The CFSE is illustrated in figure 
. This shows the
total energy calculated using VMC for diamond-structure silicon using
a finite simulation cell with periodic boundary conditions plotted for
simulation cells containing multiples, n=2,3,4,5 of primitive
lattice cells, which corresponds to 16,54,128 and 250 atoms,
respectively. For each of the calculations, a full
Hartree-Fock-Jastrow-Chi wavefunction as in Eq.()
was used where the 
and u functions were optimised using the
variance minimisation techniques described in
chapter . Six stars of 
vectors were used to
describe the function and 22 variational parameters were used
in the u function. The single particle orbitals used to construct
the Slater determinant have 
-points chosen from a reciprocal
space grid that is offset from the origin by 
sampling, where the 
are
the primitive reciprocal lattice vectors of the supercell,
i.e. L-point sampling. This choice of sampling almost totally removes
the IPFSE, leaving the CFSE as the dominant finite size effect. A
norm-conserving, non-local pseudopotential was used to describe the
silicon cores. The extensions to the variance minimisation scheme
introduced in chapter to deal with non-local
pseudopotentials were experimented with in both the simple ``fixed
non-local'' and ``full optimisation'' forms. The two resulting
wavefunctions produced variational energies that were
indistinguishable at the level of the statistical noise. This
suggests that in silicon the ``fixed non-local'' approximation in the
variance minimisation scheme is sufficient.
Figure shows the total energy per atom
asymptotically approaching a value that can be taken as the energy per
atom in the bulk solid. The CFSE can then be defined as the
difference in the total energy (once any IPFSE have been removed) at a
specific system size and this bulk value.
Figure: Total energy per atom calculated using VMC as a function
of system size. The statistical error bars are smaller than the size
of the symbols.
The motivation for reducing the CFSE is twofold. Firstly, if the
desired level of accuracy in a specific calculation can be achieved by
performing that calculation on a smaller system as a result of
reducing the CFSE, then the computational benefits will be large as
the computational time scales as approximately the third power of the
number of electrons in the system. In Figure , the
number of electrons scales as the cube of the system size, n and
hence the total computational time scales as the ninth power of the
x-axis.
Secondly, there are some problems in which even if one is able to
perform the calculation on a large system size this still fails to
reduce the CFSE. A standard problem in electronic structure theory is
to calculate the energy required to create a point defect. This is
done by subtracting the energy of the perfect crystal from that of a
large simulation cell containing a single defect. Taking the example
of the n=3 (54 atom) simulation cell of silicon, we find (see
figure .) that the CFSE error in the VMC energy of
the whole simulation cell is -5 eV. This is much larger than the
energies of interest, which are often tenths of an eV per simulation
cell. Moreover, it has been observed in previous work
[46, 33] that these CFSE are approximately
inversely proportional to the number of atoms in the simulation cell
and hence the CFSE for the whole cell is almost independent of
N
. Of course there will be a cancellation between
the CFSEs in the perfect and defective solids, which will become more
complete as the size of the simulation cell increases, so that
eventually the energy difference will converge. However, we must
expect that the incomplete cancellation of errors for finite
simulation cells will lead to a significant uncertainty in the defect
energy. Therefore, the only guaranteed way of improving the accuracy
of such calculations is to significantly reduce the CFSE at all system
sizes.