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Recently, there has been a question by Voldemort concerning different resonance structures of $\ce{NCO-}$, requesting an explanation why one resonance structure would be more preferred than another.

There has been a little discussion on that between ron and Martin - マーチン that took my interest:

ron: "[...] structure #1 ($\ce{N#C-O-}$) is more stable (preferred) than structure #2 ($\ce{^{-}N=C=O}$)."

Martin - マーチン: When talking about resonance structures, I think stable is the wrong word. I agree with your analysis in terms of contribution. Therefore #1 should have the highest contribution of all.

Now while I agree with ron that structure #1 is the most preferred resonance structure (referring to electronegativities), I regret with him concerning his attitude towards the "stability" as Martin does.

But I mainly do this, because I used NBO6 with the Natural Resonance Theory (NRT) keyword on the Natural Bond Orbitals (NBO) produced of an MP2/6-311+G(2d) optimized wavefunction to find the contributions of all relevant resonance structures - $66~\%~(\ce{N#C-O-})$, $30~\%~(\ce{^{-}N=C=O})$ and $3~\%~(\ce{^{2-}N-C#O+})$ - and not because I have a superior understanding on all underlying theories.

For me, the resonance stuff lookes a lot like MCSCF treatment of some VB wavefunction, which means that the complete molecule is described best from it's contributing resonance structures.

Now on a more complete comparison, what are the main similarities and differences between both theories and as they don't exist but are rather part of an explanation, can resonance structures indeed be described as "stable"?

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Great question - it engages with the difficult task concerning conceptualization of molecules, and how our models and jargon we use can often cause confusion while also helping to explain significant phenomena.

To begin, it is important to emphasize the fact that all these theories are models only (as you said, they are "part of an explanation") - we attempt to describe complex systems of quantum phenomena in a way that helps us explain chemical interactions in experiment (and/or what we would then expect from computational studies). Instead of focusing on all the minutiae of both resonance theory and MCSCF, I'm going to aim for using the differences between their orbital theory choice (VB or MO) to help with this distinction, and then approach the "stable" vs. "contribution" remarks. For further details on VB and MO theory, Martin - マーチン 's comment towards another question has a fantastically detailed answer, that I would recommend taking a look at here by hBy2Py.

Resonance theory relies on VB as a means to explain what is occurring in a molecular system, illustrated by the drawings seen in organic chemistry and later on with increasingly complex systems relying only on this idea of atomic orbitals (simple constructs localized around atoms). Resonance as a concept is applied to help us visualize the idea of electron delocalization in a way that also lets us select "better" structures to draw (based on electronegativities, geometric preference, etc...), which match up to chemical expectation/intuition. However, when drawn or used as the basis for a calculation, the orbitals are then limited to those around the atom, (with some hybridization) and how they overlap with neighbors, without much freedom to mold the orbital shapes (leading to computational difficulties due to non-orthogonality).

MCSCF, or other configuration interaction theories, build on the MO idea, which greatly expands the types of orbitals utilized when describing the bonding interactions between atoms, and captures the idea of delocalization much more effectively in doing so, a feature that applying the idea of "resonance" is supposed to recover in VB, but does not do quite as well as MO. Not perfect, but one of the best models so far, and continuing to be improved upon even today (or, built on, such as with new basis sets).

In essence, yes, those "resonance structures" do not exist by themselves, and thus there is no true "stability" to them, nor in that case much of a "contribution" as it is still not the whole picture. However, use of "contribution" helps to avoid implying that the resonance structures, drawn only to illustrate how VB theory handles delocalization, are localized/exact organizations of the electrons and atoms at a particular moment. In practicality, you will likely hear these interchanged, but it is best to keep in mind the idea trying to be captured: electron location is highly complex, with a range of local/delocal orbital characteristics that could be "best" depending on the system.

As further reading just in case, and as references for the topics discussed above:

Resonance Theory: if you go down to the section on misconceptions, it will show the common errors made when describing resonance, which you brought up nicely here in terms of computation as well. This is where use of "stable" may imply one of these misconceptions, and "contribution" may help a bit (but could also lead to the second misconception)

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