Q5(a). Thermally forbidden [1,3]-H shift; rules for sigmatropic reactions

Introduction. In a sigmatropic [1,j] shift a σ-bonded H moves from one end of a π system to the other. In the FMO picture the TS is treated as an H atom (1s orbital) moving over an allyl radical; the H must bond to C-1 and C-3 with lobes of the same phase.

Figure: Allyl radical HOMO with a migrating H: shaded and open lobes have opposite phase; the dot marks a node at C-2.

1. Thermal [1,3]-H. The allyl HOMO in the ground state is ψ₂ (non-bonding): node at C-2, and the lobes of C-1 and C-3 on the same face have opposite phase. An H leaving C-1 from the top face meets a lobe of opposite sign at C-3: suprafacial shift is symmetry-forbidden.
2. The allowed alternative is antarafacial: H leaves the top face of C-1 and arrives on the bottom face of C-3. Across a three-carbon span the H cannot reach both faces; the strain is impossible. So the concerted thermal [1,3]-H shift does not occur (thermal “1,3-H shifts” go by stepwise acid/base or radical paths).
3. Photochemical. The HOMO is ψ₃ (two nodes), whose terminal lobes on the same face have the same phase: the suprafacial [1,3]-H shift is allowed on irradiation.
4. In a [1,5]-H shift the HOMO of the pentadienyl radical (ψ₃) has same-phase termini: thermal suprafacial [1,5]-H shifts are allowed and common (cyclopentadienes).

Woodward–Hoffmann selection rules for sigmatropic shifts of order [i,j]

For a migrating H (i=1): [1,3] and [1,7] thermal must be antarafacial ([1,7]-H antarafacial is possible in a long, helical chain, as in previtamin D → vitamin D); [1,5] thermal is suprafacial. For a migrating carbon, a suprafacial shift with inversion at the migrating carbon is the same as an antarafacial component, so a thermal [1,3]-C shift with inversion is allowed.

Conclusion. The thermal [1,3]-H shift is forbidden because the allyl HOMO ψ₂ has opposite-phase termini on the same face, so only the geometrically impossible antarafacial path is allowed. Rule: (4n+2) shifts are thermally supra–supra, 4n shifts are thermally supra–antara; light reverses the rules.

Application. Thermal [1,5]-H shifts scramble substituents round cyclopentadiene rings; the antarafacial [1,7]-H shift converts previtamin D3 to vitamin D3 in the skin.