Introduction to Quantum Monte Carlo and Density Functional Theory : For Atoms, Molecules, Clusters, and Solids

[BOOK DESCRIPTION]

Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems to solve many body systems in order to investigate the electronic structure of many-body systems. This book presents a numeric approach to determine the electronic structure of atoms, molecules and solids. Because of the simplicity of its theoretical concept, the authors focus on the variational Quantum-Monte-Carlo (VQMC) scheme. The reader is enabled to proceed from simple examples as the hydrogen atom to advanced ones as the Lithium solid. Several intermediate steps cover the Hydrogen molecule, how to deal with a two electron systems, going over to three electrons, and expanding to an arbitrary number of electrons to finally treat the three-dimensional periodic array of Lithium atoms in a crystal. The exmples in the field of VQMC are followed by the subject of diffusion Monte-Calro (DMC) which covers a common example, the harmonic ascillator. The book is unique as it provides both theory and numerical programs.It includes rather practical advices to do what is usually described in a theoretical textbook, and presents in more detail the physical understanding of what the manual of a code usually promises as result. Detailed derivations can be found at the appendix, and the references are chosen with respect to their use for specifying details or getting an deeper understanding . The authors address an introductory readership in condensed matter physics, computational phyiscs, chemistry and materials science. As the text is intended to open the reader's view towards various possibilities of choices of computing schemes connected with the method of QMC, it might also become a welcome literature for researchers who would like to know more about QMC methods. The book is accompanied with a collection of programs, routines, and data.

[TABLE OF CONTENTS]

Preface                                            ix
    1 A First Monte Carlo Example                  1  (22)
      1.1 Energy of Interacting Classical Gas      1  (22)
        1.1.1 Classical Many-Particle              2  (16)
        Statistics and Some Thermodynamics
        1.1.2 How to Sample the Particle           18 (5)
        Density?
    2 Variational Quantum Monte Carlo for a        23 (16)
    One-Electron System
    3 Two Electrons with Two Adiabatically         39 (22)
    Decoupled Nuclei: Hydrogen Molecule
      3.1 Theoretical Description of the System    39 (3)
      3.2 Numerical Results of Moderate Accuracy   42 (4)
      3.3 Controlling the Accuracy                 46 (7)
      3.4 Details of Numerical Program             53 (8)
    4 Three Electrons: Lithium Atom                61 (60)
      4.1 More Electrons, More Problems:           63 (8)
      Particle and Spin Symmetry
        4.1.1 Antisymmetry and Decomposition of    63 (2)
        the Many-Body Wave Function
        4.1.2 Three-Electron Wave Function         65 (2)
        4.1.3 General Wave Function                67 (3)
        4.1.4 Relaxing Symmetry of Total Spin      70 (1)
      4.2 Electron Orbitals for the Slater         71 (5)
      Determinant
      4.3 Slater Determinants: Evaluation and      76 (6)
      Update
      4.4 Some Important Observables in Atoms?     82 (9)
        4.4.1 The Module "observables"             87 (4)
      4.5 Statistical Accuracy                     91 (2)
      4.6 Ground State Results                     93 (22)
        4.6.1 Results for Lithium Atom             93 (10)
        4.6.2 Code of Main Program, Modules of     103(12)
        Variables, of Statistic, of Jastrow
        Factor, and of Output
      4.7 Optimization?                            115(6)
    5 Many-Electron Confined Systems               121(26)
      5.1 Model Systems with Few Electrons         121(1)
      5.2 Orthorhombic Quantum Dot                 122(14)
        5.2.1 Confined Single-Particle Wave        122(1)
        Functions
        5.2.2 Details of Program                   123(2)
        5.2.3 Energy and Radial Density            125(6)
        5.2.4 Pair-Correlation Function            131(3)
        5.2.5 Program of the Pair-Correlation      134(2)
        Function
      5.3 Spherical Quantum Dot                    136(11)
        5.3.1 Fundamentals of DFT                  137(1)
        5.3.2 DFT Calculation of the Jellium       138(2)
        Cluster: Methodology
        5.3.3 QMC Calculation of the Jellium       140(1)
        Cluster: Methodology
        5.3.4 QMC Code for the Calculation of      141(1)
        Jellium Clusters
        5.3.5 Comparison between DFT and QMC       142(5)
        Calculations of Jellium Clusters
    6 Many-Electron Atomic Aggregates: Lithium     147(34)
    Cluster
      6.1 Clusters and Nanophysics                 147(3)
      6.2 Cubic BCC Arrangement of Lithium Atoms   150(13)
        6.2.1 Structure of the Main Program        150(1)
        6.2.2 Single-Electron Wave Functions       150(3)
        and Structure of the Determinant
        6.2.3 Geometric Setting of the Cluster     153(3)
        6.2.4 Changes in the Program               156(7)
      6.3 The Cluster: Intermediate between        163(18)
      Atom and Solid
        6.3.1 1 x 1 x 1 Cluster: Li2               164(3)
        6.3.2 2 x 2 x 2 Cluster                    167(5)
        6.3.3 3 x 3 x 3 Cluster                    172(2)
        6.3.4 4 x 4 x 4 Cluster                    174(4)
        6.3.5 Cluster Size                         178(3)
    7 Infinite Number of Electrons: Lithium        181(42)
    Solid
      7.1 Infinite Lattice                         183(25)
        7.1.1 The Lattices                         183(3)
        7.1.2 Structure of the Electrostatic       186(5)
        Potential
        7.1.3 Ewald Summation and Tabulation       191(13)
        7.1.4 Finite-Size Effects                  204(4)
      7.2 Wave Function                            208(4)
        7.2.1 Linear Combination of Atomic         208(2)
        Orbitals
        7.2.2 Plane Waves                          210(2)
      7.3 Jastrow Factor                           212(4)
        7.3.1 Standard Choice                      213(2)
        7.3.2 Principal Ideas and Extensions       215(1)
      7.4 Results for the 3 x 3 x 3 and 4 x 4 x    216(7)
      4 Superlattice Solid
    8 Diffusion Quantum Monte Carlo (DQMC)         223(14)
      8.1 Towards a First DQMC Program             224(11)
        8.1.1 Relating Schrodinger Equation to     224(4)
        Diffusion
        8.1.2 Generate Gaussian Random Numbers     228(1)
        8.1.3 Application                          229(1)
        8.1.3.1 Harmonic Oscillator                229(6)
      8.2 Conclusion                               235(2)
    9 Epilogue                                     237(2)
  Appendix                                         239(30)
      A.1 The Interacting Classical Gas: High      239(2)
      Temperature Asymptotics
      A.2 Pseudorandom Number Generators           241(6)
      A.3 Some Generalization of the Jastrow       247(2)
      Factor
      A.4 Series Expansion                         249(8)
      A.5 Wave Function Symmetry and Spin          257(2)
        A.5.1 Four Electrons                       257(2)
      A.6 Infinite Lattice: Ewald Summation        259(4)
      A.7 Lattice Sums: Calculation                263(6)
References                                         269(4)
Index                                              273