Representations - Group Theory Applied to Chemistry - page 73

Knowing the diagonal elements of the induction matrix, we can now calculate
the frequency of a given Γ irrep of the main group, using the character theorem:

χ Γ (g) Tr D

G (g)

1
| G |

H ↑

c Γ (γH ↑ G) =

G ¯ g ∈

g ∈ G ¯

χ Γ (g)
κ

P κκ (g)χ γ g − 1 gg κ

1

= (4.78)

κ

| G |

The only elements g that are allowed in the summation over κ are the ones such ˆ

H A . For other elements, P κκ (g) are zero. Let us denote by h the
subelement that allows us to express

g − κ ˆ that ( ˆ g g κ ) ˆ ∈

ˆ

g as

g κ h g − 1

κ

g ˆ =ˆ ˆ (4.79)

Introducing this substitution in Eq. ( 4.78 ) yields

κ χ Γ g κ hg − κ χ γ (h)P κκ g κ hg − κ

1

c Γ (γH ↑ G) = (4.80)

| G |

h ∈ H

The first character in this equation belongs to the full group and is the same for all
elements of a conjugacy class, and hence,
χ Γ g κ hg − κ = χ Γ (h) (4.81)

Substituting the result of Eq. ( 4.81 ) and the sum rule in Eq. ( 4.75 ) into the character
expression finally gives

κ ¯

χ Γ (h)χ γ (h)P κκ g κ hg − κ

1

c Γ (γH ↑ G) =

| G |

h ∈ H

χ Γ (h)χ γ (h)
κ

P κκ g κ hg − κ

h ∈ H ¯

1

=

| G |

1
| H |

H χ Γ (h)χ γ (h)

=

h ∈

= c γ (Γ G ↓ H) (4.82)

which concludes the proof. Armed with the subduction tables, we can now read
these at once in the opposite sense and obtain the corresponding induction frequen-
cies. As a simple example, consider a hydrogen atom in ammonia. The site symme-
try is C s , and the subduction from C 3 v reads:

A 1 → a

A 2 → b (4.83)

E → a + b