Chemistry Reference
In-Depth Information
3. Subscripts 1 and 2 are attached to
A
and
B
labels to indicate those that are symmetric
(subscript 1) and antisymmetric (subscript 2) with respect to a
C
2
axis perpendicular to
the principal axis or, if the axis is absent, to a vertical mirror plane. Symmetric means a
character
1.
4. In groups with a centre of inversion
i
, the subscripts 'g' and 'u' are added to symbols of
representations which are symmetric and antisymmetric respectively for the inversion
operation. These symbols come from the German words
gerade
meaning 'even' and
ungerade
meaning 'odd'.
5. Primes and double primes are attached to letters to indicate those that are symmetric
and antisymmetric with respect to
+
1 and antisymmetric a character of
−
σ
h
in groups that do not contain
i
.
Points 4 and 5 come about from the idea of subgroups. For example, in Chapter 2,
the multiplication table for ethane in the staggered conformation was considered, which
we now recognize as an example in the
D
3d
point group. The
D
3d
multiplication table
(Table 2.4) shows that the identity operator and the simple rotations could be taken for a
group in their own right, because no other operations result from their products. This type
of subgroup, in this case
D
3
, is called a pure rotational group, since it contains only simple
rotations.
Looking along the row of the multiplication table for the inversion centre, it is also clear
that all the other members of the full
D
3d
point group occur as a product of one of the
simple rotational operations and the inversion centre. In the
D
3d
character table there are
three different possibilities for the characters under the pure rotation subgroup: labels
A
1
,
A
2
and
E
. For each of these there are then two possibilities for the behaviour of an object
under the inversion operation:
1. If the object has even symmetry under
i
, then the product of each member of the sub-
group with
i
will lead to the same character as the subgroup member itself. These are
the even irreducible representations given the additional subscript 'g'. For the
gerade
representations in
D
3d
, the first three characters for the rotational subgroup are simply
repeated under the operations generated by their product with
i
, so that the
E
,2
C
3
,3
C
2
characters are repeated under
i
,2
S
6
and 3
σ
d
.
2. If an object has
ungerade
symmetry with respect to inversion, then it will have a neg-
ative character in the
i
column and the operations generated from combinations of the
rotational subgroup members and
i
will have the opposite sign to that of the simple
rotation involved. These are the representations labelled 'u'.
In groups without an inversion centre but containing a horizontal mirror plane, a similar
argument can be put forward based on the combination of the pure rotational subgroup
with the
σ
h
plane.
The points listed above still do not give a complete set of explanations for Mulliken's
symbols. For example, the numerical subscripts used on
E
and
T
representations are not
covered. However, these require mathematical explanations that are beyond the scope of
this introductory text, and so we shall regard them simply as standard labels.
