have been obtained from fixed-node pure diffusion quantum Monte Carlo calculations and were used for the calibration of more approximate methods. We compare our results to experimental data and briefly discuss the relation between structure and spectroscopic properties of mercury clusters. Besides total energies it is possible to calculate energy derivatives like static dipole polarizabilities using finite difference formulas as well as expectation values of local operators which are otherwise difficult to evaluate. An example are atomic charge fluctuations which are intimately related to the type of bonding and reveal the local behavior of the correlated wavefunctions. An outlook is given how adsorption of small molecules on cluster surfaces can be handled within quantum Monte Carlo methods. For this we have developed an embedding scheme which enables a selective treatment of parts of the system.

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Quantum Monte Carlo (QMC) methods such as Variational Monte Carlo, Diffusion Monte Carlo or Path Integral Monte Carlo are the most accurate and general methods for computing total electronic energies. We will review methods we have developed to perform a coupled QMC for the electrons and another MC simulation for the ions. Using QMC methods, one estimates the Born-Oppenheimer energy E(Z) where Z represents the ionic degrees of freedom. That estimate of the energy is used in a Metropolis simulation of the ionic degrees of freedom where one compares exp(–[E(Z′) – E(Z)]/k_BT) to a random number and accepts or rejects the move. We have shown that one can modify the usual Metropolis acceptance probability to eliminate the bias caused by noise in this energy difference, thus allowing more noisy estimates of the energy difference and thereby drastically reduce the sampling time of the electronic degrees of freedom. We have implemented several different QMC methods for estimating the energy change including Diffusion Monte Carlo and Variational Monte Carlo. We have also developed a correlated sampling technique symmetrical in Z and Z′, so that the variance of [E(Z′) – E(Z)] is smaller than of each energy individually. Using these methods, we have performed simulations of liquid H₂ on a parallel computer. We have developed novel methods to move the H₂ molecules (separate translations, rotations and vibrations) and ways to pre-reject the moves using an empirical potential in an effort to speed up the simulation. We discuss some possible advantages of the CEIMC method concerning how the quantum effects of the ionic degrees of freedom can be included and how the boundary conditions can be integrated over.

A stochastic path-integral method (SPIM) for chemical reaction dynamics is explored not only for a full quantum system but also for a quantum-classical coupled system, e.g., a proton dynamics in classical solution. It is shown that this technique enables the direct computation of the transition amplitude with a finite space-time range, by generating a set of classical paths subject to simultaneous stochastic partial differential equations (SPDEs). In addition, a formulation plausible to calculate the transition amplitude for a quantum-classical coupled system in equilibrium bath, is proposed within the present SPIM framework on the microscopic basis. Introducing a numerical scheme, i.e., Euler-Maruyama scheme, to solve the present SPDE and discussing the statistical properties of its numerical realizations by the t-averaged coordinate, standard deviation and correlation function, it is shown that the numerical values of the Boltzmann matrix elements are in very good agreement with the analytical ones for a typical quantum harmonic system at temperature 100 K. Further, within the quantum transition state theory, the flux-flux autocorrelation function is evaluated at 6301C for the H + H₂ exchange reaction and is found to give a satisfactory agreement with the previous studies. To appraise the influence of dimensionality, both 1-dimensional Eckart potential and a full 3-dimensional (3D) Liu-Siegbahn-Truhlar-Horowitz (LSTH) potential calculations have been performed. The 3D thermal rate constant becomes in very good agreement with the previous one. Furthermore, for an example of the quantum-classical coupled system in equilibrium bath, the proton transfer reaction of ectoine in aqueous solution is modeled with a quantum ID double well potential, which couples instantaneously with a time-dependent classical external potential field. For the numerical demonstration, taking two configurations out of 6M configurations of solvent water molecules which are generated using Metropolis sampling in Monte Carlo simulation, the corresponding transition amplitudes are estimated. In conclusion, it is conjectured that there should exist a limiting value which is convergent with respect to the number of sampling among the equilibrium configurations of solvent molecules at a temperature.

Shifted contour (SC) auxiliary field Monte Carlo (AFMC) for electronic structure is described in considerable algorithmic detail. Several applications demonstrate the high accuracy achievable in molecular electronic structure, including excited states, using the techniques of correlated sampling and variational AFMC. SC-AFMC is also applicable to large Hubbard lattices of strongly correlated electrons. Applications are shown, up to lattices of size 16 × 16 with 202 electrons. Finally, we present a method for exact coarse graining of the AFMC integrand.

The following sections are included:

• SUBJECT INDEX