Several different definitions of the average local energy, , in
Eq.(
) were experimented with. The choice which most
accurately describes the average local energy for a given set of
variational parameters is the reweighted local energy, summed over all
the configurations,
If one adopts this definition, then the optimisation process can be
pictured as minimising the variance of the energy about a mean energy
which is also slowing decreasing as the quality of the trial
wavefunction improves. This is illustrated in figure .
It was found that this choice of 
is the most unstable
due to fluctuations in the reweighting factors, 
. The
overall stability of the optimisation process was improved by choosing
the simpler definition,
Figure: Schematic
representation of the reduction in the energy and variance of the
energy during the optimisation process. The blue line shows the mean
local energy, summed over the ensemble of configurations. The black
line schematically represents the variance of the local energy of the
configurations.
Both definitions of are equivalent at the end point of the
optimisation where the reweighting factors are all unity. The simple
definition of in Eq.() has been used in all the
optimisations described in this thesis. It is worth noting at this
point that alternative choices for have also been proposed.
Umrigar[22] often chooses to keep fixed,
throughout the optimisation process, to the original VMC energy
produced by the VMC calculation used to generate the configurations.
Presumably, this is to improve the stability of the optimisation
process still further. However, in our calculations, this choice of
produced a lower quality of trial wavefunction because as the
optimisation proceeds and the average local energy decreases, one is
no longer minimising the variance of the energy about the mean value.
Finally, one could think of attempting to 'guide' the optimisation
process in the right direction by choosing to minimise the variance
about a fixed energy that was lower than the original VMC energy, such
as the DMC energy. This also produced a lower quality trial
wavefunction in our tests.
