Conjugated circuit

In the case of benzene, both Kekuld valence structures are equivalent, being identical except for their orientation in space. In naphthalene, two of the three structures are symmetry-related, but the third structure is different. The question is How different is the symmetry non-equivalent structure With a close look at the naphthalene Kekule valence structures it is easy to see that the unique Kekule structure of naphthalene has two rings each having three C=C bonds just as have Kekule structure of benzene. The other two Kekuld valence structures of naphthalene have but a single benzene ring with three C=C bonds. 

FIGURE 10.1 Three Kekule valence structures ofnaphthalene and their conjugated circuits. 

Alternant molecules are all acyclic molecules and molecules having only rings with an even number of atoms. Non-alternant molecules have rings with an odd number of atoms. Thus, in the case of an alternant molecule, all vertices of its molecular graph can be colored as black and white so that no vertices of the same color are adjacent. The division of molecules into alternant and non-alternant is desirable because of their distinct mathematical properties. Thus, in the case of alternant systems, all eigenvalues come in pairs, positive and negative. This is not the case with non-alternant systems. 

Observe that coronene, which is not so big molecule, has 380 conjugated circuits. In the case of buckminsterfuUerene Qo the number conjugated circuits is well over 150 million (to be precise, 156,237,500, easily calculated from Gutman s theorem stating that conjugated hydrocarbon having K Kekule valence structures has (K - 1) 

FIGURE 10.4 All symmetry nonequivalent Kekule valence structures of coronene. Under each structure is shown the multiplicity of the structure, that is, how many times the structure occurs among 20 Kekule valence structures of coronene. 

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Different circuit counts are designated as <4n+2) or (4,) and are obtained by the summation of all (4n + 2) or An) conjugated circuits. When the conjugated circuits model (CCM) is applied, the RE of a polycyclic conjugated molecule (CCMRE) may be determined as follows, taking into account that the REs are additive within the model in question (76CPL68),... 

A number of computational approaches to the (G) have been developed and there have been widespread applications of the conjugated-circuits model, motivated both from Herndon s and from Randic s approaches. The applications extend even much beyond benzenoids. This is reviewed elsewhere by Randic et al. [76],... 

Isoarithmicity is discussed in Cyvin SJ, Gutman I (1988) Kekule Structures in Benzenoid Hydrocarbons, Springer, Berlin Heidelberg New York, p 29 (Lecture Notes in Chemistry 46) also Balaban AT, Tomescu I (1983) Match 14 155 Croat. Chem. Acta. 57 391 The name isoconjugate is used sometimes to describe hydrocarbons which are isoarithmic and contain identical numbers of conjugated circuits Randic M (1977) Tetrahedron 33 1905... 

Fig. 4 Conjugation types in porphyrinoids. Fully conjugated circuits are shaded. For details, see text...

Several qualitative models, e.g. Platt s ring perimeter model [88], Clar s model [89] and Randic s conjugated circuits model [90-92] have either been or are frequently used for the rationalisation of their properties. All these qualitative models rationalise the properties of aromatic and anti-aromatic hydrocarbons in terms of the Hiickel [4n+2] and [4n] rules. The extra stability of a PAH, due to 7t-electron delocalisation, can also be determined, computationally or experimentally, by either considering homodesmotic relationships [36] or by the reaction enthalpy of the reaction of the PAH towards suitable chosen reference compounds [93],... 

Fig. 11. Resonance between the structures leading to benzene-like resonance in the top six 7t electron, central six 7t electron and 14 7t electron conjugated circuits, respectively.

The conjugated-circuits model is one of the simplest quantitative models that has been reasonably well studied. As already mentioned this model may be motivated from classical chemical bonding theory (extended a la Clar s classical empiricist argument) or from Simpson s existential quantum-theoretic argument [ 121 ], or from a quantum chemical derivation indicated in our hierarchy of section 3.2. But beyond derivation of the model there is the question of its solution, such as we now seek to address. 

There are different methods to evaluate the numerator (T, Hhs F,) in the conjugated-circuits expression for the resonance energy. One approach is simply to go through each Kekule structure identifying and counting up conjugated n-circuits as occur in each. In another approach this numerator is recast as... 

Thus many of the Kekule-structure enumeration methodologies of section 4 have been shown to rather neatly extend to conjugated-circuits enumerations, with but modest trouble beyond the overall Kekule-structure count K(G). 

As it turns out though even the simplified VB-theoretic formulations giving rise to conjugated-circuit theory or even just Kekule-structure enumeration, may become challenging for sufficiently large (perhaps formally infinite) systems, or for non-Kekulean (i.e., radicaloid) systems. It might oft be convenient if explicit enumeration of Kekule structures could be avoided. Notably for such cases there are some few alternative sorts of means by which to obtain some partial information about the system, within a VB-theoretic context. 

In Fig. 7 we show all Kekule valence structures of benzo[ghi]perylene and their degrees of freedom. The first eight structures have df = 3, the next five Kekule valence structures have df = 2, and the last Kekule valence structure has df = I. The individual Kekule valence structures have quite different count of conjugated circuits R, which can be as high as five (in the first Kekule valence structure) and as low as one (in the last Kekule valence structures). A close look at these... 

Conjecture The degree of freedom of a Kekule valence structure is given by the maximal number of disjoint conjugated circuits. 

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