My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
similar to a conditional probability. The exchange-correlation hole for an electron
at r ′ is then defined by the formula
n(r|r ′ ; N) = n(r; N) + n XC (r, r ′ ; N). (6.75)
In the calculations above, the external charge density was chosen to model an electron;
after fixing this electron at r ′ , the change to the density of the remaining free
electrons was estimated:
δn ind (r|r ′ ; N) ≈ n(r|r ′ ; N + 1) − n(r; N). (6.76)
Note the change from N to N + 1 in the conditional density, which arises because
an electron has been added to the system. Combining these equations then gives
n XC (r, r ′ ; N + 1) = n(r|r ′ ; N + 1) − n(r; N + 1)
≈ δn ind (r|r ′ ; N) + n(r; N) − n(r; N + 1).
(6.77)
For large N, adding an extra electron should not change the density dramatically.
The following approximation may therefore be used:
( ) N + 1
n(r; N + 1) ≈ n(r; N). (6.78)
N
The exchange-correlation hole is then
n XC (r, r ′ ; N + 1) ≈ δn ind (r|r ′ ; N) − 1 n(r; N). (6.79)
N
This function is plotted in figure 6.7, which shows the characteristic spreading-out
of the hole near the surface. As in the idealised system, the surface hole is curtailed
by the cell boundary, although the effect is weaker. This supports the idea that the
remaining finite-size error is caused by the exchange-correlation hole being squashed
into the simulation cell. This error would act to reduce the energy of the system,
and therefore has the correct sign.
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