The Point Groups Used with Molecules - Molecular Symmetry - page 56

operations around the axis. If we assume an atom is in one of the vertical mirror planes,
then it must follow that all of the symmetry-related atoms are also in mirror planes. In
the example of Figure 3.12a this leads to two

σ v . We can also position vertical planes as

the bisectors of the angles between the two σ v planes. These new planes are in a different

environment: in Figure 3.12a they contain no atoms and in Figure 3.12b they contain atoms
from the other set. Since the new planes bisect the angles between the σ v planes, they are

σ d .
If we generate an example object for an odd-order axis, such as the C 3v case shown in
Figure 3.13, then the distinction between

labelled

σ d is no longer relevant. All mirror planes
contain only one of each type of point and also bisect the angle between other planes. In
this case the vertical planes are all identical and are simply labelled

σ v and

σ v . This difference is
seen in the listing of the point group operations at the top of the corresponding point group
tables (Figure 3.14).
Equivalent mirror planes are another example of a class containing more than one sym-
metry operation in a group. They have identical arrangements of atoms around them and

1

σ v

C 3v

1

σ v

σ v

σ v

Figure 3.13 An object in the C 3v point group; in this case, all vertical mirror planes are
equivalent, as can be seen in the plan view, and so only the label σ v is used.

C 4v E 2 C 4 C 2 2 σ v 2 σ d

(a)

C 3v E 2 C 3 3 σ v

(b)

Figure 3.14 The headings used in (a) the C 4v and (b) the C 3v character tables.