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Is it always the sign of the Laplacian of the electron density a good criterion to decide on the covalent nature of a bond?

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Yes, and no, and maybe.

Oftentimes the sign of the Laplacian at the bond critical point in terms of the quantum theory of atoms in molecules (QTAIM) is taken as a measurement for the covalency of a bond. That it is not that simple is quite logical from your question alone.

Just as in bonding in general, i.e. there is a spectrum from pure covalent (ideally, but not achievable) and pure ionic (ditto),* there cannot be a clear yes or no answer here, too. I think the case of $\ce{BCl3}$ versus $\ce{AlCl3}$ and respective dimers, which I have written about, demonstrates this feature and how QTAIM can help understand bonding. As with many things it certainly is hard to draw a line. In that sense it is probably a good criterion to judge the nature of a bond. Judging only based on the sign of the Laplacian, while being a rigorous and unambiguous way to look at things, might lead to the wrong conclusions though.

What does it matter anyway? In many cases such a strict separation is not helpful. It should be clear, that there are covalent and ionic contributions in every bond, and reactivity often does not care about that. You'll never obtain a complete picture, if you are looking at a single property.

* And then there is charge-shift bonding (What is charge shift bonding?), but I personally don't see any value in that.

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  • $\begingroup$ Forgive me my incompetence but not even $\ce{H2}$ can be considered a purely covalent bond? $\endgroup$ – Jan Nov 1 '17 at 14:40
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    $\begingroup$ @Jan I would say it is probably as close as you can get. But there will always be an electrostatic contribution with it. $\endgroup$ – Martin - マーチン Nov 1 '17 at 15:03

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