Atomic Structure Theory

5.2 Atoms with One Valence Electron 143
where the sum over a extends over all core states. The core sum is easily
shown to vanish:
〈a|t k λ(r)|a〉 =
a
a
=
naκa
jama
✻
− kλ
jama
δk0δλ0
〈a||t k ||a〉 =
[ja] 〈a||t k ||a〉 =0.
✗✔ ✛
naκa ✖✕
ja
kλ
−
〈a||t k ||a〉
The expectation value of T k
λ , therefore, reduces to the valence electron expec-
(r). For a one valence electron atom we have
tation value of t k λ
a = gI 〈nvκvmv = jv|t 1 0|nvκvmv = jv〉×13074.7MHz, (5.19)
jv
b =2Q〈nvκvmv = jv|t 2 0|nvκvmv = jv〉×234.965 MHz . (5.20)
In the magnetic case, k = 1, we obtain from (5.6)
〈w|t 1 λ(r)|v〉 = −i √
dr
2
r2
(0)
−iPnwκw (r)Qnvκv (r) 〈κwmw|σ·C 10 (ˆr)|−κvmv〉
+iQnwκw (r)Pnvκv (r) 〈−κwmw|σ · C (0)
10 (ˆr)|κvmv〉
, (5.21)
where, for example,
〈κwmw|σ · C (0)
kq |−κvmv〉
= dΩ Ω † κwmw (ˆr) σ · C(0)
kq (ˆr) Ω−κvmv (ˆr) .
Often in relativistic calculations, one encounters angular matrix elements,
such as those in the above equation, of σ times a normalized vector spherical
harmonic C (ν)
kq . Such matrix elements are easily reduced to matrix elements
of normalized spherical harmonics. We find:
〈κbmb|σ · C (−1)
kq |κama〉 = −〈−κbmb|C k q |κama〉 , (5.22)
〈κbmb|σ · C (0)
kq |κama〉 = κa − κb
〈κbmb|C
k(k +1) k q |κama〉 , (5.23)
〈κbmb|σ · C (1)
kq |κama〉 = κa + κb
k(k +1) 〈−κbmb|C k q |κama〉 . (5.24)
With the aid of (5.23), we obtain
where
〈w|t 1 λ(r)|v〉 =(κv + κw) 〈−κwmw|C 1 λ|κvmv〉
1
r2
, (5.25)
wv