Naphthalenic coronoid

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). 

Assume that C is a naphthalenic tuple coronoid with (h, n and the formula Furthermore, let the associated benzenoid to C be denoted by B, of which some selected 

By means of these expressions one obtains the following special cases of eqns. (35) — (37), which apply to and B. 

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From the information on B, the associated benzenoid to B, we deduce the following properties of the naphthalenic coronoid B( i s). 

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). 

Example Let the naphthalene and phenalene corona holes in the double coronoid of Fig. 1 be identified by the subscript 1 and 2, respectively. Then... 

Assumption A smaUest multiple coronoid can be found as a catacondensed naphthalenic system, where the naphthalene holes are paraUel. 

In order to depict (one of) the smallest coronoid(s) with g holes (a) select an extremal benzenoid with P = and draw it so that the parallel edges are horizontal (b) convert each hexagon of this benzenoid into a naphthalene corona hole according to ... 

Figure 2 shows the forms of the smallest catacondensed tuple coronoids for g < 13, obtained from systematic generations by hand. All the systems fulfil the assumptions (Par. 3.5.2) of being naphthalenic with parallel holes. Those which emerge from the algorithm of Par. 3.5.3 (cf. especially Table 3) are all found in Fig. 2. In addition one "off-algorithm" system each for g = 3 and p = 7, which account for the above examples, are also found in Fig. 2, and finally one ofr—algorithm system for = 8, one for = 9, and four for g = 12.

In connection with Observation 1, suppose that a coronoid C has a corona hole larger than naphthalene. Then we can imagine that a closer packing of the hexagons of C is possible by a partial filling of the corona hole so that the total number of internal vertices increases. In Observation 2 the crucial term ("perfect extremal coronoid") conforms with Definition 3.5 of Par. 3.3.4. It is reasonable to imagine that there is a critical smallest size for an extremal benzenoid, say A, so that A can be perforated with g naphthalene holes, which is taken to be the necessary condition for creating a perfect extremal tuple coronoid. 

Consider two extremal coronoids AJ h) and A (/i), along with their associated benzenoids, say B and B, respectively. Here B may be identical with B, namely if it is possible to perforate B = B by naphthalene holes in two different ways so as to create the two nonisomorphic systems A (h) and A h), In general B and B must have the same invariants as indicated by B(H,iNT), B (H,A ). This is a consequence of Observation 1. It foUows that B and B also have the same formula, say C yH, and the same perimeter length, say N. The connections... 

It was stated (Par. 3.6.2) that n - = 0 is realized in a y-polyhex for every g and h. It should also be clear that, if a catacondensed y-tuple extremal coronoid exists, then it can only occur for h = h g). It is always possible, namely, to add a hexagon to a catacondensed coronoid so that it becomes pericondensed. (The same property is also found for benzenoids with one exception benzene. Accordingly, both benzene H = 1) and naphthalene H = 2) are catacondensed extremal benzenoids.) The above discussion, along with the last lines of Par. 3.5.5, lead to the following conjecture. 

Here it is found by inspection that three naphthalene holes cannot be introduced in any of these benzenoids. Hence it is inferred that no triple coronoid with the formula C64H26 can be constructed. We must therefore look for the possibilities with h > h. Then the first case to be inspected is h = 18, corresponding to C68H28 and the associated benzenoids C68H22( =24, =30). These benzenoids are nonextremal and count 789 isomers (Stojmenovic et al. 1986 BrunvoU and Cyvin SJ 1990), a too large number to be inspected by sight, even if the depictions were available. However, two of these isomers perforated by three naphthalene holes, have actually been constructed (see Fig. 2) as the smallest catacondensed triple coronoids, in consistency with the present analysis. 

It is found by inspection that dicumovalene is nonperforable (by three naphthalene holes). Notice that the heavy-line cycle in the above depiction cannot be trespassed by any corona hole, (iv) = 1, B = Ce7H2i(iTs=24, 31) represents 43 isomers, which were generated and depict probably for the first time. The depictions are not reproduced here, but they were diecked in order to verify that exactly two of these isomers were perforable and led to the two C67H2 (h=18, n l) extremal triple coronoids, which are shown in Fig. 3. (v) h = 19, = 3, A... 

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 ... 

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. 

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... 

It has been mentioned (cf. e.g., Par. 4.3.1) that an A coronoid is naphthalenic and perfect extremal it is obtained from an A benzenoid by perforating it vdth 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. 

A coronoid P is a naphthalenic perforated P. Hence the relations for (n s ) may be deduced alternatively from (Cyvin SJ, Brunvoll and Oyvin 1991a Oyvin SJ 1992c Brunvoll, Oyvin BN and Cyvin 1992b)... 

A coronoid E is not necessarily naphthalenic. However, for every formula (n 5 ) a... 

Figure 6 shows the ten smallest X coronoids generated in the way described above. The chemical formulas are indicated, but only one of the several existing isomers for each formula is displayed. Some of the additional isomers are found in Fig 7 this figure demonstrates at the same time the existence of X systems which are not naphthalenic. 

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

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. 

The situation for coves is different every cove (inner or outer) of C is evidently available for addition of a hexagon. A fjord of C is always available for addition except for the inner fjords of a naphthalene hole it is not allowed to immerse a hexagon into a fjord of a naphthalene hole of C when a new coronoid is supposed to be generated. 

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. 

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. 

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