6.10 Application: Bonding Schemes for Polyhedra
157
Fig. 6.10 The leapfrog
extension consists of two
operations: first, place an
extra atom in the centres of
all the polygons (middle
panel), then, take the dual.
The result is indicated by the
solid lines in the right panel
It involves two operations, which are carried out consecutively, as illustrated in
Fig. 6.10. One first places an extra capping atom on all pentagons and hexagons.
This leads to a cage which consists only of triangles, and this is a deltahedron. By
taking the dual one restores a trivalent cage. As can be seen, all vertices of the primitive have been turned into hexagons, while the original pentagons and hexagons
are recovered, but in a rotational stagger. The edges of the primitive are also recovered, but rotated 90 ◦ . In summary, the leapfrog operation inserts 6 vertices in
the hexagons of P , and 5 vertices in the pentagons, which, according to the final
expression in Eq. (6.142), multiplies the number of atoms by 3. The first and best
known leapfrog is Buckminsterfullerene, C 60 , which is the leapfrog of the dodecahedron itself. Each carbon atom contributes, besides the sp 2 orbitals, which build
the σ -frame, one radial p r -orbital. These orbitals form π -bonds which control the
frontier orbitals of fullerenes. In the case of the leapfrog, this frontier MO region
is always characterized by six low-lying almost non-bonding orbitals, which, moreover, always transform as Γ T + Γ R . This can be explained with the help of the Euler
rules [27].
As in the case of C 60 , all leapfrogs can be considered to be truncations of the
primitive fullerenes, in the sense that all the faces of the primitive have become isolated islands, surrounded by rings of hexagons. With every bond of the primitive
is associated a perpendicular bond, which always forms a bridge between these islands. Based on this neat bond separation, two canonical valence-bond frames can
be constructed for the leapfrog, which in a sense are the extremes of a correlation
diagram, with the actual bonding somewhere in between. These bond schemes are
known as the Fries and the Clar structures. The Fries structure is an extreme case
where all bridges are isolated π -bonds. The induced representation for these bonds
corresponds to Γ σ (e P ). On the other hand, in the Clar structures the bonding is
completely redistributed to aromatic sextets on the hexagonal and pentagonal islands. The corresponding representation is the fibre bundle Γ σ (f P ) × Γ T .W enow
compare the representations of both bonding schemes, using the symmetry theorems. We start with the main theorem, applied to the primitive P , and multiply left
and right with Γ R .
Γ σ
v
P
× Γ R − Γ
e
P
× Γ R + Γ
f
P
× Γ R = Γ T + Γ R
(6.144)
157
Fig. 6.10 The leapfrog
extension consists of two
operations: first, place an
extra atom in the centres of
all the polygons (middle
panel), then, take the dual.
The result is indicated by the
solid lines in the right panel
It involves two operations, which are carried out consecutively, as illustrated in
Fig. 6.10. One first places an extra capping atom on all pentagons and hexagons.
This leads to a cage which consists only of triangles, and this is a deltahedron. By
taking the dual one restores a trivalent cage. As can be seen, all vertices of the primitive have been turned into hexagons, while the original pentagons and hexagons
are recovered, but in a rotational stagger. The edges of the primitive are also recovered, but rotated 90 ◦ . In summary, the leapfrog operation inserts 6 vertices in
the hexagons of P , and 5 vertices in the pentagons, which, according to the final
expression in Eq. (6.142), multiplies the number of atoms by 3. The first and best
known leapfrog is Buckminsterfullerene, C 60 , which is the leapfrog of the dodecahedron itself. Each carbon atom contributes, besides the sp 2 orbitals, which build
the σ -frame, one radial p r -orbital. These orbitals form π -bonds which control the
frontier orbitals of fullerenes. In the case of the leapfrog, this frontier MO region
is always characterized by six low-lying almost non-bonding orbitals, which, moreover, always transform as Γ T + Γ R . This can be explained with the help of the Euler
rules [27].
As in the case of C 60 , all leapfrogs can be considered to be truncations of the
primitive fullerenes, in the sense that all the faces of the primitive have become isolated islands, surrounded by rings of hexagons. With every bond of the primitive
is associated a perpendicular bond, which always forms a bridge between these islands. Based on this neat bond separation, two canonical valence-bond frames can
be constructed for the leapfrog, which in a sense are the extremes of a correlation
diagram, with the actual bonding somewhere in between. These bond schemes are
known as the Fries and the Clar structures. The Fries structure is an extreme case
where all bridges are isolated π -bonds. The induced representation for these bonds
corresponds to Γ σ (e P ). On the other hand, in the Clar structures the bonding is
completely redistributed to aromatic sextets on the hexagonal and pentagonal islands. The corresponding representation is the fibre bundle Γ σ (f P ) × Γ T .W enow
compare the representations of both bonding schemes, using the symmetry theorems. We start with the main theorem, applied to the primitive P , and multiply left
and right with Γ R .
Γ σ
v
P
× Γ R − Γ
e
P
× Γ R + Γ
f
P
× Γ R = Γ T + Γ R
(6.144)