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Figure 3. Möbius and Hückel Orbital Arrays (Möbius Left and Hückel Right) |
Figure 3. Möbius and Hückel Orbital Arrays (Möbius Left and Hückel Right) |
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[[Image:MH-Orb_Array.gif| |
[[Image:MH-Orb_Array.gif|thumb|Figure 3. Möbius and Hückel Orbital Arrays (Möbius Left and Hückel Right)]] |
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==The Butadiene to Cyclobutene example== |
==The Butadiene to Cyclobutene example== |
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Figure 4. Butadiene - Cyclobutene interconversion; Disrotation (Hückel) Left, Conrotation (Möbius) Right. |
Figure 4. Butadiene - Cyclobutene interconversion; Disrotation (Hückel) Left, Conrotation (Möbius) Right. |
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[[Image:ButaDienene_M-H.gif| |
[[Image:ButaDienene_M-H.gif|thumb|Figure 4. Butadiene - Cyclobutene interconversion; Disrotation (Hückel) Left, Conrotation (Möbius) Right. |
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Revision as of 16:19, 9 July 2010
The Möbius-Hückel (M-H) Concept for reaction allowedness and forbiddeness. One year following the Woodward Hoffmann[1] and Longuet-Higgins[2] publications, it was noted by Zimmerman that both transition states and stable molecules sometimes involved a Möbius array of basis orbitals[3][4] 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.[5] In contrast, the Möbius-Hückel (M-H) 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[3] was advanced which provides the MO energies of the system; this was the counterpart of the Frost-Musulin mnemonic[6] 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 ordinary Hückel systems.
The Möbius-Hückel Circle Mnemonic
The cyclopentadienyl annulenes are used here as an example; but in general whatever cyclic annulene is desired is inscribed in the circle of radius 2 beta and centered at zero (the energy of an isolated p-orbital). For every intersection of the annulene with the circle a molecular orbital energy is predicted at that vertical displacement. For Hückel Systems the vertex is positioned at the circle bottom at suggested by Frost. But 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, followng the ordinary Hückel rule. However, in contrast, the Möbius Systems have degenerate pairs of MO's 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 forbiddeness. Additionally, the M-H mnemonics give the MO’s at part reaction. At each degeneracy there is a crossing of MO’s. Thus one can determine if the highest occupied MO becomes anti-bonding with a forbidden reaction resulting. Finally, the M-H parity of sign inversions was utilized in the 1970 W-H treatment of alloweness-forbiddeness. 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.[7]
Simple Tabular Corrlation of Allowedness-Forbiddeness on 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. Dependeng 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.
Figure 2.Prediction of Allowed vs Forbidden Reactions; Aromatic vs Anti-Aromatic Molecules
Center
One further relevant point is that the first organic correlation diagrams were in a 1961 publication on carbanion rearrangements.[8] It has been noted that when an occupied molecular orbital becomes antibonding the reaction is inhibited and this phenomenon was correlated with a series of rearrangements.
The generalized Möbius-Hückel orbital arrays; Recognizing each variety
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 (even vs oddness) is not changed. One choice of signs leads to zero plus-minus overlaps for the Hückel array on the right.
Figure 3. Möbius and Hückel Orbital Arrays (Möbius Left and Hückel Right)
The Butadiene to Cyclobutene example
Figure 4 show 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. 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.
Figure 4. Butadiene - Cyclobutene interconversion; Disrotation (Hückel) Left, Conrotation (Möbius) Right.
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 eveness or oddness of the orbital array.
With a controtation 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 forbiddeness. Hence a very large number of organic reactions can be understood. Even where symmetry is present, the Möbius-Hückel analysis proves simple to employ.
References
- ^ “Stereochemistry of electrocyclic reactions”, Woodward, R. B.; Hoffmann, Roald. J. Amer. Chem. Soc. 1965, 87, 395-397.
- ^ "The Electronic Mechanism of Electrocyclic Reactions" Longuet-Higgins, H. C.; Abrahamson, E. W. J. Am. Chem. Soc., 1965, 87, 2045-2046.
- ^ a b c "On Molecular Orbital Correlation Diagrams, the Occurrence of Möbius Systems in Cyclization Reactions, and Factors Controlling Ground and Excited State Reactions. I," Zimmerman, H. E. J. Am. Chem. Soc., 1966, 88, 1564-1565.
- ^ "On Molecular Orbital Correlation Diagrams, Möbius Systems, and Factors Controlling Ground and Excited State Reactions. II," Zimmerman, H. E. J. Am. Chem. Soc., 1966, 88, 1566-1567.
- ^ "Hueckel molecular orbitals of Moebius-type conformations of annulenes," Heilbronner, E. Tetrahedron Letts 1964, 1923-1928.
- ^ Frost, A. A.; Musulin, B. "Mnemonic device for molecular-orbital energies." J. Chem. Phys. 1953, 21 572-573.
- ^ "The Möbius-Hückel Concept in Organic Chemistry. Application to Organic Molecules and Reactions," Zimmerman, H. E. Acc. Chem. Res., 1971, 4, 272-280.
- ^ "Carbanion Rearrangements. II," Zimmerman, H. E.; Zweig, A. J Am. Chem. Soc., 1961, 83, 1196-1213.