My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
density tends to zero in the idealised surface system, but to some non-zero positive
value in the more realistic slab model.
This is because the net change in the electron density is forced to be zero in the
slab system (which is finite); since the electron density is reduced in the vicinity of
the charge disturbance, it must be increased elsewhere. This constraint does not
apply to the idealised surface system (which is infinite).
cell:
To see this, consider integrating the original differential equation (6.29) over the
∫
cell
∫
∫
∇ 2 δφ tot (r) d 3 r − 4π δn ind (r) d 3 r = −4π δρ ext (r) d 3 r. (6.71)
cell
cell
The external charge density was chosen to be neutral overall, giving
∫
δρ ext (r) d 3 r = 0. (6.72)
cell
When the potential δφ tot is forced to be periodic, the first integral in equation (6.71)
also gives zero. This is evident on substitution of the Fourier series representation:
∫
∫
∇ 2 δφ tot (r) d 3 r = ∇ ∑ 2 δ ˜φ tot (k)e −ik·r d 3 r
cell
cell
k
= − ∑ ∫
k 2 δ ˜φ tot (k) e −ik·r d 3 r (6.73)
k
cell
= 0.
Thus, for a finite system with periodic boundary conditions, the net change to the
electron density is zero:
∫
cell
δn ind (r) d 3 r = 0. (6.74)
This effect becomes smaller as the system size increases, as can be seen in figure 6.6;
in an infinite system like the idealised surface, it disappears.
Since the aim is to investigate the exchange-correlation hole, it is helpful to
establish a link between this entity and δn ind . For this, a new notation is required.
The electron density in a system of N electrons at the point r is labelled n(r; N).
When one electron is fixed at the position r ′ , the density at r is n(r|r ′ ; N); this is
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