Kekule valence-bond structure

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-... 

Graphical modeling can also be useful in representing the elements of the transfer matrix, J], adopted by Klein et ah, [13] Fig 3 shows the five Kekule valence-bond structures of enanthrene and their local states In this case one needs a directed graph with weighted edges and loops ... 

A more striking case which illustrates the advantage of the equivalent alternative forms was discovered sometime ago by one of the authors [33] An alternative form of a KekulS valence-bond structure, K, is its factor graph, F(K), c f Fig 4 In Fig 6 all Kekul structures of picene are hown (in the form of their Clar notation) along with the corresponding factor graphs One recalls that an F(K) is... 

Each Kekule valence-bond structure corresponds to a perfect match of the edges of the polyhex graph, see ref- 38-J- Riordan, An Introduction to Combinatorial Analysis Wiley, New York (1958) C-L- Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York (1968)- A chemical version on rook boards may be found in C-D- Godsil and 1- Gutman, Croat- Chem-Acta- 54, 53 (1981). 

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... 

It is wrong (but common) to see a reversible reaction written with a double-headed arrow, as A B. Such an arrow implies resonance, e.g. between the two extreme valence-bond structures of Kekule benzene. 

These three structures, the most stable valence-bond structures that can be formulated for naphthalene, are seen to have about the same energy and to correspond to about the same molecular configuration. It is to be expected then that they will be combined to represent the normal state of the naphthalene molecule, to which they should contribute about equally. Resonance among these three stable structures should stabilize the molecule to a greater extent than does the Kekul resonance in benzene, involving two equivalent structures it is seen from Table 6-2 that the resonance energy of naphthalene, 75 kcal/mole, is indeed much greater than that of benzene. 

The method that commonly is used is to draw a set of structures, each of which represents a reasonable way in which the electrons (usually in p orbitals) could be paired. If more than one such structure can be written, the actual molecule, ion, or radical will have properties corresponding to some hybrid of these structures. A double-headed arrow <—> is written between the structures that we consider to contribute to the hybrid. For example, the two Kekule forms are two possible electron-pairing schemes or valence-bond structures that could contribute to the resonance hybrid of benzene ... 

The resonance theory is very useful in accounting for, and in many cases predicting, the behavior of substances with tt bonds. However, it is not omnipotent. One example where it fails is cyclobutadiene, for which we can write two equivalent valence-bond structures corresponding to the Kekule structures for benzene ... 

Exercise 22-31 Draw the Kekule-type valence-bond structures for napthalene, anthracene, and phenanthrene. Estimate the percentage of double-bond character for the 9,10 bond of phenanthrene, assuming that each of the valence-bond structures contributes equally to the hybrid structure. 

IThe term Kekule structure will refer to any valence-bond structure for unsaturated compounds in which single and double bonds alternate. 

Because the three-electron bond structure is equivalent to resonance between structures Xa and Xb, it follows from eq.(3) that resonance between the Lewis structures XIX (or XVII) and XVIII is equivalent to use of the VB structure XX. In XX, a thin Y-A bond line is used to indicate that the Y-A bond-number in this structure is fractional, i.e. its bond-number is less than the value of unity that obtains for the Kekule Lewis structure XVIII [4-6,46]. (The fractionality is a consequence of the absence of a Y-A bond in structure XVII [4-6,46].) Valence bond structure XX is an example of an increased-valence structure for a four-electron three-centre bonding unit [4-6,37,38]. It may always be generated from the Lewis structure of type XVIII by delocalizing a non-bonding b electron into the bonding MO v fab, as is indicated in XXI — XX... 

In the hands of Pauling this treatment has been successful in correlating many facts in terms of a number of possible structural i ormulae, and it has been widely developed by Wheland and many others. Sometimes the valence bond structures betv een which resonance is considered to take place are natural and plausible, such as the two Kekule forms of benzene on other occasions it is tiecessary to postulate less plausible structures of liigh energy such as the ionic forms of carbon dioxide see Long i ). The... 

Returning, then, to the expansion of Equation (2), we note that the terms represent different valence bond structures. Why should they all have the same amplitude and phase This situation is very similar to the problem of determining the "resonance energy" of ben-zenoid molecules (25,26,27). In that case, of all the possible valence bond structures which might contribute, only the Kekule structures are used. For large benzenoid systems this is only a small fraction of the total number of structures. Furthermore, it is assumed that they all enter with equal expansion coefficients (i.e., equal amplitude and phase). In addition, the matrix elements which convert one structure into another are set equal to a common value, determined empirically. Thus, the energy lowering associated with "resonance" in benzenoid molecules has a mathematical structure which maps onto the discussion in the Introduction. However, there are some important differences. 

El-Basil S, Randic M (1992) Equivalence of Mathematical Objects of Interest in Chemistry and Physics. Adv Quant Chem 24 239-290 Graovac A, Gutman I, Randic M, Trinajstic N (1973) Kekule Index for Valence Bond Structures of Conjugated Polycyclic System. J Am Chem Soc 95 6267-6273... 

Fig. 7.2. The dimerized geometrical structures and associated Kekule valence bond...

The effect of substituents on colour in substituted anthraquinones may be rationalised using the valence-bond (resonance) approach, in the same way as has been presented previously for a series of azo dyes (see Chapter 2 for details). For the purpose of explaining the colour of the dyes, it is assumed that the ground electronic state of the dye most closely resembles the most stable resonance forms, the normal Kekule-type structures, and that the first excited state of the dye more closely resembles the less stable, charge-separated forms. Some relevant resonance forms for anthraquinones 52, 52c, 52d and 52f are illustrated in Figure 4.3. The ground state of the parent compound 52 is assumed to resemble closely structures such as I, while charge-separated forms, such as structure II, are assumed to make a major contribution to the first excited state. Structure II is clearly unstable due to the carbocationic centre. In the case of aminoanthraquinones 52c and 52d, donation of the lone pair from the... 

Since a calculation of the resonance energy of benzene by the valence bond method shows that the greater part of it is a result of the resonance between the two Kekule structures shown, we might suppose that its homologs would also have significant resonance stabilization energies. Such conclusions are at variance with experimental fact, however, since cyclobutadiene appears to be too unstable to have any permanent existence, and cyclooctatetraene exists as a nonplanar tetraolefin, having no resonance stabilization of the sort considered. 

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