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Preface
Contents
Linear-scaling methods in
ab initio
quantum-mechanical calculations
A dissertation submitted for the degree of
Doctor of Philosophy
at the University of Cambridge
Peter David Haynes
Christ's College, Cambridge
July 1998
Gzipped PostScript
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version of the complete dissertation (1.5 Mb)
Preface
Psalm 8
Acknowledgements
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
7.2.1.1 Kinetic and pseudopotential energies
7.2.1.2 Hartree and exchange-correlation energies
7.2.1.3 Total energy
7.2.2 Support function derivatives
7.2.2.1 Kinetic and pseudopotential energies
7.2.2.2 Hartree and exchange-correlation energies
7.2.2.3 Total energy
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
7.7.2.1 Density-kernel variation
7.7.2.2 Support function variation
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
Peter Haynes
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