Analysis in Local Symmetry

Woodward and Hoffmann [1, p. 33] state that although the occupied orbitals correlate along the pathway of least motion, the reaction is forbidden because two of the occupied it orbitals of COT correlate at the MO level of approximation with two unoccupied a orbitals of cubane that have the same symmetry labels, and vice versa. This apparent non-correlation, which is not evident when the reaction in analyzed in global symmetry, induced the authors to invoke local symmetry instead and describe the reaction as comprising two forbidden [2-f2]-cycloadditions. 

The source of the offending bonding-antibonding correlations is revealed when the relevant bond-orbitals in Equations 5.1-5.8 are rewritten as combinations of their contributing AOs. For COT  

Let us mimic the pathway that retains T)2d symmetry Rotate Cj and C2 of COT about the x axis in one sense while rotating C5 and Cg in the other. 

Note that in order for two p AOs to form a t bond they have to point towards each other, so they must have opposite signs. 

Substituting the transformed AOs into Equations 5.10-5.13 and rearranging them in the same order as in Equations 5.14-5.17, we obtain  

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It is clear from the two examples just given that all of the mechanistic information obtainable from an analysis in local symmetry can be deduced from correlation or correspondence diagrams formulated in the global symmetry of the reactant and product. As we will see in the following sections, the converse is not true the arbitrary separation of a molecular system into moieties with different local symmetries can be misleading. 

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