By Trjitzinsky W. J.
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Extra resources for A Property of Indefinitely Differentiable Classes
Using known values for KA and KB, one can calculate the “end” constants A and B and then derive KN for a polymer having N units. For more details, consult refs 480-485. I. 486-489 In such situations, the Kekule´ structure counts follow a recursion, KN+1 ) AKN + BKN-1 from which one can quickly find KN+1 once the constants A and B (which depend on the “end” group for the polymers considered) are known. For example, for the benzenoid polymer shown in Figure 40, with K1 ) 20, K2 ) 198, K3 ) 1960, and K4 ) 19 402, one obtains a set of linear equations,483 198A + 20B ) 1960 1960A + 198B ) 19 402 which gives A ) 10 and B ) -1 and leads to the recursion KN+1 ) 10KN - KN-1 The coefficients A and B differ only in sign from the coefficients of the secular equation det| T - λI | ) 0 given earlier.
Types to connecting CC bonds (top of Figure 39), the K for the monomer unit is that of a phenanthrene fragment with K ) 5, but with DS,SD assignment the monomer unit becomes a diphenylene fragment with K ) 4, etc. In this way we obtain the transfer matrix shown at the bottom of Figure 39. For any finite molecule which is a member of a family that is based on the same repeating monomer unit, the number of Kekule´ valence structures is given by the formula KN ) A(λ1)N + B(λ2)N Figure 39. Construction of the 2 × 2 transfer matrix for a tetrabenzanthracene polymer.
The conjugated circuits model, as has been recognized by Herndon,540 and Schaad and Hess,541,542 can be cast in a format that is mathematically equivalent to the Simpson-Herndon VB approach.