7.6 Conclusions

The one-body density matrix for the valence electrons of bulk silicon was calculated using QMC techniques.

In real space, the VMC and LDA density matrices are very similar, the greatest differences being due to the differing charge densities in each case. The natural orbitals, obtained by diagonalising the density matrix, very closely resemble the LDA orbitals. The occupation numbers of the natural orbitals differ significantly from the non-interacting values, reducing linearly with increasing energy for the LDA occupied bands. The occupations are about 3% lower than the non-interacting value near the top of the valence band, and above the Fermi level, the occupation numbers fall slowly to zero.

A DMC calculation for the ground state energy of silicon using a trial wave function containing a determinant of natural orbitals gives an energy which is almost identical to that obtained using a determinant of LDA orbitals. This shows that the quality of the nodal surfaces is almost identical in each case. A DMC calculation comparing the quality of the nodal surfaces obtained with LDA and HF orbitals, found that for the system studied the nodal surface of a determinant of LDA orbitals is marginally better.

The relationship between Compton profiles, momentum densities and density matrices was discussed. The zero momentum component of the momentum density was computed from the density matrix. Its value was found to be in good agreement with other calculations, supporting the conclusion that the impulse approximation used in the analysis of Compton profiles is the most likely source of the current discrepancy between calculations and experiment.

A Monte Carlo formulation of the Extended Koopmans' theorem was applied to bulk silicon, and the resulting band energies are in reasonable agreement with the available experimental data. The success of the VMC-EKT relies on a cancellation of errors between the ground and excited state energies. The wave functions for the excited states contain the variational freedom of the orbitals $u_v$ and $u_c$. This variational freedom in the orbitals reduces the energy of the $N$-1 and $N$+1 electron states and improves the agreement with experiment. In the diagonal approximation to the EKT (DEKT), the $u_v$ and $u_c$ orbitals are fixed and there is no variational freedom in the excited state wave functions. We have found that the DEKT works quite well for silicon using LDA orbitals for $u_v$ and $u_c$. The diagonal approximation is exact within HF theory, so that it is expected to be a good approximation for weakly correlated systems.

Greater accuracy could be obtained with more accurate trial functions, or using DMC in the calculation of $V^v_{ij}$, $V^c_{ij}$, and $\rho_{ij}$. In comparison with direct methods of calculating excitation energies,[42,101,103] the EKT has the advantage that only a single calculation involving the ground state is required to obtain many excitation energies. Direct methods methods require that accurate energies are computed for each state of interest and excitation energies are then computed via energy differences, which is very inefficient.

The EKT involves assumptions about the nature of the excited state wave functions, but nevertheless is a practical method for calculating excitation energies including correlation effects for weakly correlated systems. The diagonal approximation to the EKT has a very advantageous scaling with system size compared with direct QMC calculations. This scaling thereby allows the study of excitation energies in larger systems with a greater efficiency than is possible with direct techniques.