Chemistry Reference
In-Depth Information
Table 12.5
Reducible representation
G
and symmetry-adapted AOs for H
2
O
s
0
v
C
2v
I
C
2
s
v
Symmetry basis
1
A
1
1
1
1
1
k
;
s
;
z
;
h
z
¼
ð
h
1
þ
h
2
Þ
p
A
2
1
1
1
1
1
1
B
1
1
1
x
B
2
1
1
1
1
y
;
h
y
¼
1
ð
h
1
h
2
Þ
p
A
1
B
1
B
2
G
7
1
5
3
G
¼
4
;
G
¼
1
;
G
¼
2
properties and theorems introduced in the formal group theory of the
preceding section.
Just to be clear, we give below the construction of the matrix repre-
sentative
D
(C
2
) for the reducible representation
in the original ba-
sis (12.31). We must first construct the transformation table of the basis
functions (12.31) under the symmetry operations of C
2v
. Using the active
transformation, we obtain Table 12.6.
Following the recipe (12.9), we then immediately obtain
G
0
@
1
A
1
1
1
1
Dð
C
2
Þ¼
;
tr
Dð
C
2
Þ¼
1
ð
12
:
35
Þ
1
1
1
Table 12.6
Transformation table of the AO basis for
H
2
O under the operations of C
2v
R
x
I
C
2
s
v
s
0
v
k
k
k
k
k
s
s
s
s
s
z
z
z
z
z
x
x
x
x
x
y
y
y
y
y
h
1
h
1
h
2
h
1
h
2
h
2
h
2
h
1
h
2
h
1
