Chemistry Reference
In-Depth Information
Property 2:
The
order of the group
is the
total number of operations
that are in the
group and is given the symbol
h
.
Finding the order of a point group is simply a matter of totalling the number of unique
operations. In the previous section we considered the
C
2v
point group with H
2
Oasan
example; this contains the operations
E
,
C
2
,
σ
v
(
XZ
) and
σ
v
(
YZ
), and so
h
=
4.
In Chapter 4 we also considered the
D
4h
group, which has the following operations:
E
,2
C
4
,
C
2
,2
C
2
,2
C
2
,
i
,2
S
4
,
σ
h
,2
σ
v
,2
σ
d
Notice that in this list the equivalences discussed in Chapters 1 and 2 have been accounted
for so that only unique operations are listed. Multiple operations in the same class are
written as a single entry in the list; the number of operations within each class
g
c
is noted
by a number in front of the symbol for the operation. This is the list that appears at the top
of the character table in Appendix 12.
Counting up the total number of operations shows that the order of the
D
4h
group is
16. From a standard character table, the value of
h
can be calculated from the total number
of operations given in the column headings. Mathematically, this summation process can
be written:
h
=
g
c
(5.3)
C
where the
C
under the summation sign indicates that the sum is over all classes of oper-
ations within the point group. Equation (5.3) means that the order of a group is the sum
of the number of operations in each class. This makes calculating the order of the group
when we have the point group table a simple matter of adding the numbers at the head of
each column. For example, the character table for the
D
3h
group in Appendix 12 has the
class headings
E
,2
C
3
,
σ
v
, making the
g
c
values 1, 2, 1, 2 and 3 respectively.
Summing these gives the order of the group,
h
σ
h
,2
S
3
and 3
=
9.
Property 3:
The number of irreducible representations in a point group is equal to the
number of classes, i.e. all character tables are square.
Property 3 can be confirmed by inspecting any of the character tables in Appendix 12.
For example, the
D
3d
point group for molecules such as ethane in the staggered conforma-
tion has the headings
E
,2
C
3
,3
C
2
,
i
,2
S
6
,3
σ
d
so that there are six classes.
The irreducible representations listed for
D
3d
are
A
1g
,
A
2g
,
E
g
,
A
1u
,
A
2u
,
E
u
so there are six of these as well.
