The Point Groups Used with Molecules - Molecular Symmetry - page 52

Groups containing only S n axes must be of even order. This is because point groups
containing odd-order improper rotation axes will always have the associated horizontal
mirror plane as an additional symmetry element through the equivalence, S n n

= σ h ( n odd),
defined in Section 2.3.3. The S 2 group is synonymous with the C i group, since S 2 1 i , and

so the group character table listings in Appendix 12 begin with S 4 . The top line of the S 4
character table is reproduced in Figure 3.7b. The equivalences for the improper rotation
axes also lead to the inclusion of simple rotations in these character table headings, since
an even-order S n axis implies the presence of a C n / 2 rotation axis.
An example of an object belonging to the point group S 4 is constructed from our four
generating points in Figure 3.6b. The plan view in Figure 3.6b is drawn looking from above
the object, and the lower atoms are drawn on the inner circle for clarity. In this example,
we have placed the grey atoms on the S 4 axis and put the black and white generating points
at the same distance from the axis at opposite ends of the object. The S 4 operation involves
arotationby90 ◦ followed by a reflection through a mirror plane perpendicular to the axis.
The mirror plane is at the midpoint on the axis joining the two grey atoms. This means
they must be equivalent to one another, and so they are shaded the same and one is marked
as a generator. The black and white points are off axis, and so generate three images each.
The reflection part of the S 4 operation results in atoms being swapped from one end of the
'molecule' to the other, so that the two ends have an equal number of each type. This also
means that the related substituents at either end of the molecule must be equidistant from
the axis.

=

3.5.1 Examples of Molecules for the Cyclic Groups: C n , S n
There are several examples of simple C 2 symmetric molecules, including hydrogen
peroxide (H 2 O 2 , Figure 3.8a) and (1 S ,2 S )-1,2-dimethylcyclopropane (Figure 3.8b). By
definition, C n groups do not contain improper rotation axes, and so molecules in these

(a)

H

O

O

H

C 2

(b)

C 2

(1 S ,2 S )-1,2-dimethylcyclopropane

Figure 3.8 Example molecules from the C 2 point group. (a) The O H bonds of hydrogen
peroxide are not coplanar; the only symmetry operations for this molecule are E and C 2 .
(b) (1S,2S)-1,2-Dimethylcyclopropane. Three-dimensional models of each molecule are
shown to the right of the chemical drawings.