My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
Substituting equations (7.49) and (7.45a) into equation (7.48) and solving the
resulting differential equation gives
( )
ψ i (p i ) = exp −
p2 i
. (7.50)
2ɛ 0 ω i
The full ground-state solution is therefore
(
)
Ψ({p i }) = exp − 1 ∑ 1
p 2 i . (7.51)
2ɛ 0 ω i
Alternatively, equation (7.39) may be used to obtain the solution in terms of the
Fourier components of the plasmon charge density:
(
Ψ({ρ k }) = exp − 1
(
∑
) )
M
−1/2
kk ′
ρ
2ɛ 0 kk ′ k ′ . (7.52)
k,k ′ ρ ∗ k
Equation (7.2) gives the relationship between the plasmon charge density and the
electron density in real space. In Fourier space, this becomes
where n k and ¯n k are defined analogously 1 to ρ k .
i
ρ k = −e(n k − ¯n k ) (7.53)
Substituting for ρ k allows the
ground-state wave function to be written in terms of the electron density:
(
Ψ({n k }) = exp − e2 ∑( ) ( M −1/2) ( ) )
n
∗
2ɛ 0 k − ¯n ∗ kk ′
k
nk
kk ′ ′ − ¯n k . (7.54)
k,k ′
The electron density operator is
n(r) = ∑ i
δ(r − r i ), (7.55)
or, in k-space,
n k = √ 1 ∑
e ik·r i
. (7.56)
V
The final step is to write the ground-state wave function in terms of the electron
coordinates:
(
Ψ({r i }) = exp − 1 ∑
u(r i , r j ) + ∑ )
χ(r i )
(7.57)
2
i,j
i
i
1 See equation (7.22).
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