My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 6. THE MODIFIED PERIODIC COULOMB INTERACTION IN
QUASI-2D SYSTEMS
Since the transforms can be pre-calculated, the one-body term is not computationally
costly. The use of 3D transforms means that the simulation cell may no longer
have infinite extent in the non-periodic direction. To avoid overlapping, the electron
density must be restricted to a range w, where the size of the simulation cell in this
direction is at least 2w. 1 The requirement of finite extent is not unreasonable for
quasi-2D systems: the electron density usually tends exponentially to zero beyond
a certain point.
6.3 Results
In this section, the quasi-2D versions of the MPC and Ewald interactions will be
compared; the test system is the jellium slab described in chapter 5. The chosen
density parameter is 2.07, which corresponds to aluminium; this density is very
frequently studied.
The simulation cell is square in the xy-plane, with the size
determined by the number of electrons being used:
√
4πNr
3
L =
s
. (6.20)
3s
The number of electrons is N, while s is the slab width, chosen to be 18.63 in
these investigations. At this width, LDA calculations reveal that six sub-bands are
occupied.
The trial wave function is defined by the following set of equations:
Ψ(X) = e J(X) D ↑ (R ↑ )D ↓ (R ↓ ) (6.21)
J(X) = − ∑ u σi σ j
(r ij ) + ∑ χ(z i ) (6.22)
i>j
i
u σi σ j
(r ij ) = A ( [1 − exp − r )] ( )
ij
exp − r2 ij
(6.23)
r ij F σi σ j
L 2 c
χ(z i ) = ∑ k
c k sin kz i . (6.24)
The motivation for choosing this form for the trial wave function was detailed in
section 3.4.1. The variational parameters are A and c k ; F σi σ j
is related to A by the
1 This enforced zero-padding is standard when calculating a convolution [62].
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