Convergent results were obtained using 8 parameters for both the and u functions. The energy obtained with the optimised wavefunction was -103.22 0.01 eV and the standard deviation was 0.52 eV. The functions for up- and down-spin electrons were found to be very similar. This result is perhaps surprising because there is a single down-spin electron in an s-orbital, while there are three up-spin electrons, one in an s-orbital and two in p-orbitals. This point was investigated further by performing calculations using a single function, which gave an energy of -103.20 0.01 eV, and a standard deviation of 0.54 eV, which are almost identical to the values obtained using separate functions for up- and down-spin electrons. The resulting and u functions were almost unchanged.
Figure shows LDA and QMC charge densities for the pseudo-atom, calculated by Alan James from Imperial College. In figure a the VMC charge density from a wavefunction consisting of a determinant of LDA orbitals and a u function, but no function, is compared with the LDA density. This shows that the inclusion of the correlation factor, u, smears out the charge density considerably. In Fig. b we plot the charge density from an optimised wavefunction, containing both u and . This plot shows that the variance minimisation procedure results in a wavefunction whose charge density is very close to the LDA form. In figure c we plot the DMC charge density calculated with the optimised wavefunction as guiding function. The DMC charge density is very close to both the VMC and LDA charge densities, which shows that the true charge density is close to the LDA form. It seems that the physical idea behind the original Fahy prescription for , i.e. returning the charge density to the LDA form, is extremely good. However, the number of configurations required to fit the variational parameters in a well parameterised wavefunction via variance minimisation is much less than the number required to obtain an accurate charge density. Therefore it appears that for a given computational effort it is more efficient to generate via variance minimisation than to construct the charge density and generate from it. Moreover, the method of variance minimisation is more general in the sense that it does not rely on the separate determination of an accurate charge density from, for example, the LDA.
Figure: Comparison of QMC densities (coloured lines) and the LDA density (black lines) of the germanium pseudo-atom. Fig.a shows the VMC density (red line) calculated using a wavefunction consisting of a determinant of LDA orbitals and a u function, but with no function. Fig.b shows the VMC (blue line) density calculated using a wavefunction containing optimised u and functions. Fig.c shows the DMC density (green line) calculated using the same optimised wavefunction as the guiding wavefunction.