Contents

• Contents
• List of Figures
• List of Tables
• 1. Introduction
  • 1.1 Quantum Mechanics
  • 1.2 Electronic structure calculations
  • 1.3 Outline of thesis
• 2. Electronic structure methods
  • 2.1 Introduction
  • 2.2 The Hamiltonian
    • 2.2.1 The Born-Oppenheimer approximation
  • 2.3 Hartree Fock theory
  • 2.4 Post Hartree-Fock techniques
    • 2.4.1 Configuration interaction
    • 2.4.2 Other methods
    • 2.4.3 Limitations
  • 2.5 Density functional theory
    • 2.5.1 The Hohenberg-Kohn theorem
    • 2.5.2 The Kohn-Sham equations
    • 2.5.3 The local density approximation
    • 2.5.4 Limitations
  • 2.6 Summary
• 3. Quantum Monte Carlo methods
  • 3.1 Introduction
  • 3.2 Monte Carlo methods
    • 3.2.1 Monte Carlo integration
    • 3.2.2 Importance sampling
    • 3.2.3 The Metropolis algorithm
  • 3.3 Variational Monte Carlo
    • 3.3.1 The variational principle
    • 3.3.2 Monte Carlo integration
    • 3.3.3 The local energy
    • 3.3.4 Trial wavefunctions
    • 3.3.5 The VMC algorithm
  • 3.4 Diffusion Monte Carlo
    • 3.4.1 Outline of method
    • 3.4.2 Importance sampling
    • 3.4.3 Transformation to integral form
    • 3.4.4 The fixed-node approximation
    • 3.4.5 Improving the fixed-node energy
    • 3.4.6 The DMC algorithm
  • 3.5 Summary
• 4. Implementation
  • 4.1 Introduction
  • 4.2 History and background
  • 4.3 Wavefunctions
    • 4.3.1 Form of trial wavefunction
    • 4.3.2 Form of Jastrow factor
  • 4.4 Algorithms and practicalities
    • 4.4.1 Calculation and update of the Slater determinants
    • 4.4.2 Evaluation of the local energy
    • 4.4.3 Calculation of the kinetic energy
    • 4.4.4 Calculation of the potential energy
    • 4.4.5 Parallelisation of VMC
    • 4.4.6 Parallelisation of DMC
  • 4.5 Pseudopotentials
    • 4.5.1 Non-local pseudopotentials
    • 4.5.2 Core polarisation potentials
  • 4.6 Supercell calculations
    • 4.6.1 Long-range interactions
  • 4.7 Wavefunction optimisation
    • 4.7.1 Linearity of the Jastrow factor
    • 4.7.2 The non-local energy during wavefunction optimisation
  • 4.8 Statistics
    • 4.8.1 VMC applications
    • 4.8.2 DMC applications
    • 4.8.3 Systematic errors
  • 4.9 Summary
• 5. Wavefunction optimisation
  • 5.1 Introduction
  • 5.2 The importance of wavefunction optimisation
  • 5.3 Objective functions
  • 5.4 Numerical instabilities
  • 5.5 Analysis of objective functions
  • 5.6 Further effects of finite sampling
  • 5.7 Tests of minimisation procedures
    • 5.7.1 The model system
    • 5.7.2 Generation of test ensemble
    • 5.7.3 Results
  • 5.8 Limitation of outlying energies
  • 5.9 Other objective functions
  • 5.10 Conclusions
• 6. Finite-size errors in supercell calculations
  • 6.1 Introduction
  • 6.2 Finite size errors in many-body calculations
  • 6.3 The Hamiltonian within periodic boundary conditions
  • 6.4 Finite size correction and extrapolation formulae
    • 6.4.1 Previous work
  • 6.5 General approach
  • 6.6 Independent particle finite size effects
  • 6.7 Coulomb finite size effects
    • 6.7.1 Systems of electrons and nuclei
  • 6.8 Tests of the MPC interaction
    • 6.8.1 Application within HF theory
    • 6.8.2 Application within VMC
    • 6.8.3 Application within DMC
  • 6.9 Excitation energies
    • 6.9.1 HF theory of excitation energies
    • 6.9.2 QMC theory of excitation energies
    • 6.9.3 Modified interaction for excitation energies
  • 6.10 Conclusions
• 7. The one-body density matrix and excitation energies of silicon
  • 7.1 Introduction
    • 7.1.1 Orbital optimisation
    • 7.1.2 Excited states
  • 7.2 Silicon
    • 7.2.1 The supercell
    • 7.2.2 The trial wavefunction
    • 7.2.3 VMC and DMC calculations
  • 7.3 The density matrix and natural orbitals
    • 7.3.1 Symmetry and efficiency considerations
    • 7.3.2 Density matrix results
    • 7.3.3 Natural orbital results
    • 7.3.4 Relation to Compton profiles
  • 7.4 Tests of orbitals
    • 7.4.1 LDA and natural orbitals compared
    • 7.4.2 LDA and HF orbitals compared
  • 7.5 The Extended Koopmans' theorem
    • 7.5.1 Valence band energies
    • 7.5.2 Conduction band energies
    • 7.5.3 VMC formulation of the EKT
    • 7.5.4 Results
    • 7.5.5 Approximations to the EKT
  • 7.6 Conclusions
• 8. The energetic stability of small carbon clusters
  • 8.1 Introduction
  • 8.2 Background
  • 8.3 Previous work
  • 8.4 Methodology
  • 8.5 Candidate geometries
  • 8.6 DFT calculations
  • 8.7 DMC calculations
  • 8.8 Results and analysis
    • 8.8.1
    • 8.8.2
    • 8.8.3
    • 8.8.4
    • 8.8.5 Ring and fullerene binding energies
  • 8.9 Discussion
  • 8.10 Conclusions
• 9. Conclusions
  • 9.1 Summary
  • 9.2 Future developments
• Bibliography