Clar-structure

Like phenanthrene, some PAHs have a unique Clar structure, whereas several alternative Clar structures are possible for other PAHs [21]. Thus, Clar s rule does not designate the resonance structure mainly responsible for the aromaticity of anthracene. Clar s model cannot differentiate between its outer and inner ring (Scheme 28.2). 

In the theory known nowadays as the Clar theory of the aromatic sextet [12] a benzenoid system is represented by a Clar structure which is obtained by drawing circles in some of the hexagons of the corresponding benzenoid graph. These circles represent the aromatic sextets in the hydrocarbon. We consider here only Clar structures containing the maximum number of circles which are some times referred to as proper (or correct) Clar formulas. The rules for constructing such Clar structures are as follows [13]. 

This type of Clar structure can be generated for a limited number of classes of catacondensed benzenoid hydrocarbons by defining an operation which resembles coloring [14] of the vertices of certain caterpillars but allows more flexibility. Namely, we define a Clar coloring [ 15] of the vertices of a caterpillar by the following steps ... 

Fig. 3 depicts all five Clar structures of a branched hydrocarbon and the five Clar colorings of the corresponding caterpillar, in this case, a comb tree . It has been... 

It is worthy of observation that for both classes of benzenoid hydrocarbons represented in Figs. 2, 3 the corresponding trees (whose number of colorings = the number of Clar structures) coincide with the inner duals [ 19] of the benzenoid graphs. 

Fig. 3. All five Clar structures of a branched benzenoid hydrocarbon and the corresponding five colored Gutman trees. The indicated coloring generates F4, the fourth Fibonacci number...

Phenanthrene has also a reactive 9,10-double bond, in agreement with the Clar structure having two aromatic sextets and a C=C fixed double bond in the median ring. On co-ozonolysis with formaldehyde, acetyl cyanide, or benzoyl cyanide, phenanthrene reacted accordingly, affording an aldehydic ozonide 112, which in a separate co-ozonolysis with vinyl acetate that produced formaldehyde oxide (H2C-0-0) gave rise to a diozonide 113 (Scheme 35 and Table 14). 

Fig. 1 Benzenoid hydrocarbons that have a single Clar structure consisting of only...

Fig. 2 Cata- and peri-condensed benzenoid hydrocarbons with Clar structure...

Kekul6 valence structures, which contribute to so called Clar structures, Clar valence structures are depicted with inscribed circles in some benzene rings obeying the following rule. Inscribe as many circles as possible subject to two restrictions (i) No adjacent rings should have inscribed circles and (ii) CC bonds not involved in rings with inscribed circles should allow completion of valence structure by assignment of CC double bonds wherever possible. 

In Fig. 1 we illustrate Clar structures for several smaller benzenoids in which only inscribed circles appear. These are the unusually stable 6n Jt-electron benzenoids which Clar called fully benzenoid. In Fig. 2 we illustrate Clar structures for smaller benzenoids in which besides inscribed circles representing aromatic sextets also appear one or more CC double bonds. Finally in Fig. 3 we illustrate benzenoid hydrocarbons for which one can draw several Clar valence structures. In such cases the overall Clar structure is obtained as a superposition of the component valence structures in an analogous way as one represents molecules by superposition of several Kekul6 valence structures. Clar represented the final superposition of component structures by drawing a single structure to which one or more arrows are added to suggest delocalized Jt-sextets. 

The novel definition of Clar structures to be outlined in the next section, is based on the concept of the degree of freedom (df) of a... 

Because Clar structures can be viewed as a superpositions of selected Kekule valence structures we will briefly examine pair-wise superposition of the five Kekule valence structures of phenanthrene (shown in Fig. 5). In all we can construct ten combinations illustrated in Fig. 8. First,... 

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 ... 

Conjecture The mathematical definition of Clar structure and the geometrical (constructive) definition are equivalent. 

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. 

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. 

We will now investigate the role of the new definition of Clar structures for Clar-type structures that do not posses the maximal number of inscribed re-aromatic sextets. Already Clar, in discussing properties of benzo[a]coronene, evoked a structure that has less than the maximal number of re-sextets [10]. While the proper Clar structure of benzo[a]coronene has four re-sextets, which is the maximal number for benzo[a]coronene, Clar considered also a structure having three sextets (Fig. 12). Such imperfect Clar structures were introduced into Clar Sextet Theory systematically by Hosoyaand Yamaguchi [11]. 

Hosoyaand Yamaguchi generalized Clar structures by removing the restriction that Clar structures have to have the maximal number of rearomatic sextets. We will refer to these as HY-Clar structures, or briefly HY-structures (HY for Hosoya-Yamaguchi) in order to differentiate them from other generalizations of Clar structures that will be discussed later. In Fig. 13 we show the set of generalized HY-Clar formulas for... 

Fig. 12 Clar structure of benzo[a]coronene and a Clar-type structure that has less than the maximal number of re-sextets considered by Clar...

Fig. 13 Generalized Clar structures introduced by Hosoya-Yamaguchi illustrated on...

Generalized Clar structures introduced by Herndon and Hosoya illustrated on benzol ghi]perylene. 

We have inserted in their definition the word uniquely for added precision. In other words, once the positions of inscribed circles have been selected the rest of the molecule should have the residual degree of freedom equal one, that is, positions of all CC double bonds are unique. In Fig. 14 we illustrate the five HH-Clar structures of benzo[ghi]perylene. In the case of benzo[ghi]perylene the first structure is the proper Clar structure that has the maximal number of Jt-sextets. Of the four generalized HH-Clar structures three have two it-sextet and one has but a single Jt-sextet. 

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... 

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