488
9 The Unification of Structure and Reaction Frameworks
The general plane wave expansion, in which q has an arbitrary direction, is
recovered by using the addition theorem of spherical harmonics:
P L (cos(θ )) =
4π
2L + 1
L
M L =−L
Y
∗
LM L
(Ω q ) Y LM L (Ω) .
Multiplying the general plane wave expansion by Y LM L (Ω q ) on left- and righthand sides and integrating on Ω q provide with the integral expression of
J LM L (q, r).
B. One will express J LM L (q, r) as a finite linear combination of functions depending on either relative or center-of-mass coordinates. Let us write J LM L (q, r)
using its Bessel function expansion:
J LM L (q, r) = (4π i
L )
−1
e
iq·r int e
iq·R CM Y LM L (Ω q ) dΩ q ,
where the equality r int + R CM = r has been used. The exponential functions in
this equation, depending only on r int or R CM , can be expanded with relative and
center-of-mass Bessel functions, respectively. One then obtains the demanded
result:
J LM L (q, r) = (4π i
L )
−1
Y LM L (Ω q ) dΩ q
× 4π
int m int
i
int Y
∗
int m int
(Ω q ) Y int m int (Ω int ) j int (qr int )
× 4π
L CM M L CM
i
L CM Y
∗
L CM M L CM
(Ω q ) Y L CM M L CM (Ω CM ) j L CM (qR CM )
= 4π
int m int
L CM M L CM
i
int +L CM −L J int m int (q, r int ) J L CM M L CM (q, R CM )
×
Y LM L (Ω q ) Y
∗
int m int
(Ω q ) Y
∗
L CM M L CM
(Ω q ) dΩ q . .
(9.196)
C. The integral of three spherical harmonics is analytical, and its value is called the
Gaunt coefficient [63]:
Y LM L (Ω q ) Y
∗
int m int
(Ω q ) Y
∗
L CM M L CM
(Ω q ) dΩ q
= (−1)
int +L CM
(2 int + 1)(2L CM + 1)
4π
×
int L CM L
0 0 0
int m int L CM M L CM |L M L .