Chemistry Reference
In-Depth Information
5.9
Linear Molecules: Groups of Infinite Order
When we try to find the irreducible representations for vibrations or orbitals in a group
containing a
C
∞
axis there is a stumbling block. The infinite axis gives rise to an infinite
number of operations, since a rotation by any angle about the axis of a linear molecule is
a symmetry operation. This means that the order of these groups,
h
, is infinity and so the
1/
h
term in the reduction formula is always zero. There are several approaches in the lit-
erature to coping with this problem. The most straightforward is to deduce the irreducible
representations by inspection of the character set in the reducible representation. In this
section we will see how this allows the elimination of sets of irreducible representations
that are not consistent with the reducible character set. This process will result in only one
sum which is able to give all the characters of the reducible representation.
The reducible representation in these groups can be assigned in the normal way. We
consider the effect of an example operation on each member of the basis to assign a char-
acter for each class of operations in the group. To proceed with the reduction into the set of
standard irreducible representations we return to the basic idea from which the reduction
formula was derived in Section 5.5.
Equation (5.15) states that, within each class, the sum of the characters from the set of
irreducible representations which make up a given reducible representation
sum to the
character obtained for that
χ
(
C
). This sum must work for every class; so, once a par-
ticular alternative combination of irreducible representations is shown to be inconsistent
with the
,
χ
(
C
) in any class, we need not consider that mix again.
For example, the linear molecule CO
2
belongs to the point group
D
∞
h
.Ifwewishto
analyse the C O stretching modes in this molecule then we can use the basis of the
two bond vectors shown in Figure 5.25. There are two basis vectors, and so under the
E
operation of the reducible representation we must have a character of 2. Both vectors are
on the axis of the molecule, and so any rotation around the
C
∞
axis will have a character of
2 also. Likewise, the vertical mirror planes in the group each contain the molecular axis,
leading to a character of 2 again. The remaining operations in the group,
i
,
S
∞
or any
C
2
axis, will exchange the basis vectors, giving a character of 0. The resulting reducible
representation is given in Table 5.20.
C
∞
Figure 5.25
The D
∞h
molecule CO
2
showing a suitable basis for the analysis of the C O
stretching modes.
Table 5.20
The reducible representation for the C O stretch-
ing modes of CO
2
.
D
∞h
E
∞
i
∞
...
σ
v
(
XZ
)
...
C
2
(
X
)
2
2
...
2
0
0
...
0
Now the standard character table is shown in Table 5.21, and so we have to look for
combinations of irreducible representations that correspond to
. The first restriction is
that the irreducible representations must give only two objects, because we have used
