Structure counting

In Fig. 8 the preferred sites for the unpaired electron are estimated from the numbers of resonance structures counted when the electron is fixed at that site. The structural differences between Figs. 6 and 7 can now be understood. In addition the similarity between type uj and aj is explained and also its marginally more stable centre a.i. Also the similar properties of and 63 are rationalised as is the expectancy that is a more likely structure than c. 

A review on the topological indices can hardly be complete. We had to omit here some indices like structure count (SC) and algebraic structure count of Herndon 91) which are related with the characterization of -electron molecules. 

Herndon s structure count ratio (SCR) [24] derived from VBSRT [25] is defined as... 

More recently, correlations of a values with purely topological reactivity indices, vie. structure count ratio and Dewar reactivity number have been extensively studied by v. Szentpaly and Herndon [33, 58], Rather satisfying correlations were obtained the correlation coefficients for obvious reasons (see Sect. 4) being nearly identical for both indices (0.959 and 0.960 respectively, sample size 27). A significant improvement was achieved with Dewar reactivity numbers calculated according to the free electron version of the PMO treatment (correlation coefficient r = 0.973). 

To cast some light on the relative importance of steric effects on the positional reactivities of benzenoid hydrocarbons, correlations of experimental a values of phenanthrene (4), tetrahelicene (5), pentahelicene (<5), and hexahelicene (7) with purely topological reactivity indices (Huckel cation localization energy, Dewar reactivity number and Herndon structure count ratio) have been studied [59],... 

Experimental log k2 values were correlated with Brown para-localization energies, Dewar reactivity numbers, Herndon structure count ratios, Hess-Schaad resonance energy differences, indices of free valence, and second-order perturbation stabilization energies. The latter are based on Fukui s frontier orbital theory [67] which classifies the Diels-Alder reaction of benzenoid hydrocarbons with maleic anhydride as mainly HOMO (aromatic hydrocarbon)-LUMO (maleic anhydride) controlled. However, the corresponding orbital interaction energy given by... 

Correlations of experimental rate constants of benzogenic Diels-Alder reactions with Polansky (butadienoid) character orders [74] and Dewar reactivity numbers [75] have been observed. Herndon structure-count ratios, however, proved to be superior [76]. The standard deviation of the linear correlation between log k2 and structure count ratio is 0.483, i.e. goodness of fit is much less compared to that of plots obtained... 

Yokono et al. [85] have suggested that the results obtained by Lewis and Edstrom [84] can be understood in terms of the maximum value of the index of free valence as calculated by the HMO method. However, as Herndon [30] has shown, some discrepancies occur when the free valence approach is applied to the experimental findings. He found that the structure count ratio for the single position in each compound that would give rise to the most highly resonance stabilized radical is a reliable reactivity index to correlate and predict the qualitative aspects of the thermal behaviour of benzenoid hydrocarbons. 

As usual K B is used to denote the Kekule structure count of a (generalized) benzenoid system B. 

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 Kekule structure count (or number of Kekule structures) is denoted by K. More specifically, when B is a benzenoid, then K B is used to identify its Kekule structure count. Thus, for instance, K (Rf(3,3) = 64andK Rj(3,3) = 650 cf. Fig. 1. 

A prolate rectangle, R (m, n), is an essentially disconnected benzenoid [1-3]. Hence the Kekule structure counts are easily obtained by... 

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]... 

The famous John-Sachs theorem [26] gives the Kekule structure count of a benzenoid in terms of an np x np determinant, where np is the number of peaks [27], equal to the number of valleys [27]. As pointed out by Gutman and Cyvin [28] the elements of this John-Sachs determinant may be identified with K numbers of certain benzenoids, occasionally degenerated to an acyclic chain (polyene), or zero. An application to the oblate rectangles gave the result [2,13] ... 

Powers of the transfer matrix account for propagation between local states that are more distant. That is, (g T p) gives the number of ways of propagating from p) across / cells to q). To count Kekule structures on Rj(/n, n) we note that there are m — 1 cells and that any one of the local states may occur at the boundaries of the initial and final cells. Thus the total Kekule structure count is... 

A study of the Kekule structure counts for perforated rectangles gave the result ... 

Essay scorers are trained to use a holistic approach, meaning they consider the essay as a whole, rather than word-by-word. Big issues, such as organization and structure, count more than little ones, such as an errant spelling mistake or extraneous comma. That means essays receiving a twelve may have a couple of mechanics errors. 

The recursions of the preceding section can be alternatively cast into an especially elegant form for polymer graphs. The Kekule-structure count KL for a polymer chain of length L monomers can [139-144] quite generally be cast into the form of a trace... 

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... 

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). 

In [tpCBj, ix)/cp(B2, ix)] tends to 2 In [K B1 /K B2 ]. Nevertheless, the assertion [68] that the energy difference of benzenoid isomers is proportional to the difference between the logarithms of their Kekule structure counts was not confirmed by later investigations (see Sect. 7.2). 

The results of enumerations and classifications of polyhexes are reviewed and supplemented with new data. The numbers are collected in comprehensive tables and supplied with a thorough documentation from an extensive literature search. Numerous forms of the polyhexes are displayed, either as dualists or black silhouettes on the background of a hexagonal lattice. In the latter case, the Kekule structure counts for Kekulean systems are indicated. Emphasis is laid on the benzenoid systems (planar simply connected polyhexes). 

Another important quantity for a benzenoid is the Kekule structure count or K number. A Kekule structure, being a typical concept from (mathematical) chemistry, corresponds to a 1-factor or perfect matching in mathematics. 

In Fig. 18 the forms of the Dik catacondensed benzenoids up to h = 25 are displayed. They have been given previously [79], Being catacondensed, all these systems are normal and therefore Kekulean. The Kekule structure counts (K) are given in the figure. 

A substantial amount of additional enumeration data for normal benzenoids and some data for essentially disconnected benzenoids are available, but shall not be reproduced here. They were produced in the course of the extensive studies of the distribution of K, the Kekule structure count. 

Much information can be extracted from Knop et al. [5], where all benzenoids with h < 9 are depicted. These computer-generated pictures are ordered according to the numbers of internal vertices (n,) within each h value. The Kekule structure counts are indicated (K > 0 for Kekulean and K = 0 for non-Kekulean systems). In Tables 2 and 3 this reference is quoted in appropriate places for some total Kekulean and total non-Kekulean systems. We have not taken into account the corresponding mammoth listing for h = 10, on which it was informed by Knop et al. [44]. It was stated that a very limited number of copies were available for distribution in 1984. We are not in the possession of any of these copies. 

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