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Preface
Contents
Techniques and Applications of Quantum Monte Carlo
A dissertation submitted for the degree of
Doctor of Philosophy
at the University of Cambridge
Paul Richard Charles Kent
Robinson College, Cambridge
August 1999
Preface
Acknowledgements
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
2.3.0.1 Basis set expansion
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
4.3.2.1 Mitás' approach
4.3.2.2 Williamson's approach
4.3.2.3 Atom centred approach
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
4.5.1.1 Evaluation of the non-local energy
4.5.1.2 The locality approximation
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
7.5.4.1 Full EKT
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
©
Paul Kent
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Paul Richard Charles Kent
