Chemistry Reference
In-Depth Information
Ta b l e 7 . 1
Complex and cubic real forms of the spherical harmonics for
=
0
,
1
,
2
,
3. The con-
stants
N
are the common normalizing factors over the
θ
and
φ
coordinates
N
|
LM
|
Γγ
4
π
s
|
00
=
1
|
A
1
g
=
1
4
π
r
−
1
1
√
2
(x
+
iy)
p
|
1
+
1
=−
|
T
1
u
x
=
x
1
√
2
(x
−
iy)
|
1
−
1
=
|
T
1
u
y
=
y
|
10
=
z
|
T
1
u
z
=
z
15
8
π
r
−
2
1
2
(x
1
√
6
(
3
z
2
iy)
2
r
2
)
d
|
2
+
2
=
+
|
E
g
θ
=
−
1
2
(x
−
iy)
2
1
√
2
(x
2
−
y
2
)
|
2
−
2
=
|
E
g
=
|
T
2
g
ξ
=
√
2
yz
|
2
+
1
=−
(x
+
iy)z
|
T
2
g
η
=
√
2
xz
|
2
−
1
=
(x
−
iy)z
|
T
2
g
ζ
=
√
2
xy
√
6
(
3
z
2
1
−
r
2
)
|
20
=
35
2
xyz
8
π
r
−
3
1
2
√
2
(x
iy)
3
f
|
3
+
3
=−
+
|
A
2
u
=
1
2
√
2
(x
iy)
3
√
10
x(
5
x
2
1
3
r
2
)
|
3
−
3
=
−
|
T
1
u
x
=
−
√
3
2
z(x
+
iy)
2
√
10
y(
5
y
2
1
3
r
2
)
|
3
+
2
=
|
T
1
u
y
=
−
√
3
2
z(x
−
iy)
2
1
√
10
z(
5
z
2
−
3
r
2
)
|
3
−
2
=
|
T
1
u
z
=
2
x(z
2
√
3
2
√
10
(x
+
iy)(
5
z
2
3
r
2
)
−
y
2
)
|
3
+
1
=−
−
|
T
2
u
ξ
=
2
y(x
2
√
3
2
√
10
(x
iy)(
5
z
2
3
r
2
)
z
2
)
|
3
−
1
=
−
−
|
T
2
u
η
=
−
2
z(y
2
√
10
z(
5
z
2
1
−
3
r
2
)
−
x
2
)
|
30
=
|
T
2
u
ζ
=
spherical harmonics is given by
1
√
2
π
Φ
m
(φ)
=
exp
(im
φ)
(7.5)
A rotation
C
α
about the
z
-direction affects this function in the following way:
C
α
Φ
m
(φ)
=
Φ
m
(φ
−
α)
=
exp
(
−
im
α)Φ
m
(φ)
(7.6)
The trace over the entire function space is then given by
+
sin
(
1
/
2
)α
sin
(α/
2
)
χ
(C
α
)
=
−
=
exp
(
im
α)
(7.7)
m
=−
