Atomic Structure Theory

190 6 Radiative Transitions
The resulting potentials reduce to the length-form multipole potentials in the
nonrelativistic limit. We refer to this choice of gauge as the length gauge in the
sequel. Let us examine the nonrelativistic limit of the length-gauge transition
operator
α · a (1) 1
JM (r) −
c φJM(r).
Since the vector potential contribution is smaller than the scalar potential by
terms of order kr, the interaction can be approximated for small values of kr
by
lim α · a
k→0
(1) 1
JM (r) −
c φJM(r)
= i
(2J + 1)(J +1)
4πJ
k J
(2J + 1)!! QJM(r),
where
QJM(r) =r J CJM(ˆr)
is the electric J-pole moment operator in a spherical basis.
In either gauge, the multipole-interaction can be written in terms of a
dimensionless multipole-transition operator t (λ)
JM (r) defined by
α · a (λ) 1
JM (r) −
c φJM(r)
(2J + 1)(J +1)
= i
t
4πJ
(λ)
JM (r) . (6.129)
The one-electron reduced matrix elements 〈i||t (λ)
J ||j〉 are given by
Transverse (Velocity) Gauge:
〈κi||t (0)
J ||κj〉
∞
= 〈−κi||CJ||κj〉
0
dr κi + κj
J +1 jJ(kr)[Pi(r)Qj(r)+Qi(r)Pj(r)] ,
∞
dr −
0
κi
− κj
j
J +1
′ J(kr)+ jJ(kr)
×
kr
[Pi(r)Qj(r)+Qi(r)Pj(r)] + J jJ(kr)
[Pi(r)Qj(r) − Qi(r)Pj(r)] .
kr
〈κi||t (1)
J ||κj〉 = 〈κi||CJ||κj〉
and
Length Gauge:
〈κi||t (1)
J ||κj〉 = 〈κi||CJ||κj〉
∞
0
dr
jJ(kr)[Pi(r)Pj(r)+Qi(r)Qj(r)]+
κi − κj
jJ+1(kr)
J +1 [Pi(r)Qj(r)+Qi(r)Pj(r)] + [Pi(r)Qj(r) − Qi(r)Pj(r)]
.
The functions Pi(r) andQi(r) in the above equations are the large and small
components, respectively, of the radial Dirac wave functions for the orbital
with quantum numbers (ni,κi).