Physical Chemistry , 1st ed.

due to the symmetry operation. For a molecule having Natoms, this will require

a 3N 3 Nmatrix—but there are some major simplifications, as we shall see.

Example 13.1

Show how the equation yxhas a center of inversion at x0.

Solution

From Table 13.1, the complete matrix definition ofiis

i

Consider any point in the upper right (x,y) quadrant of a standard 2-D

graph. The value ofzin this case is zero, so the 3-D coordinate set is (x,y, 0).

Operating on this point with the inversion symmetry operation:



Is the point (x,y, 0) a point on the line yx? Yes, it is, for any value of

x. Such points are in the lower left quadrant. Therefore, this equation con-

tains a center of inversion. To convince yourself of this, plot the graph and

repeat the example.

13.3 The Mathematical Basis of Groups

We have established two things about symmetry operations. First, they are op-

erators and expressed mathematically in terms of a 3 3 matrix for opera-

tions on a point in 3-D space. Second, we have stated that only certain collec-

tions of symmetry elements, called point groups, are possible for real objects.

The area of mathematics that deals with symmetry and point groups is

called group theory.A groupis a certain collection of operations that satisfies

the following conditions:

• The group must have an identity operationsuch that operation by iden-
  tity does not change the object. This operation must be commutative

x

y

0

x

y

0

0

0

 1

0

 1

0

 1

0

0

0

0

 1

0

 1

0

 1

0

0

13.3 The Mathematical Basis of Groups 423

y

y

x

Figure 13.8 The rectangle’s symmetry opera-
tions can also be defined mathematically. In this
case, the symmetry operation operates on the
rectangle to produced an equivalent rectangle.

Table 13.1 General matrix representations of the five types of symmetry operationsa

E Cn,  (^36) n 0°
z i
Sn,  (^36) n 0°
aMatrices are in 3 3 form. The rotations are assumed to occur about the z-axis, and the plane is assumed to be
the xyplane so that only the zcoordinate is affected.
0
0
 1
sin 
cos 
0
cos 
sin 
0
0
0
 1
0
 1
0
 1
0
0
0
0
 1
0
1
0
1
0
0
0
0
1
sin 
cos 
0
cos 
sin 
0
0
0
1
0
1
0
1
0
0