November 5, 2004
PHY 741 – Notes on Multiplet splittings
Reference: E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra,
Cambridge University Press, 1967. (First printed in 1935.)
We are concerned with evaluating the “atomic terms” which approximate the possible
states of a two-electron open shell system such as np2 , nd2 , nf 2 , etc.
Suppose that the spatial part of the two-electron wavefunction can be written as an
eigenstate of total L2 and Lz :
X
Ψ(r1 , r2 ) = Rnl (r1 )Rnl (r2 )
hLM |lma lmb iYlma (rˆ1 )Ylmb (rˆ2 ),
(1)
ma mb
where hLM |lma lmb i denotes a Clebsch-Gordon coefficient. The multiplet energy will be
determined by evaluating the expectation value of
+
e2 Ψ Ψ .
r1 − r2 *
∆E =
(2)
We can define the radial integral
Fλ ≡
Z
r12 dr1
Z
r22 dr2
λ
r<
2
2
λ+1 (Rnl (r1 )) (Rnl (r2 )) .
r>
(3)
We also need to use the Gaunt coefficients:
GLM
l1 m1 l2 m2 ≡
√
4π
Z
∗
dΩYl1∗m1 (r̂)YLM
(r̂)Yl2 m2 (r̂) =
√
4π
Z
dΩYl1 m1 (r̂)YLM (r̂)Yl2∗m2 (r̂)
(4)
As discussed in Cohen-Tannoudji’s text, the Gaunt coefficent can be expressed in terms of
a product of two Clebsch-Gordan coefficients:
GLM
l1 m1 l2 m2
v
u
u
m2 t (2l1 + 1)(2l2 + 1)
= (−1)
hl1 0 l2 0|L 0ihl1 m1 l2 − m2 |L M i.
(2L + 1)
(5)
In terms of these quantities, the multiplet energy can be calculated according to:
∆E = e2
X
F λ AλL ,
(6)
λ
where the angular term is given by
AλL =
X
1
λ(mc −ma ) λ(mb −md )
hLM |lma lmb ihlmc lmd |LM iGlma lm
Glmd lmb .
c
2λ + 1 ma mb mc md
(7)