Physical Chemistry , 1st ed.

Characters of the irreducible representations of a particular point group can
be multiplied by each other. The reason for doing this will be obvious shortly.
However, only characters of the same symmetry operationsare multiplied to-
gether. For example, within a certain point group, we will multiply the char-
acter of, say,C 3 of the A 2 representation by the character ofC 3 of the E 1 rep-
resentation, but we won’t multiply the character ofC 3 with the character for,
say,h. By performing these multiplications for all classes, it is easy to show
that the irreducible representations themselves constitute a mathematical
group. For example, the closure requirement is easy to illustrate. Further, if the
products of all the symmetry operations of two different irreducible represen-
tations were summed up, this sum would be exactly zero. (When doing this,
one must include the number of symmetry operations in each class. Otherwise,
the form of the character table in Example 13.6 can be used.) A way of stating
this is that the irreducible representations of a point group are orthogonal to
each other. This is a very useful property of the irreducible representations.
The product of the characters of any irreducible representation with itself
equals h, the order of the point group, which equals the number of symmetry
operations in the group. By using the order of the group as a “normalization
constant,” we can also say that each irreducible representation is normalized.
Mathematically, using the symbol ato represent any irreducible representa-
tion and the symbol ito represent the individual characters ofa:

h

1

 

all classes

Nij0 if1 ifiijjfor all for all ii,,jj (13.5)

ofa

where his the order of the point group and Nis the number of symmetry op-
erators in each class. This “orthonormality condition” can be applied to more
than two irreducible representations multiplying each other, which we will
shortly add as a powerful tool to apply to wavefunctions. Although symmetry
operations can be represented by operators, characters are numbers (eigenvalues,
actually), and so their multiplication is commutative.

Example 13.7

a.Show that the individual characters for any two different irreducible rep-

resentations for C3vsatisfy the closure property of groups.

b.Show that the sums of the products of the irreducible representations for

C3vare orthonormal.

Solution

(a)For the individual characters ofE,2C 3 , and 3v, respectively:

A 1 A 2  1  11  11  1  11  1

the A 2 irreducible representation

A 1 E 1  21  11  0  2  10

the Eirreducible representation

A 2 E 1  21  1  1  0  2  10

the Eirreducible representation

These are the only three combinations, since multiplication of characters is

commutative. (A more complete test should include products of the irre-

ducible representations with themselves. Although it is easy to see that A 1 

A 1 A 1 and A 2 A 2 A 1 , the product EEis not as easy to see. We will

13.5 Character Tables 435