Kekule Structures Counting Method

The most general Kekule-structure-count method of the present type was devised by Kasteleyn [146], though there is slightly earlier work for different special cases [33,147]. This too involves certain matrices, most simply the graph adjacency matrices /4(G) with rows columns that are labelled by the sites of G and elements that are all 0 except those Aab=+ with a b adjacent sites in G. Then Kastelyn shows how for "planar" graphs to set up a "signed" version (G) of this matrix with half of its +1 elements replaced by -1 such that... 

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. 

The fully computerized method is actually a numerical coefficient fitting for polynomials in general, but was developed in connection with Kekule structure counts. It was used to reproduce the K formula (10) for Rj(4, n) [9]. Furthermore, it allowed to proceed to the m value one unit larger, yielding [10]... 

Linear Recursive Methods - Kekule Structure Counting. - A rather... 

Method of the Pfaffian to a Skew-Symmetric Matrix, Consider [12]annulene (see Chart 2), which is a [4k]—cycle, in contrast to [lOjannulene (Chart 1) being a [4k + 2]-cyde. The Kekule structure count is again K = 2, but the adjacency matrix (as well as its submatrices according to the partitioning shown in Chart 2) appear to be singular. Hence eqn. (13) would give zero. A correction is needed. 

There are different methods to evaluate the numerator (T, Hhs F,) in the conjugated-circuits expression for the resonance energy. One approach is simply to go through each Kekule structure identifying and counting up conjugated n-circuits as occur in each. In another approach this numerator is recast as... 

Gordon and Davison136 initiated the path counting method. In the path counting method the number of Kekule structures Kof a benzenoid B is equal to the number of the mutually self-avoiding directed peak-to-valley paths 138,139... 

The first step in the SRT method is the determination of the structure count (SC), which is the number of Kekule structures for a given molecular structure. In one procedure, a polycyclic molecule is constructed by drawing a single chain or ring containing all atoms of the molecule, followed by insertion of lines one at a time until the molecule is complete. With each line insertion the SC is obtained from equation... 

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. 

Anyway, beginning with the 1970s, a number of approaches was put forward, all based on counting or examination of Kekule structures (and only Kekule structures ), all offering reasonably good quantitative prediction of various molecular properties. We provide a short survey of these methods in the subsequent section. 

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