Enumeration of Kekule Structures

The computational methods developed to deal with the various (explicitly correlated) resonance-theoretic models turn out to be most powerful for the more highly simplified models, of the preceding section 2. It is emphasized that these more highly simplified schemes need not necessarily entail significant approximation or loss of accuracy when properly parameterized. Moreover, these more simplified schemes often allow general conclusions for whole sequences or sets of molecules. For the higher-level models the manner of solution turns out to 

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The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

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. 

In the book [3] the role of both Kekule and Clar structures in various (contemporary) chemical theories as well as their relevance for practical chemistry were outlined in detail. A recent book [46] by Cyvin and the present author is devoted to the enumeration of Kekule structures of benzenoid molecules. In addition to this, the first volume of Advances in the Theory of Benzenoid Hydrocarbons contains several review articles [47-51] dealing with topics of relevance for our considerations. In order to avoid repetition and overlapping we will just briefly mention the work on the elaboration and application of the John — Sachs theorem for the enumeration of Kekule structures [52-55], the search for concealed non-Kekulean benzenoid systems [2, 56-59], examination of fully benzenoid (=all-benzenoid) systems [60-62, 135] as well as the enumeration of Kekule structures in long and random benzenoid chains [63-65]. 

The neo classification divides all benzenoids into normal (n), essentially disconnected (e) and non-Kekuleans (o), where the n and e systems cover all the Kekuleans. Cyvin and Gutman [26] have advocated for this classification by saying From the point of view of the enumeration of Kekule structures the classification. . . [neo]. . . seems to be a rather appropriate one [94,87] . However, the distinction between Kekulean (closed-shell, non-radicalic) and non-Kekulean (radicalic) benzenoid hydrocarbons was made long before the explicit definition of the neo classification. This practice started with the first (substantial) enumeration of benzenoids in the chemical context by Balaban and Harary [13]. 

Herndon, W.C. (1974b) Resonance theory and the enumeration of Kekule structures./. Chem. Educ., 51, 10-15. 

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. 

Figure 2 The Gordon-Davison scheme for enumeration of Kekule structures in catacondensed benzenoids...

Babid D, Graovac A (1986) Enumeration of Kekule Structures in One-dimensional Polymers. Croat Chem Acta 59 731... 

Chen RS, Cyvin SJ (1989) Enumeration of Kekul Structures - Perforated Rectangles. J Mol Struct (Theochem) 200 251... 

Cyvin SJ (1989) Enumeration of Kekule Structures for Some Coronoid Hydrocarbons — "Waffles". Monatsh Chem 120 243... 

D. Babic and A. Graovac, Enumeration of Kekule structures in one-dimensional polymers, Croat. Chem. Acta 59 (1986) 731.744. 

Sheng" outlined an economical two-vertex elimination method for deciding whether a polycyclic ben-zenoid hydrocarbon has Kekule valence structure, and brothers He Wenjie (mathematician) and He Wenchen (chemist) discuss enumeration of Kekule structures using a matrix corresponding to peak and valley carbon atoms." ° We should also mention the quick and robust method of Kearsley" for assigning CC double bonds in a Kekule valence structure on the basis of an ordered fragmentation of the molecular skeleton. 

Characteristic polynomial and related, refs 950— 957 graph spectra and related, refs 958—960 automorphism, refs 961—966 enumerations of Kekule structures, refs 967, 968 enumeration of walks and related, ref 969 more on Kekule structures, refs 970— 977 Pauling bond orders, refs 978—980 more on Clar structures, refs 981—990 aromaticity, refs 991—994 and fullerenes, refs 995, 996. 

One area in which the novel definition of Clar structures may have an advantage over the geometrical counterpart is in computer manipulations with Kekule and Clar structures. There are several algorithms and computer programs that enumerate, and even construct, all Kekule valence structures for benzenoid hydrocarbons [9]. These programs can now be combined with evaluation of the degree of freedom of Kekule structure, and such information can be combined into a scheme to produce list of Clar valence structures. 

The ElS-concept was much used for the enumeration and classification of benzenoid systems [18-21]. It has been demonstrated recently [22] that EIS contains information about the existence/nonexistence of Kekule structures in the respective benzenoid molecule. An application of EIS is found in Theorem 13. 

This is the number of Kekule structures in an aromatic system [Trinajstic, 1992], It can be calculated by extensive enumeration of the structures or by using appropriate algorithms. 

Individual formal valence structures of conjugated hydrocarbons are excellent substrates for research in chemical graph theory, whereby many of the concepts of discrete mathematics and combinatorics may be applied to chemical problems. The lecture note published by Cyvin and Gutman (Cy-vin, Gutman 1988)) outlines the main features of this type of research mostly from enumeration viewpoint. In addition to their combinatorial properties, chemists were also interested in relative importance of Kekule valence-bond structures of benzenoid hydrocarbons. In fact, as early as 1973, Graovac et al. (1973) published their Kekule index, which seems to be one of the earliest results on the ordering of Kekule structures These authors used ideas from molecular orbital theory to calculate their indices... 

Overall the general recursion of eqn. (2) is applicable beyond the case of Kekule structures here elaborated for illustration. The related so-called conju-gated-circuit method turns out to have quite neat (related) linear re-cursions. Generally many sub-graph enumeration problems turn out to be of a linear recursive nature. 

Thus Ising-type graph enumeration on is equivalent to Kekule-structure enumeration on and indeed such correspondences have sometimes been used. But again correspondences between different types of enumerations seem to be quite frequent. The correspondence of Kekule structures to sets of mutually self-avoiding walks as noted in passing in Section 2.2 in connection with the John-Sachs method of enumeration provides yet another example of such a correspondence. 

Counts of Resonance Structures and Related Items. - Resonance-theoretic based enumerations seem to have been somewhat less studied dining the last two years, though a deeade or two ago there were tremendous niunbers of papers. For the case of Kekule-structure enumeration the methods we deem more powerfijl or elegant have been briefly indieated in Seetions 3.2 and 3.3. Perhaps the eurrent relative quietness of the field indieates that now developed methods are near optimum. StiU there has been some work. 

Other Enumerations. - Evaluations of permanents have been pursued by Cash (such permanents being involved in several different enumerations, including that of Kekule structures as mentioned in Section 2.5). He finds efficient means for their evaluation for matrices of up to 80 rows and columns, at least if the matrices have some sparsity. 

The permanent (also referred to as the positive determinant) of the vertex-adjacency matrix per A can be used to enumerate the number of Kekule structures K, or in the graph-theoretical terminology 1-factors (Harary, 1971 Cvetkovid et al., 1995) or dimers (Percus, 1969, 1971 Cvetkovid et al., 1995), of alternant structures (Mine, 1978 Cvetkovid et al., 1972, 1974a Kasum et al., 1981 Schultz et al., 1992 Cash, 1995 Torrens, 2002 Jiang et al., 2006) ... 

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

Kekule valence structures in lattices of peri-condensed benzenoids. Enumeration of Kekule valence structures in benzenoid lattice structures requires a modification of the algorithm of Gordon and Davison, which was introduced in chemical literature by Randic [22], Let us consider dibenzocoronene, the first structure in Figure 2.6. 

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