Kekule structures, benzenoids peri-condensed

The concept of numeric Kekule structures is used for coding and ordering Kekule structures of cfltfl-condensed benzenoids. In the case of peri-condensed benzenoids, the numeric code of Kekule structures is not discriminative enough since there are Kekule structures with identical numeric codes. In that case, an additional code is needed—Milicevic et al. (2004) tested the use of the Wissweser coding system for benzenoid hydrocarbons, and Lucid et al. (2011) tested the perimeter codes. 

FIGURE 2.8 The completion of the count of Kekule valence structures for peri-condensed benzenoid lattices of Figure 2.6. 

It should be noted that the numeric code is a linear code in which the codes for peri-condensed benzenoids are built in the same way as the Wiswesser codes. For example, the numeric code for the Kekule structure VI (see Fig. 9.17) is 45335. 

However, it appears that there exist peri-condensed benzenoids with distinct geometric Kekule structures but with the identical numeric Kekule structures (Gutman et al. 2004 Vukicevic et al. 2004). The smallest example of such a case is pyrene. In Fig. 9.18, we give its six geometric Kekule stmctures and in Fig. 9.19 its six numeric Kekule structures. Note, stmctures denoted by B and C possess identical numeric codes. Therefore, there is no one-to-one correspondence between geometric and numeric Kekule stmctures for some peri-condensed benzenoids. In such cases the recovery of geometric stmctures requires the amendment of the numeric codes. 

In Fig. 9.21, we show how a pair of peri-condensed benzenoid with identical numeric codes can be recovered from their perimeter codes. As an illustrative example, we selected numeric code 5 5 2 2 5 5 which is identified to belong to a pair of Kekule structures of dibenzo[, op]naphthacene. 

Kekule valence structures in lattices of peri-condensed benzenoids. Enumeration of Kekule valence structures in benzenoid lattice structures requires a modification of the algorithm of Gordon and Davison, which was introduced in chemical literature by Randic [22], Let us consider dibenzocoronene, the first structure in Figure 2.6. 

FIGURE 2.9 The Kekule valence structure counts for several smaller peri-condensed benzenoid non-convex lattices considered by Clar in his booklet The Aromatic Sextet. 

The procedure outlined applies to all nonconvex lattices, the inner dual of which is a nonconvex polygon. In Figure 2.9, we have illustrated the Kekule valence structure counts for peri-condensed benzenoid lattices considered by Clar in his booklet The Aromatic Sextet [29]. This time, we only show the result in the top benzene ring leaving it to the readers to flu in the empty benzene rings and verify, as an exercise, that the count shown is correct In the next section, we will briefly outline the count of Kekule valence structures for a more general class of benzenoid hydrocarbons. 

This count includes benzenoid hydrocarbons that would be unstable by having eight or more linearly fused benzene rings. For comparison, the total number of peri-condensed benzenoids with n = benzene rings is about twice the number of cata-condensed benzenoids (A/ = 41 764), to which one should add 78 350 non-Kekulean structures (systems for which one cannot write Kekule valence structures), giving a total of just over 120 000 possible structures for analysis. 

We continue with outlining selected algorithms for counting Kekule valence structures separately for cata-condensed and peri-condensed benzenoids. 

---