9.1 Summary

In this thesis, Quantum Monte Carlo (QMC) techniques have been developed and applied to real electronic systems with the aim of obtaining greater accuracy and insight than possible with other current electronic structure methods.

In chapters 3 and 4 the details and subtleties of implementing a viable and accurate QMC computational scheme were discussed. The importance of developing QMC techniques was stressed. Two developments were presented:

Firstly, the poor behaviour of many wavefunction optimisation schemes was investigated, and the increased stability of objective functions based on variance minimisation over energy minimisation was demonstrated numerically. The poor behaviour of the variance sometimes observed during optimisation was shown to be due to the presence of configurations of extreme ``outlying'' energies which result from commonly used trial wavefunctions. Limiting these outlying energies was shown to greatly improve the statistical properties of common objective functions.

Secondly, the compensation of finite size effects in periodic supercell calculations was demonstrated. The finite size effects were shown to result from a combination of two effects, and methods were presented which minimise both. A modified Coulomb interaction was introduced which minimise the finite size effects. Extensive calculations of ground and excited states demonstrate the efficacy of this approach.

Substantial parallel codes implementing the methods and techniques described have been written or rewritten.

Two large-scale applications of QMC methods are presented:

Firstly, the one-body density matrix of the valence electrons in bulk silicon is studied. A QMC method for the calculation of the density matrix from accurate many-body wavefunctions is presented. The matrix obtained by QMC methods is found to greatly resemble density function results. Natural orbitals, which diagonalise the density matrix, are compared with Hartree-Fock (HF) and local density approximation-density functional theory (LDA-DFT) orbitals. Only small differences in nodal quality are found between the different orbitals. A QMC formulation of the extended Koopmans' theorem is developed, making use of the density matrix. The band structure of silicon is obtained from a single variational Monte Carlo calculation, and the scaling of and approximations to the theory are considered.

Secondly, the stability of small carbon clusters is studied by diffusion Monte Carlo. The zero temperature relative energies of many energetically competitive structures are determined and the smallest stable fullerenes identified as the $\mathrm{C}_{2v}$ symmetry $\mathrm {C}_{26}$ and $\mathrm{T}_{d}$ symmetry $\mathrm {C}_{28}$ fullerenes. Comparisons with density functional results show that most current, popular, density functionals are of low predictive power for the systems studied. These results also demonstrate the applicability of diffusion Monte Carlo using the atom-centred trial wavefunctions described in chapter 4.

These successful results demonstrate the viability of a statistical approach to the many-body problem. Using current resources and wavefunctions of similar quality, i.e. without further technical or resource developments, accurate energy differences between systems of several hundred electrons can be determined. The current obtainable energy resolution, $\sim 0.1$ eV for several hundred electrons, is sufficient to determine the formation and relative energies of many solid-state and chemical systems and is smaller than the error of other applicable first-principles methods.

QMC calculations of silicon interstitials [166] have already been performed and calculations of the neutral vacancy in diamond are currently underway.[167] Possible future applications could include studies of interstitials and defects in other group $\mathrm{IV}$ materials and $\mathrm{III}-\mathrm{V}$ compounds. A challenging molecular application would be to accurately compute the energy barriers and heats of formation in a reaction with excited state intermediates.