**"Monte Carlo Methods in Ab Initio Quantum Chemistry"
Brian L. Hammond, William A. Lester, Jr.
and Peter J. Reynolds **

**An accurate and consistent treatment of electron correlation is one of the great challenges currently confronting physics and chemistry. In the context of interest here such a treatment is critical in many aspects of the ab initio determination of atomic and molecular structure. In such electronic structure problems, the reference point from which degree of correlation is typically judged is the Hartree-Fock (HF) limit (which by convention is said to have no correlation). Often the exact, Born-Oppenheimer (clamped nuclei) solution is the desired result. To this end, a number of ``post-HF" methods have been developed, i.e., methods which go beyond the mean field approximation, yet retain the simple molecular orbital picture. Importantly, these methods add many-body correlation at various levels of approximation. The success of HF and post-HF methods for treating electronic structure benefits considerably from the cancellation of errors between two or more computations in the calculation of obserables. In many other cases, in particular in the calculation of excited states and classical barrier heights, these errors do not cancel reliably. Sometimes quantitative accuracy can be obtained by including vast numbers of terms in various expansions, but these techniques then strain computational resources.
Post-HF methods fall mainly into two categories: those based on configuration interaction (CI) which expand the wavefunction in a double basis set expansion, and those based on many-body perturbation theory (MBPT) which treat electron-correlation as a perturbation. In both cases, correlation corrections begin from a HF wave funtion. Other important formalisms which treat the correlation energy are those of density functional theory (DFT) and electron propagator methods. The former aproach is rapidly gaining momentum and useful in a variety of contents, but suffer in others. DFT like HF remains a mean field theory.
In this book we present Monte Carlo methods for the treatment of electron correlation. Monte Carlo methods are one of the standard approaches in many-body physics. Yet these methods are still mostly unfamiliar in the study of electronic structure of atoms and molecules.
Our aim is to present the fundamental background and current status of the application of Monte Carlo to the determination of correlation in electronic structure. We will use the term quantum Monte Carlo (QMC) to refer to methods which directly solve the Schrodinger equation, as contrasted with variational Monte Carlo (VMC) methods that simply evaluate expectation values by Monte Carlo procedures. QMC differs from post-HF approaches in that it is a fully correlated method from the outset, rather than a modification of a mean-field approach. We also distinguish QMC here from a large class of other MC methods used in the treatment of quantum problems, often also referred to as QMC in the literature. The distingusihing feature of all these QMC methods is their essentially exact nature, and the lack of any special attention that needs to be paid to correlation per se. Their differences lie in the class of problems they were originally designed to treat. Among these other methods we mention Hubbard Stratanovich approaches designed for lattice models of strongly interating electrons and path integral MC designed for continuum simulators of condensed matter. Both of these methods are also finite temperature approaches, important for treating macroscopic systems.
Though the QMC we discuss here also has roots in condensed matter physics and statistical mechanics, it is a zero temperature or ground state approach. As such it is the most appropriate starting point for electronic structure. Nevertheless, work on adapting other QMC approaches to electronic structure, as well as to adapting the kinds of QMC we discuss here to other realms are currently in progress.
All QMC methods are currently computationally demanding. Nevertheless, as experimental advances push the limits of accuracy required of computational approaches, and as computers continue to achieve greater speed, emphasis must increasingly be placed on the simplicity, consistency and accuracy of a method. QMC holds high promise here. Moreover, Monte Carlo methods are the most natural for massively parallel computation, ultimately providing them with a great advantage.
The subject matter is divided conceptually into two parts. The first five chapters present the basic concepts in detail. The latter part of the book, chapters 6-9, covers important extensions of these basic methods. Chapter 1 provides background on random numbers, probability distributions, integral estimation, and simulation of simple differential equations. Those readers already familiar with statistics and use of random numbers may wish to skip this chapter. Chapter 2 treats VMC, describing methods to evaluate the energy and properties derived from variational trial wavefunctions. The power of VMC is that Monte Carlo integration methods enable one to exploit functional forms that are not analytically integrable for many-electron systems, such as those due to Hylleraas which explicitly include electronic correlation terms. In addition, concepts central to both VMC and QMC, such as importance sampling and optimization are introduced here. QMC solution of the Schrodinger equation is the topic of Ch. 3 which focuses on the use of Green's functions to sample the exact ground state energy. Chapter 4 continues the discussion of QMC with the treatment of Fermi statistics. In Ch. 5 we discuss the important issue of choice of trial function. Though QMC methods are not based on trial functions or basis set expansions like many other ab inito approaches, these functions do play an important role in importance sampling and the fixed node method.
These chapters provide the necessary information and methods to evaluate ground state energies of atoms and molecules. The remainer of the book is dedicated to a number of extensions of this which are necessary for most important chemical applications. The determination of excited-states is treated in Ch. 6 while Ch. 7 describes how to evaluate properties with an emphasis on dipole and quadrupole moments, and multistate properties such as the transition dipole moment. Chapter 8 discusses the determination of equilibrium geometries and interaction potentials by means of explicit energy differences and through calculation of energy gradients. Finally, in Ch. 9 we explore new directions that aid the usefulness of QMC to heavy atoms. To facilitate understanding of unfamiliar terms we have included a glossary of symbols and terms (Appendix 1). Several special mathematical topics, as well as a detailed QMC algorithm are also presented in the appendixes.
We have organized the subject matter with the intent that it may be understood at the entering graduate student level while also being of interest to researchers familiar with the topic. For graduate school courses the material in the first four chapters is fundamental. If computational projects are to be undertaken, then Ch. 5 will also be germane. The final chapters may be treated as special topics on the basis of time and interest. For those already familiar with electronic structure theory and Monte Carlo methods in general, chapters 2 through 5 will provide background and breadth, whereas the final chapters will be of more topical interest.
Finally, we wish to thank the many people who have provided input and criticism during the writing of this manuscript: James Anderson, Dario Bressanini, Willard Brown, David Ceperley, William Glauser, Eric Johnson, Ade Odutola, and Maria Soto. We also acknowlege the support of the Department of Energy, Office of Naval Research and the SuperComputer Group of Fujitsu America in the preparation of the manuscript.
**