Kekule Valence Structures Weights

Occasionally, one comes across benzenoid hydrocarbons with several Kekule valence structures that have the same number of benzene rings with alternating double and single bonds however, some of those structures are distinct, that is, symmetry non-equivalent. Hence, such Kekule valence structures need not be equally important for the aromaticity of these compounds. In order to find out how important individual Kekuld valence structures are, we propose a model in which the relative weight of Kekuld valence structures is given by their contribution to molecular resonance energy (RE). 

FIGURE A21.1 Contributions of linearly independent conjugated circuits of individual Kekule valence structures of benzanthracene to molecular resonance energy. 

With this notation, the count of the conjugated circuits in the seven Kekule valence structures of benzanthracene is as follows  

As one can see, all seven expressions are different. The same is true for the eight cata-condensed non-iso-Kekulean benzenoid built from five fused benzene rings. 

We looked also at non-iso-Kekulean cata-condensed benzenoids built from six fused benzene rings and found two pairs having the same count of all K(K -1) conjugated circuits, but involving R3 rings of different shape. As is well known, conjugated circuits R3 are the smallest conjugated circuits of different shape, which in the case of cata-condensed benzenoids can be the periphery of anthracene and periphery of phenanthrene, which have different shape. 

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A simple new weighting scheme, for example, could be to take the count of rings having three C=C bonds as the weight of Kekule valence structures. The model can... 

FIGURE 3.9 The weights of Kekule valence structures of smaller benzenoid hydrocarbons based on the count of benzene rings with three C=C bonds. 

FIGURE 3.10 The bond contributions to the CC bond orders based on the weighted Kekule valence structures. Bond orders are obtained by dividing the bond contributions by K. ... 

We see from Table 12 that, although initially all Kekule valence structures were assumed to have the same weight, they make visibly different contribu-... 

Figure 120. The four Kekule valence structures of benzoic] phenanthrene, having the maximal weight based on the smallest Pauling bond of CC double bonds.

Graphical modeling can also be useful in representing the elements of the transfer matrix, J], adopted by Klein et ah, [13] Fig 3 shows the five Kekule valence-bond structures of enanthrene and their local states In this case one needs a directed graph with weighted edges and loops ... 

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