Benzenoids circulene

We have here given a precise interpretation of the Lunnon numbers, perhaps for the first time since they were published in 1972 [10] but see also Muller et al. [36]. These numbers pertain to all planar (non-helicenic) polyhexes (benzenoids + circulenes, including dicirculenes). Using this interpretation we have actually reproduced these numbers, also for h = 11 and 12. In other words, we have provided an independent check of the Lunnon numbers for the first time. 

These numbers, when taken together, account for all the classes of polyhexes defined above, viz. benzenoids, helicenes, and circulenes including helicirculenes (Fig. 2). In Sect. 3 the distinction between catacondensed and pericondensed systems is the only one considered for the different classes in question (Fig. 2). Section 4 contains some additional definitions. 

In one of the classical papers on the enumeration of poly hexes, Lunnon [10] found the numbers of all geometrically planar polyhexes with h (number of hexagons) up to 12 see Table 4. In these numbers the helicenic systems are excluded, but all non-helicenic circulenes are included. Hence the numbers pertain to benzenoids plus planar circulenes. The contribution of Balaban and Harary [13] to the Lunnon numbers (cf. footnotes to Table 4) is commented in the next section. The Lunnon numbers have been extended to h = 13 and h = 14 by means of the data from Muller et al. [36]. 

The planar (non-helicenic) circulenes have also been considered separately, first by the Diisseldorf-Zagreb group [16,17, 35] see Table 5. He and He [33] depicted the forms of these systems for h <, 9 from these figures we have extracted separate numbers for the catacondensed and pericondensed systems up to this h value. For the catacondensed planar monocirculenes with h < 11 the numbers may be deduced from a work of Brunvoll et al. [54], who enumerated the benzenoid systems composed of coronene with catacondensed appendages up to h = 12. On... 

When looking at Tables 1, 4, 7 and 9, for instance, we find that the numbers for h < 5 are coincident, although their documentations by footnotes vary, since they are viewed in different contexts. They are also identical with the Diisseldorf-Zagreb numbers. This is of course so because no systems other than benzenoids occur at these low h values. For h = 6 the pericondensed polyhexes are still only benzenoids, while the numbers of catacondensed polyhexes with h = 6 varies from 36 to 38. This is explained by the inclusion of hexahelicene or [6]circulene, occasionally both of them or none. Notice that from the number 37 alone, which is accompanied by the total of 82 (h = 6), one can not deduce which interpretation is the actual one. 

Only benzenoid systems are considered. According to the adopted definition they are the planar, simply connected polyhexes. Consequently all circulenes (including coronoids, multiple coronoids and polycirculenes) are excluded, and all helicenic systems (helicenes, helicirculenes) are also excluded. 

Since the convention for constructing the Wiswesser codes for polyhexes is understood, one can easily produce the Wiswesser code of any benzenoid by inscribing the appropriate number in each hexagon of a properly oriented polyhex. In Figs. 9.4 and 9.5, we give as examples Wiswesser codes for two structures in which the starting numeral is not unity one representing perylenes, and the other circulenes. 

A radialene-type core also appears in the appropriate bond resonance structures of the so-called [ ]circulenes, where the central -sided ring is surrounded by n fused benzenoid rings (Figure 4.4) [107]. As the first members of the homologous series, the parent structures comprise the [3]circulene (probably too strained to be accessible by synthesis), the so far unknown [4]circulene (153), [5]circulene or corannulene (154) which is a substructure of buckminsterfullerene (Cgg), the long-known [6]circulene or coronene (155), and [7]circulene (156). 

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