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The Claisen rearrangement is a [3,3]-sigmatropic rearrangement. Count the number of atoms from the old to the new sigma bond and you will see there are 3 atoms in both directions round the ring. [3,3] sigmatropic rearrangements happen though a chair-like transition state. If there is a substituent on the saturated carbon next to the oxygen, then trans (E) geometry is strongly favoured in the product. This a consequence of the chair transition state favouring equatorial substituents and the concerted nature of pericyclic reactions.
Richard Windsor - Undergraduate Final Year Project 2008
Other pages of interest to do with Animated Molecular Orbitals are