Nonlocal behavior of an electron in the ring-opening of cyclobutene

The nonlocal behavior of an electron in the ring-opening of cyclobutene is analyzed using the sharing amplitude and tools based thereupon. The sharing amplitude is a generalization of the absolute value squared of the wave function (a probability density according to the Born interpretation) to a measure that gives, for a single particle in a many particle system, both the relative phase of a wavelike quantity between any two space/spin points and a measure of the distribution of that quantity between those points. The sharing indices are related to the absolute value squared of the amplitude, thereby ensuring that the amplitude and the indices are consistent. To provide prototypical behavior of the sharing quantities in single and double bonds and to identify nonbonding and antibonding behavior, these tools are used to describe single electron behavior in ethane and ethylene, in the latter including separate σ and π contributions and two excited states. Similar analyses of the single and double carbon-carbon bonds in the reactant cyclobutene and in the product s-cis-butadiene (a planar transition state allowing for σ and π separation) are carried out. Comparisons are made to the prototypical molecules. The sharing quantities in the locally stable forms of s-gauche-butadiene are then considered. A remnant of the bond that is broken in cyclobutene is found in a form that is consistent with a conrotatory ring-opening, a behavior suggested by Woodward and Hoffmann on the basis of the symmetry of the highest occupied molecular orbital in the product, but here without appeal to only that molecular orbital. Finally, the nonlocal behavior of an electron during the breaking and making of bonds in the reaction is discussed for several geometries along the reaction path.