In chemistry, the Möbius–Hückel treatment is a methodology used to predict whether a reaction is allowed or forbidden. It is often used along with the Woodward–Hoffmann approach. The description in this article uses the plus and minus sign notation for parity as shorthand while proceeding around a cycle of orbitals in a molecule or system, while the Woodward–Hoffmann methodology uses a large number of rules with the same consequences.
Introduction 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-sign ↔ 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.
The theory and concept 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.
The Möbius–Hückel circle mnemonic
To determine the energy levels, the polygon corresponding to the cyclic annulene is desired is inscribed in the circle of radius 2β and centered at α (the energy of an isolated p orbital). The y-coordinate of the vertices of the polygon are the simple Hückel theory orbital energies. For systems with Hückel topology, the vertex is positioned at the circle bottom as suggested by Frost; for systems with Möbius topology, a polygon side is positioned at the circle bottom. In other words, for an N carbon system, the Möbius Frost circle is rotated by π/N radians compared to the Hückel system. 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 for pericyclic processes.
Application to molecules and pericyclic reactions Thus it was noted that along the reaction coordinate of pericyclic processes one could have either a Möbius or a Hückel array of basis orbitals. With 4n or 4n + 2 electrons, one is then led to a prediction of allowedness or forbiddenness. Additionally, the M–H mnemonics give the MOs at part reaction. At each degeneracy there is a crossing of MOs. Thus one can determine if the highest occupied MO becomes antibonding with a forbidden reaction resulting. Finally, the M–H parity of sign inversions was utilized in the 1970 W–H treatment of allowedness and forbiddenness. The parity of sign inversions between bonds and atoms was used in place of the M–H use of atoms; the two approaches are equivalent.
Simple tabular correlation of allowedness and forbiddenness of Möbius versus Hückel and 4n + 2 versus 4n electrons
The table in Figure 2 summarizes the Möbius–Hückel concept. The columns specify whether one has a Möbius or a Hückel structure and the rows specify whether 4n + 2 electrons or 4n electrons are present. Depending on which is present, a Möbius or a Hückel system, one selects the first or the second column. Then depending on the number of electrons present, 4n + 2 or 4n, one selects the first or the second row.
The generalized Möbius–Hückel orbital arrays
The two orbital arrays in Figure 3 are just examples and do not correspond to real systems. In inspecting the Möbius one on the left, plus–minus overlaps are seen between orbital pairs 2-3, 3-4, 4-5, 5-6, and 6-1, corresponding to an odd number (5), as required by a Möbius system. Inspection of the Hückel one on the right, plus–minus overlaps are seen between orbital pairs 2-3, 3-4, 4-5, and 6-1, corresponding to an even number (4), as required by a Hückel system. The plus–minus orientation of each orbital is arbitrary since these are just basis set orbitals and do not correspond to any molecular orbital. If any orbital were to change signs, two plus–minus overlaps are either removed or added and the parity (evenness or oddness) is not changed. One choice of signs leads to zero plus–minus overlaps for the Hückel array on the right.
The butadiene to cyclobutene example
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