By Randic M.

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410 A question can be raised: When does a chemical formula CnHm represent a benzenoid hydrocarbon? J. R. Dias developed an algorithm for deciding when the formula CnHm is compatible with a benzenoid hydrocarbon. He then arranged the possible entries in a table called the “Formula Periodic Table”, that we have already mentioned,328-336 which is illustrated in Table 11 for the case of benzenoid hydrocarbons. Similar tables have been constructed for non-benzenoid polycyclic hydrocarbons,411-414 for coronoid hydrocarbons,415 etc.

Use of Signed Matrices Before we outline Kasteleyn’s490,491 ingenious method for counting the number of Kekule´ structures that is based on the signed matrix of a graph, that is, matrices of graphs which may have positive and negative entries, we have to introduce a few basic definitions of graph theory. We have already defined a bipartite graph as a graph in which one can partition the vertex set into two subsets, V* and V 0, so that only vertices belonging to different sets are connected. We can now consider equitable bipartite graphs, defined as follows: Definition: A bipartite graph is equitable if the number of vertices in the two subsets V* and V 0 is equal, that is, n* ) n0.

A. Semi-empirical Valence Bond Approaches for Benzenoid Hydrocarbons 1. Pauling−Wheland Valence Bond Approach The Pauling-Wheland valence bond approach is based on considering a set of canonical valence structures for a molecule, known also as Rumer diagrams,130 in which no crossing of valence lines representing pairing of π-electrons occurs. In the case of benzene, Rumer diagrams are, in fact, the two Kekule´ structures and three Dewar structures (Figure 48). 524 Matrix elements between different canonical structures then reduce to the form 1/2n(qQ + rR) where Q and R are Coulomb and exchange integrals, respectively, and q and r count the occurrence of Q and R terms.

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