With the new expression in Eq. (
) for the
electron-electron interaction proving so successful for calculations
on the HEG, it was decided to attempt to produce a more general
expression for use in inhomogeneous solids. Now the long range
interaction term is non-trivial because the charge density is no
longer uniform. In a VMC calculation the electronic charge density,
, appearing in Eq. () could be
accumulated during the simulation and the interaction energy evaluated
afterwards. In a DMC calculation this is not possible because the
total energy needs to be evaluated at the end of each step of the
simulation. Therefore, one must know the charge density before
starting the calculation. To overcome this problem calculations were
performed with the LDA charge density, 
, as the
reference density in Eq. () i.e.
It would be possible to update the input charge density afterwards and
perform a self-consistent calculation. However, LDA charge densities
are normally remarkably good and, moreover, the interaction energy is
insensitive to the quality of the charge density used because
differs significantly from zero only
when 
is large. This approximation
was
found to be so successful that for convenience it was also used for
VMC calculations and in the variance minimisation procedures for
optimising wavefunctions.
To implement the new interaction into the QMC code for solids
described in chapter the following changes were made
This extra one-body potential is stored in reciprocal space and
accumulated on the same set of primitive cell reciprocal lattice
vectors that the 
function is evaluated on and the density is
accumulated on.
The interaction leading to the energy expression in
Eq. () is illustrated in Fig. .
This depicts a rhombohedral simulation cell containing two electrons,
on one of which is centred a hexagonal window corresponding to the
Wigner-Seitz cell of the simulation cell. The (red) electron at the
centre of the window experiences a 1/r interaction with all the
other electrons within the window (one (blue) in this case) and an
electrostatic interaction with the electronic charge density of the
shaded region outside of the window.
Figure: An illustration of the new interaction for a rhombohedral
simulation cell containing two electrons (crosses). The hexagonal
clear window centred on one of the electrons has the shape of the
Wigner-Seitz cell of the simulation cell.
