In the solid QMC calculations described in chapter , is expressed as a Fourier expansion in reciprocal lattice vectors,

These Fourier coefficients provide an obvious set of variational parameters to optimise the wavefunction with respect to. A total of 2554 -vectors was originally chosen [33] for the number of 's used in the expansion of in the germanium solid. This is far too large a number of parameters for the optimisation procedure and so the first simplification is to reduce the number of free parameters by forcing all the coefficients of a star of 's to have the same magnitude. A star of 's is a group of 's related by the point group symmetry of the lattice. By grouping the 's in stars we can impose the full space group symmetry of the crystal on the function. This grouping allows Eq.() to be rewritten as

For a crystal with the origin of coordinates at a centre of inversion symmetry, the phase factors in Eq.() are simply . Recently, Fahy et al.[67] have studied the Boron Nitride crystal within VMC. In this case the inversion symmetry no longer holds, but one can still choose a set of `generalised stars' with which the function can be represented by an independent, real, variational coefficients for each `generalised star'. The grouping of vectors into stars reduces the number of free parameters to around 130. This is still a very large space in which to perform an accurate optimisation, so it was decided to investigate the effect of reducing the number of stars in the expansion of , as it was suspected that stars for larger vectors contained only noise. It was found that for solid germanium the function could be described using 5-10 stars of vectors.

Tue Nov 19 17:11:34 GMT 1996