Polyhedral three-dimensional aromatic hydrocarbons

In recent years the fundamental ideas of Huckel molecular orbital theory, the Huckel rule, and other aspects of aromaticity have been extended to polyhedral three-dimensional inorganic structures regarded as aromatic like the two-dimensional aromatic hydrocarbons. Such an extension of Huckel molecular orbital theory requires recognition of its topological foundations so that they can be applied to three-dimensional structures as well as two-dimensional structures. In this connection graph theoretical methods can be used to demonstrate the close analogy between the delocalized bonding in two-dimensional planar aromatic systems such as benzene and that in three-dimensional deltahedral boranes, and carboranes. Related ideas can be shown to be applicable for metal carbonyl clusters, bare post-transition metal clusters, and polyoxometallates. ... 

Here TV is the number of vertices in a graph (which corresponds to the number of atoms in a conjugated molecule), Xj are the roots of the characteristic polynomial of the aromatic system, and xf are the roots of the acyclic polynomial of the polyene-like reference system. In essence this corresponds to the procedure of the Hiickel method to solve for the eigenvalues Xj of the Hiickel matrix in units of Finally, gj is the orbital occupancy number. The method was applied to a large number of conjugated hydrocarbons - with results for TREPE that usually show a similar trend as the HSREPE values. Aiha-ra76.77 extended the concept to three-dimensional systems, in particular polyhedral boranes. However, soon afterward, controversial difficulties arose with this approach. ... 

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