The Conjugated Circuit Model

The theory outlined in the present chapter has to be associated with the name of Milan Randid who discovered it (Randic 1976) and eventually elaborated it (Randic 1977a,b) and applied to numerous classes of conjugated molecules (Randic 1980,1982, Randic et al. 1987b, and the references cited therein). In what follows we expose only the conjugated circuit model for benzenoid hydrocarbons. One should, however, note that the model covers a much wider class of conjugated systems (Randic 1977a,b, 1982)... 

It takes some imagination, according to E. B. Wilson, to appreciate novelty, and some recognized the merits of conjugated circuits immediately as soon as they were discovered. Thus Professor D. Hellwinkel of Heidelberg in a letter to the author of the paper on aromaticity and conjugation wrote (in March of 1977)  

I was pleased to receive your letter and your paper, which I have examined with interest. I agree with you that it is better to make rather simple calculations, such as yours, than the very complicated ones. 

Your work on conjugated circuits reminds me of a paper that I wrote on the diamagnetic anisotropy of aromatic molecules. Journal of Chemical Physics 4, 673 (1936)... Again let me thank you for writing to me. 

So apparent confusion about what is aromaticity is not due to a lack of explanation of aromaticity in terms of the presence, or lack of the presence, of conjugated circuits of 4 -I- 2 and 4n size in the set of Kekule valence structures of compounds, but due to (i) lack of attention given to articles describing conjugated circuits aud their use for characterization of aromaticity and (ii) not recognizing that these explanations, in fact, represent a generalization of the famous Hiickel rule to polycycUc systems, which Hellwinkel recognized just as the paper was published, almost 40 years ago. 

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Different circuit counts are designated as <4n+2) or (4,) and are obtained by the summation of all (4n + 2) or An) conjugated circuits. When the conjugated circuits model (CCM) is applied, the RE of a polycyclic conjugated molecule (CCMRE) may be determined as follows, taking into account that the REs are additive within the model in question (76CPL68),... 

A number of computational approaches to the (G) have been developed and there have been widespread applications of the conjugated-circuits model, motivated both from Herndon s and from Randic s approaches. The applications extend even much beyond benzenoids. This is reviewed elsewhere by Randic et al. [76],... 

The conjugated-circuits model is one of the simplest quantitative models that has been reasonably well studied. As already mentioned this model may be motivated from classical chemical bonding theory (extended a la Clar s classical empiricist argument) or from Simpson s existential quantum-theoretic argument [ 121 ], or from a quantum chemical derivation indicated in our hierarchy of section 3.2. But beyond derivation of the model there is the question of its solution, such as we now seek to address. 

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]. 

All the resulting partial orders are consistent with the conjugated-circuits model of Randic (1977). 

However, in the 4, 8 semiregular planar lattice (Fig. 2) both the angle strain and the antiaromaticity contribute to destabilization in the conjugated circuits model there are 10- and 14-membered circuits with small positive contributions R2, Rsas well as 4-, 8- and 12-membered rings with appreciable negative coefficients Qi, Q2, 03. 

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. 

Plavsic D, Nikolic S, Trinajstic N (1992) The Conjugated-Circuit Model - Application to Non—Alternant Hydrocarbons and a Comparison with Some Other Theoretical Models of Aromaticity. J Mol Struct (Theochem) 277 213... 

We introduce the topic of renormalization in chemistry by considering calculation of ring currents in polycyclic conjugated hydrocarbons based on the conjugated circuit model. 

The simple /LA-approach was eventually extended to the conjugated circuit model [66-70]. 

According to the conjugated circuit model, for m = 1, 2, 3,. .., one has to determine the number p of conjugated circuits of size 4 + 2 in all Kekule stmctures of the underlying benzenoid system, and compute the resonance energy as... 

Randid M, Nikolid S, Trinajstic N (1987) The conjugated circuit model on the selection of parameters for computing the resonance energies. In King RB, Rouvray DH (eds) Graph theory and topology in chemistry. Elsevier, Amsterdam, pp 429 147... 

Although the conjugated-circuit model [33] suggested that the linear [N]phenylenes are more stable than their angular isomers, the application of ah initio methods proved the opposite [126]. Schulman and Disch s examination of the problem by modem DFT methods placed the stabilization of 15 vs. 9b at 2.4 kcal mol [53]. Branched [4]phenylene 21b is the most stable of the five... 

Simpson-Herndon model, and the conjugated-circuits model. The calculation of the KSC can be performed with recurrence relationships, matrix methods, and explicit combinatorial expressions derived for a large number of classes of conjugated hydrocarbons various classes of cata-condensed benzenoid hydrocarbons honeycomb lattice strips polymers. 

The conjugated-circuit model is a simple valence-bond resonance-theoretic model introduced for the study of aromaticity and conjugation of polycyclic conjugated systems which can be used to compute their r-resonance energies, RE. This model is a measure of the aromaticity in conjugated systems. Developed initially for polycyclic conjugated... 

On the other hand, if a model is simple, empirical, and parametric, that does not necessarily means that it has no firm, apparently hidden link to generally accepted fundamental laws, such as those of quantum chemistry. Just as the opposite may be the case, a model that is thought to be based on basic axioms may turn out not to reflect this deep connection with quantum chemistry. It thus was found, mostly through the work of D. J. Kleln, that the conjugated circuits model has a quite firm foundation in quantum chemical principles, while as we all know, the Hiickel molecular orbital (MO) model that started as a quantum chemical model turned out to be a consequence of molecular topology, rather than an intricate interaction of r-electrons governed by the Schrodinger equation. 

The conjugated circuits model, as will be seen, offers insight into aromaticity that has been so far missing, and in that sense in our view it will be found... 

According to the statement made by Gutman and Cyvin in their book. Introduction to the Theory of Benzenoid Hydrocarbons, on p 80 in a section entitled The Conjugated Circuit Model The Theory outlined in this chapter has to be associated with the name of Milan Randic who discovered it and eventually elaborated it and applied it to numerous classes of conjugated molecules. ... 

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