List of Figures

• 2.1. Indistinguishable particles in quantum mechanics.
• 3.1. Supercell approximation.
• 3.2. Pseudopotential approximation.
• 4.1. Purifying transformation.
• 5.1. Test of analytic kinetic energy matrix elements.
• 5.2. Green's function method for non-local pseudopotential.
• 5.3. Kleinman-Bylander method for non-local pseudopotential.
• 6.1. Kohn's penalty functional.
• 6.2. Schematic illustration of Kohn's variational principle.
• 6.3. Failure of quadratic interpolation for Kohn's penalty functional.
• 6.4. Convergence properties of Kohn's penalty functional.
• 6.5. One possible choice of analytic penalty functional.
• 6.6. Schematic illustration of the analytic penalty functional.
• 6.7. Variation of the occupation number errors with $\alpha$.
• 6.8. Total energy, total functional and corrected energy versus $\alpha$.
• 6.9. Two further examples of analytic penalty functionals.
• 7.1. Rate of convergence for different numbers of inner cycles.
• 7.2. Performance of the conjugate gradients algorithm.
• 9.1. Convergence of total energy with respect to support region radius.
• 9.2. Convergence of total energy with respect to density-kernel cut-off.
• 9.3. Electronic density in the (110) plane.
• 9.4. Diamond structure of silicon, highlighting a {110} plane.
• 9.5. CASTEP electronic density.
• 9.6. Electronic density difference.
• 9.7. Energy-volume curve for silicon.
• 9.8. Variation of computational effort with system-size.
• B.1. Steepest descents method.
• B.2. Conjugate gradients method.