6.10 Application: Bonding Schemes for Polyhedra
155
Fig. 6.9 Edge bonding in electron-precise trivalent cages. The valence shell splits into an occupied
set of localized edge-bonds, and a matching virtual set of edge-anti-bonds. The sets may be further
differentiated by use of the symmetry theorems
objects. Hence, this is not only the mechanical representation with three displacements on each vertex, but it is equally well the symmetry of a set of sp 2 hybrids on
every vertex, directed along the three edges. Along each edge the hybrids at either
end can be combined in a local bonding and anti-bonding combination. The corresponding induced representations are respectively: Γ σ (e) and Γ (e); hence, the
symmetry extension of Eq. (6.140) reads:
Trivalent: Γ σ (v) × Γ T = Γ σ (e) + Γ (e)
(6.141)
Edge Bonding in Trivalent Polyhedra
The understanding of the bonding schemes in polyhedra is based on the correct
identification of the local hybridization scheme on the constituent fragments. Trivalent polyhedra are often electron-precise: this means that the fragment has three
electrons in three orbitals, which are available for cluster bonding and give rise to
edge-localized σ -bonds. Such is the case for the methyne fragment, CH, forming
polyhedranes, but equally well for the isolobal [24] organo-transition-metal fragments such as M(CO) 3 , where M is a d 9 metal such as Co, Rh or Ir. Figure 6.9
shows the bonding pattern based on such electron-precise fragments. As indicated
before, the orbital basis corresponds to the fibre representation Γ σ (v)×Γ T , and contains 3n orbitals. Local interactions along the edges will split this orbital basis into
an occupied σ -bonding half and a virtual σ -anti-bonding counterpart, transforming
as Γ σ (e) and Γ (e), respectively. This is precisely the result of Eq. (6.141). Now,
155
Fig. 6.9 Edge bonding in electron-precise trivalent cages. The valence shell splits into an occupied
set of localized edge-bonds, and a matching virtual set of edge-anti-bonds. The sets may be further
differentiated by use of the symmetry theorems
objects. Hence, this is not only the mechanical representation with three displacements on each vertex, but it is equally well the symmetry of a set of sp 2 hybrids on
every vertex, directed along the three edges. Along each edge the hybrids at either
end can be combined in a local bonding and anti-bonding combination. The corresponding induced representations are respectively: Γ σ (e) and Γ (e); hence, the
symmetry extension of Eq. (6.140) reads:
Trivalent: Γ σ (v) × Γ T = Γ σ (e) + Γ (e)
(6.141)
Edge Bonding in Trivalent Polyhedra
The understanding of the bonding schemes in polyhedra is based on the correct
identification of the local hybridization scheme on the constituent fragments. Trivalent polyhedra are often electron-precise: this means that the fragment has three
electrons in three orbitals, which are available for cluster bonding and give rise to
edge-localized σ -bonds. Such is the case for the methyne fragment, CH, forming
polyhedranes, but equally well for the isolobal [24] organo-transition-metal fragments such as M(CO) 3 , where M is a d 9 metal such as Co, Rh or Ir. Figure 6.9
shows the bonding pattern based on such electron-precise fragments. As indicated
before, the orbital basis corresponds to the fibre representation Γ σ (v)×Γ T , and contains 3n orbitals. Local interactions along the edges will split this orbital basis into
an occupied σ -bonding half and a virtual σ -anti-bonding counterpart, transforming
as Γ σ (e) and Γ (e), respectively. This is precisely the result of Eq. (6.141). Now,