It has already been shown in chapter 
that the choice
of electron-electron interaction is fundamental, if one is to control
the Coulomb finite size effects introduced when attempting to simulate
a bulk system using a model system consisting of a few 10's of
electrons in a simulation cell with periodic boundary conditions.
Therefore, when performing QMC calculations for the gap energy as
given in Eq.(), the new electron-electron interaction
described in section was used both in the
wavefunction optimisation procedure for each of the systems and then
in the subsequent QMC calculations. The new energy expression for an
N-electron system as stated in Eq.() naturally
extends to systems containing 
electrons in the following
way,
where 
is the LDA charge density of a simulation cell
containing X=N or electrons. The sum in the first term now
extends over all electron-electron pairs including those involving the
additional electron. The Hamiltonian associated with this energy
expression is
and the electron-electron energy in a VMC calculation is then given by
