QUANTUM MONTE CARLO CALCULATIONS OF ELECTRONIC EXCITATIONS

By
Andrew James Williamson
Robinson College, Cambridge

Theory of Condensed Matter Group
Cavendish Laboratory
Madingley Road
Cambridge
CB3 0HE

A dissertation submitted for the
degree of Doctor of Philosophy

September 1996

• Contents
• List of Figures
• List of Tables
• Introduction
  • The Many-Electron Problem
    • One-Electron Methods
      • The Hartree Approximation
      • The Hartree-Fock Approximation
    • Configuration Interaction
    • Density Functional Methods
      • Kohn-Sham Equations
  • Quantum Monte Carlo Calculations
  • Layout of Thesis
• Quantum Monte Carlo Methods
  • The Variational Principle
  • Monte Carlo Methods
  • Importance Sampling
    • The Metropolis Algorithm
  • Variational Quantum Monte Carlo
    • Trial Wavefunctions
    • Evaluating the Local Energy
      • Evaluating the Kinetic Energy
      • Electrostatic Energies
    • Accumulating Averages
    • Performing VMC calculations of Parallel Computers
  • Diffusion Quantum Monte Carlo
    • The Method
    • A DMC Algorithm
    • The Fixed-Node Approximation
    • DMC With Non-Local Pseudopotentials
    • Performing DMC Calculations on Parallel Computers
• Quantum Monte Carlo Calculations on Solids
  • Supercell Calculations
  • Wavefunctions for Solid Calculations
  • Germanium and Silicon - The Diamond Structure
    • Choice of Slater Determinant
  • Coulomb Interactions in Supercell Calculations
    • Isolated Simulation Cell
    • Periodic Boundary Conditions
    • Ewald Summation
    • Electron-Ion Coulomb Interactions
• Optimising Trial Wavefunctions
  • Motivation
    • VMC Calculations
    • DMC Calculations
  • Optimisation Method
    • Why Minimise the Variance of the Energy?
    • Previous Applications of Variance Minimisation
    • The Variance Minimisation Method
    • Control of the Reweighting Factors
    • Choice of Average Local Energy,
    • Generating Configurations
  • Optimising the function
    • Which Part of to Optimise?
    • Choice of Parameters
    • Adding a new Function to Chi
    • Results of Optimising the Function in Germanium Solid
  • Optimising the u Function
    • Choice of Functional Form for new term in the Jastrow Factor
    • Implementation of the new Jastrow Function
    • Results of Optimising the new Jastrow Factor
    • Removing the Prefactor in
  • A New u function
    • Form of the New u function
    • Tests on Jellium
  • Applying the New u Function to Solids
  • Optimising Wavefunctions for Atoms
    • Choice of Atomic Wavefunction
    • Results
    • Cohesive Energies
  • Variance Minimisation on Parallel Computers
  • Variance Minimisation with Non-Local Pseudopotentials
    • Keeping the Non-Local part Fixed during Optimisation
    • Evaluating the non-local Integral during Optimisation
  • Limits of Variance minimisation
    • VMC Calculations
    • DMC Calculations
• Finite Size Effects
  • Introduction
  • Motivation
  • Previous Methods of Removing Finite Size Effects
  • Analysis of e-e energy
    • Comparison of Hartree-Fock and LDA Results
    • Electron-Electron Interaction in More Detail
  • New Proposed Energy Expression
    • Short range of Exchange-Correlation Hole
    • Use Interaction
    • Choices for the f function
  • Tests on the Homogeneous Electron Gas
    • VMC Results
  • Finite Size Effects in Inhomogeneous Solids
    • Electron-Ion and Ion-Ion Interactions
  • Results for Silicon
    • VMC Results
    • HF Results
    • DMC Results
  • Variance Minimisation with the New Interaction
  • Conclusions
• QMC Calculations of Excitation Energies
  • Previous Work on Excitation Energies
    • QMC Calculations
    • Hartree-Fock Calculations
    • Density Functional Calculations
    • Experimental Determination of Excited States
  • QMC Methods for Calculating Excitation Energies
  • Addition and Subtraction of Electrons
    • Trial Wavefunction
    • Electron-Electron Interaction
    • New Electron-Electron Interaction
    • Hartree-Fock Analysis of New Interactions
      • New Interaction, Eq.()
      • Enhanced Version from Eq.()
    • Addition and Subtraction of Electrons in VMC
      • VMC Results
    • Addition and Subtraction of Electrons in DMC
      • DMC Results
  • Promoting Electrons
    • Trial Wavefunctions for Promoted States
      • Spin Contamination
      • One and Two-body Functions for Promotion Calculations
    • Electron-Electron Interaction for Promoted States
    • DMC Results
      • Calculating Band Widths
      • Excitonic Effects
  • Summary and Comparison of the Methods
  • Alternative Methods for Calculating Excited States within QMC
    • Spectrum Folding
    • DMC Decay Curves
• Conclusions
• Updating the Slater Determinant
• References
• About this document ...