Sigmatropic Rearrangements: Notation and Principles

Introduction

A sigmatropic rearrangement moves a σ bond while nearby π bonds shift. Unlike a cycloaddition, it need not join two separate molecules; the starting and product structures often have the same formula and similar-looking unsaturation. The [i,j] notation reveals how far the two ends of the migrating σ bond travel and helps identify the orbital array involved.

Core explanation

In a sigmatropic reaction, an old σ bond breaks and a new σ bond forms at different positions while π bonds reorganise in a connected system. In the ideal pericyclic description, these changes occur concertedly through a cyclic array of interacting orbitals. To label the shift, take the two fragments joined by the original σ bond. Number the atom bonded at each old end as position 1 on its respective fragment, then count along each fragment to the atom that receives the new σ connection. Those destination positions give [i,j] , conventionally listing the smaller number first. The IUPAC sigmatropic definition specifies this counting and includes [3,3] and [1,5] examples.

For a [3,3] shift , the new bond joins position 3 on each of the two fragments formerly attached by the σ bond. The Cope rearrangement of a 1,5-diene is a classic carbon-only example: a central C–C σ bond is replaced by a bond between the distant terminal carbons while the two π bonds move. The Claisen rearrangement of an allyl vinyl ether is another [3,3] process; after rearrangement and tautomerisation, it yields an unsaturated carbonyl product. These names describe different substrate types but share a six-electron, six-membered cyclic transition-state topology.

For a [1,5] hydrogen shift , the hydrogen is one fragment with only one atom, so the first label is 1. It moves from one end of a pentadienyl system to the fifth position of that conjugated fragment as its two π bonds shift. This does not mean the hydrogen physically jumps over five isolated atoms by a free-radical path; the notation describes the old and new bonding positions in a conjugated array. Thermal [1,5] H migration can be suprafacial on the pentadienyl component when the relevant orbital phases and geometry permit it.

Bracket numbers for a sigmatropic shift are not the same bookkeeping question as [4+2] cycloaddition shorthand. The former describes positions along fragments attached by a migrating bond; the latter is usually introduced through participating π components. Always draw the old and new σ bonds to make the classification unambiguous. A simple tautomerisation may also move a hydrogen and π bond, but if its actual mechanism is acid/base-mediated and stepwise, it should not be called a concerted sigmatropic shift merely because the net formula resembles one.

Step-by-step reasoning

Compare reactant and product and mark the σ bond that disappears. Highlight the new σ bond and the π bonds that move. Split the reacting array conceptually at the old σ bond into two fragments. Starting from each old attachment atom as position 1, count to each new attachment atom and write the ordered [i,j] label. Finally ask whether a continuous cyclic orbital pathway and appropriate geometry are plausible for a concerted interpretation.

Visual explanation

Draw a 1,5-diene chain of six carbons. Color the central old σ bond blue and the new terminal σ bond red. Number three carbons outward from each end of the blue bond to the atoms joined by the red bond, yielding [3,3]. Beside it draw a five-carbon conjugated chain with a hydrogen attached at carbon 1 and then at carbon 5, yielding [1,5].

Real-world analogy

Imagine two teams holding a rope at their closest members. The teams rearrange positions and the rope is passed to a member three places down each team: that is like [3,3]. If one team is a single person holding the rope and the other passes it to its fifth member, the index resembles [1,5]. The labels record endpoints, not the distance traveled through empty space.

Real-world example

Thermal Claisen rearrangements are used to form carbon–carbon bonds from allyl vinyl ethers while generating carbonyl functionality after tautomerisation. The reaction can set useful relative stereochemistry through a compact six-membered transition-state geometry, making it valuable in multistep synthesis planning.

Why?

Identifying the migrating σ bond exposes a cyclic six-electron interaction that is hard to see in a product-only drawing. The [i,j] label tells a chemist which orbital-symmetry and geometric analysis is relevant. It also prevents confusing a bond migration with a cycloaddition that would add two new σ bonds between separate π components.

Common misconception

The [3,3] label does not mean a σ bond migrates three whole bonds in a single direction. It means the new attachment is at position 3 on each fragment when counted from the old σ-bond ends. Similarly, a [1,5] hydrogen shift is not necessarily a free hydrogen atom crossing five carbons.

Worked example

Question: A rearrangement breaks the central σ bond of a 1,5-diene and forms a new σ bond between the outer termini while shifting both π bonds. Classify it. Reasoning: Each old σ-bond end is position 1 on an allylic three-atom fragment. The new bond joins position 3 of one fragment to position 3 of the other. Answer: It is a [3,3] sigmatropic rearrangement, specifically a Cope-type bond map for an all-carbon 1,5-diene.

Quick check

1. What does the first number 1 represent in a [1,5] hydrogen shift? Answer: The migrating hydrogen is a one-atom fragment, so its attachment position is necessarily 1.

Exam focus

Mark the old and new σ bonds with different colors or line styles, then number from both old endpoints. State the π-bond changes and do not assign a pericyclic mechanism solely from product connectivity if acid, base or radical alternatives are plausible.

Advanced insight

The orbital-symmetry allowance of a [i,j] shift depends on the total participating electron count and whether the migrating group and π framework react suprafacially or antarafacially. A formally allowed topology can still be slow if the required geometry is strained. Stereochemical retention or inversion at a migrating carbon group can provide mechanistic information beyond the bracket label.

Summary

Sigmatropic rearrangements relocate a σ bond while π bonds reorganise through a cyclic array. The [i,j] label counts destination positions on the two fragments joined by the old bond. Cope and Claisen are [3,3] examples; migration of hydrogen across a pentadienyl array is [1,5]. Atom mapping and mechanism evidence remain essential.

Practice questions

1. Which σ bond should be marked first when classifying a sigmatropic reaction? Answer: The bond present in the reactant that disappears as the new σ bond forms.

2. What [i,j] order describes a Cope rearrangement of a 1,5-diene? Answer: [3,3], because the new bond joins position 3 on each original fragment.

3. Why is a simple acid-catalysed proton transfer not automatically a [1,5] sigmatropic shift? Answer: Its actual mechanism may be stepwise through external acid/base transfer rather than a concerted cyclic orbital path.

4. What must change along with the σ bond in a typical sigmatropic rearrangement? Answer: The adjacent π-bond arrangement also reorganises.