My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX B. THE CUSP CONDITIONS
may be written as
[− 2
Ĥ({r i }) = ∑ ( 1
∇ 2 i + V (r i ) +
2m
i>2 e 4πɛ 0 2
]
+
[− 2
∇ 2 R + 2V (R) +
4m e
e2
∑
j≠i, j>2
[− 2
m e
∇ 2 r +
1
r ij
+
)]
2
|R − r i |
]
.
e2
4πɛ 0 r + O[r2 ]
(B.5)
For any eigenstate of Ĥ, the divergence in the Coulomb energy must cancel out
exactly; the only other term which can give rise to a matching divergence is
− 2
m e
∇ 2 r.
(B.6)
Therefore, when two electrons are close together, the Schrödinger equation is entirely
dominated by these two terms and is approximately separable; the part of the wave
function depending on r is a solution of the reduced Schrödinger equation
)
(− 2
∇ 2 r +
e2
Ψ r (r) = EΨ r (r).
m e 4πɛ 0 r
(B.7)
This hydrogenic equation has the well-known solution in spherical polar coordinates
Ψ r (r) = ∑ l, m
f lm (r)Y lm (θ, φ)
where the Y lm are spherical harmonics and f lm satisfies the radial equation
(
− 2 d
r 2 df ) ( )
lm e
2
l(l + 1)2
+ + f
m e r 2 dr dr 4πɛ 0 r m e r 2 lm = Ef lm .
Substituting the power series solution
f lm = r p
∞
∑
j=0
a lmj r j
(B.8)
(B.9)
(B.10)
into equation (B.9) gives
r p
∞
∑
j=0
( )
2 [ ]
a lmj l(l + 1) − (p + j)(p + j + 1) r j−2 +
e2
r j−1 − Er j = 0 (B.11)
m e 4πɛ 0
so that 1 p = l and
( )
me e 2 1
a lm1 = a lm0
4πɛ 0 2 2(l + 1) . (B.12)
1 The alternative result, p = −l − 1, is rejected on physical grounds.
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