Corona Holes

Consider a perforated oblate rectangle as depicted in Fig. 9. Like a rectangle it is a system with two parameters (m, n). A perforated rectangle, Q(m, ), belongs to the multiple coronoids it has m — 1 corona holes. 

A specification of the five addition modes is useful for some of the subsequent discussions and definitions. The modes can be vizualized as shown in Fig. 1 (i)-(v). An example of corona-condensation is included (vi) here the addition of a hexagon creates a corona hole. 

Another instructive example is found in Fig. 2 The first three systems therein (r + he+ e) illustrate the same features as Fig. 1 with respect to the Kekule structure counts (/iQ. The systems are again obtained by successive additions of hexagons, but four hexagons are added each time in this case. The last system (Fig. 2), obtained by adding four hexagons into the corona hole of the e system, is a concealed non-Kekulean (o). This set of single coronoids illustrate nicely the rheo classification. 

The HED coronoid depicted in Sect. 2.2, as well as the systems of the same category (he) in Figs. 1 and 3, are all examples of half essentially disconnected coronoids with isolated (nonadjacent) internal vertices. Notice that the colors of the internal vertices alternate successively as white-black—white—black—. when going around the corona hole cf. the left-hand colunm of the below diagram. 

On the other hand, if a Kekulean single coronoid with isolated internal vertices has two black and two white internal vertices in succession around the corona hole, then it is essentially disconnected. This feature is illustrated for the e systems from Figs. 1 and 3 in the right-hand column of the above diagram. 

Our term corofusene is not directly interchangeable with "coronafusene" of Balaban (1982). We shall also speak about single corojusene and single corohelicene when referring to the respective systems with one corona hole each. 

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). 

This chapter deals with certain nonhelicenic (geometrically planar) polyhexes, which may be simply connected (without holes) or multiply connected with a varying number of corona holes. It is recalled that a corona hole by definition should have a size of at least two hexagons. 

In a tuple coronoid, the perimeter length n can be split into n for the outer perimeter and (wj") ) ( 5")2> perimeters of the g corona holes so that... 

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 implied that the above equations (11) — (15) are valid individually for every corona hole. Let us now take the summations over all k and introduce the notations... 

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... 

Additional Definitions, Terminology and Relations 3.3.1 Corona Holes... 

Let the g corona holes of a tuple coronoid define a set of benzenoids 82, ., ... 

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

Definition S.l The associated benzenoid to is defined by the benzenoid which emerges when the g corona holes of are completely filled by hexagons. 

Usually a benzenoid B is compatible with different corona holes in different constellations. In other words, two or more nonisomorphic coronoids may be associated with the same benzenoid B. In fact the term "perforated B" may or may not characterize a coronoid unambiguously. The ambiguity depends highly on different restrictions, which may be imposed on C. Firstly, g may be fixed or not. Moreover, definite shapes of the holes may be assumed. If g is not fixed and "perforated B" is unambiguous, then must clearly be a single coronoid (C ). Namely, if with > 1 is associated with B, then we can always construct another coronoid which adso is associated with B. 

The assumption about naphthalene holes is very plausible since, loosely speaking, a larger corona hole would imply a waste of hexagons. It is also clear that the holes should be packed... 

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 ... 

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. 

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... 

Below we summarize additional formulas for single coronoids, which are consistent with the expressions of Sect. 3.3. The corona hole (cf. Par. 3.3.1) for a single coronoid is represented... 

Fig. 4.1. The naphthalene corona hole characterized by its invariants the corresponding benzenoid (CioHg naphthalene) is included (right-hand drawing).

Here = 0, 1, 2, 3, 4, 5, and each of these values is associated with a characteristic shape of the circular coronoid. These shapes were already known at least to Balaban (1971), who investigated their important role in the studies of annulenes cf. also a later work in this area (Cyvin SJ, Brunvoll and Gutman 1990). The same shapes are also encountered under the studies of certain primitive single coronoids called hollow hexagons (Cyvin SJ, Brunvoll and Cyvin 1989d Cyvin SJ, Brunvoll, Cyvin, Bergan and Brendsdal 1991), where the extremal property = hOj ax( ) corona hole is of interest. A detailed treatment of this topic is offered in Vol. 

Bowever, aU benzenoids with the hP values prescribed above need not be inspected (for h > 8). Take for instance h = 12, for which the maximum value is 7 in accordance with eqn. (1). Here only one out of the 331 benzenoids with seven hexagons is of interest, viz. coronene. The corresponding corona hole h = 7, = 6) occurs in kekulene h = 12), whUe all single... 

Then, according to the last equation in Volume I, I—(9.9), a primitive single coronoid around the hole (ho, nfl) has h = h (and = 0). This of course presupposes that the corona hole benzenoid, which is characterized by (h , can be circumscribed. The foUowing restrictions are vaUd for primitive single coronoids. 

It is also needed to consider corona hole benzenoids which cannot be circumscribed. Such corona holes are associated with the non-primitive basic single coronoids (Vol. 1-8, especially 1-8.3). The well known smallest non-primitive basic coronoid (I-8.2.2 and I—Fig. 8.2) has h = 12. Its corona hole (benzo[c]phenanthrene) has ho = 4, 71 0 = 0. Here again h = h, where h is defined in (2), but n-= 1. The next-smallest non-primitive basic coronoid, which has h = 13, is also well known (references as above). Its corona hole (pentahelicene) has ho = 5, nfl = 0. In this case h = h — 1 and n = 0. The two smallest non—primitive basic coronoids are depicted in the following (cf. also the top row of I-Fig.8.1). 

Fig. 7.1. Corona holes (represented as benzenoids) of basic single coronoids with h < 14. All these benzenoid forms are found in Brunvoll, Cyvin BN and Cyvin (1992b). Correction for C28H14 (corresponding to = 8, = 6), two forms in the cited reference should be interchanged in...

These two cases exemplify corona hole benzenoids with a cove and a fjord, respectively. They are representative for all the non—primitive basic single coronoids with h = 12, 13 and 14 (I-Fig. 8.2), which are those of the prime interest in this section. For the sake of clarity we repeat the relevant rules in strict formulations. For 12 < h < 14 the non-primitive basic single coronoids are of two kinds, (a) The corona hole benzenoid possesses exactly one cove then h = h n = 1. (b) The corona hole possesses exactly one fjord then h = h — 1, = 0. 

The corona holes, represented as benzenoids, of all basic single coronoids for 8 < h < 14 (primitive and non-primitive) are depicted in Fig. 1. The encircled numerals identify the different classes with characteristic combinations of nfi and h. Table 3 includes a listing of the corona holes in question or the appropriate basic single coronoids, which amounts to the same. 

Here A° = 6, nfi = 0. The pertinent corona hole benzenoid has one fjord, but when a hexagon is immersed a cove is created. As a result, one obtains A = A - 1, = 1 (the internal vertex is... 

The occurrence of corona holes is cumulative in the following sense. All the holes which occur in the basic (primitive and non-primitive) single coronoids with h hexagons each, are also found in single coronoids with more than h hexagons. A detailed account on the numbers of single coronoid isomers with the different classes of holes (identified by endrded numerals), is furnished by Table 4. 

Table 7.4. Numbers of single coronoid isomers, classified according to corona holes complete data.

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