My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX B. THE CUSP CONDITIONS
with
J = − 1 ∑
u cusp (x i , x j ) = − 1 ∑ m e e 2
2
2 4πɛ 0 · 1 ( 2 δσi σ
4
j
− 2 ) r ij . (B.19)
i≠j
i≠j
However, while this form gives the correct divergence when two electrons are close
together, it spoils the wave function when the separation is large. A more sensible
choice for u cusp is
u cusp (x i , x j ) =
(
me e 2
4πɛ 0 2 )
β σi σ j
e −ασ i σ j r ij
(B.20)
with the parameters chosen so that
⎧
⎪⎨ 1/4 when σ i = σ j
α σi σ j
β σi σ j
=
⎪⎩ 1/2 otherwise.
(B.21)
As |r i − r j | → ∞, u cusp (r i , r j ) → 0 and the original form of the wave function is
restored; the rate at which u cusp decays in this limit is determined by α σi σ j
.
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