Three-Dimensional Aromaticity in Polyhedral Boranes and Related ...

1126 Chemical Reviews, 2001, Vol. 101, No. 5 King
Figure 5. Types of vertex orbitals participating in the
delocalization of aromatic systems of various types as
classified by their nodalities: (a) The anodal sp hybrid
unique internal (radial) orbitals of a B-H vertex in the
deltahedral boranes; (b) the uninodal p orbital of a C-H
vertex in planar polygonal aromatic hydrocarbons such as
benzene.
in eqs 6a and 7 as applied to surface bonding. This
portion of the chemical bonding topology can be
described by a disconnected graph Gs having 2n
vertices corresponding to the 2n twin internal orbitals
and n isolated K2 components; a K2 component
has only two vertices joined by a single edge. The
dimensionality of this bonding of the twin internal
orbitals is one less than the dimensionality of the
globally delocalized system. Thus, in the case of the
two-dimensional planar polygonal systems, such as
benzene, the pairwise overlap of the 2n twin internal
orbitals leads to the σ-bonding network, which may
be regarded as a set of one-dimensional bonds along
the perimeter of the polygon using adjacent pairs of
polygonal vertices. The n bonding and n antibonding
orbitals thus correspond to the σ-bonding and σ*antibonding
orbitals, respectively. In the case of the
three-dimensional deltahedral systems, the pairwise
overlap of the 2n twin internal orbitals results in
bonding over the two-dimensional surface of the
deltahedron, which may be regarded as topologically
homeomorphic to the sphere. 54
The equal numbers of bonding and antibonding
orbitals formed by pairwise overlap of the twin
internal orbitals are supplemented by additional
bonding and antibonding orbitals formed by the
global mutual overlap of the n unique internal
orbitals. The bonding topology of the n unique
internal orbitals, whether the uninodal p orbitals in
the planar polygonal aromatic hydrocarbons (Figure
5b) or the anodal sp hybrids in the three-dimensional
deltahedral boranes (Figure 5a), can be described by
a graph Gc in which the vertices correspond to the
vertex atoms of the polygon or deltahedron, or
equivalently their unique internal orbitals, and the
edges represent pairs of overlapping unique internal
orbitals. The energy parameters of the additional
molecular orbitals arising from such overlap of the
unique internal orbitals are determined from the
eigenvalues of the adjacency matrix Ac of the graph
Gc using c as the energy unit where c refers to the
Figure 6. Benzene and the corresponding spectrum of Gc
(the C6 cyclic graph).
parameter in eqs 6a and 7 as applied to core
bonding. In the case of the two-dimensional aromatic
system benzene, the graph Gc is the C6 cyclic graph
(the 1-skeleton 55 of the hexagon) which has three
positive (+2, +1, +1) and three negative (-2, -1, -1)
eigenvalues corresponding to the three π-bonding and
three π*-antibonding orbitals, respectively (Figure 6).
The spectra of the cyclic graphs Cn all have odd
numbers of positive eigenvalues 50 leading to the
familiar 4k + 2(k ) integer) π-electrons 56 for planar
aromatic hydrocarbons. The total benzene skeleton
thus has 9 bonding orbitals (6σ and 3π) which are
filled by the 18 skeletal electrons which arise when
each of the CH vertices contributes three skeletal
electrons. Twelve of these skeletal electrons are used
for the σ-bonding and the remaining six electrons for
the π-bonding.
Figure 7a illustrates how the delocalized bonding
in benzene from the C6 overlap of the unique internal
orbitals, namely, the p orbitals, leads to aromatic
stabilization. In a hypothetical localized “cyclohexatriene”
structure in which the interactions between
the p orbitals on each carbon atom are pairwise
interactions, the corresponding graph G consists
of three disconnected line segments (i.e., 3 × K2). This
graph has three +1 eigenvalues and three -1 eigenvalues.
Filling each of the corresponding three bonding
orbitals with an electron pair leads to an energy
of 6 from this π bonding. In a delocalized “benzene”
structure in which the delocalized interactions between
the p orbitals on each carbon atom are described
by the cyclic C6 graph, filling the three
bonding orbitals with an electron pair each leads to
an energy of 8. This corresponds to a resonance
stabilization of 8 - 6 ) 2 arising from the
delocalized bonding of the carbon p orbitals in
benzene corresponding to the two-dimensional aromaticity
in benzene. We will see below how similar
ideas can be used to describe the three-dimensional
aromaticity in deltahedral boranes.
An important question is the nature of the core
bonding graph Gc for the deltahedral boranes BnHn 2- .
The two limiting possibilities for Gc are the complete
graph Kn and the deltahedral graph Dn, and the
corresponding core bonding topologies can be called
the complete and deltahedral topologies, respectively.
In the complete graph Kn, each vertex has an edge
going to every other vertex leading to a total of
n(n - 1)/2 edges. 57 For any value of n, the corresponding
complete graph Kn has only one positive
eigenvalue, namely, n - 1, and n - 1 negative
eigenvalues, namely, -1 each. The deltahedral graph
Dn is identical to the 1-skeleton 55 of the borane
deltahedron. Thus, two vertices of Dn are connected
by an edge if, and only if, the corresponding vertices
of the deltahedron are connected by an edge.