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In amine inversion, there is a planar $sp^2$ intermediate.

The electron pair transferred from the blue lobe to the red one during the inversion. It is natural to assume, for a layman like me, that in the planar intermediate the electron pair is present in both lobes- one electron in each. That would impart paramagnetic character to the molecule. Does this really happen? I couldn't find any such phenomenon on internet.

Picture: http://chemistryjee.blogspot.in/2015/02/amines.html

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For somebody who is not well-versed in quantum mechanics, it might be natural to guess that this is the case. But if I were you, I wouldn't assume so.

Both electrons share the same p-orbital, and are therefore still paired. They both occupy the entire volume of the p-orbital. What they do not do, is to divide the p orbital up into two half-orbitals, with each electron taking one half.

There is a somewhat related question here, which might address some issues: How do Electrons Cross Nodes in Orbitals?

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