DMC Results

As DMC calculations are considerably more computationally expensive than VMC calculations, only a selected set of the VMC results were repeated in DMC. It was decided to use the addition and removal of electrons at the -point as a test for comparing the DMC with VMC and experiment. Two separate modifications to the DMC algorithm were experimented with,

1. DMC calculations were performed using the original Ewald expression for the electron-electron interaction and the new version of the interaction described in section .
2. DMC calculations were performed using single-particle orbitals in the Slater determinant from LDA calculations where the orbitals are kept fixed when an electron is added or removed and where the orbitals are allowed to relax when an electron is added or removed. This relaxation will change the nodal structure of the guiding wavefunction.

The first set of DMC calculations were performed using the new electron-electron interaction described in section  and a Slater determinant containing fixed LDA orbitals. The value of the gap at the -point obtained from adding and removing an electron under these conditions was 3.95  0.4 eV. This is almost within error bars of the experimental gap of 3.4 eV.

The same set of three calculations were repeated, but this time allowing the LDA orbitals used in the Slater determinant to relax. The value of the gap at the -point was reduced to 3.59  0.4 eV.

Finally, the calculations using the relaxed LDA orbitals were repeated using the original Ewald expression for the electron-electron interaction. The value of the gap at the -point was reduced further to 3.34  0.4 eV.

From these results it appears that

1. The DMC performs at least as well, if not better than the VMC. This is to be expected because the DMC calculations involving electrons do not suffer from the limitations in the one- and two-body terms in the trial wavefunction.
2. Relaxing the LDA orbitals used to construct the Slater determinant appears to have a small beneficial effect on the DMC results. This could be due to the relaxed orbitals reproducing the nodal surface of the true many-body wavefunction more accurately than fixed orbitals.
3. There appears to be very little difference in DMC results for gaps obtained using the new interaction described in section  and the original Ewald interaction. As the new interaction is designed to remove the long range finite size errors this suggests that the finite size errors present in the gap are a short range effect not a long range effect as suggested in Ref.[79]. The considerable finite size errors present in the individual N, N+1 and N-1 calculations performed with the Ewald interaction almost entirely cancel from the gap.

   The relative insensitivity of the gap energy to the choice of electron-electron interaction is confirmed by the HF results. These show a small but consistent improvement when using the new interaction in preference to the Ewald interaction. For example in an n=2 system, the gap at the point is improved by 15% when using the new interaction.