Following the method of Ref.[26], we choose to represent the ionic cores in our germanium and silicon supercell calculations with pseudopotentials. This enables the number of valence electrons that are explicitly handled by the QMC algorithm to be reduced to four per atom in both cases.
The pseudopotential used to represent the Ge 
ions in the
germanium calculations described in chapter 
was a
local pseudopotential of the Starkloff-Joannopoulos
form[58]. The pseudopotential used to represent the
Si ions in the silicon calculation described in
chapters and is a
norm-conserving, non-local pseudopotential generated using the method
described by Kerker[59]. In this pseudopotential, the s
and p potentials were generated from an 
atomic groundstate
and the d potential was generated from an 
atomic configuration as in Ref.[60]. In our calculations we
chose the p potential to be the local potential as this results in a
smaller contribution from the remaining non-local potential to the
total energy than choosing either s or d to be local. A small
non-local energy is desirable as the non-local energy is evaluated by
a statistical integration within the QMC code. This integration is
expensive to evaluate and can be evaluated more approximately (and
cheaply) if the overall contribution from the non-local potential is
small. Also, in DMC calculations, we would like the non-local energy
to be as small as possible to reduce the effect of the ``locality
approximation''.
Both these pseudopotentials feature a cutoff radius, beyond which the
pseudopotential reduces to the full Z/r potential due to a +Z
point charge, where Z is the valence of the ion. To deal with the
long ranged tails of the ionic potential the Ewald prescription, as
described in the previous section, is used to evaluate the interaction
energy between the lattice of charges representing the ionic core and
all its periodic images and the lattice representing an electron and
all its periodic images. This is illustrated in
figure . For each electron-ion pair, if the
electron is outside the cutoff radius of the pseudopotential (position
1 in figure ), then the Ewald interaction is
directly applied to calculate the Coulomb energy between the two
corresponding lattices of charged particles and their screening
background charges. If the electron falls within the cutoff radius of
the pseudopotential, (position 2 in figure ),
then the Ewald interaction is still used to evaluate the interaction
between the electron and all the periodic images of the ion, but a
correction is applied to include the effect of the pseudopotential
from the ion in the simulation cell on the electron in the
simulation cell and identical ``in cell'' effects in all the periodic
images of the simulation cell.
Figure: Schematic representation of the electron-ion
interaction. The red point represents an ionic core with a cutoff
radius, 
, marked by the red circle. The blue circles marked 1 and 2
represent two different positions for an electron in the simulation
cell. Only a central simulation cell and one periodic image in each
direction is shown.