My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 8. APPLYING THE PLASMON NORMAL MODE THEORY TO
SLAB SYSTEMS
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
∆r ll 5
6
7
8 -4
-3
-2
-1
0
1
z
2
3
4
Figure 8.8: The fixed electron is now just outside the slab at z ′ = −1.0; the bulk plasmons no
longer contribute to u ∞ pl .
range correlation between electrons; this must therefore be introduced artificially.
The behaviour of the wave function when two electrons are in close proximity is
analysed in appendix B; the divergence in the potential energy in this limit must be
compensated by a cancelling divergence in the kinetic energy, which can only arise
as a consequence of a cusp (gradient discontinuity) in the wave function. The cusp
conditions, originally due to Kato [37], specify only the gradient with respect to the
separation of the two electrons in the limit of this separation tending to zero; they
do not determine the value of the wave function.
In appendix B, it is shown that a wave function with the desired short-range
behaviour is
[
Ψ = exp − 1 2
∑
i≠j
]
u cusp (r i , r j ) Ψ smooth , (8.104)
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