My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
or, using equations (7.75) and (7.72),
(
Ψ[ f] ˙ = exp − ɛ ∑
) ) 2
0 1
∇f i · ∇f 2 ω
i i
(∫V
˙ d 3 r
(
Ψ[ρ] = exp − 1 ∑
) ) 2
1
f i ρ d
2ɛ 0 ω
i i
(∫V
3 r
⎛
(
Ψ({r i }) = exp ⎝− e2 ∑ 1
− ∑ ∫ ) ⎞ 2
f i (r m ) + f i¯n d 3 r ⎠ .
2ɛ 0 ω i
m
V
i
(7.80a)
(7.80b)
(7.80c)
Several previous results have been used in equation (7.80), including equations
(7.2), (7.18) and (7.55).
The wave function has now been reduced to the standard form given in equation
(7.57), with
and
χ(r) = e2 ∑
∫
1
f i (r) f i (r ′ )¯n(r ′ ) d 3 r ′ (7.81)
ɛ 0 ω
i i V
u(r, r ′ ) = e2 ∑ 1
f i (r)f i (r ′ ). (7.82)
ɛ 0 ω
i i
Note that φ and f satisfy the same equation of motion. The boundary conditions
used in determining the normal modes for these two quantities are also identical;
in fact, the normal modes differ only by a numerical factor, and not in functional
form. The numerical factor may be eliminated by suitable normalisation, with the
result that f i can be replaced by φ i in equations (7.81) and (7.82).
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