Contents

• Contents
• List of Figures
• List of Tables
• 1. Introduction
  • 1.1 Quantum mechanics
  • 1.2 Computer simulations
  • 1.3 Dissertation outline
• 2. Many-body Quantum Mechanics
  • 2.1 Principles of quantum mechanics
    • 2.1.1 Wave-functions and operators
    • 2.1.2 Expectation values
    • 2.1.3 Stationary states
  • 2.2 The Born-Oppenheimer approximation
  • 2.3 Identical particles
    • 2.3.1 Symmetries
    • 2.3.2 Spin and statistics
  • 2.4 Variational principles
• 3. Quantum Mechanics of the Electron Gas
  • 3.1 Density-functional theory
    • 3.1.1 The Hohenberg-Kohn theorems
    • 3.1.2 The constrained search formulation
    • 3.1.3 Exchange and correlation
    • 3.1.4 The Kohn-Sham equations
    • 3.1.5 The local density approximation
  • 3.2 Periodic systems
    • 3.2.1 Bloch's theorem
    • 3.2.2 Brillouin zone sampling
  • 3.3 The pseudopotential approximation
    • 3.3.1 Operator approach
    • 3.3.2 Scattering approach
    • 3.3.3 Norm conservation
    • 3.3.4 Kleinman-Bylander representation
• 4. Density-Matrix Formulation
  • 4.1 The density-matrix
  • 4.2 Partial occupation of the Kohn-Sham orbitals
  • 4.3 Density-matrix DFT
  • 4.4 Constraints on the density-matrix
    • 4.4.1 Trace
    • 4.4.2 Idempotency
    • 4.4.3 Penalty functional
    • 4.4.4 Purifying transformation
    • 4.4.5 Idempotency-preserving variations
  • 4.5 Requirements for linear-scaling methods
    • 4.5.1 Separability
    • 4.5.2 Spatial localisation
  • 4.6 Non-orthogonal orbitals
• 5. Localised basis-set
  • 5.1 Introduction
  • 5.2 Origin of the basis functions
  • 5.3 Fourier transform of the basis functions
  • 5.4 Overlap matrix elements
  • 5.5 Kinetic energy matrix elements
  • 5.6 Non-local pseudopotential
    • 5.6.1 Green's function method
    • 5.6.2 Kleinman-Bylander form
  • 5.7 Computational implementation
• 6. Penalty Functionals
  • 6.1 Kohn's method
    • 6.1.1 Variational principle
    • 6.1.2 Implementation problems
  • 6.2 Corrected penalty functional method
    • 6.2.1 Derivation of the correction
    • 6.2.2 Further examples of penalty functionals
    • 6.2.3 Minimisation efficiency
• 7. Computational implementation
  • 7.1 Total energy and Hamiltonian
    • 7.1.1 Kinetic energy
    • 7.1.2 Hartree energy and potential
    • 7.1.3 Exchange-correlation energy and potential
    • 7.1.4 Local pseudopotential
    • 7.1.5 Non-local pseudopotential
  • 7.2 Energy gradients
    • 7.2.1 Density-kernel derivatives
    • 7.2.2 Support function derivatives
  • 7.3 Penalty functional and electron number
  • 7.4 Physical interpretation
  • 7.5 Occupation number preconditioning
  • 7.6 Tensor properties of the gradients
  • 7.7 Practical details
    • 7.7.1 Expansion coefficient derivatives
    • 7.7.2 Normalisation constraint
    • 7.7.3 General outline of the scheme
• 8. Relating linear-scaling and plane-wave methods
  • 8.1 Wave-functions from density-matrices
  • 8.2 Density-matrices from Kohn-Sham orbitals
    • 8.2.1 Projecting plane-wave eigenstates onto support functions
    • 8.2.2 Obtaining auxiliary matrices
    • 8.2.3 Optimising the support functions
  • 8.3 Density-matrix initialisation
• 9. Results and discussion
  • 9.1 Bulk crystalline silicon
    • 9.1.1 Convergence with density-matrix cut-off
    • 9.1.2 Electronic density
    • 9.1.3 Predictions of physical properties
  • 9.2 Scaling
    • 9.2.1 System-size scaling
    • 9.2.2 Scaling with density-matrix cut-off
• 10. Conclusions
  • 10.1 Summary
  • 10.2 Further work
• A. Bessel function identities
• B. Conjugate gradients
• Bibliography