Naphthalenic single coronoid

A naphthalenic single coronoid (cf. Par. 3.3.3) has exactly one naphthalene hole. Figure 1 summarizes the characteristic features of this hole and the corresponding benzenoid (naphthalene). The special cases of Eqs. (9) - (11) for naphthalenic single coronoids read ... 

Let C(n s) be a naphthalenic single coronoid, and apply the same notation for its circumscribed and excised systems as well as their invariants as in Sect. 5.4.6. Then the transformation (4.13) applied to eqns. (26) and (30) gives straightforwardly... 

The coefficients of N 5) were transferred to those of a naphthalenic single coronoid, say B(h, n s, whose associate benzenoid is B N 5). Then the expressions (14) read ... 

Proposition 6.1 For every single coronoid formula there exists (at least) one naphthalenic single coronoid C 6 which is reproducible. 

It is seen (Table 5.3 and Fig. 5.8) that X 6 C53H21 are the smallest h = 16) X isomers where coves possibly may occur. Then comes X C65H23 h = 21) and X C72H24 h = 24). These coronoid isomers are at the same time those where a phenalene hole, which implies inner coves, cannot be excluded. Indeed, such isomers with phenalene holes can be constructed, as is exemplified by Fig. 5.7. Naphthalenic single coronoid isomers with the same formulas and outer coves are also possible ... 

All extremal single coronoids A are naphthalenic. The circular single coronoids 0 are also extremal they form a subclass of the A systems. Some expressions for circumscribing naphthalenic single coronoids, C, are given below. 

Asshme that C(n 5) is a naphthalenic single coronoid. If C can be circumscribed and C (7ip5 ) = circum-C(7i s), then... 

Assume that 0 N S) is a circular benzenoid which can be perforated by a phenalene hole to produce a single coronoid 0(n s). Then one has for the formula coefficients l T=n+l, 5=s — 3. All the isomers of the class 0 were enumerated. This is not a complete determination of the cardinalities because there always exist at least naphthalenic single coronoids with the... 

Finally we give some supplements to the numbers of naphthalenic single coronoids. The 321409 systems of this category with /i = 15 (Vol.I—Table 8.3 I—Table 8.4) were found to be classified as is shown in Table 18. This table furnishes a supplement to Vol.I—Table 8.5. 

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). 

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). 

The circular single coronoids are represented by the dots on the stippled curve in Fig. 5. It is an important feature that the circular coronoids form a subclass of the extremal coronoids (of the same genus). Consequently, the circular coronoids are naphthalenic. Furthermore, a circular single coronoid is a circular benzenoid perforated by a naphthalene hole. 

Some explanations should be attached to this somewhat complicated definition. It is dear that a drcumscribed extremal (single) coronoid is circumextremal write circum—A = P. Then it is also dear that excis-P = A. Assume now that the naphthalene hole of P is moved as near as possible to the outer perimeter, and call the new coronoid P. Then P and P are isomers, but P cannot be excised to give another coronoid excis—P becomes a degenerate system. Nevertheless, P falls under the definition of drcumextremal coronoids. An illustration is given below. 

Fig. 5.7. Examples of nonextremal extreme single coronoids which are not naphthalenic.

Property 6.S An extremal single coronoid has never a cove and never a fjord in addition to the inner fjords of its naphthalene hole. 

Property 6,4- A nonextremal extreme single coronoid X has never a fjord except for the inner fjords of its naphthalene hole if the system is naphthalenic. 

Altogether we have generated all the nonisomorphic C42H20 single coronoid isomers 82 (5971+26+2102) in consistency with the entries in Table 2. Also the distribution of corona holes is consistent with Table 4 72 naphthalene (h = 2, np = 0) holes, 9 phenalene (h<> = 3, ti-o = 1) holes, and 1 phenanthrene hP = 3, = 0) hole. 

A circular single coronoid, 0, (Par. 4.6.2) is a circular benzenoid perforated by a naphthalene hole (cf. also Par 5.5.3). Cyvin SJ (1991b) presented a complete mathematical solution in terms of combinatorial formulas for the numbers of 0 isomers. 

Exactly all the circular benzenoids 0 N S) with 5 > 18 can be perforated by a coronene hole to produce single coronoids 0(n 5). Then one has i T=n+6, 5=5 — 6. A detailed treatment on a complete enumeration of the isomers of the class 0 has been published (Cyvin BN, Brunvoll, Chen and Cyvin 1993). The smallest system of this class is kekulene. In the cited reference a coronoid with coronene hole(s) was referred to as "coronenic . We quote a nice sentence therefrom "A coronenic coronoid has a coronene corona hole. We shall not use the term "coronenic in the following, although it is analogous with our term "naphthalenic (Definition 3.3). A brief account on the title systems and their enumeration is given in the next two paragraphs. 

Table 7.18. Enumeration and detailed classification of single coronoids with the naphthalene hole and h = 15.

Coronoids with exclusively naphthalene holes play important roles in different contexts, as we shaU see in the foUowing. This is true for both single and multiple coronoids. 

Definition S.S A naphthalenic coronoid is a coronoid (single or multiple) with only naphthalene hole(s). 

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