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.
1 Introduction 2 The theory and concept
2.1 The Möbius–Hückel circle mnemonic
3 Application to molecules and pericyclic reactions
4 Simple tabular correlation of allowedness and forbiddenness of
Möbius versus Hückel and 4n + 2 versus 4n electrons
5 The generalized Möbius–Hückel orbital arrays
6 The butadiene to cyclobutene example
7 MO degeneracies leading to correlation diagrams
8 Relation of the
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
Figure 1. The Möbius–Hückel Circle Mnemonic applied to the example of cyclopentadienyl
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. 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
Figure 2. Prediction of allowed versus forbidden reactions; aromatic versus antiaromatic molecules.
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
Figure 3. Möbius (left) and Hückel (right) 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
Figure 4. Butadiene–cyclobutene interconversion; disrotation per Hückel (left), conrotation per Möbius (right).
Figure 4 shows the orbital array involved in the butadiene to cyclobutene interconversion. It is seen that there are four orbitals in this cyclic array. Thus in the interconversion reactions orbitals 1 and 4 overlap either in a conrotatory or a disrotatory fashion. Also, it is seen that the conrotation involves one plus-minus overlap as drawn while the disrotation involves zero plus-minus overlaps as drawn. Thus the conrotation uses a Möbius array while the disrotation uses a Hückel array. 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. Were one p-orbital in either reaction mode to be written upside-down, this would change the number of sign inversions by two and not change the evenness or oddness of the orbital array. 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. Although in these two examples symmetry is present, symmetry is not required or involved in determination of reaction allowedness versus forbiddenness. Hence a very large number of organic reactions can be understood. Even where symmetry is present, the Möbius–Hückel analysis proves simple to employ. MO degeneracies leading to correlation diagrams
Figure 5. Möbius–Hückel correlation diagram; two modes of butadiene to cyclobutene conversion.
It has been noted that for every degeneracy along a reaction
coordinate there is a molecular orbital crossing. Thus for the
butadiene to cyclobutene conversion, the two Möbius (here
conrotatory) and Hückel (here disrotatory) modes are shown in Figure
5. The starting MOs are depicted in the center of the correlation
diagram with blue correlation lines connecting MOs. It is seen that
for the Möbius mode the four electrons in MOs 1 and 2 end in the
bonding MOs (i.e. σ and π) of cyclobutene. In contrast, for the
Hückel mode, there is a degeneracy and thus an MO crossing leading to
two electrons (drawn in red) are headed for an antibonding MO. Thus
the Hückel mode is forbidden while the Möbius mode is allowed.
One further relevant point is that the first organic correlation
diagrams were in a 1961 publication on carbanion rearrangements. 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.
Relation of the
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^ Longuet-Higgins, H. C.; Abrahamson, E. W. (1965). "The Electronic
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^ Heilbronner, E. (1964). "Hückel molecular orbitals of Möbius-type
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^ Frost, A. A.; Musulin, B. (1953). "Mnemonic device for
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^ a b Zimmerman, H. E. (1971). "The Möbius–Hückel Concept in
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