Chemistry Reference
In-Depth Information
which again has a total character of 0, and so we may consider grouping the
C
2
and
C
3
operations. However, selecting the C atom positions as a basis, the two operations give
different characters:
C
3
1
:C
1
→
C
2
,
C
2
→
C
3
,C
3
→
C
1
total character 0
C
3
2
:C
1
→
C
3
,
C
2
→
C
1
,C
3
→
C
2
total character 0
C
2
1
:C
1
→
C
1
,
C
2
→
C
3
,C
3
→
C
2
total character 1
So, while the
C
3
operations still give the same character as each other, for the
C
2
1
rotation
the C atom on the axis is unaffected and we find a total character of 1. This means that the
C
2
operation cannot be in the same class as
C
3
1
and
C
3
2
.
The
C
2
axis is one of a set of three (each passes through a C atom and the centre of
the opposite C C bond). Rotation around any of these axes leaves one C unaffected but
swaps around all the H atoms; so, in either basis set, any of the three axes gives the same
total character. The three
C
2
axes form another class within the
D
3h
point group; so, in
the corresponding character table heading we see 3
C
2
and would assign
g
c
=
3 when
considering this class.
In Section 5.5 it will be shown that any set of characters we obtain for a basis of our
choice can be reduced to a summation of the standard irreducible representations from the
character tables. This means that a sufficient condition for putting a given set of operations
in the same class is that they have the same character for all irreducible representations.
In general, operations that can be collected together fall into one of the two types we have
discussed here.
The first type has operations that are linked to the same symmetry element, such as
C
3
1
and
C
3
2
. However, operations linked to the same element will not always fall into the
same class; for example, in
D
4h
the
C
4
1
and
C
4
3
rotations associated with the principal
axis are in the same class, but the
C
4
2
operation is listed separately in the character table
as
C
2
. The second types of operation that fall into the same class are those for sets of
symmetry-equivalent elements, such as the three equivalent mirror planes in
D
3h
.
It is also possible that symmetry-equivalent elements each give more than one operation
to a class. For example, in the octahedral point group
O
h
there are four equivalent
C
3
axes,
each of which contributes two operations which are in the same class (
C
3
1
and
C
3
2
), and so
the heading in the character table reads 8
C
3
. One of these
C
3
axes is marked on the paper
model of an octahedron from Appendix 4.
Problem 5.3:
Show that each operation below belongs to a class with
g
c
>
1 and assign
the value of
g
c
. Illustrate your answer with sketches of the result for each operation and
derive the relevant total characters using the suggested structure and basis:
1.
σ
d
,inthe
D
3d
point group, using a basis of the hydrogen atoms in staggered ethane
(e.g. see Figure 2.5c).
2.
S
4
1
,inthe
D
4h
point group, using a basis of
x
,
y
,
z
vectors on the central atom of a
square planar complex (e.g. see Figure 4.10);
3. all
C
3
1
and
C
3
2
operations in the
T
d
point group, using as a basis the C
H bond
vectors in methane (see also the model tetrahedron from Appendix 3).
