elements such that operation on that molecule by the corresponding symme-
try operations will produce a “new” orientation in space that is indistinguish-
able from the original orientation. Figure 13.5 shows the symmetry elements
for a simple molecule, H 2 O. The ability to recognize symmetry elements in a
molecule will be an important skill to develop.
Real objects, including molecules, do not have any random set of symme-
try elements. Only certain specific groupsof symmetry elements are possible
for any physical object. Because all of the symmetry elements of such a group
intersect at a single point in the object, such groups are called point groups.A
point group is usually referred to by a label to indicate that an object contains
that certain set of symmetry elements. For example, the C2vpoint group, which
describes the symmetry of the water molecule, consists ofE,a C 2 proper rota-
tion, and two planes of symmetry . Consider the symmetry elements for H 2 O
in Figure 13.5. All objects that have C2vsymmetry contain these four and no
othersymmetry elements. Examples of point groups and their symmetry ele-
ments are listed in the Appendix 3 character tables. The tabulations of point
groups in Appendix 3 contain additional information whose utility will be-
come clearer in the next few sections. For now, you should learn to identify all
the symmetry elements of any given point group.
A real object can possess more than one of the same type of symmetry ele-
ment. For example, benzene has several rotational axes, as shown in Figure
13.6a. In real objects, the proper axis of rotation that has the largest n(an
n-fold axis) is called the principal axis.It is conventional to consider the prin-
cipal axis to be the z-axis in 3-D space. In the identification of axes of rotation,
both directions of rotation (clockwise and counterclockwise) need to be con-
sidered independently, so that a rotation of 90° in a clockwise direction is not
the same as a 90° rotation in the counterclockwise direction.
Finally, there are different types of planes of symmetry. For example, in the
rectangle are three different planes of symmetry, including the plane that the13.2 Symmetry Operations and Point Groups 421S 2C (^2)
Figure 13.4 The scroll pictured has an S 2 symmetry element. It is the combination of a C 2 ro-
tation and a reflection through a plane that is perpendicular to the C 2 axis. The C 2 operation
switches the curls on the scroll, and the resulting reflection returns the curls’ positions back to
their original orientation. The improper axis of rotation does this as a single symmetry element.
Note that although the scroll has an S 2 symmetry element, neither the C 2 axis nor a plane as
indicated are, by themselves, symmetry elements of the object.
and E
H
O
H
v
C 2 v'
Figure 13.5 The four symmetry operations
present in H 2 O. The molecule has the Esymme-
try element by virtue of its existence. Collectively,
these four symmetry operations define the C2v
point group.