Kekule structures, benzenoids cata-condensed

We first consider the numeric Kekule structures of cata-condensed benzenoids. As an example, we consider dibenzo[b, ]phenanthrene. In Fig. 9.14 we give geometric Kekule stmctures of dibenzo[b,g]phenanthrene and in Fig. 9.15 their numeric counterparts. Kekule structures of dibenzo[b,g]phenanthrene can be generated in... 

From the numbers coding the numeric Kekule structures of cata-condensed benzenoids, a given Kekule structure can be easily recovered because the number of digits in the code determines the number of hexagons in the cata-condensed benzenoid, and the sum of digits gives the total rc-electron count in the benzenoid. 

ENUMERATION OF KEKULE VALENCE STRUCTURES 2.3.1 Cata-Condensed Benzenoid Hydrocarbons... 

FIG U RE 2.6 The count of Kekule valence structures in smaller convex benzenoid lattices by extending the algorithm for cata-condensed benzenoids of Gordon and Davison. 

Clar proposed that the maximum number of localized aromatic sextets (and thus the number of Kekule structures) that can be drawn for benzenoid hydrocarbons correlates well with several properties of the compounds. For example, phenanthrene contains two localized sextets (five Kekule resonance structures), while its isomer anthracene has only one localized sextet (four Kekule resonance structures), and so might be considered to be more aromatic . Of the 4-ring benzenoid isomers, naphthacene has the fewest sextets (one), triphenylene has the most (three), and the others have two apiece. Generally, the cata-condensed species which have more phenanthrene subunits, and thus have greater angularity , also have more localized Clar sextets. How well does the Clar model correlate with the enthalpies of formation ... 

Which isomer among cata-condensed benzenoids has the largest number of Kekule structures ... 

This count includes benzenoid hydrocarbons that would be unstable by having eight or more linearly fused benzene rings. For comparison, the total number of peri-condensed benzenoids with n = benzene rings is about twice the number of cata-condensed benzenoids (A/ = 41 764), to which one should add 78 350 non-Kekulean structures (systems for which one cannot write Kekule valence structures), giving a total of just over 120 000 possible structures for analysis. 

We continue with outlining selected algorithms for counting Kekule valence structures separately for cata-condensed and peri-condensed benzenoids. 

Figure 27. Cata-condensed benzenoid hydrocarbons with the maximal number of Kekule valence structures for a given number of fused benzene rings.

Figure 29. Illustration of the elegant algorithm of Gordon and Davison for the count of Kekule valence structures in non-branched cata-condensed benzenoids.

Figure 31. Fibonacci numbers as the count of Kekule valence structures for families of cata-condensed benzenoids in which every benzene ring is a kink ring.

For families of cata-condensed benzenoids in which every benzene ring is a kink ring (illustrated in Figure 31), the GD algorithm immediately yields Fibonacci numbers 2, 3, 5, 8, 13, 21, 34,. .. as the number of Kekule valence structures for the corresponding cata-condensed benzenoids. Fibonacci numbers first appeared in a book on algebra written by Fibonacci (in the 13 century ) to illustrate the fast growth of a population of rabbits. ... 

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