Hiickel topology

Thermally, it will proceed via a Mobius topology involving one antarafacial component. Both methyl groups will rotate in the same direction conrotation) leaving one endocyclic and one exocyclic. However, under photochemical conditions, a [4n] ir-reaction is predicted to proceed via a Hiickel topology with suprafacial (disrotation) bond formation. 

Similarly, for hexatriene, the Mobius and Hiickel topologies are shown in Fig. 8.50. It is clear that in a [b-rr]-electrons system, the Hiickel topology is aromatic and the reaction is symmetry allowed by disrotation cyclization. 

There is another aspect of cycloaddition TS structure that must be considered. It is conceivable that some systems might react through an arrangement with Mobius rather than Hiickel topology (see p. 716). Mobius systems can also be achieved by addition to opposite faces of the tt system. This mode of addition is called antarafacial and the face-to-face addition is called suprafacial. In order to specify the topology of cycloaddition reactions, subscripts s and aare added to the numerical classification. For systems of Mobius topology, as for aromaticity, 4n combinations are favored and 4n- -2 combinations are unfavorable... 

Figure 15.17 B shows the aromatic transition state analysis of these reactions. We draw a picture of an opening pathway with the minimum number of phase changes and examine the number of nodes. The four-electron butadiene-cyclobutene system should follow the Mobius/conrotatory path, and the six-electron hexatriene-cyclohexadiene system should follow the Hiickel/disrotatory path. As such, aromatic transition state theory provides a simple analysis of electrocyclic reactions. The disrotatory motion is always of Hiickel topology, and the conrotatory motion is always of Mobius topology.

Occasionally, though, you will run across a more exotic pericyclic process, and will want to decide if it is allowed. In a complex case, a reaction that is not a simple electrocyclic ringopening or cycloaddition, often the basic orbital symmetry rules or FMO analyses are not easily applied. In contrast, aromatic transition state theory and the generalized orbital symmetry rule are easy to apply to any reaction. With aromatic transition state theory, we simply draw the cyclic array of orbitals, establish whether we have a Mobius or Hiickel topology, and then count electrons. Also, the generalized orbital symmetry rule is easy to apply. We simply break the reaction into two or more components and analyze the number of electrons and the ability of the components to react in a suprafacial or antarafacial manner. 

Fig. 11 The edge-localized bonding model for Os3(CO)i2 showing the three 2c-2e bonds along with the a-aromatic model showing the 3c-2e core bond with Hiickel topology and the 3c-4e perimeter bond with MSbius topology. Reprinted with permission from [123], Copyright-. Elsevier...

Fig. 12 Isotropic NICS (NICS given in parentheses) contributions at the ring center for the 3c-2e radial orbitals with Hiickel topology and the 3c-4e peripheral orbitals with Mobius topology in a) cyclopropane andb) [Fe3(CO)i2], Z>3h [Ru3(CO)i2], and >3h [Os3(CO)i2]. Values...

Fig. 16. Observed and calculated carbon-carbon bond lengths for tetrahaptometal-olefin complexes (156), based on a topological Hiickel ir-bonding model.

Finally, if we abandon Hiickel s topological approach altogether and consider more elaborate quantum-mechanical approaches, the concept of aromaticity derived purely from a consideration of -electrons becomes blurred and tends to disappear completely. In fact, allelectron methods allow the calculation of aromatic properties (Section V,B) of a given substance without introducing explicitly the concept of aromaticity. Certain authors, notably Dewar,19 have published resonance energies derived from self-consistent field molecular-orbital (SCF-MO) calculations, and these could be used as a measure of aromaticity. 

Topological orbitals... Probably the smartest of our readers have already appreciated the close relationship between the graph theory and the Hiickel method. Actually in quantum chemistry there is a great variety of problems in which the Hamiltonian of a molecule can be written in a matrix form as a one-valued function of the topological matrix of that molecule ... 

With Eq. (21) in mind, compare the Hiickel equation (9) with Eq. (16) defining the ectrum of topological graph of molecule ... 

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