It is theoretically possible to use the fact that the variance of the
local energy of an eigenstate of the Hamiltonian is zero to calculate
excitation energies within QMC. The zero variance property of the
ground state has already been utilised within the variance
minimisation procedure described in chapter 
. In
that case, the zero variance provides a useful lower bound to the
quantity being minimised, namely the variance.
It is possible to extend this principle to search for the
excited eigenstates of the same Hamiltonian. An ensemble of independent
configurations were sampled from the ground state wavefunction using
the procedure described in chapter . The variance
of the local energy of this ensemble was then evaluated using the
following expression,
where 
is the energy about which the
variance is to be evaluated. The variance about was
then minimised with respect to the variational parameters in the
wavefunction for a series of values of (i.e. scan over
the energy range). The hope was that around each eigenvalue of the
Hamiltonian, the variance should decrease considerably hence
indicating the presence of such an eigenvalue. This is represented
schematically in figure .
Figure: Schematic representation of
the relationship between the local energy and its variance.
Unfortunately, we found that any small changes in the variance due
to changes we made in the value of were not distinguishable
from the statistical noise present. To make this procedure effective,
a considerably larger number of configurations would be required than
are normally used in the variance minimisation procedure. This makes
the technique prohibitively expensive.
