Having studied the HEG, the newly developed u function was applied
to a crystalline solid. To enable direct comparison with the previous
results, germanium in the diamond structure was used as a test
material. The same fcc simulation cell of diamond structure
germanium containing 16 atoms was studied. The same single-particle
orbitals were used to construct the Slater determinant. The 
function was chosen to have the full symmetry of the diamond
structure. Again the function was expanded in a Fourier
series, grouping the 
vectors into stars as in Eq.(
).
For the u function, the functional form of
Eqs.(-) which was developed for the HEG was
chosen. The u and functions were optimised simultaneously
because they are strongly coupled. Typically 6 non-zero coefficients
in Eq.() for the function and 8 parameters
for both the parallel- and antiparallel-spin u functions in
Eq.() were used, giving a total of 22 parameters in the
minimisation problem. Variance minimisations were carried out using
10,000-100,000 independent N-electron configurations, which were
regenerated several times. The final energy of -107.69 
0.01 eV
per atom is 0.08 eV lower than the result obtained using the (Ewald
summed) Yukawa potential of Eq.() and the variance
minimisation procedure for , and 0.20 eV lower than the result
obtained in our previous work using the Yukawa potential and Fahy's
original prescription for [48, 26]. The energy of
-107.69 eV per atom is only 0.34 eV per atom higher than the DMC
result for this system of -108.03 0.07 eV per atom quoted in
Table I of Ref. [50]. (As discussed in
Refs. [33, 50], we estimate that about 0.12 eV
of this energy difference is due to the basis set incompleteness error
in the single-particle orbitals, which affects the VMC much more than
the DMC result, and which could be eliminated by the use of a larger
basis set or a smoother pseudopotential.)
The optimised spin-parallel and spin-antiparallel u functions for
germanium are similar to the Yukawa form in all directions. However,
they have a smaller derivative at intermediate distances, exactly as
observed in the HEG (see figure ). The optimised
function differs significantly from the original Fahy form, with some
parameters changing by an order of magnitude. Altering the number of
parameters in the optimisation scheme revealed that 6 non-zero
coefficients was again sufficient to converge the function.