3.3 Scattering by a Particle with a Shifted Nucleus
37
Scalar multiplication of expression (3.29) by M
3
ki and integration using orthogonality
of spherical harmonics yield
c
j
ki 2 o
=
∞
n=0
d
j
ni 2
A
i,3
nk + c
j
ni 2
B
i,3
n,k
,
(3.37)
This expression can be represented as
c
j
nm 2 o
=
∞
n =0
d
j
n m 2
A
m,3
n n + c
j
n m 2
B
m,3
n n
,
(3.38)
Similar relationships are obtained for remaining expressions (3.30)–(3.32):
c
j
nm 2 o
=
∞
n =0
d
j
n m 2
A
m,3
n n + c
j
n m 2
B
m,3
n n
,
(3.39)
d
j
nm 2 o
=
∞
n =0
c
j
n m 2
A
m,3
n n + d
j
n m 2
B
m,3
n n
,
(3.40)
f
j
nm 2 o
=
∞
n =0
f
j
n m 2
A
m,4
n n + g
j
n m 2
B
m,4
n n
,
(3.41)
g
j
nm 2 o
=
∞
n =0
g
j
n m 2
A
m,4
n n + f
j
n m 2
B
m,4
n n
.
(3.42)
With allowance for formulas (3.23) and (3.24), expressions (3.40) and (3.42) are
represented as
d
j
nm 2 o
=
∞
n =0
d
j
n m 2
A
m
n n + c
j
n m 2
B
m
n n
,
(3.43)
c
j
nm 2 o
=
∞
n =0
c
j
n m 2
A
m
n n + d
j
n m 2
B
m,3
n n
,
(3.44)
f
j
nm 2 o
=
∞
n =0
f
j
n m 2
A
m
n n + g
j
n m 2
B
m,4
n n
,
(3.45)
g
j
nm 2 o
=
∞
n =0
g
j
n m 2
A
m
n n + f
j
n m 2
B
m
n n
.
(3.46)