Inner perimeter

It may seem unnecessary to bring in such artifacts as the above examples of non-polyhexes. Yet we find the discussion to be warranted. Ege and Vogler [28, 29], for instance, treated pyrene in a group together with coronene ([6]circulene) and six coronoids. Implicitly, the authors considered pyrene as [4]circulene and ascribed an inner perimeter of two edges to it. That leads to vertices of degree four as in one of the forbidden examples above. Therefore we claim that the interpretation of pyrene as [4]circulene is not justified. 

There is a close relationship between polycyclic compounds such as coronene 81198) or Staab s kekulene 82199) and the macrocyclic cyclophanes such as 83200). They can be described as perimeter structures in which an outer and inner perimeter can be considered. 

Figure 23.17 A phospholipid has a polar head and two nonpolar tails. The membranes of living cells are formed by a double layer of lipids, called a bilayer. The polar heads are on the outer and inner perimeter of the membrane and the tails are on the inside of the bilayer. 

The ambitious aim of Kauffmann to synthesize a supercyclopolythiophene as represented in 4.10 (Scheme 1.36) could not be achieved due to the failure of a second lithiation of 4.6 [365]. The inner perimeter of hypothetical 4.10 corresponds to an a-conjugated cyclo[8]thiophene (in blue), which was synthesized nearly 20 years later by Bauerle et al. in a completely different way (see below). 

In the case of special degenerate single coronoids, as well as in single (nondegenerate) coronoids, one speaks about the outer and inner perimeter, viz. C and C", respectively, but now some edge(s) belong to both C fl C" 0. 

Having the graph—theoretical planarity in mind, it is clear that the outer and inner perimeters of a single corohebcene always can be identified. However, an identification of the corona hole with a benzenoid may be obscured. Therefore we must rephrase the requirement that the corona hole should have a size of at least two hexagons (cf. Sect. 2.1). The following formulation is valid for corofusenes any inner perimeter should consist of at least ten edges (and vertices). 

The poly hex under consideration (Fig. 6) is embeddable in the hexagonal lattice, and it has overlapping edges or, more precisely, two overlapping hexagons. Hence it is reasonable to consider this system as helicenic, and that has actually been done previously (Randid, Nikolid and Trinajstid 1988 Randic, Gimarc et al. 1989 Cyvin BN, Brunvoll and Cyvin 1992b). However, cyclohelicene is not a corohelicene, and of course not a coronoid. It is true that two perimeters are present, but they are symmetrically equivalent none of them qualifies to be identified especially as the outer— or inner perimeter. There are two enantiomers of the structure in question (Fig. 6). 

The hydrocarbon which corresponds to a polyhex under consideration, has the chemical formula C H. Here the number of carbon atoms (n) corresponds to the number of vertices (see the above listing). The number of hydrogens (5) is equivalent to the number of secondary carbon atoms and corresponds to the number of vertices of degree two. These vertices are exclusively on the perimeters (inner and outer) of the polyhex. The total number of tertiary carbon atoms on the perimeters is t (see above). The total number of the boundary vertices (on the perimeter), viz. is also the total number of edges on the perimeters. This number (n is the combined perimeter length (i.e. the sum of the lengths for the outer and all the inner perimeters). 

When it comes to the inner perimeters it is expedient to invoke the interpretation of the corona holes as benzenoids. The benzenoid which corresponds to a corona hole is just defined by the inner perimeter of the hole as its unique perimeter. Let the numbers of external vertices of degree two and of degree three on the perimeters of these benzenoids be identified by the symbols SgO, SgO,. , s o and 2° respectively. Now we have... 

It is especially interesting to compare these two equations, viz. (18) and (19), or alternatively (15) and (14), with eqns. (9) and (10), respectively. These relations emphasize the different properties of an outer perimeter from those of an inner perimeter. This feature has been described in detail by Hall (1988). Also Polansky and Rouvray (1977) have offered an elaborate treatment of the perimeter lengths of coronoids. 

The last part of eqn. (27) expresses the fact that the inner perimeter for a corona hole coincides... 

For the outer perimeter, eqns. (3.9) and (3.10) are immediately applicable to single coronoids for the inner perimeter one has... 

Firstly, if an excised coronoid Cq, viz. c(Cq), is not itself a coronoid, then the circumscribing of c(Cq) is not defined, and Cq becomes automatically a core coronoid. The two last systems of the top and bottom rows of Fig. 4.2 (viz. C32H16 and C48H20) examples. In particular, every catacondensed coronoid is a core coronoid belonging to the category under consideration. If Q is a catacondensed single coronoid vdthout an inside feature (cf. Vol. I), then e(Q) consists of the inner perimeter of Q. 

Assume now that Q is a catacondensed single coronoid without inside feature(s) cf. Vol. I-8.2.1. Then, as mentioned in Par. 5.4.5, = e(Q) consists of the inner perimeter. We shall... 

Case 1. If r = then obviously both the outer and inner perimeter are M-alternating cycles. 

Three Kekule structures are indicated, each by their n/2 = 20 double bonds. In the structures (a) and (b), r= n 2 = 18. The outer and inner perimeters are alternating cycles in both these structures. This condition (Case 1) does not exclude the possibility for aromatic sextets to be present, as is demonstrated here in (a) there is no aromatic sextet, but in (b) there are three (marked by small circles). In the structure (c), r = 16 and therefore r < n /2 (Case 2). This condition prescribes with certainty the presence of an aromatic sextet there are four of them in the structure at hand. 

Consider now a single coronoid G and assume that A is a set of a nonadjacent vertices, all of them on the outer or on the inner perimeter of G. The same restriction is imposed on A as above. 

Case 1. If r = (n /2) — a, then one of the perimeters (outer or inner) of G is already an M-altemating cyde. More predsdy, if the vertices of A are on the inner perimeter of G, then the outer perimeter is an M-altemating cyde, or vice versa. 

In the following we prove that the inner perimeter (C ) of G is an M—alternating cyde for... 

In both subcases (Subcase 3.1 and Subcase 3.2) it was demonstrated that a coronoid G is generated from G by a normal tearing down so that both the outer and inner perimeters of G are M —alternating cycles for some Kekule structure M of G. Repeat the argument for G. Eventually one will arrive at a coronoid with a hexagon of mode L2 or A2, whereby the case is reduced to Case 1. Thus the proof of sufficiency is completed. 

Case 2 In G-C there is no fixed double bond incident to any vertex of C . Thus C must be contained in an effective unit G which is a generalized coronoid and contains C as its inner perimeter. By Theorem 8.3 or 8.3, G -C has a Kekule structure M. Let the other effective units of G-C be Gg,. , , which have Kekule structure M2, . , M, respectively. Clearly... 

Nevertheless, the coronoid of this example is not HED because U Kg does not constitute the total of 82688 Kekule structures for this system in fact, the formidable amount of 74624 Kekule structures do not belong to K U Kg. One of them is shown in the right-hand drawing above. This Kekule structure was selected among those where both the outer and inner perimeter are alternating cycles, confirming that the system is a regular coronoid. 

Ap. Therefore G is disconnected. This implies that w is on one of the perimeters of G. Since e and e are on the outer perimeter of G by assumption, the inner perimeter of G entirely... 

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