Perforated benzenoid

Associated Benzenoid and Perforated Benzenoid Let denote a tuple coronoid. 

The designation "perforated rectangles" for certain single and multiple coronoids, of which the Kekul6 structures have been studied (Cyvin SJ, Cyvin and BrunvoU 1989a Chen and Cyvin 1989 Cyvin SJ, Cyvin, BrunvoU and Chen 1989 Chen, Cyvin et al. 1990), is compatible with the above definition of perforated benzenoids. 

Example. The method of perforating benzenoids shall presently be exemplified by the C46H20 (h = 13, 71 = 6) single coronoid isomers. The application of the fundamental building-up principle would be somewhat laborious in this case it would imply the appropriate additions to (i) 2 C42H18, (ii) 37 C43H19 and (iii) 142 C44H20 coronoid isomers. 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

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. 

This derivation is based on the assumption that the number of internal vertices in a polyhex with h hexagons, viz. n h, g), cannot be larger than it would be in an extremal benzenoid perforated by g naphthalene holes. At this stage, however, it is an open question whether such an... 

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... 

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. 

For g = Aj h = 21. One obtains as a starting point the formula C78H30, which corresponds to the associated benzenoids C78H22( =29, JV=40). The actual four isomers (Cyvin SJ, BrunvoU and Cyvin 1991d) of these extremal benzenoids were inspected, but none of them could be perforated by four naphthalene holes. Consequently, we turn to h = 22, the formula C82H32 and associated benzenoids C82H24(f7=30, —40). There are 1799 isomers of these nonextremal benzenoids. In accord with Fig. 2 it is claimed that only one of them can be perforated by four naphthalene holes in a unique constellation. 

The algorithm stops when a benzenoid A(h+2y, n +10y) is perforable for the first time, i.e. for the smallest h = oi h= h Then A (/i,n ) represents the smallest perfect extremal tuple coronoid(s) for the chosen g value. 

In Fig. 4 the smallest perfect extremal tuple coronoids for = 1, 2, 3, 4 and 5 are shown they occur for h = 8, 13, 19, 23 and 28, respectively. The analysis was pursued, by perforating the appropriate extremal benzenoids, in order to generate the systems with a few higher h values for each g the results are included in Fig. 4. 

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. 

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. 

Only in three cases the normal benzenoids become essentially disconnected coronoids when perforated. They are shown in the below diagram, where hexagons containing fixed bonds are black. 

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. 

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... 

Fig. 7.4. The smallest circular benzenoids perforated by a phenalene hole, augmented by three degenerate coronoids (in the top row).

Table 7.8. Formulas for circular benzenoids perforated by one phenalene hole each. ...

---