Novel Definition of Clar Structures

With use of the concept of the innate degree of freedom, or shortly degree of freedom, we can characterize Clar s structures as follows Clar 

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  

Definition (mathematical) Clar valence structure is obtained by superposition of k-sextet structures having the maximal k possible. 

Definition (geometrical) Clar valence structure is constructed by placing the maximal number of disjoint circles inside benzene rings of benzenoid system subject to completion of valence structure by assigning CC double bonds if necessary. 

Conjecture The mathematical definition of Clar structure and the geometrical (constructive) definition are equivalent. 

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The novel definition of Clar structures to be outlined in the next section, is based on the concept of the degree of freedom (df) of a... 

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. 

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