Physical Chemistry , 1st ed.

rectangle is lying in. In real objects, planes that contain the principal axis are

considered vertical planesof symmetry and are given the symbol v. Planes of

symmetry that are perpendicular to the principal axis are called horizontal

planesand are symbolized h. If a vertical plane lies exactly in the middle be-

tween two C 2 axes that are perpendicular to the principal axis, it is called a di-

hedral planeand is given the symbol d. (The presence of dihedral planes im-

plies a “higher” symmetry than regular vertical planes.) Figure 13.6b shows

some of the different types of planes in the benzene molecule; not all of the

planes of symmetry are shown, for the sake of clarity. In order to standardize

the labeling of multiple symmetry elements of the same kind in highly sym-

metric systems, different axes and planes are sometimes differentiated with sin-

gle and multiple primes.

It is easy to think of symmetry operations as spatial motions of objects that

reproduce the original object. However, symmetry operations can be defined

mathematically. Consider the point (3, 4) in 2-D Cartesian coordinates shown

in Figure 13.7. Reflection through the y-axis moves this point to (3, 4). The

xcoordinate has changed sign, as shown in Figure 13.7. Although we can con-

clude that reflection through the y-axis, labeled y, acts to change the sign of

the xcoordinate, we need a more general mathematical definition ofy.Matrix

algebra is useful for this. A simple diagonal matrix is used to define the sym-

metry operation so that the multiplication of the symmetry operation and the

original coordinates generates the new coordinates. In this case, we would have



where the coordinates are written in columns (“column matrices” or “column

vectors”). Standard matrix multiplication generates the new coordinates. This

implies that the particular symmetry operation yis defined as

y (13.1)

and that this symmetry operation is operating on the original coordinates to

generate new coordinates. That is, symmetry operations act as mathematical

operators.In general, the performance of any symmetry operation on a point

can be represented as the matrix multiplication of the symmetry operation

(written in square-matrix form) on that point (written in column vector

form).

This example considers a single point on a graph. Now consider our rec-

tangle, superimposed about the origin in two-dimensional Cartesian space.

Operate on every pointof the rectangle with yand consider the new shape. It

is the same as the original shape, as shown in Figure 13.8. Therefore we can say

that the rectangle has the symmetry element y, whose operation is defined by

the above two-dimensional expression.

For real systems, three-dimensional space is considered. All symmetry op-

erations can be defined by a specific 3 3 matrix. As such, all symmetry op-

erations can act as operators on a set of points to generate a new set of points.

If the new set of points is in exactly the same position as the original, then that

set of points is said to contain the corresponding symmetry element. Table 13.1

lists the matrices that define the symmetry operations. In molecules, the atomic

positions will represent our points in three-dimensional space. Instead of us-

ing a 3 3 matrix to describe the symmetry operation, each atom will require

a 3 3 “block” of a larger matrix to describe its change in position in space

0

1

 1

0

 3

4

3

4

0

1

 1

0

422 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

C 6

C 2

C 2

C 6

h

v

(b)

(a)

y

y

x

 1

0

0

[ 1

 3

[] 4

3

4 ]][

Figure 13.7 Symmetry operations can also be

defined mathematically. Here, the reflection of the

point (3, 4) through the y-axis is equivalent to

the product of two matrices, one representing

the point and one representing the symmetry

operation.

Figure 13.6 (a) The benzene molecule has one

sixfold principal axis of symmetry,C 6. It also has

several C 2 axes of symmetry perpendicular to the

principal axis, in the plane of the molecule. Only

two are shown. Can you find all of the other axes

of symmetry? (b) Reflection planes are vertical

planes if they contain the principal axis of sym-

metry, and horizontalplanes if they are perpen-

dicular to the principal axis. There are also dihe-

dralplanes, which are vertical planes that bisect

two intersecting C 2 axes of symmetry. In the ben-

zene molecule, the vertical planes are actually di-

hedral planes.