Conclusions

1. The choice of is very important in determining both the energy and variance of energy in a VMC calculation for solid germanium in the diamond structure.
2. The previously used form of ¸, based on a Fourier expansion of 2554 , whose coefficients were determined according to the method of Fahy [9], did indeed produce an improved trial function, , compared with the with no ¸. This can be seen in Figure . where the energy with no function in the trial wavefunction is shown for comparison. However, this method is extremely computationally inefficient because 100-200 are adequate to describe ¸. The values of the parameters being used were also far from optimal, e.g the optimised value for the first star of was 2 orders of magnitude greater than that previously being used.
3. Optimising the Fourier coefficients of by the minimisation of variance scheme produces a with both a lower energy and variance of energy than the unoptimised ¸. The function can be further optimised by including an extra short range function, expressed as a Chebyshev expansion, whose parameters are also optimised by minimisation of variance.
4. Even after fully optimising by the method described above, the DMC energy is still 0.37 eV per atom lower than the VMC energy. Attempts to lower this gap still further by improving the Jastrow function in , by adding an additional short range function with optimised coefficients, proved futile.
5. Despite the fact that the whole optimisation procedure is based on minimising the variance in the energy of a set of configurations, the variance is only reduced by a small fraction in the above optimisations.