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. . (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.