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.