Enumeration of Conjugated Circuits

C.-D. Lin, Efficient method for calculating the resonance energy expression of benzenoid hydrocarbons based on the enumeration of conjugated circuits, J. Chem. Inf. Comput. Sci. 40 (2000) 778-783. 

Use of Signed Matrices. The transfer-matrix method, the approaches using recursive relations, and the method based on signed matrices all can be suitably modified to yield the count of conjugated circuits. We will mention only the use of signed matrices for enumeration of conjugated circuits. 

Because Kekule valence structures are central to the concept of conjugate circuits, we will follow with a discussion of the properties of Kekule valence structures, their construction, and their enumerations. We will be relatively brief and would like to direct the attention of readers to the book by Cyvin and Gutman on Kekule structures in benzenoid hydrocarbons, where the enumeration and properties of benzenoid structures are discussed on over 300 pages. Kekule valence structures have been around... 

Figure 46. Enumeration of linearly independent conjugated circuits for 10 symmetry-non-equivalent Kekule valence structures of benzo[ / i]perylene by inscribing integers that indicate the size of conjugated circuits involving individual rings. Under each structure is given the number of Ri, Rz, Rs, and R4 conjugated circuits.

Thus many of the Kekule-structure enumeration methodologies of section 4 have been shown to rather neatly extend to conjugated-circuits enumerations, with but modest trouble beyond the overall Kekule-structure count K(G). 

As it turns out though even the simplified VB-theoretic formulations giving rise to conjugated-circuit theory or even just Kekule-structure enumeration, may become challenging for sufficiently large (perhaps formally infinite) systems, or for non-Kekulean (i.e., radicaloid) systems. It might oft be convenient if explicit enumeration of Kekule structures could be avoided. Notably for such cases there are some few alternative sorts of means by which to obtain some partial information about the system, within a VB-theoretic context. 

The successful accomplishments of Miillen and coworkers [22-25] who synthesized several giant benzenoid hydrocarbons will undoubtedly stimulate further theoretical interest in benzenoid hydrocarbons. It is not surprising that all the giant benzenoids that have been synthesized have 6n jt-electrons, which Clar predicted to be unusually stable. Now that the inverse problem of Clar structures has been solved we may expect novel theoretical developments in this area that may continue to expand experimentally beyond expectations. For example, the Conjugated Circuit Model, that has already been applied to giant benzenoids [26-28], may have to be modified so to take into account the prominent role of the Clar structures of benzenoids rather then considering all Kekule valence structures as equally important. Construction and enumeration of giant benzenoids and their Kekule valence structures has also received some attention [29, 30]. 

This particular class of hydrocarbons has lead to numerous investigations and probably deserves an entire chapter to be properly reviewed. Here we summarize only the major findings related to covmting. The reader further interested by polyhexes and benzenoids can consult the books of Gutman and Cyvin " as well as the books of Dias. These books, as well as that by Trinajstic," ° provide valuable information regarding the counting and enumeration of Kekule structures and the conjugated-circuit model, neither of which is reviewed here because of space limitations. 

In the next step, conjugated circuits of size 4n + 2 are assigned counterclockwise current circulation, and conjugated circuits of size 4n are assigned clockwise current circulation. When all such currents are enumerated and added for all Kekuld valence structures, one obtains the current pattern shown at Figure 12.1b. 

Find a mathematical rationale for the close parallelism for the ring currents in benzenoid and non-benzenoid systems calculated using classical physics and obtained by enumeration of contributions from conjugated circuits. 

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