Recent results
[3, 74, 68, 69, 75] have
provided strong numerical evidence that not only does the shape
of the exchange-correlation hole converge rapidly with simulation cell
size, it is also a relatively short ranged quantity. This is
illustrated in figure 
, which shows the
exchange-correlation hole calculated using VMC
[74], for diamond-structure silicon, using a simulation
cell containing 54 atoms. One electron is placed at the centre of a
silicon-silicon covalent bond and the other electron position is
within the (110) plane. Figure shows a slice
through the QMC charge density in the same (110) plane. The position
of the central electron from figure has been marked
with a large white circle.
Figure: Exchange-Correlation hole in diamond-structure silicon from
Ref.[74], with 
at the bond centre and 
ranging over the (110) plane. The black circles represent the
positions of the nuclei.
Figure: VMC Charge density calculated for 3x3x3 diamond structure
silicon plotted in the (110) plane through the centre of a
silicon-silicon covalent bond.
The short range nature of the exchange-correlation hole ensures that
the exchange and correlation energy associated with each electron as
written in Eq.() is well described purely within the
simulation cell surrounding each electron. The use of the Ewald
interaction to try and describe the exchange-correlation interaction
between electrons in different simulation cells therefore appears
unnecessary. The essential requirements of the electron-electron
interaction are simply that (i) it correctly describes the Hartree
energy and (ii) each electron interacts with its exchange-correlation
hole via the full 1/r interaction. In fact it appears that it is the
extra terms in the expansion of the Ewald interaction,
Eq.() that are introducing finite size effects
into the calculations.