Möbius-Hückel concept

The Möbius–Hückel treatment is one of two predicting reaction allowedness versus forbiddenness. The concept is the counterpart of the Woodward–Hoffmann approach. The methodology in this treatment utilizes the plus-minus sign parity in proceeding around a cycle of orbitals in a molecule or reaction while the Woodward–Hoffmann methodology uses a large number of rules with the same consequences.

One year following the Woodward–Hoffmann and Longuet-Higgins–Abrahmson publications, it was noted by Zimmerman that both transition states and stable molecules sometimes involved a Möbius array of basis orbitals The Möbius–Hückel treatment provides an alternative to the Woodward–Hoffmann one. In contrast to the Woodward–Hoffmann approach the Möbius–Hückel treatment is not dependent on symmetry and only requires counting the number of plus-minus sign inversions in proceeding around the cyclic array of orbitals. Where one has zero or an even number of sign inversions there is a Hückel array. Where an odd-number of sign inversions is found a Möbius array is determined to be present. Thus the approach goes beyond the geometric consideration of Edgar Heilbronner. In any case, symmetry may be present or may not.

Edgar Heilbronner had described twisted annulenes which had Möbius topology, but in including the twist of these systems, he concluded that Möbius systems could never be lower in energy than the Hückel counterparts. In contrast, the Möbius–Hückel concept considers systems with an equal twist for Hückel and Möbius systems.

For Möbius Systems there is an odd number of plus-minus sign inversions in the basis set in proceeding around the cycle. A circle mnemonic was advanced which provides the MO energies of the system; this was the counterpart of the Frost–Musulin mnemonic for ordinary Hückel systems. It was concluded that 4n electrons is the preferred number for Möbius moieties in contrast to the common 4n + 2 electrons for Hückel systems.

To determine the energy levels, the polygon corresponding to cyclic annulene is desired is inscribed in the circle of radius 2β and centered at α (the energy of an isolated p orbital). For every intersection of the polygon with the circle a molecular orbital energy is predicted with the energy corresponding to the vertical displacement. For Hückel Systems the vertex is positioned at the circle bottom as suggested by Frost; for Möbius systems a polygon side is positioned at the circle bottom. It is seen that with one MO at the bottom and then groups of degenerate pairs, the Hückel systems will accommodate 4n + 2 electrons, following the ordinary Hückel rule. However, in contrast, the Möbius Systems have degenerate pairs of molecular orbitals starting at the circle bottom and thus will accommodate 4n electrons. For cyclic annulenes one then predicts which species will be favored. The method applies equally to cyclic reaction intermediates and transition states.

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