Physical Chemistry , 1st ed.

describe molecular wavefunctions more easily (like the example of the quadri-

lateral and square above). Symmetry considerations are important in approx-

imating molecular orbitals as linear combinations of atomic orbitals. Finally,

we will consider the shapes of molecules, how their shapes are dictated by the

atomic orbitals of the atoms, and how such orbitals can be better described as

combinations called hybrid orbitals. Symmetry is also applicable to such hybrid

orbitals in a natural way.

13.2 Symmetry Operations and Point Groups

Consider the rectangle in Figure 13.1. When you rotate the rectangle by 180°

or radians, the resulting figure looks identical to the original. Imagine an axis

through the center of the rectangle and the shape rotating about that axis by

180°. Such an axis is called an axis of symmetry.A shape like a rectangle has

several axes of symmetry; for each one the rotation occurs about a different

spatial axis. The rotation of the rectangle to generate an equivalent rectangle is

an example of a symmetry operation.A symmetry operation is any movement

of an object that leaves the object looking as it did originally. The axis about

which the rotation occurs is an example of a symmetry element.A symmetry

element is a point, line, or plane (or combination thereof ).

Rotations are not the only simple symmetry operation. Imagine a plane at

right angles to the rectangle, cutting it in half, as shown in Figure 13.2. Reflect

every point on the rectangle through the plane, as if it were a mirror. The orig-

inal shape is present after the reflection. Such a symmetry operation is called

a reflection plane of symmetry.In this case, the plane is the symmetry element

through which reflection occurs.

Consider the point at the center of the rectangle, Figure 13.3. Take every

point on the rectangle and pass it through the center and place it on the op-

posite side of the center, the same distance away. The resulting rectangle looks

the same as it did originally. This symmetry operation is called inversion,and

a center of inversionis the corresponding symmetry element.

The three examples above represent general types of symmetry operations.

Each general type of symmetry operation is given a symbol to represent it.

An axis of symmetry is denoted Cn,where nis the number of times the oper-

ation has to be repeated in order for the object to return to its originalstart-

ing position. It can be shown that n360°/,where is the angle of rotation

needed for the object to look like it originally did. The axis for the rectangle

above is a C 2 axis. Planes of symmetry are given the symbol , and the center

of inversion is indicated by i. (This should not be confused with i, the square

root of1.)

There are two other types of symmetry operations. The first is called the

identity element,represented by E. Everything has Eas a symmetry operation;

it is the symmetry operation due to the object’s very existence. The last sym-

metry operation is an improper axis of symmetry,indicated by Sn.(Cnis more

specifically called a proper rotation.) It is a combination of a Cnrotation (that

is, turning on an axis by 360°/n) followed by reflection through a plane that is

perpendicular to the axis. Figure 13.4 illustrates the Snsymmetry operation.S 1

is equivalent to a symmetry operation, and S 2 is equivalent to a center of in-

version,i. The rotational part of the Snsymmetry element may or may not cor-

respond with an existing axis of symmetry.

Molecules also possess symmetry elements. Depending on the identities and

positions of atoms in a molecule, any molecule will have certain symmetry

420 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

180 °



i

Figure 13.3 A rectangle has a center of inver-
sion, labeled i. Reflection of every point of the
rectangle through the center of inversion pro-
duces a rectangle that is indistinguishable from
the original object. Reflection of only one point
on the rectangle is shown. An object can have
only one center of inversion.

Figure 13.1 A rectangle is a simple example of
an object that has symmetry. For example, rotat-
ing the rectangle 180°, an operation labeled C 2 ,
produces a rectangle that is indistinguishable
from the original object. Can you find two other
axes of rotation for this rectangle?

Figure 13.2 A rectangle has reflection planes
of symmetry, labeled . Upon reflection of all
points of the rectangle through the plane of sym-
metry, the original object is reproduced. The re-
flection of only one point on the rectangle is
shown. Can you find two other reflection planes
of symmetry for the rectangle?