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, 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  and  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.
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  direction (red line) and the  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 , (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 formula to update the matrix of cofactors.