The direction of -sigmatropic rearrangements

One new aspect of orbital symmetry has appeared in this diagram—how did we deduce a or s symmetry in the way the 7 bond reacted For k bonds it is simple—if both bonds are formed on the same side of the old n bond, it has reacted suprafacially if on opposite sides, antar a facially. 

With a a bond the symmetry is not so obvious. We want to know if it does the same thing at each end (s) or a different thing (a). But what is the thing it does It reacts using the large lobe of the sp3 orbital (retention) or the small lobe (inversion). If it reacts with retention at both ends or inversion at both ends, it reacts suprafacially, while if it reacts with retention at one end and inversion at the other, it reacts arcfarafacially. There are four possibilities. 

In the routine above, we chose to use our 7 bond so that we got inversion at one end and retention at the other. That was why we identified it as an antarafacial component. If we had chosen another style we should have got different descriptions of the components, but the reaction would still have been allowed—for example, changing just one connecting line. 

This changes the symmetry of the a bond so that it becomes a CT2S component but it also changes the symmetry of one of the k bonds so that it becomes a TC2a component. The net result is still only one component of the Woodward-Hoffmann symmetry, the sum is still one, and the reaction still allowed. 

Orbital symmetry tells us that [3,3]-sigmatropic rearrangements are allowed but says nothing about which way they will go. They are allowed in either direction. So why does the Claisen-Cope rearrangement always go in this direction  

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