Physical Chemistry , 1st ed.

have to wait until section 13.7 to show that EEdoes satisfy the closure

requirement.)

b.For sums of the products of characters:

h

1

 

all classes

NA 1 A 1 

1

6

(1  1  1  2  1  1  3  1 1)



1

6

(1  2 3)  0

h

1

 

all classes

NA 1 E

1

6

(1  1  2  2  1  1  3  1 0)



1

6

(2  2 0)  0

h

1

 

all classes

NA 2 E

1

6

(1  1  2  2  1  1  3  1 0)



1

6

(2  2 0)  0

This shows that the irreducible representations are orthogonal. In addition:

h

1

 

all classes

NA 1 A 1 

1

6

(1  1  1  2  1  1  3  1 1)



1

6

(1  2 3)  1

h

1

 

all classes

NA 2 A 2 

1

6

(1  1  1  2  1  1  3  1 1)



1

6

(1  2 3)  1

h

1

 

all classes

NEE

1

6

(1  2  2  2  1  1  3  0 0)



1

6

(4  2 0)  1

This shows that the irreducible representations are normalized. Irreducible

representations therefore have some of the same properties as wavefunctions.

All point groups have one irreducible representation that has all 1’s for char-

acters. This representation is called the totally symmetric irreducible representa-

tionand is very important in spectroscopy. By convention, it is the first irre-

ducible representation listed in all character tables. Some irreducible

representations have 2 or 3 for the character of the Eelement. (This can be re-

lated to degeneracies, in some cases.) Irreducible representations are labeled af-

ter a system devised by Robert S. Mulliken. Irreducible representations that

have a character for E,E, of 1 are given Aor Blabels; those that have Eequal

to 2 are labeled E(not to be confused with the symmetry operation E), and

those having Eof 3 are labeled T. Subscripts and superscripts are also used

to indicate the character with respect to other symmetry operation. The char-

acter tables in Appendix 3 use the Mulliken system. Another system, using

436 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics