Subduction and Induction - Group Theory Applied to Chemistry - page 205

C zv
2 v

C zd
2 v

C 2 v C 2 v D 4 h D 2 d C 4 h

C 2 , 2 C 2 , 2 C 2 , 2 C 2 , 2 C 2 , C 2 ,

σ d ˆ σ v ˆ σ d ˆ σ h , ˆ σ v ˆ σ h , ˆ σ d ˆ

A 1 g A 1 A g A 1 A 1 A 1 A 1

A 2 g A 2 A g A 2 A 2 B 1 B 1

B 1 g B 1 B g A 1 A 2 A 1 B 1

B 2 g B 2 B g A 2 A 1 B 1 A 1

E g E E B 1 + B 2 B 1 + B 2 A 2 + B 2 A 2 + B 2

A 1 u B 1 A u A 2 A 2 A 2 A 2

A 2 u B 2 A u A 1 A 1 B 2 B 2

B 1 u A 1 B u A 2 A 1 A 2 B 2

B 2 u A 2 B u A 1 A 2 B 2 A 2

E u E E B 1 + B 2 B 1 + B 2 A 1 + B 1 A 1 + B 1

D 3 d S 6 C 3 v C 2 h

A 1 g A g A 1 A g

A 2 g A g A 2 B g

E g E g E A g + B g

A 1 u A u A 2 A u

A 2 u A u A 1 B u

E u E u E A u + B u

T d D 2 d C 3 v D 2 C 2 v S 4

A 1 A 1 A 1 A A 1 A

A 2 B 1 A 2 A A 2 B

E 1 + B 1 E 2 A A 1 + A 2 A + B

T 1 A 2 + E 2 + E 1 + B 2 + B 3 A 2 + B 1 + B 2 A + E

T 2 B 2 + E 1 + E 1 + B 2 + B 3 A 1 + B 1 + B 2 B + E

T h D 2 h S 6 C 2 v C 2 h

A g A g A g A 1 A g

A u A u A u A 2 A u

E g 2 A g E g 2 A 1 2 A g

E u 2 A u E u 2 A 2 2 A u

T g B 1 g + B 2 g + B 3 g A g + E g A 2 + B 1 + B 2 A g + 2 B g

T u B 1 u + B 2 u + B 3 u A u + E u A 1 + B 1 + B 2 A u + 2 B u