Recent Advances in Quantum Monte Carlo Methods

The following sections are included:

• PREFACE

• CONTENTS

We present a method for extracting analytical wavefunctions for atoms and molecules from the sampling of the exact wavefunction provided by quantum Monte Carlo (QMC) methods. The parameters in the trial wavefunctions are optimized minimizing the total square deviation between the trial and the exact wavefunction. The least squares optimized wavefunction gives energy and other properties in good agreement with the exact values. The optimized wavefunction can be used to compute properties not easily accessible to QMC simulations.

The difference δ between a true wavefunction Ψ and a trial wavefunction Ψ₀ may be determined directly in quantum Monte Carlo calculations. For an analytic trial function from any source the difference δ may be calculated and used to correct the trial function to obtain a wavefunction of higher accuracy and a more accurate eigenvalue. Successive corrections offer the possibility of unlimited accuracies. In this chapter we review the difference method, its applications to date, and its prospects for applications to large systems.

The following sections are included:

• Introduction

• Generating configurations with few electrons

• Generating configurations with many electrons

• Optimizing the parameters in the trial wavefunction

• Choosing the form of the trial wavefunction

• Calculating the ground state energy

• Simple properties

• Future prospects

• Acknowledgments

• Appendix A Nonlinear Minimization

• References

A hybrid QMC-VMC approach similar to a previously used “damped core” method for separating valence and core time scales is used here to perform nonadiabatic calculations. A number of simple trial wave functions are constructed. Variational and hybrid energies are calculated for the system HD⁺. An energy of −0.59784 (4) hartree is obtained with relatively little computational effort, to be compared to the exact result of −0.5978979… hartree. The Monte Carlo result is essentially exact, despite the hybrid approximation. Additional precision is obtainable by running longer simulations. Also of note is how much better the hybrid result is than the variational energy obtained from the same wave function. This variational energy is -0.59652 (2) hartree.

The following sections are included:

• Introduction

• Generation of CSFs for QMC calculations

• Selection of the dominant CSFs for the ground states of the atoms B to F

• Fixed-node PDMC calculations with different types of trial wave functions

• Computational details

• Comparison with previous QMC and other ab initio calculations

• Acknowledgment

• Appendix A. Coupling schemes for the CSFs used in QMC calculations

• Appendix B. The explicit construction of spin-free CSFs.

• References

The following sections are included:

• Quantum Monte Carlo Method

• QMC Calculations on Atoms
  • Calculations on Small Atoms
  • Model Potential Method for Large Atoms
  • Application to chemical reaction

• QMC Calculations on Molecules
  • GFMC with Nodal Release
  • Maximum Entropy Method
  • Simplification of Released-node GFMC

• Atoms and Molecules Containing a Positron
  • Binding Energies of Positronium Halides
  • Binding Energy of P_sCH₃

• References

Use of non local pseudopotentials to eliminate core electrons in quantum Monte Carlo calculations is described, including approximate localization of the non-local potential for use in diffusion Monte Carlo. Results are presented for DMC calculations on atoms and small molecules. The pseudopotential is localized with a trial function that incorporates correlations through a generalized Jastrow factor including electronelectron, electron nuclear, and electron-electron-nuclear correlations. We present selected atomic ionization potentials and electron affinities, spectroscopic constants for and results on the binding of A1₂. Comparisons of our results with experiment and accurate all electron calculations indicate that the pseudopotentials are capable of representing the effects of core electrons to high accuracy.

We present an overview of our recent quantum Monte Carlo (QMC) electronic structure calculations of carbon and silicon molecular systems. For comparison, these systems are also studied by a multitude of other methods including Local Density Approximation (LDA), Generalized Gradient Approximations (GGAs), Hartree-Fock (HF) and correlated ' quantum chemistry approaches. The results show a significantly higher accuracy of QMC when compared with mean-field approaches and much more favorable scaling in the number of particles when compared with correlated quantum chemistry methods. The high quality treatment of many-body effects enabled us to, for example, obtain a unique insight into the correlation energy dependence as a function of Si cluster size, find the most stable isomer of C₂₀, study the performance of various approaches for C₁₀ isomers and reproduce the experimental binding energies of a series of hydrocarbons to within ≈ 0.1 eV. Our results demonstrate that quantum Monte Carlo can currently treat up to 100-200 valence electrons on the level of about 95% of the correlation energy and consistent agreement with experiment within a few percent. QMC thus provides an extremely promising framework for large-scale correlated electronic structure studies.

The following sections are included:

• Introduction
  • Positrons and Positronium
  • Appropriate Quantum Mechanics
  • Connecting Experiment and Theory

• QMC vs. Basis Expansions of Wave Functions
  • Electron-Positron Correlation
  • Examples from the Older Literature
  • Review of Modern Bound State Calculations

• Prospects for the Future
  • Experiments
  • Calculations
  • Resonances

• Acknowledgements

• References

The following sections are included:

• Challenge of heavy-atom systems

• Monte Carlo calculations on heavy atom systems

• Some sampling methods for all electron simulations on large-Z systems
  • “split-tau’ technique
  • Diffusion-only Metropolis sampling

• Core-valence partitioning schemes

• Spectroscopic properties

• Acknowlegements

• References

The following sections are included:

• Introduction

• Quantum Monte Carlo method

• Model potential method

• Model potential-quantum Monte Carlo method

• Ionization potentials of Mg, Ca, and Sr

• Electron affinity of Cl

• Quadratic accuracy

• Concluding Remark

• Acknowledgment

• Bibliography

This review covers applications of quantum Monte Carlo methods to quantum mechanical problems in the study of electronic and atomic structure, as well as applications to statistical mechanical problems both of static and dynamic nature. The common thread in all these applications is optimization of many-parameter trial states, which is done by minimization of the variance of the local energy or, more generally for arbitrary eigenvalue problems, minimization of the variance of the configurational eigenvalue.

The following sections are included:

• SUBJECT INDEX