In chapter we have developed a technique for producing optimised trial wavefunctions based on the method of minimising the variance of the local energy. This technique has been successfully applied to both atoms and periodic solids. Additional enhancements to the technique enabled wavefunctions to be optimised with respect to different electron-electron interactions and with Hamiltonians containing non-local pseudopotentials.
The variance minimisation algorithm has been extended to work on parallel computer architectures and this allows optimisations to be performed over very large ensembles of up to one million configurations. These large ensembles of configurations enable wavefunctions to be optimised, within their parameter spaces, to an order of magnitude more accuracy than those currently described in the literature. New functional forms of one- and two-body functions have been introduced that are both more suited to optimisation and considerably faster to evaluate with the QMC code.
VMC calculations performed using these optimised trial wavefunctions have considerably lower variational energies. In most cases the energy difference between VMC and DMC has been approximately halved. When calculating cohesive energies, VMC calculations using the optimised trial wavefunctions yield almost identical results to the more sophisticated DMC technique.
In chapter we have identified the source of the Coulomb finite size error present in QMC calculations as the Ewald interaction. We introduce a new electron-electron interaction which dramatically reduces the Coulomb finite size effects present in QMC and HF calculations. This interaction has been successfully tested in both a homogeneous system (HEG) and an inhomogeneous system (silicon). The interaction is equally successful when used in VMC and DMC calculations.
In chapter we combined the technical advances made in the QMC methodology of chapters and to tackle the problem of calculating excitation energies within QMC.
Two separate methods of calculating excitation energies within QMC were introduced, (i) the addition and subtraction of electrons and (ii) the promotion of electrons. In both cases an enhanced version of the new electron-electron interaction, introduced in chapter , was used to reduce the effect of using a finite sized supercell to calculate excitation energies.
The results indicate that the VMC technique is not sufficiently accurate to resolve the 1/N changes in the energy when an electron is added(removed) from the system. However, within DMC the results are extremely encouraging. Using DMC we have calculated excitation energies in diamond-structure silicon using both of the above techniques. In both cases the results show a significant improvement over results obtained using the LDA.
As QMC is a non-perturbative many-body technique (in contrast to, for instance, the GW approximation) it is hoped that these preliminary calculations on the weakly correlated silicon system can be extended to more strongly correlated systems. A challenging group of systems to study would be the Mott insulating 3d monoxides such as MnO, FeO, CoO, and NiO.