Single coronoid

The below scheme of classification applies to benzenoids and coronoids, and to the single coronoids in particular. It is consistent with the classifications adopted previously, which are detailed in Vol. I and elsewhere (Gutman and Cyvin 1989 Brunvoll, Cyvin SJ and Cyvin 1991a). 

Classification of Single Coronoids in Rdation to Kekule Structures... 

Whereas the scheme in Sect. 2.1 applies to both benzenoids and single coronoids, a finer classification for the Kekulean single coronoids was found to be needed during the studies of Kekul6 structure counts for these systems. In this way the concept half essentially disconnected (single) coronoids was introduced by Cyvin SJ, Cyvin and Brunvoll (1987). This concept is explained in the book of Gutman and Cyvin (1989) and in Vol. 1-3.3. 

In the first place, the Kekulean single coronoids are classified into normal (n) and essentially disconnected (e) systems. Obviously, n + e accounts for all the Kekulean systems. 

Together with the non-Kekul ans (o), the total number of single coronoids is accounted for. This classification, referred to as the neo classification, applies to benzenoids as well as (single) coronoids. 

It remains to define the regular single coronoids. In order to give an idea of the approach which was adopted (precise definitions are saved for a subsequent chapter), consider the regular coronoid ... 

The purpose of this diagram is twofold (1) It should recall the notation of modes of hexagons cf. Vol. 1-3.2.1, especially I-Fig. 3.1, and elsewhere (Cyvin SJ and Gutman 1988 Gutman and Cyvin 1989). (2) It should illustrate the definition of a regular single coronoid, which implies... 

It was conjectured that the provisional definition of HED single coronoids as the normal systems which are not regular, always would lead to systems reasonable to be dassified as HED ... 

All the non-Kekulean coronoids with h < 14 (for their numbers, see Table 1) are known to be obvious non—Kekuleans. The smallest concealed non—Kekul an (single) coronoids have h = 15, and there are exactly 23 nonisomorphic systems of this category (Cyvin SJ, BrunvoU and Cyvin 1989c Vol. 1-7.3), each of them possessing the naphthalene hole. For the subclass of single coronoids with the phenalene hole it was found that the smallest concealed non—Kekuleans have h = 16, and there are exactly 21 such systems (Cyvin SJ, BrunvoU and Cyvin 1990b). 

Table 2.1. Numbers of single coronoids classified according to rheo ...

Sometimes it is convenient to formulate theorems for a broader class of systems than the single coronoids, but containing the single coronoids as a subclass. 

Definition 2.2 A special degenerate single coronoid is a d enerate single coronoid where the edge(s) not belonging to hexagons are common to the outer boundary and the hole. 

In I—Fig. 2.4 the three systems of the upper row are not special d enerate coronoids, but this is the case for the bottom system. Here are four additional examples of degenerate single coronoids ... 

Two of them ( 2 22 26 22 special degenerate single coronoids, while the other... 

In the case of special degenerate single coronoids, as well as in single (nondegenerate) coronoids, one speaks about the outer and inner perimeter, viz. C and C", respectively, but now some edge(s) belong to both C fl C" 0. 

Definition 2.S A generalized single coronoid is either a single coronoid or a special degenerate single coronoid. 

If one wants to avoid the term "special degenerate single coronoid", then one has the synonymous designation "degenerate generalized single coronoid" at hand. 

The classifications normal and essentially disconnected are immediately applicable to degenerate single coronoids. 

In the bdow diagram, three essentially disconnected single coronoids and their effective units are depicted. The junctions are painted black. 

In the first case (i) of the above diagram, the effective units are four benzenoids (two pyrenes and two benzenes) in case (ii) the effective units are two benzenoids (anthracenes) and one coronoid (kekulene) in case (iii) the effective units are two benzenoids (naphthalenes) and one special degenerate single coronoid (condensed [18]annulene, viz. hexabenzo[6cdc/,fcim7io][18]-annulene). 

Fig. 2.1. One regular (r), one half essentially disconnected (he), and one essentially disconnected (c) single coronoid. Numbers K of Kekule structures are indicated.

Another instructive example is found in Fig. 2 The first three systems therein (r + he+ e) illustrate the same features as Fig. 1 with respect to the Kekule structure counts (/iQ. The systems are again obtained by successive additions of hexagons, but four hexagons are added each time in this case. The last system (Fig. 2), obtained by adding four hexagons into the corona hole of the e system, is a concealed non-Kekulean (o). This set of single coronoids illustrate nicely the rheo classification. 

Figures 1 and 2 emphasize the anomalous behaviour of Kekule structure counts for irregular single coronoids. However, it is not always so that the K numbers decrease through similar sets of coronoids as in these figures. The opposite situation is observed in Fig. 3, where again one he and one e system is produced by successive additions of hexagons to a regular (r) coronoid six hexagons are added each time. In this case the K numbers happen to increase udth increasing h in the "normal" way.

Irregular Single Coronoids With Isolated Internal Vertices... 

On the other hand, if a Kekulean single coronoid with isolated internal vertices has two black and two white internal vertices in succession around the corona hole, then it is essentially disconnected. This feature is illustrated for the e systems from Figs. 1 and 3 in the right-hand column of the above diagram. 

SuTv of Kekol Stractuie Counts for Single Coronoids... 

Here h = (2fl+l)6 for a > 1. Unity must be added for a = 1 in order to comply with the benzenoid drcumcoronene h = 19). All the other systems (a > 1) are regular pericondensed single coronoids. The result for the Kekule structure counts reads (Cyvin BN, BrunvoU and Cyvin 1988) ... 

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