Möbius–Hückel concept

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.

One year following the Woodward–Hoffmann[1] and Longuet-Higgins–Abrahmson[2] publications, it was noted by Zimmerman that both transition states and stable molecules sometimes involved a Möbius array of basis orbitals.

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.

For Möbius systems there is an odd number of plus–minus sign inversions in the basis set in proceeding around the cycle.

However, in contrast, the Möbius Systems have degenerate pairs of molecular orbitals starting at the circle bottom and thus will accommodate 4n electrons.

The method applies equally to cyclic reaction intermediates and transition states for pericyclic processes.

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.

[3] But it is important to note, as described for the generalized orbital array in Figure 3, that the assignment of the basis-set p-orbitals is arbitrary.

With a conrotation giving a Möbius system, with butadiene's four electrons, we find an "allowed" reaction model.

With disrotation giving a Hückel system, with the four electrons, we find a "forbidden" reaction model.

[4] Thus for the butadiene to cyclobutene conversion, the two Möbius (here conrotatory) and Hückel (here disrotatory) modes are shown in Figure 5.

[8] It had been noted that when an occupied molecular orbital becomes antibonding the reaction is inhibited and this phenomenon was correlated with a series of rearrangements.

But in 1969 and 1970 a general formulation was published,[9][10] namely, A ground-state pericyclic change is symmetry-allowed when the total number of (4q + 2)s and (4r)a components is odd.

Figure 1. The Möbius–Hückel Circle Mnemonic [ 3 ] applied to the example of cyclopentadienyl
Figure 3. Möbius (left) and Hückel (right) orbital arrays.
Figure 4. Butadiene–cyclobutene interconversion; disrotation per Hückel (left), conrotation per Möbius (right).
Figure 5. Möbius–Hückel correlation diagram; two modes of butadiene to cyclobutene conversion.