A sigmatropic rearrangement is a concerted pericyclic reaction in which a σ bond migrates across a conjugated π system. The σ bond breaks at its original position and re-forms at a new position, while the π electrons reorganize to accommodate the new connectivity.

The total number of σ and π bonds stays constant throughout the reaction. No intermediates form, no atoms are gained or lost, and the overall molecular connectivity is preserved. What changes is where the σ bond sits relative to the π framework.

Because these are concerted (single-step) reactions, they follow the Woodward-Hoffmann symmetry rules, which predict whether a given rearrangement is thermally or photochemically allowed.

Notation for Sigmatropic Rearrangements

Sigmatropic rearrangements are classified with [i,j] notation:

i = the number of atoms the migrating group traverses from its starting point (counting the atom it was bonded to as atom 1)
j = the number of atoms traversed along the other fragment to reach the new bonding site (again counting from atom 1)

A few common examples:

[1,5] shift: The σ bond migrates from atom 1 to atom 5 on the same fragment. A classic case is the [1,5]-hydrogen shift in 1,3-pentadiene, where an H moves across a pentadienyl system.
[3,3] shift: Both ends of the σ bond migrate three atoms. The Cope rearrangement (1,5-hexadiene → a new 1,5-hexadiene isomer) and the Claisen rearrangement (allyl vinyl ether → γ,δ-unsaturated carbonyl) are the most important [3,3] examples.
[1,7] shift: The σ bond migrates from atom 1 to atom 7, involving a larger conjugated system.

The sum $i + j$ tells you the total number of atoms in the cyclic transition state. For a [1,5] shift, that's 6 atoms; for a [3,3] shift, also 6 atoms.

Suprafacial vs. Antarafacial Modes

The migrating bond can stay on the same face of the π system (suprafacial) or cross from one face to the other (antarafacial). This distinction controls the stereochemical outcome.

The Woodward-Hoffmann rules for sigmatropic rearrangements depend on the electron count in the cyclic transition state:

$i + j$ electrons = $4n + 2$ (e.g., 6 electrons for a [1,5] or [3,3] shift)
- Thermally allowed: suprafacial on both components
- Photochemically allowed: antarafacial on one component
$i + j$ electrons = $4n$ (e.g., 4 electrons for a [1,3] shift)
- Thermally allowed: antarafacial on one component (suprafacial on the other)
- Photochemically allowed: suprafacial on both components

Practical consequence: Thermal [1,5]-H shifts are suprafacial and occur readily because the 6-membered transition state is geometrically comfortable. Thermal [1,3]-H shifts would need to be antarafacial, which requires the hydrogen to reach the opposite face of a small π system. That's geometrically impossible for hydrogen (it has only a spherical 1s orbital), so thermal [1,3]-H shifts are effectively forbidden. They can occur photochemically via a suprafacial path.

For carbon migrations, the story is slightly different. Carbon has a p orbital with two lobes, so it can undergo inversion of configuration at the migrating center to achieve an antarafacial interaction. A thermal [1,3]-alkyl shift is symmetry-allowed if the migrating carbon inverts its stereochemistry (retention of configuration at carbon would require a suprafacial path, which is forbidden for the $4n$ case thermally).

Orbital Symmetry Considerations

The Woodward-Hoffmann rules for sigmatropic rearrangements come from analyzing the symmetry of the HOMO of the π system:

Identify the π system across which the σ bond migrates and count the π electrons.
Determine the HOMO for the reaction conditions. For thermal reactions, use the ground-state HOMO. For photochemical reactions, use the first excited-state HOMO.
Check the phase relationship at the terminal lobes of the HOMO. If the lobes on the same face have the same sign, a suprafacial shift is allowed. If they have opposite signs, an antarafacial shift is required.

For example, in a [1,5]-H shift the relevant π system is a pentadienyl fragment (4 π electrons in the transition state framework). The ground-state HOMO ( $\psi_2$ ) has lobes of the same phase at C-1 and C-5 on the same face, so a suprafacial, thermally allowed shift results.

This orbital analysis is fully consistent with the electron-counting shortcut above and gives you a deeper understanding of why certain shifts are allowed or forbidden.

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