Wavefunctions were optimised for the HEG at a range of densities from
to 
. Excellent results were obtained at all
densities, but for brevity only the results for 
=1 are presented
here. A wavefunction of Slater-Jastrow type (cf. Eq.(
))
was used, where the determinants, 
, were constructed from the lowest energy plane waves at
zero wavevector within the simulation cell Brillouin zone. The
one-body 
function was set to zero and the u function of
Eqs.(-) was used. Separate u functions for
parallel and antiparallel spins were used for fcc simulation
cells containing N=30, 54, 178 and 338 electrons. In each case the
numbers of up- and down-spin electrons were equal. Typically 10,000
electron configurations were sampled from a VMC run of sufficient
length to ensure that the chosen configurations are statistically
independent as described in section .
The variance minimisation procedure is stable for small N, but
gradually becomes unstable as N increases. The technique described
in section , was used to control this instability by
fixing the reweighting factors to unity and regenerating
configurations several times. This proved to be completely successful
for all the system sizes studied. All calculations used 9 Chebyshev
polynomials to represent f(r), which tests show to give essentially
complete convergence for the systems studied. The minimisation problem
then has 20 parameters.
Table shows the energy and standard deviation, 
, of the energy as a function of system size[2],
comparing our u function Eqs.(-) with that
of Eq.(), which includes a sum over simulation cells, and
with DMC results. For the DMC calculations we used a time step of 0.01
au and an average population of 640 configurations. After
equilibration the averages were collected over 5000 moves of all the
electrons. The results obtained using our new u function are of
similar quality to those obtained with the u function of
Eqs.(-), but the new u function is much
faster to evaluate. In figure we show the optimised
spin-parallel u function for N=338 , together with the u
function of Eqs.(-) which is plotted in
the [100] and [110] directions (for all other directions the u
function lies between the values in these directions). In
figure the derivatives of the functions are shown. These
figures show that the two functions are similar, but the optimised u
function exhibits a slightly smaller derivative at intermediate
distances. The optimised spin-antiparallel u function shows similar
behaviour.
| N |  | | | |  |
| Eqs.(-) | Eqs.(-) | Eqs.(-) |
Eqs.(-) | ||
| 30 | 0.4657 | 0.2 | 0.46979 | 0.22 | |
| 54 | 0.6085 | 0.24 | 0.6110 | 0.22 | 0.6069 |
| 178 | 0.6161 | 0.17 | 0.61679 | 0.17 | 0.6141 |
| 338 | 0.5772 | 0.14 | 0.57707 | 0.17 |
Figure: Comparison of spin-parallel u functions for the HEG at
. The optimised function (black line) is shown along with the
Ewald summed Yukawa form along the [100] direction (red line)
and the [110] direction (blue line). Fig. a shows
the u functions themselves while Fig. b shows the first
derivatives.
The reduction in computing cost from using the new u function is
very significant. It is particularly effective when combined with our
recently developed technique for evaluating the expectation value of
Coulomb interactions in homogeneous systems [3], (see
chapter ). This combination of techniques entirely
eliminates the need for time-consuming sums over simulation cells, and
the resulting algorithm is extremely fast, with the most costly
remaining operation being the calculation of determinants which are
evaluated for each electron move using the standard
Sherman-Morrison[18] formula to update the matrix
of cofactors.