Symmetry - Methods of Molecular Quantum Mechanics - page 256

squares of the moduli of the characters be equal to the order h of the
group:

X
R x

ð R Þxð R Þ¼ X
R

2

jxð R Þj ¼ h ð 12 : 25 Þ

i and j of a group, we

2. Given any two irreducible representations G G

have the orthogonality theorem for the characters:

X
R x

ð R Þ

i j

x ð R Þ¼ h d ij ð 12 : 26 Þ

3. This theorem is a particular case of the more general orthogonality
theorem for the components of the representative matrices of the h
elements of the group:

X

h
' i d ij d mm 0 d nn 0

D i ð R Þ mn D j ð R Þ m 0 n 0 ¼ ð 12 : 27 Þ

R

where ' i is the dimensionality of the ith irrep.

12.2.5 Number of Irreps in a Reducible Representation

j occurs in the reducible representa-

The number of times a j a given irrep G

tion G follows from the orthogonality theorem and is given by

h X

1

ð R Þ

j

x G ð R Þ a j ¼

R x

ð 12 : 28 Þ

12.2.6 Construction of Symmetry-adapted Functions

Symmetry-adapted functions transforming as the l -component of the jth

j are obtained by use of the projector

irreducible representation G

h X

ll ¼ ' j

P j

ll R

ð R Þ

j

R D

ð 12 : 29 Þ