Q5(a). [1,5]-Hydrogen shift in penta-1,3-diene: FMO treatment

Introduction. A sigmatropic rearrangement moves a σ bond, with its group, along a π system. In a [1,j] hydrogen shift the H keeps its point of attachment on itself (“1”) and moves from C-1 to C-j. In cis-penta-1,3-diene the H on C-5 (the CH₃) moves to C-1: a [1,5]-H shift, which involves six electrons (the C–H σ pair and two π pairs).

Woodward–Hoffmann rule for [1,j] H shifts

A thermal [1,j] shift with (4q+2) electrons is allowed suprafacially; with 4q electrons it must be antarafacial. Light reverses the rule.

FMO approach

Figure: Sketch: transition state as H atom + pentadienyl radical (ψ₃ has nodes at C-2 and C-4). Shaded lobes = one phase; C-1 and C-5 have the same phase on the top face.

1. Divide the transition state into a hydrogen atom (1s) and a pentadienyl radical (5 π electrons). The frontier orbital of the radical is its singly occupied MO, ψ₃ (non-bonding: nodes at C-2 and C-4, lobes on C-1, C-3 and C-5).
2. In ψ₃ the lobes at the two termini, C-1 and C-5, have the same phase on the same face. The H 1s orbital can therefore stay bonded to C-5 while it starts bonding to C-1 on the same face: suprafacial shift, thermally allowed.
3. In the excited state the frontier orbital is ψ₄, whose termini have opposite phases on the same face; the shift would have to be antarafacial, which a single H cannot do across a five-carbon chain. Photochemical [1,5]-H shifts are not observed.
4. Same result by electron count: 6 electrons (4q+2) ⇒ thermal suprafacial (aromatic, Hückel-type transition state).

Scheme: Deuterium label moves from C-5 to C-1 of (Z)-penta-1,3-diene (s-cis, termini close), and the double bonds shift.

Conclusion. A thermal [1,5]-H shift is suprafacial and allowed (six electrons, 4q+2): in the SOMO ψ₃ of the pentadienyl radical C-1 and C-5 have the same phase on one face. Photochemically the shift would have to be antarafacial.

Application. Thermal [1,5]-H shifts scramble substituents in cyclopentadienes (isomerisation of 5-substituted to 1- and 2-substituted cyclopentadienes) and convert previtamin D into vitamin D ([1,7]-H, antarafacial) in the skin.