Chemistry Reference
In-Depth Information
Table 12.1
The molecular point group C
2v
s
0
v
C
2v
I
C
2
s
v
A
1
1
1
1
1
A
2
1
1
1
1
1
1
B
1
1
1
1
1
B
2
1
1
(v)
In general, the commutative law does not hold for the symbolic
multiplication; namely: G
r
G
s
=
G
s
G
r
.IfG
r
G
s
¼
G
s
G
r
, then the
group is said to be Abelian (a commutative group).
Almost all textbooks on group theory contain tables of molecular point
groups, their relation to the chemical structure of some polyatomic
molecules being given for instance in Eyring et al. (1944). We give below
two examples of molecular point groups and their multiplication tables
(Tables 12.1-12.4): the Abelian group C
2v
of the H
2
O molecule and
the groupC
3v
of NH
3
. Apoint group of order h has h symmetry operations
characterizing it. So, C
2v
is a group of order 4, while C
3v
is a group of
order 6.
Tables 12.1 and 12.3 are called character tables of the point groups. In
the first column of both tables, below the denomination of the point
group, are the symbols denoting the symmetry-defined types or, in the
Table 12.2
The multiplication table of the point group C
2v
s
0
v
C
2v
I
C
2
s
v
s
0
v
I
I
C
2
s
v
s
0
v
C
2
C
2
I
s
v
s
0
v
s
v
s
v
I
C
2
s
0
v
s
0
v
s
v
C
2
I
Table 12.3
The molecular point group C
3v
þ
3
3
C
3v
I
s
1
s
2
s
3
C
C
A
1
1
1
1
1
1
1
A
2
1
1
1
1
1
1
cs
s
c
10
0
c s
sc
cs
sc
s
10
01
E
1
c
s
c
