Pericondensed

The structureless fluorescence spectra of crystalline pericondensed hydrocarbons, pyrene, perylene, and 1,12-benzperylene, on the other hand, are red-shifted by 6000 cm-1 from the 0"-0 molecular fluorescence band,103-105 and in the case of pyrene is virtually identical with the excimer band. As shown schematically in Figure 13, the molecular orientation in a... 

In a third type of crystal structure (Type B2, Fig. 13) exhibited by the larger pericondensed hydrocarbons (e.g., coronene and ovalene), the characteristic excimer fluorescence band may originate from a small concentration of... 

Using refined X-ray dilfraction techniques and the extraction of the radial distribution function from molecular X-ray scattering it has been possible to develop a model for a graphene layer [31]. This model is free of the difficulties mentioned above and predicts cluster sizes of between n = 3 and n — 5 for pericondensed rings (coronene, hexaperibenzo coronene) in full agreement with electron microscopic [25] and NMR [131] data which led to construction of Fig. 19. The model material [31] was a coal sample before carbonization containing, as well as the main fraction of sp2 centres, about 20% carbon atoms in aliphatic connectivity. This one-dimensional structure analysis represents a real example of the scheme displayed in Fig. 9(C). 

In pericondensed benzenoid systems a Hamiltonian cycle may, but need not exist. The problem of the existence of a Hamiltonian cycle is solved by the following result [11]. 

Theorem 2. A (pericondensed) benzenoid system B possesses a Hamiltonian cycle if and only if there exists an e-transformation, B => B, such that B is catacondensed. 

Eq. (7) does not hold for all pericondensed benzenoids and its range of validity was established in a series of investigations [90, 92-96]. The work of Ohkami [96] can be considered as a complete solution of this problem. 

It is really surprising that the Wiener indices have such number-theoretical properties. This kind of modular behavior was observed never before for any of the numerous topological indices studied in chemical graph theory. Results analogous to those given in Theorem 20 were later found for other classes of (non-benzenoid) graphs [130, 131], but their extension to pericondensed benzenoids was never accomplished. 

Theorem 3 [56]. If B is a benzenoid systems and Z its cycle, such that the size of Z is divisible by four, then ef(B, Z) < 0. (Note that in this case B must be pericondensed [1].)... 

The distinction between catacondensed and pericondensed systems is applicable to all the classes of polyhexes treated above (Fig. 2). A catacondensed polyhex is defined by the absence of internal vertices. An internal vertex is a vertex shared by three hexagons. A pericondensed polyhex possesses at least one internal vertex. In terms of dualists a pericondensed polyhex reveals itself by the presence of at least one three-membered cycle (triangle). A dualist of a circulene has a cycle larger than a triangle. 

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. 

So far we have only been speaking about the numbers of single and double coronoids. Triple coronoids do not occur before h = 18, in which case there are 2 eatacondensed and 2 pericondensed forms of these systems [43,44],... 

Numbers for benzenoids separately can be determined as the differences between appropriate numbers in Tables 1 and 2, but this is not the way they were deduced originally. The Dusseldorf-Zagreb numbers (or DZG numbers) is the designation which has been used [16,17] about the data for eatacondensed and pericondensed benzenoids, and benzenoids in total with h < 10. These data have later been supplemented by other investigators [19,46, 47], as well as the Dusseldorf-Zagreb... 

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... 

Once the numbers in Table 5 are known all the Lunnon numbers for h < 12 can be split into those for catacondensed and pericondensed systems. One only has to add the known numbers of benzenoids (see Sect. 3.1). 

Figure 6 includes 35 pericondensed helicenes with ft = 8, obtained by a systematic search. This number disagrees with Table 8, which prescribes 36 such systems. The controversy represents an open problem an error in the computer-enumeration [17] can not be ruled out. 

It seems, strangely enough, that a list of all simply connected polyhexes (catacondensed and pericondensed together), as in the last column of Table 7, has not been given before explicitly. 

All the 4 helicirculenes in Fig. 7 are catacondensed. The number of helicirculenes increases rapidly with increasing h. For h = 11 we have generated 58 systems by hand, 32 catacondensed and 26 pericondensed, but we are not sure that we have not missed any. 

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. 

In Sect. 2.6 the concepts of catacondensed and pericondensed polyhexes are defined. It is implied that these notions are applicable to benzenoids (cf. Sect. 2.1) in particular. In the following we shall only speak about benzenoids, although all the concepts are applicable to other polyhexes as well. 

Another subdivision of benzenoids (apart from catacondensed/pericondensed) distinguishes between Kekulean and non-Kekulean systems. A Kekulean benzenoid system possesses Kekule structures (K > 0). A non-Kekulean benzenoid has no Kekule structure (K = 0). The shorter designations Kekuleans and non-Kekuleans are often used. All catacondensed benzenoids are Kekulean therefore all non-Kekuleans are pericondensed. Any Kekulean benzenoid has a vanishing color excess A = 0. 

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. 

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