Essentially disconnected

A prolate rectangle, R (m, n), is an essentially disconnected benzenoid [1-3]. Hence the Kekule structure counts are easily obtained by... 

A Kekulean benzenoid may be normal or essentially disconnected. In an essentially disconnected benzenoid there are fixed double and/or single bonds. A fixed single (resp. double) bond refers to an edge which is associated with a single (resp. double) bond in the same position of all the Kekule structures. A normal (Kekulean) benzenoid has no fixed bond. All catacondensed benzenoids are normal therefore all essentially disconnected benzenoids are pericondensed. But a pericondensed Kekulean may be either normal or essentially disconnected. 

The neo classification takes into account all benzenoids they can be either normal (n), essentially disconnected (e) or non-Kekulean (o). 

The neo classification divides all benzenoids into normal (n), essentially disconnected (e) and non-Kekuleans (o), where the n and e systems cover all the Kekuleans. Cyvin and Gutman [26] have advocated for this classification by saying From the point of view of the enumeration of Kekule structures the classification. . . [neo]. . . seems to be a rather appropriate one [94,87] . However, the distinction between Kekulean (closed-shell, non-radicalic) and non-Kekulean (radicalic) benzenoid hydrocarbons was made long before the explicit definition of the neo classification. This practice started with the first (substantial) enumeration of benzenoids in the chemical context by Balaban and Harary [13]. 

A substantial amount of additional enumeration data for normal benzenoids and some data for essentially disconnected benzenoids are available, but shall not be reproduced here. They were produced in the course of the extensive studies of the distribution of K, the Kekule structure count. 

As far back as 1968, Balaban and Harary [13] were aware of the unique position of zethrene, which was placed in a class of its own under the pericondensed benzenoids with h = 6. It is an essentially disconnected benzenoid. However, these authors did not sort out the other two essentially disconnected benzenoids (annelated perylenes) with the same numbers of hexagons. Neither did they sort out perylene itself, which is the unique essentially disconnected benzenoid with h = 5. In the table we are referring to [13], the entry for the classified pericondensed system with h = 3 is misplaced. 

Knop et al. [91] published a list of the numbers of Kekulean benzenoids for h < 9 with a misprint in the number for h = 9. Brunvoll et al. [101] reported a wrong number (with reference to private communication from He and He) for the Kekulean benzenoids with h = 12. The curve of the distribution of K for essentially disconnected benzenoids with h = 9 [26,95] in imperfect. 

Essentially disconnected benzenoids are, by definition, the Kekulean benzenoids (with K > 0, A = 0) which are not normal. All of them are pericondensed. 

Table 28. Numbers of essentially disconnected benzenoids classified according to symmetry +...

The smallest essentially disconnected benzenoids with D6h and with C6h symmetries (one each) occur at h = 25 [71]. 

In Fig. 23 the forms of all essentially disconnected benzenoids up to h = 8 are displayed as black silhouettes. Such figures have been given previously for h < 7 [26] and h < 8 [105], for h = 7 also as dualists [55]. In one of these works [105] a number of (bizarre) forms of larger essentially disconnected benzenoids are included. 

Fig. 23. All essentially disconnected benzenoids with h < 8. K numbers are given...

All snowflakes have vanishing color excess A = 0. Therefore they can be either normal, essentially disconnected or concealed non-Kekulean (but not obvious non-Kekulean). 

Fig. 30. Snowflakes all essentially disconnected benzenoids with hexagonal symmetry, Dbh (one system) or C6h, and h < 37 2 systems with h = 25 and 8 with h — 31. K numbers are given...

Here we give a re-edited selection of the forms of snowflakes. Figures 29, 30 and 31 display the forms of normal, essentially disconnected and concealed non-Kekulean snowflakes, respectively, both proper (D6h) and improper (C6h). Similarly, the forms of proper snowflakes in particular are displayed in Figs. 32, 33 and 34, pertaining to the normal, essentially disconnected and concealed non-Kekulean systems, respectively. 

Abbreviations e essentially disconnected n normal o non-Kekulean. a Cyvin, Brunvoll and Cyvin (1988) [78] b Gutman and Cyvin (1988) [102] c Cyvin, Brunvoll and Cyvin (1989) [110]... 

In Figs. 35, 36 and 37 the smallest benzenoids with trigonal symmetry are illustrated the figures pertain to the normal, essentially disconnected and non-Kekulean systems, repectively. The same collection of forms has been displayed elsewhere [78]. 

In the work of Cyvin et al. [78] the two first (silhouette) drawings of essentially disconnected benzenoids should be switched in order to match the given K numbers. 

The classification referred to as neo takes into account all benzenoids. They are either normal (n), essentially disconnected (e) or non-Kekulean (o) cf., e.g., the multi-author report of Balaban et al. [30] and references cited therein. Among the Kekulean systems (n + e) the Kekule structures possess fixed bonds in the case of essentially disconnected benzenoid, while those of the normal benzenoids do not. 

Here we refer to the enumeration data of benzenoid isomers as complete if, for a given h, all the numbers of C HS isomers are given at least for the Kekulean and non-Kekulean systems separately. Such data are known for h values up to 14 cf. Tables 1 -4. In addition, we have specified the numbers of normal (n) and essentially disconnected (e) benzenoids among the Kekulean systems in the tables. The smallest essentially disconnected benzenoid, viz. perylene, occurs at h = 5. Furthermore, the non-Kekulean systems are classified according to the A values (Tables 1-4). 

For a documentation pertaining to the normal pericondensed and essentially disconnected benzenoids, as well as the classification according to A values, references are made to Brunvoll et al. [25, 33]. For h < 9 the information on A values could, of course, be extracted from Knop et al. [5] by studying all the forms of the benzenoids depicted therein. 

Table 6 is a compilation of data for numbers of benzenoid isomers with h > 15. They are incomplete in the sense that, for each h, the list of n values is not complete. Furthermore, in many cases the subdivisions into normal and essentially disconnected benzenoids are unknown. 

I. Gutman, J. Brunvoll, B. N. Cyvin, E. Brendsdal, and S. J. Cyvin, Essentially disconnected fully acenoids, and fully-arenoid hydrocarbons, Chem. Phys. Lett. 219, 355-359 (1994). 

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... 

Chem., 24, 51 (1989). Essentially Disconnected Benzenoids Distribution of K, the Number of Kekule Structures, in Benzenoid Hydrocarbons. VIII. 

The Kekulean and non-Kekulean systems are those which possess Kekule structures K > 0) oi do not possess Kekule structures K = 0), respectively. In an essentially disconnected system there are fixed bonds, viz. edges which correspond to double and/or single bonds in the same positions of all Kekule structures. Kekulean systems without fixed bonds are called normal Obvious- and concealed non—Kekulean systems have A > 0 and A = 0, respectively, where A is the color excess, viz. the absolute magnitude of the difference between the numbers of peaks and valleys (1-3.2.4). Finally, the catacondensed (unbranched or branched) and the pericondensed systems have n- = 0 and n-> 0, respectively, where n- designates the number of internal vertices (I-3.3.1). 

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. 

Notice that the top and bottom L3 hexagons have changed their modes to P2 (and L5 to A4). The new system is an HED coronoid and cannot be tom down in the prescribed way. If one of the Lr-or A 2-mode hexagons are deleted, one obtains an essentially disconnected benzenoid, also if the Li-mode hexagon was deleted first. This example shows that the sequence of hexagons which are deleted, is not arbitrary. Now it is instructive to demonstrate the two characteristic schemes of Kekule structures for the above HED coronoid. The hatched parts (in the below diagram) indicate hexagons with fixed bonds. 

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

The terms "junction and "effective unit", which were introduced for essentially disconnected benzenoids (BrunvoU, Cyvin BN, C rvin and Gutman 1988), are transferable to coronoids (cf. Vol. 1-3.3.4). 

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

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.

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