158
6 Interactions
Fig. 6.11 Correlation diagram for C 60 . The Fries and Clar structures are bonding extremes, where
double bonds are either localized on the 30 bonds between the pentagons (Fries), or form isolated
aromatic sextets on the twelve pentagons. The true conjugation scheme is found in between, and is
characterized by six unoccupied levels, which are anti-bonding in the Fries structure and bonding
in the Clar structure, and which transform as rotations and translations. Buckminsterfullerene has
low-lying LUMO and LUMO+1levelsoft 1u (Γ T ) and t 1g (Γ R ) symmetry
Since Γ R = Γ T × Γ ǫ , we could already simplify the face term to a form which
precisely corresponds to the Clar representation:
Γ
f
P
× Γ R = Γ σ
f
P
× Γ T = Γ Clar
(6.145)
The vertex term can be expressed with the help of Eq. (6.141):
Trivalent: Γ σ
v
P
× Γ R = Γ σ
e
P
× Γ ǫ + Γ
e
P
× Γ ǫ = Γ
e
P
+ Γ ⊥
e
P
(6.146)
where the pseudo-scalar irrep turns a σ -object into a circular current, and rotates the
parallel edge current over 90 ◦ . Note that this is applied in the primitive cage, to the
edges of P only. To complete the derivation one final fibre bundle is needed, which
applies to all convex polyhedra:
Γ σ (e) × Γ T = Γ σ (e) + Γ (e) + Γ ⊥ (e)
(6.147)
This result is based on the C 2v site-symmetry of an edge. The translation in this
site has a radial σ -component of a 1 symmetry, and two tangential π -components
of b 1 + b 2 symmetry. The fibre bundle will thus correspond to the induction of
6 Interactions
Fig. 6.11 Correlation diagram for C 60 . The Fries and Clar structures are bonding extremes, where
double bonds are either localized on the 30 bonds between the pentagons (Fries), or form isolated
aromatic sextets on the twelve pentagons. The true conjugation scheme is found in between, and is
characterized by six unoccupied levels, which are anti-bonding in the Fries structure and bonding
in the Clar structure, and which transform as rotations and translations. Buckminsterfullerene has
low-lying LUMO and LUMO+1levelsoft 1u (Γ T ) and t 1g (Γ R ) symmetry
Since Γ R = Γ T × Γ ǫ , we could already simplify the face term to a form which
precisely corresponds to the Clar representation:
Γ
f
P
× Γ R = Γ σ
f
P
× Γ T = Γ Clar
(6.145)
The vertex term can be expressed with the help of Eq. (6.141):
Trivalent: Γ σ
v
P
× Γ R = Γ σ
e
P
× Γ ǫ + Γ
e
P
× Γ ǫ = Γ
e
P
+ Γ ⊥
e
P
(6.146)
where the pseudo-scalar irrep turns a σ -object into a circular current, and rotates the
parallel edge current over 90 ◦ . Note that this is applied in the primitive cage, to the
edges of P only. To complete the derivation one final fibre bundle is needed, which
applies to all convex polyhedra:
Γ σ (e) × Γ T = Γ σ (e) + Γ (e) + Γ ⊥ (e)
(6.147)
This result is based on the C 2v site-symmetry of an edge. The translation in this
site has a radial σ -component of a 1 symmetry, and two tangential π -components
of b 1 + b 2 symmetry. The fibre bundle will thus correspond to the induction of