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 8.9: The fixed electron is now further outside the slab (z ′ = −2.0), and the strength of the
correlation is reduced.
where Ψ smooth is the unmodified cuspless wave function and
( )
me e 2
u cusp (x i , x j ) = β
4πɛ 0 2 σi σ j
e −ασ i σ j r ij
. (8.105)
The analysis depends on the original wave function Ψ smooth being smooth (hence the
subscript): more precisely, it must be possible to expand this function in a Taylor
series about the point r i = r j , irrespective of the positions of the other electrons.
This is true for the two-determinant wave function Ψ sr , and almost true for the
product wave function Ψ pr . The problem is that the plasmon contribution Ψ pl has
cusps at the slab boundaries.
In fact, the cusps in the plasmon wave function create a more serious problem
in QMC. These extended gradient discontinuities lead to singularities in the second
derivative of Ψ pl which should make a finite contribution to the energy expectation
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