Benzene isomer counting

The Mills-Nixon hypothesis that small ring annelation on benzene would induce bond fixation (bond alternation) by trapping out one Kekul6 tautomer is a casualty of early twentieth century structural chemistry. Due to a lack of direct methods for analyzing molecular structure, structural postulates of that time were often supported by an analysis of product distributions. An experimental observable such as product selectivity or isomer count was correlated to an unobservable structural feature derived on the basis of a chemical model. Classical successes of this method are van t Hoff s proof of the tetrahedral carbon atom and Fischer s proof for the configuration of sugars. In the case of Mills and Nixon, however, the paradigm broke down. 

The technique of isomer counting used by Wemer to prove the configuration of cobalt-ammines and other complexes did not originate with him, but had been used earlier by Wilhelm Kdmer for benzene derivatives in 1874, and was also suggested by Jacobus Henricus van t Hoff in 1875. However, the technique of comparing the number and type of isomers actually prepared with the number and type theoretically predicted for different configurations probably attained its zenith with Wemer. Not only did he use this method to discredit completely the chain theory bnt also to... 

Figure 5 Count of k-chloro-benzene isomers with Polya s theorem.

The application of P6lya s theorem can be illustrated by its use for the counting of substituted benzene isomers. The cycle index (equation S) of the planar hexagonal skeleton can be written as follows using the permutation group ... 

In order to obtain the number of benzene isomers with one type of substituent (e.g., the numbers of isomeric mono-, di-, and trichlorobenzenes) the following equation is used for the figure-counting series ... 

Interestingly, cyclopropane and benzene have the same number of isomers for a given number of identical substituents ignoring optical isomers, cyclopropane and benzene have one type of monosubstituted derivative, three types of disubstituted, three types of trisubstituted, three types of tetrasubstituted and one type apiece of penta- and hexasubstitution. But does this seemingly accidental counting equivalence have any thermochemical consequences ... 

A more complicated example is to count the isomers of substituted benzene. Here the problem of symmetry becomes sound. We start now to elucidate the symmetries of the planar hexagon. 

The number of benzene substitution isomers with formula C6Hg, tXt is obtained by substituting equation (87) in the cycle index for benzene, equation (89). This substitution gives the counting polynomial ... 

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