The monopole moment is:
l
= 0,
m
= 0:
q
00
=
0
This makes sense because the monopole moment is just the total charge and the total charge in this case
is zero.
The dipole moments are:
l
= 1,
m
= -1:
q
1,
−
1
=
0
l
= 1,
m
= 0:
q
1,0
=
0
l
= 1,
m
= 1:
q
1,1
=
0
This also makes sense. A close inspection of the geometry reveals that the symmetry would lead to no
overall dipole moment.
The quadrupole moments are:
l
= 2,
m
= -2:
q
2,
−
2
=
qa
2
15
2
l
= 2,
m
= -1:
q
2,
−
1
=
0
l
= 2,
m
= 0:
q
2,0
=
0
l
= 2,
m
= 1:
q
2,1
=
0
l
= 2,
m
= 2:
q
2,2
=
qa
2
15
2
The potential in terms of these moments is:
r ,
,
≈
1
4
0
∑
l
=
0
2
∑
m
=−
l
l
4
2
l
1
q
l m
Y
lm
,
r
l
1
r ,
,
≈
1
4
0
4
5
q
2,
−
2
Y
2,
−
2
,
r
3
1
4
0
4
5
q
2,2
Y
2,2
,
r
3
r ,
,
≈
1
4
0
4
5
1
4
15
2
sin
2
1
r
3
[
q
2,
−
2
e
−
i
2
q
2,2
e
i
2
]
r ,
,
≈
3
qa
2
4
0
sin
2
cos
2
r
3