Sextet rotation

Define the sextet rotation (R) as a simultaneous rotation of all the proper sextets in a given Kekule pattern k into the improper sextets to give another Kekule pattern k... 

Similarly the counter-sextet rotation (FL) is defined as follows ... 

The sextet rotation to the set of the Kekule patterns k, gives a directed tree graph with a root, or the root Kekule pattern, representing the hierarchical structure of kj, where each point corresponds to a Kekule pattern. 

In Figs. 3 and 4 are given the rooted directed trees derived by joining all the entries of the Kekule patterns of XVI and V with the sextet and counter-sextet rotations. One can realize how all the Kekule and sextet patterns are related to each other. Theorem 3 can be proved by showing that there is one and only one root Kekule pattern, a Kekule pattern without any proper sextet, (see Lemma) and no cyclic relation among the Kekule patterns with respect to the sextet rotation. This graph-theoretical discussion does not necessarily mean that the root Kekule pattern is the most chemically-important in the family of the Kekule patterns, but that all the patterns are mathematically related with each other. 

Fig. 3. Directed rooted trees of the Kekule patterns of benzanthracene derived by (R) performing the sextet and counter-sextet rotations. The double (JR) circle represents the root Kekule pattern...

The sextet rotation operation defines our adjacency relation among the set of Kekule structures and the vertex-transitive graphs generated are nothing else but posets of Kekule structures. 

Our investigations on the 2 A state can provide some help in positioning the part of the spectrum coming from doublet states, known experimentally from rotational analyses or perturbations but whose absolute energies were only approximately evaluated [21]. The sextet states of VO are described only approximately since the potential energy curves are very flat and more points would be desirable in order to perform more exact analysis. Nevertheless, their positions and geometric properties are consistent with a simple picture given by a qualitative molecular orbital scheme. 

The doublet (3) at ca. 1.23 ppm is the side chain, a methyl group. The sextet (2) at 5,25 ppm is the chiral carbon atom in the backbone with 5 vicinal hydrogens and near to the oxygen of the ester bond. The CH2 in the backbone gives tihe signal at 2,5 ppm. The vicinal coupling of the proton resonance is due to the rotation of the CH2 - CH backbone bond. 

Fig. 6.11 Correlation diagram for Cgo- The Fries and Qar structures are bonding extremes, where double bonds are either locahzed on the 30 bonds between the pentagons (Fries), or form isolated aromatic sextets on the twelve pentagons. The true conjugation scheme is found in between, and is characterized by six unoccupied levels, which are anti-bonding in the Fries structure and bonding in the Clar structure, and which transform as rotations and translations. Buckminsterfullerene has low-lying LUMO and LUMO-l-1 levels of huiFr) and t g(FR) symmetry...

The Clar structure thus has six extra bonding orbitals as compared with the Fries structure. When both bonding schemes are correlated, as illustrated in Fig. 6.11, this sextet must correlate with the anti-bonding half of the Fries stmcture. It will thus be placed on top of the Clar band, and actually be nearly non-bonding, forming six low-lying virtual orbitals, which explains the electron deficiency of the leapfrog fullerenes. Moreover, as the derivation shows, they transform exactly as rotations and translations. 

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