Chemistry Reference
In-Depth Information
Property 4:
The sum of squares of the characters of an irreducible representation for
all operations is equal to the order of the group. Since
g
c
gives the number of operations
in a class, this can be written
χ
i
(
C
)]
2
g
c
[
=
h
(5.4)
C
where
χ
i
(
C
) is the character for the
i
th representation and
C
th class of operators.
To illustrate Property 4, we begin with the
C
2v
point group; since there are no equivalent
operations, all the values of
g
c
are 1 and the totals are straightforward to calculate:
for
A
1
:
C
g
c
[
χ
A
1
(
C
)]
2
=
1
×
1
2
+
1
×
1
2
+
1
×
1
2
+
1
×
1
2
=
4
for
A
2
:
C
g
c
[
χ
A
2
(
C
)]
2
=
1
×
1
2
+
1
×
1
2
+
1
×
(
−
1)
2
+
1
×
(
−
1)
2
=
4
for
B
1
:
C
g
c
[
χ
B
1
(
C
)]
2
=
1
×
1
2
+
1
×
(
−
1)
2
+
1
×
1
2
+
1
×
(
−
1)
2
=
4
for
B
2
:
C
g
c
[
χ
B
2
(
C
)]
2
=
1
×
1
2
+
1
×
(
−
1)
2
+
1
×
(
−
1)
2
+
1
×
1
2
=
4
from class:
E
C
2
σ
v
(
XZ
)
σ
v
(
YZ
)
(5.5)
The order of the group is 4, and so this property is confirmed for all of the irreducible
representations of the
C
2v
point group.
For groups with equivalent sets of operations the corresponding values of
g
c
will be
greater than 1. For example, in the tetrahedral point group,
T
d
, the character table in
Appendix 12 states that there is one operation in the identity class, 8 operations in the
C
3
class, 3 in the
C
2
and so on. If we sum the number of operations in all classes we obtain
the order of the group, i.e.:
h
=
g
c
=
1
+
8
+
3
+
6
+
6
=
24
(5.6)
C
Taking the irreducible representations
A
2
and
E
as examples, the sums for Property 4 are
for
A
2
:
C
g
c
[
χ
A
2
(
C
)]
2
=
1
×
1
2
+
8
×
1
2
+
3
×
1
2
+
6
×
(
−
1)
2
+
6
×
(
−
1)
2
=
24
for
E
:
C
g
c
[
χ
E
(
C
)]
2
1
×
2
2
+
8
×
(
−
1)
2
3
×
2
2
6
×
0
2
6
×
0
2
=
+
+
+
=
24
from class :
E
8
C
3
3
C
2
6
S
4
6
σ
d
(5.7)
In both cases the sum gives the order of the group, and so Property 4 is confirmed again.
