C.2 Induction: H ↑ G
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C.2 Induction: H ↑ G
Ascent in symmetry tables have been provided by Boyle [4]. Fowler and Quinn
have listed the irreps that are induced by σ -, π -, and δ-type orbitals on molecular sites [5]. These tables are reproduced below. They are useful for the construction of cluster orbitals. Γ reg always denotes the regular representation. Γ σ corresponds to the positional representation. The mechanical representation is the sum
Γ σ + Γ π .
GH
G
H
Γ σ
Γ π
Γ δ
D 2
C
z
2
2
A + B 1
2B 2 + 2B 3
2Γ σ
D 3
C 2
3
A 1 + E
2A 2 + 2E
2Γ σ
D 4
C 4
2
A 1 + A 2
2E
2B 1 + 2B 2
C ′
2
4
A 1 + B 1 + E
2A 2 + 2B 2 + 2E
2Γ σ
C ′′
2
4
A 2 + B 2 + E
2A 2 + 2B 1 + 2E
2Γ σ
D 5
C 5
2
A 1 + A 2
2E 1
2E 2
C 2
4
A 1 + E 1 + E 2
2A 2 + 2E 1 + 2E 2
2Γ σ
D 6
C 6
6
A 1 + A 2
2E 1
2E 2
C ′
2
4
A 1 + B 1 + E 1 + E 2 2A 2 + 2B 2 + 2E 1 + 2E 2 2Γ σ
C ′′
2
4
A 1 + B 2 + E 1 + E 2 2A 2 + 2B 1 + 2E 1 + 2E 2 2Γ σ
C 2v C xz
s
2
A 1 + B 1
Γ reg
Γ reg
C 3v C s
3
A 1 + EΓ reg
Γ reg
C 4v C v
s
4
A 1 + B 1 + EΓ reg
Γ reg
C d
s
4
A 1 + B 2 + EΓ reg
Γ reg
C 5v C s
5
A 1 + E 1 + E 2
Γ reg
Γ reg
C 6v C v
s
6
A 1 + B 1 + E 1 + E 2 Γ reg
Γ reg
C d
s
6
A 1 + B 2 + E 1 + E 2 Γ reg
Γ reg
C 2h C 2
2
A g + A u
2B g + 2B u
2Γ σ
C s
2
A g + B u
Γ reg
Γ reg
C 3h C 3
2
A ′ + A ′′
E ′ + E ′′
E ′ + E ′′
C s
3
A ′ + E ′
Γ reg
Γ reg
C 4h C 4
2
A g + A u
E g + E u
2B g + 2B u
C s
4
A g + B g + E u
Γ reg
Γ reg
C 5h C 5
2
A ′ + A ′′
E ′
1 + E ′′
1
E ′
2 + E ′′
2
C s
5
A ′ + E ′
1 + E ′
2
Γ reg
Γ reg
C 6h C 6
2
A g + A u
E 1g + E 1u
E 2g + E 2u
C s
6
A g + B u + E 2g
Γ reg
Γ reg
+ E 1u
S 4
C 2
2
A + B
2E
2Γ σ
S 6
C 3
2
A g + A u
E g + E u
E g + E u