An analysis of reaction possibilities in Huckel-Mobius theory requires us to construct a "fully interacting basis set", which is simply a sketch of the transition state, drawn with the maximum possible bonding character (the fewest nodes possible). For butadiene and hexatriene, the fully interacting basis sets are:
Next draw a line to connect the reactant orbitals as they would be connected by conrotatory and disrotatory processes. This gives a complete model of the transition state. For butadiene, the result is:
Now inspect the transition state model. If the number of nodes (apart from the nodal plane of the p-orbitals) is zero or any even number, the transition state has a Huckel topology, and will be aromatic if a Huckel number (4n + 2) of electrons is present.
If the number of nodes is one, or any odd number, the transition state has a Mobius topology, and will be aromatic if a Mobius number (4n) of electrons is present.
In the butadiene example, the conrotatory transition state has Mobius topology, and will be aromatic if four electrons are present, which is the number available. The conrotatory process is allowed. The disrotatory transition state is a Huckel system, and would require two or six electrons for aromaticity. Four are present, so this reaction is forbidden.
As shown in the sketch:
the two possible transition states for hexatriene have the same topologies as those of butadiene, but the additional pair of electrons means that the disrotatory transition state now is aromatic.