Kekule and Clar Structures

In the book [3] the role of both Kekule and Clar structures in various (contemporary) chemical theories as well as their relevance for practical chemistry were outlined in detail. A recent book [46] by Cyvin and the present author is devoted to the enumeration of Kekule structures of benzenoid molecules. In addition to this, the first volume of Advances in the Theory of Benzenoid Hydrocarbons contains several review articles [47-51] dealing with topics of relevance for our considerations. In order to avoid repetition and overlapping we will just briefly mention the work on the elaboration and application of the John — Sachs theorem for the enumeration of Kekule structures [52-55], the search for concealed non-Kekulean benzenoid systems [2, 56-59], examination of fully benzenoid (=all-benzenoid) systems [60-62, 135] as well as the enumeration of Kekule structures in long and random benzenoid chains [63-65]. 

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One area in which the novel definition of Clar structures may have an advantage over the geometrical counterpart is in computer manipulations with Kekule and Clar structures. There are several algorithms and computer programs that enumerate, and even construct, all Kekule valence structures for benzenoid hydrocarbons [9]. These programs can now be combined with evaluation of the degree of freedom of Kekule structure, and such information can be combined into a scheme to produce list of Clar valence structures. 

The article reports investigations of the topological properties of benzenoid molecules which the author has performed in the last 20 years. Emphasis is given on recent developments and other scientists contributions to these researches. Topics covered in recent books and reviews are avoided. The article outlines spectral properties, some aspects of the study of Kekule and Clar structures, the Wiener index as well as a number of graphs derived from benzenoid systems (inner dual, excised internal structure, Clar graph, Gutman tree, coral and its dual). 

The investigations outlined in this article are grouped into four sections (a) questions concerned with the structure of benzenoid systems, (b) spectral properties and graph polynomials, (c) works related to Kekule and Clar structures and (d) topological indices. It will become clear, however, that all these researches are intimately interrelated and that there exist quite a few unexpected connections between them. 

This characterization suffice to determine Clar structure if there is but one such structure (the cases illustrated in Fig. 1 and Fig. 2). In order to define Clar structure in a general case we have to find which Kekule valence structure are used more than once in a superposition. The concept of k-sextet structure considered in the previous section allow us to arrive at the mathematical definition of Clar s structure shown below which we are comparing with the geometrical definition of Clar s structure ... 

We will leave the proof of the conjecture to mathematically inclined readers interested in this problem and will focus attention on consequences of the novel definition of Clar structures. If the two definitions of Clar structure are equivalent (as we conjecture) there should be identical consequences and the new definition is not to make a difference. However, the new definition does offer a mathematical characterization of Kekule valence structures involved in construction of Clar structure, something that has been hitherto missing. 

As we can see by comparing the HH-Clar structures and k-Clar structures they offer a distinctive basis for representation of benzenoid hydrocarbons. Each of the two approaches have their own merits, and as we will see in the next section, although the structures arising in the two models are quite different, the two approaches can be related in some respect. Both approaches use the same basis Kekule structures and differ... 

Kekule valence structures all the combination of Kekule valence structures of interest. The same is also true for finding combinations of Kekule valence structures involved in HH-Clar structures. A better way to obtain the correct combinations of Kekule valence structures instead of construction of various superpositions is to first write down k-Clar structures (or HH-Clar structures) and then decompose them into underlying Kekule valence structures. The apparent difficulty that remains in such an approach, that cannot be avoided, is the case of benzenoids having a large number of Kekuld valence structures which results in large number of decomposition. 

In Fig. 23 we have illustrated several HH-Clar structures of smaller benzenoids and have indicated their vulnerable rings. Each of the Kekule valence structure having a single vulnerable ring will produce the corresponding O-Clar structure (i.e., a Kekule valence structure that qualifies as Clar structure). In Fig. 24 are shown additional HH-Clar structures having besides vulnerable ring additional ji-sextets. These structures will reduce to k-Clar structures when Ji-sextet of the vulnerable... 

The successful accomplishments of Miillen and coworkers [22-25] who synthesized several giant benzenoid hydrocarbons will undoubtedly stimulate further theoretical interest in benzenoid hydrocarbons. It is not surprising that all the giant benzenoids that have been synthesized have 6n jt-electrons, which Clar predicted to be unusually stable. Now that the inverse problem of Clar structures has been solved we may expect novel theoretical developments in this area that may continue to expand experimentally beyond expectations. For example, the Conjugated Circuit Model, that has already been applied to giant benzenoids [26-28], may have to be modified so to take into account the prominent role of the Clar structures of benzenoids rather then considering all Kekule valence structures as equally important. Construction and enumeration of giant benzenoids and their Kekule valence structures has also received some attention [29, 30]. 

Fig. 2. Kekule structures and generalized Clar structures of benzo[a]anthracene...

The identity (7) is just a consequence of a one-to-one correspondence between generalized Clar and Kekule structures. In particular, the i-th Clar structure in... 

The study of generalized Clar structures and their relations to the Kekule structures led to the introduction of a new concept [92] which was eventually named coral [114]. 

The mathematical chemist will recognize a as one of the Kekule valence-bond structures of the hydrocarbon, while is the corres nd-ing molecular graph Model is called an inner dual or dualist [2], is a caterpillar tree [3], and is called a Clar graph [4]. The latter two models are apparently quite different from the original skeleton, however, as it will turn out later, the topological properties of this benzenoid system are best modeled by either d or e-... 

Recent calculations did further reveal that also the length of a finite nanotube has essential influence on its structure, which may adopt one out of three basic variants The Kekule-structure, the incomplete, and the complete Clar-structure (Figure 3.8). The first does not contain isolated double bonds, but is rather completely conjugated. The Clar-structures both feature p-phenylene moieties which, in the case of the incomplete Clar-structure, are flanked by isolated double bonds. This structural variety causes differences in electronic properties and thus in the... 

Figure 3.8 (a) Kekule structure, (b) incomplete, and (c) complete Clar structure. 

In Figure 2.12, we have illustrated determinants for an additional half a dozen benzenoid hydrocarbons that have three top and three bottom vertices. They have also been mentioned in Clar s booklet except for the last structure, which has no Kekule valence structure and iUnstrates a hypothetical system, referred to as concealed non-Kekulean benzenoids [32] becanse, in some cases, it is not immediately apparent that they have no F. A. Kekule (1829-1896) structures. For some of the benzenoid hydrocarbons shown in Figures 2.11 and 2.12 later in the chapter, there is an even simpler way of finding K, but the approach of John and Sachs is quite general and is often the simplest way of calculating K. 

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