Kekule valence structures

Kekule valence structure. The degree of freedom has been defined as the smallest number of choices to be made in the assignment of CC double bonds which determine a Kekule valence structure completely. For example, all three Kekule valence structures of naphthalene have df = 1, because if one assigns CC double bond character to any of the three vertical CC bonds in the naphthalene diagram (Fig. 4) the location of the remaining four CC bonds is completely determined. The same is true for anthracene, tetracene, and other linearly fused benzenoids. 

Fi g. 4 The Kekule valence structures of naphthalene and their innate degrees of... 

In Fig. 7 we show all Kekule valence structures of benzo[ghi]perylene and their degrees of freedom. The first eight structures have df = 3, the next five Kekule valence structures have df = 2, and the last Kekule valence structure has df = I. The individual Kekule valence structures have quite different count of conjugated circuits R, which can be as high as five (in the first Kekule valence structure) and as low as one (in the last Kekule valence structures). A close look at these... 

Fig. 6 A selection of Kekule valence structures of smaller benzenoid hydrocarbons...

Conjecture The degree of freedom of a Kekule valence structure is given by the maximal number of disjoint conjugated circuits. 

Because Clar structures can be viewed as a superpositions of selected Kekule valence structures we will briefly examine pair-wise superposition of the five Kekule valence structures of phenanthrene (shown in Fig. 5). In all we can construct ten combinations illustrated in Fig. 8. First,... 

All pair-wise combinations of the five Kekule valence structures of phenanthrene. 

In Fig. 9 we illustrate all 1-sextets for benzo[ghi]perylene. Under each structure are the labels indicated the Kekule valence structures involved. In all there are twelve 1-sextet structures to be used in the next step for construction of 2-sextet structures shown in Fig. 10. The first... 

This characterization suffice to determine Clar structure if there is but one such structure (the cases illustrated in Fig. 1 and Fig. 2). In order to define Clar structure in a general case we have to find which Kekule valence structure are used more than once in a superposition. The concept of k-sextet structure considered in the previous section allow us to arrive at the mathematical definition of Clar s structure shown below which we are comparing with the geometrical definition of Clar s structure ... 

We will leave the proof of the conjecture to mathematically inclined readers interested in this problem and will focus attention on consequences of the novel definition of Clar structures. If the two definitions of Clar structure are equivalent (as we conjecture) there should be identical consequences and the new definition is not to make a difference. However, the new definition does offer a mathematical characterization of Kekule valence structures involved in construction of Clar structure, something that has been hitherto missing. 

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. 

Kekule valence structures all the combination of Kekule valence structures of interest. The same is also true for finding combinations of Kekule valence structures involved in HH-Clar structures. A better way to obtain the correct combinations of Kekule valence structures instead of construction of various superpositions is to first write down k-Clar structures (or HH-Clar structures) and then decompose them into underlying Kekule valence structures. The apparent difficulty that remains in such an approach, that cannot be avoided, is the case of benzenoids having a large number of Kekuld valence structures which results in large number of decomposition. 

In Fig. 23 we have illustrated several HH-Clar structures of smaller benzenoids and have indicated their vulnerable rings. Each of the Kekule valence structure having a single vulnerable ring will produce the corresponding O-Clar structure (i.e., a Kekule valence structure that qualifies as Clar structure). In Fig. 24 are shown additional HH-Clar structures having besides vulnerable ring additional ji-sextets. These structures will reduce to k-Clar structures when Ji-sextet of the vulnerable... 

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]. 

Figure 3. Conjugated circuits for three symmetry nonequivalent Kekule valence structures of pyrene.

The so-called factor graphs (Figure 4) depict the individual Kekule valence structures by considering C=C bonds as the vertices of a graph that are connected only when separated from other C=C bonds by a single CC bond. " This approach can be viewed as a translation of the familiar Kekule valence structures to an... 

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