A 2
00
# C 3 ¼ A and E
00
# C 3 ¼ 2E:
ð19:122Þ
Then, corresponding to (18.147) we have
P
A
ð Þ
ϕ 1 ¼
1
3
X
g
χ
A
ð Þ g
ð Þ
h
i Ã
gϕ 1 ¼
1
3
1 Á ϕ 1 þ 1 Á ϕ 2 þ 1 Á ϕ 3
½
Š
¼
1
3
ϕ 1 þ ϕ 2 þ ϕ 3
ð
Þ :
ð19:123Þ
Also, we have
P
E
1
ð Þ
½ Š ϕ 1 ¼
1
3
X
g
χ
E
1
ð Þ
½ Š g
ð Þ
h
i Ã
gϕ 1 ¼
1
3
1 Á ϕ 1 þ ε
Ã
ϕ 3 þ ε
Ã
ð Þ
à ϕ 2
½
Š
¼
1
3
ϕ 1 þ εϕ 2 þ ε
Ã
ϕ 3
ð
Þ :
ð19:124Þ
Also, we get
P
E
2
ð Þ
½ Š ϕ 1 ¼
1
3
X
g
χ
E
2
ð Þ
½ Š g
ð Þ
h
i Ã
gϕ 1 ¼
1
3
1 Á ϕ 1 þ ε
Ã
ð Þ
à ϕ 3 þ ε
Ã
ϕ 2
½
Š
¼
1
3
ϕ 1 þ ε
Ã
ϕ 2 þ εϕ 3
ð
Þ :
ð19:125Þ
Here is the best place to mention an eigenvalue of a symmetry operator. Let us
designate SALCs as follows:
ξ 1 ¼ ϕ 1 þ ϕ 2 þ ϕ 3 , ξ 2 ¼ ϕ 1 þ εϕ 2 þ ε
Ã
ϕ 3 , and ξ 3 ¼ ϕ 1 þ ε
Ã
ϕ 2 þ εϕ 3 : ð19:126Þ
Let us choose C 3 for a symmetry operator. Then we have
Table 19.5 Characters for
individual symmetry
transformations of 2p z orbitals
in cyclopropenyl radical
D 3h
E
2C 3
3C 2
σ h
2S 3
3σ v
Γ
3
0
À1
À3
0
1
Table 19.6 Character table
of C 3
C 3
E
C 3
C
2
3
ε ¼ exp (i2π/3)
A
1
1
1
z; x
2 + y
2
, z
2
E
1
1
&
ε
ε
Ã
ε
Ã
ε
'
(x, y); (x
2 À y
2
, xy), (yz, zx)
760
19 Applications of Group Theory to Physical Chemistry