The minimisation of variance scheme (
) was applied to the function in solid germanium in the diamond structure.
The function was expanded in 10 stars
of 
as in () and the 10 coefficients that multiply
the stars were the parameters used in the optimisation, ().
The optimisation was performed with 1,000 and 10,000
configurations of electron positions.
The results for optimisation using 10,000 configurations
are illustrated in Figure . Results for optimisation using
1000 configurations are similar but the optimisation procedure was less
stable.
Figure: Reduction in energy and variance after optimisation.
The energy values illustrated in Figure were
obtained by taking the optimised values for the coefficients in the
expansion of and performing a VMC calculation for 50,000 moves. In
fact these energies show very good agreement with the value of 
in
() at the end of the optimisation. This implies that the 10,000
configurations adequately span the space in which the optimisation is
being performed. This is not surprising when recalling that 10,000
independent configurations is equivalent to a 30,000 move VMC
calculation.
The variance in the energy was reduced from 2.58 to 2.56 by the
optimisation procedure.
From these results it is clear that although the optimisation
scheme adopted is set to
reduce the variance of the energy of a set of configurations, it
actually has very little success in doing this. It is however, able to
significantly reduce the total energy of the system. The reason for
this apparent residual variance and the remaining
difference in energy between the VMC and DMC results are discussed
later.