Benzenoid Lattices

FIG U RE 2.6 The count of Kekule valence structures in smaller convex benzenoid lattices by extending the algorithm for cata-condensed benzenoids of Gordon and Davison. 

FIGURE 2.7 Non-convex peri-condensed benzenoid lattices, for which the count of Kekule valence structures cannot be completed because of the presence of bay regions on the molecular periphery. 

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. 

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The four-ring pyrene, as usually drawn, represents the smallest benzenoid lattice if we exclude cata-condensed benzenoid hydrocarbons. The peri-condensed benzenoids shown in Figure 2.6 illustrate additional smaller benzenoid lattices, which we use to introduce a generalization of the Gordon and Davison algorithm for counting... 

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. 

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. 

We can illustrate on a benzenoid lattice an important one-to-one correspondence between the count of Kekule valence structures and the count of monotone paths. 

A benzenoid system can be defined as follows let C be a cycle on the hexagonal lattice then the vertices and edges lying on C and in the interior of C form a benzenoid system B. 

Q consists of benzenoid subunits joined by acyclic lines on the hexagonal lattice ... 

The results of enumerations and classifications of polyhexes are reviewed and supplemented with new data. The numbers are collected in comprehensive tables and supplied with a thorough documentation from an extensive literature search. Numerous forms of the polyhexes are displayed, either as dualists or black silhouettes on the background of a hexagonal lattice. In the latter case, the Kekule structure counts for Kekulean systems are indicated. Emphasis is laid on the benzenoid systems (planar simply connected polyhexes). 

Doslic135 proved that there is a one-to-one correspondence between the number of Kekule structures in a benzenoid parallelogram and the number of all square-lattice paths from (0,0) to (n,m) with steps (1,0) and (0,1). As illustrative examples we give all Kekule structures of anthanthrene in Figure 7 and the corresponding square-lattice paths in Figure 8. 

Therefore, both approaches, the Pascal recurrence algorithm and the approach based on counting the square-lattice paths are similar. However, the Pascal recurrence algorithm is applicable to a wider range of benzenoids and not only to benzenoid parallelograms. 

El-Basil S (1993) Generation of Lattice Graphs An Equivalence Relation on Kekule Counts of Catacondensed Benzenoid Hydrocarbons. J Mol Struct (Theochem) 288 67-84... 

Two different values of REC for particular rings of benzene and phenanthrene result from different measurements of their molecular geometry. The different REC values for symmetrically equivalent rings in other benzenoid hydrocarbons result from the method of X-ray structure determination. If the symmetrical molecule does not lie at the symmetry element in the crystal lattice, each of the rings is measured independently and its geometry (and in turn its REC value) is biased by the another error of measurement. The differences in REC values for symmetrically equivalent... 

The absolute value of the tail coefficient of the characteristic polynomial (which is known to give K for benzenoids ), however, is only 3069 (Table 3), and this apparent discrepancy is another reminder that these 3D structures differ in important ways from simple planar benzenoids. All conjugated eireuits in the latter are stabilizing,but for tori, the closure of the lattice around the tube allows the presenee of destabilizing circuits in some cases. 

Diffraction curves of EB-II are presented in Figure 1,48. The authors have found the orthorhombic space group Pben to be compatible with the reflections observed (Figure 1.49 this cell is similar to that of PPS, described in Section 7.1.1). In this assigiunent, the difference between aromatic (benzenoid) and quinoid sequences of the emeraldine chain is ignored. These differences may actually show up in the diffraction, as pointed out in a later paper [137]. The lattice constants found for EB-11 powder prepared from EB-1 by successive extractions with THF and NMP are 0 = 7.65 A, 6 = 5.75 A, c= 10.20 A and P=445 A Films cast from NMP and subsequently stretched to... 

Although the alkane and alkane-like substances are the most important, no series of compounds has received as much interest in generating isomer series as has the polyhexes, with much being done on various classes of benze-noids. For example, isomer series are available for benzenoids of a variety of classes, including peri-condensed, cata-condensed, essentially disconnected,helicenes, ° resonant sextets, quinones, coronenes, "" and pyrenes.Recently, with the aid of a new lattice... 

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