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.
