Atomic Structure Theory

E (1)
J
=
vw,xy
4.7 Relativity and Fine Structure 133
cvwcxyVvw,xy +
c 2 vw[(VHF − U)vv +(VHF − U)ww] . (4.131)
The interaction potential Vvw,xy in (4.131) is given by
Vvw,xy = ηvwηxy
vw
k
+(−1) jw+jx+k
(−1) jw+jx+J+k
jv jw J
Xk(vwyx)
jx jy k
jv jw J
Xk(vwxy)
jy jx k
, (4.132)
where, as usual, the normalization factor ηvw =1/ √ 2 for identical particle
configurations (nw = nv and κw = κv) andηvw = 1, otherwise. It can be
easily seen that Vvw,xy = Vxy,vw.
As in the mixed-configuration case described previously for heliumlike ions,
diagonalizing the quadratic form in (4.131) leads to the algebraic eigenvalue
equation for the energy:
[ɛx +(VHF − U)xx + ɛy +(VHF − U)yy] δvw,xy + Vvw,xy cxy = Ecvw .
xy
4.7.3 Particle-Hole States
(4.133)
Because of the relatively large separation between energies of subshells with a
given value of l and different values of j in closed shell atoms (even an atom as
light as neon), the fine-structure splitting of particle-hole states is particularly
important. The arguments in the preceding paragraphs apply with obvious
modifications to the particle-hole states as well.
First, we construct an angular momentum eigenstate as a linear combination
of those relativistic particle-hole configurations (nvlvnala) withjv =
lv ± 1/2 andja = la ± 1/2 that couple to a given value of J:
|JM〉 =
cva|va, JM〉 , (4.134)
va
where the expansion coefficients satisfy the normalization constraint
va c2va =
1. Again, the first-order energy is a quadratic form in the expansion coefficients.
Diagonalizing this quadratic form leads to an algebraic eigenvalue
problem for the energy and the expansion coefficients. In the particle-hole
case, the eigenvalue problem takes the form
[ɛv +(VHF − U)vv − ɛa − (VHF − U)aa] δvwδab + Vwb,va cva = Ecwb ,
va
where the (symmetric) interaction matrix is given by
(4.135)