Enumeration of Kekule Valence Structures

ENUMERATION OF KEKULE VALENCE STRUCTURES 2.3.1 Cata-Condensed Benzenoid Hydrocarbons... 

Kekule valence structures in lattices of peri-condensed benzenoids. Enumeration of Kekule valence structures in benzenoid lattice structures requires a modification of the algorithm of Gordon and Davison, which was introduced in chemical literature by Randic [22], Let us consider dibenzocoronene, the first structure in Figure 2.6. 

Because Kekule valence structures are central to the concept of conjugate circuits, we will follow with a discussion of the properties of Kekule valence structures, their construction, and their enumerations. We will be relatively brief and would like to direct the attention of readers to the book by Cyvin and Gutman on Kekule structures in benzenoid hydrocarbons, where the enumeration and properties of benzenoid structures are discussed on over 300 pages. Kekule valence structures have been around... 

Finally let us mention the approach known as the transfer-matrix method, which is particularly useful for systems that have repeating sub-units, e.g., systems with high symmetry and polymers. Besides applications to benzenoid and non-benzenoid polymers, the transfer-matrix method is suitable for systems with rotational symmetry, such as buckmin-sterfullerene, Ceo- In fact, the first reported value for the number of Kekule valence structures in buck-minsterfullerene Ceo, which is 12 500, was obtained using the transfer-matrix method. The transfer-matrix method has been used for some time in statistical mechanics. " Klein and co-workers ° pioneered the application of the transfer-matrix method for chemical enumerations. 

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 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]. 

Individual formal valence structures of conjugated hydrocarbons are excellent substrates for research in chemical graph theory, whereby many of the concepts of discrete mathematics and combinatorics may be applied to chemical problems. The lecture note published by Cyvin and Gutman (Cy-vin, Gutman 1988)) outlines the main features of this type of research mostly from enumeration viewpoint. In addition to their combinatorial properties, chemists were also interested in relative importance of Kekule valence-bond structures of benzenoid hydrocarbons. In fact, as early as 1973, Graovac et al. (1973) published their Kekule index, which seems to be one of the earliest results on the ordering of Kekule structures These authors used ideas from molecular orbital theory to calculate their indices... 

Sheng" outlined an economical two-vertex elimination method for deciding whether a polycyclic ben-zenoid hydrocarbon has Kekule valence structure, and brothers He Wenjie (mathematician) and He Wenchen (chemist) discuss enumeration of Kekule structures using a matrix corresponding to peak and valley carbon atoms." ° We should also mention the quick and robust method of Kearsley" for assigning CC double bonds in a Kekule valence structure on the basis of an ordered fragmentation of the molecular skeleton. 

Figure 46. Enumeration of linearly independent conjugated circuits for 10 symmetry-non-equivalent Kekule valence structures of benzo[ / i]perylene by inscribing integers that indicate the size of conjugated circuits involving individual rings. Under each structure is given the number of Ri, Rz, Rs, and R4 conjugated circuits.

According to Klein and co-workers, one can enumerate not only the Kekule valence structures but also conjugated circuits of different size using the transfer matrix approach (briefly mentioned in the introductory part of this review). In the case of Ceo, the expression for the RE is ... 

---