Q2(b). Hückel–Möbius (PMO) analysis of cycloadditions

Introduction. In the Dewar–Zimmerman (PMO) approach the cyclic array of p orbitals in the transition state is treated like a ring. The array is Hückel if it has zero (or an even number of) phase inversions between neighbouring orbitals, and Möbius if it has an odd number. Each suprafacial component adds no inversion; each antarafacial component adds one.

Figure: Signs = phase of the lobe that points into the ring; the dashed bond links lobes of opposite phase.

1. Rule. A Hückel array is aromatic with 4n+2 electrons and antiaromatic with 4n. A Möbius array is the reverse: aromatic with 4n and antiaromatic with 4n+2.
2. Selection. An aromatic transition state is low in energy, so the reaction is thermally allowed. An antiaromatic transition state is forbidden thermally but becomes photochemically allowed, because in the excited state the energy ordering is reversed.
3. I [2s+2s]: two ethylenes face to face, 4 electrons, no inversion: Hückel, 4n, antiaromatic, photochemical (as in cyclobutane photodimerisation) ⇒ D.
4. II [4s+2a]: 6 electrons, one antarafacial component, one inversion: Möbius with 4n+2, antiaromatic ⇒ A.
5. III [4s+2s]: the Diels–Alder reaction, 6 electrons, Hückel, aromatic (like benzene), thermal ⇒ B.
6. IV [2a+2s]: 4 electrons with one inversion: Möbius, 4n, aromatic, thermal (ketene and allene [2+2] cycloadditions use this geometry) ⇒ C.

Conclusion. I–D, II–A, III–B, IV–C. Count the electrons and the phase inversions: Hückel arrays are aromatic with 4n+2 electrons and Möbius arrays with 4n; aromatic transition states are thermally allowed, antiaromatic ones photochemically allowed.

Application. The same PMO count predicts all pericyclic reactions at once: thermal Diels–Alder synthesis (aromatic Hückel TS), photochemical [2+2] routes to cyclobutanes, and thermal [2+2] of ketenes used to make cyclobutanones and β-lactams.