The VMC results for the energy gaps at all possible -points in an n=2 simulation cell are summarised in Table . For each of the -points then=2 LDA results and n=2 Hartree-Fock results have been included for comparison. The Hartree-Fock results were obtained using the fixed orbitals from the corresponding LDA calculation rather than performing a fully relaxed Hartree-Fock calculation. The LDA bandstructure exhibits virtually no finite size effect (<0.2 eV at all points across the band), whereas the HF bandstructure exhibits a large finite size effect (>3.0 eV at all point across the band).
|Band||VMC (eV)||HF (eV)||LDA (eV)||Experiment (eV)|
|-1.6 0.4||-1.36||-1.23||-1.2 0.2|
The results for VMC, LDA, HF and experiment have all been aligned by setting the value of in each case. The values at each of the points in the bandstructure can then be obtained relative to this point, for example
where refers to the ground state energy of a system with an extra electron added into an orbital that describes the bottom of the conduction band at the -point and refers to the ground state energy of a system with an electron removed from an orbital that describes the top of the valence band at the X-point.
It can be clearly seen that the LDA gives the valence band energies (relative to ) at each of the -points more accurately than the conduction band energies. The LDA significantly underestimates all the band gaps.
The quality of the VMC results is mixed. At the - and L-points, the results are broadly in agreement with experiment. At the X-point the VMC overestimates the size of the gap. The errors in the VMC calculations are due to the quality of the trial wavefunction used and finite size effects. As mentioned in section , the function for the excited states has a higher symmetry than the excited state charge density and the u function takes no account of any changes in the shape of the exchange-correlation hole due to the addition/removal of electrons. All the VMC trial wavefunctions contain a Slater determinant constructed using orbitals from a groundstate LDA calculation. The effect of allowing these orbitals to relax were suspected to be too small to resolve in VMC, but has been examined in DMC in the following section.