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Valence electrons are associated with molecular orbitals and hybridizations. Do core electrons have molecular/hybridized orbitals, or the original atomic orbitals?

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tl;dr: If you are talking about VSEPR for an undergraduate level class, core orbitals do not mix. If you are talking about Hartree-Fock theory for a graduate level thesis, core orbitals most certainly do mix.

The answer to this question depends greatly on the level of theory you are looking at.

High-level Quantum Mechanics calculations will typically involve mixing of core orbitals as well as valence orbitals, and give the most realistic picture.

That being said, by making the simplifying assumption that core orbitals do not hybridize (a situation which is almost assuredly untrue), one might make many useful predictions about geometry and reactivity. For example, the entire discipline of Organic Chemistry is based (in part) upon the central assumption that the carbon 1s orbital does not participate meaningfully in hybridization, and yet the discipline still provides many useful tools for discussing structure and reactivity.

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    $\begingroup$ What is your definition of core orbitals? Can you give examples where the mixing of core orbitals plays any role in chemical bonding? How would you justify the use of effective core potentials for heavier elements? What is your definition of hybridisation? $\endgroup$ – Martin - マーチン Mar 26 '15 at 7:20
  • $\begingroup$ Core orbitals are non-valence orbitals. I don't justify anything QM does with effective core potentials, semiempirical approximations or the rest; Ron's answer handles this part well enough, I see no need to extend this one. Hybridization is the weighted linear combination of an s and some number of p (and occasionally d orbitals). which are useful for predicting the geometry of compounds. Given that no exact solution for QM wavefunctions exist, no-one can say with certainty the core orbitals do or do not participate in bonding. Answer to comment depends on context (also noted in my answer) $\endgroup$ – Lighthart Mar 26 '15 at 17:05
  • $\begingroup$ I am sorry I did not express myself clearly. You state "[...] core orbitals most certainly do mix." and "High-level Quantum Mechanics calculations will typically involve mixing of core orbitals [...]". Could you please give (an) example(s), where this is true, even if it is just a marginally effect. I am dealing with high level quantum chemical calculations everyday and I have not encountered that to any notable effect. But I am curious to know where I should pay more attention. $\endgroup$ – Martin - マーチン Mar 27 '15 at 3:05
  • $\begingroup$ I've been retired from physical organic chemistry, focussed on kinetics, for about 6 years now, so my knowledge is getting rusty, as well as this topic being on the edge of my expertise. I'd have to unpack GAMESS and get some of those verbose coefficient charts to show it, but my dim memories point to some contribution from the 1s; I will happily take some correction if I've misinterpreted the effects, and I will happily concede that such contributions to MO's were smaller for inner orbitals than valence orbitals. $\endgroup$ – Lighthart Mar 27 '15 at 6:15
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    $\begingroup$ Not including the inner orbitals is a crude simplification and a mistake. And they do of course form molecular orbitals (linear combinations), after all, they have to represent the geometry and symmetry of the molecule. But they are mainly just symmetry adapted and somewhat polarised. Notable sp mixing (at the same centre = hybridisation) I highly doubt, since energy levels of the core orbitals should be too far apart to contribute significantly. We nowadays only use small core ECPs mainly for describing some relativistic effects for the heavy elements. $\endgroup$ – Martin - マーチン Mar 27 '15 at 6:50
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When valence atomic orbitals start to interact with orbitals on another atom or molecule (e.g. bond formation), the valence electrons respond by forming molecular orbitals with the electrons on the other atom or molecule. The core electrons also respond to this bonding interaction. Their energy levels and spatial distributions are perturbed. The more deeply buried the core electrons, the smaller the perturbation - but the core electrons will also be affected by the bonding interaction. If one were able to write out the exact wave function to describe the new molecule that was formed, the wave function would include the core electrons. Their importance would be much less than the valence electrons, but they would be involved in describing the total bonding situation.

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When the electronic structure of a molecule is represented by molecular orbitals, all electrons occupy molecular orbitals.

Molecular orbitals can be approximated as linear combinations of atomic orbitals. A complete basis set for the molecular orbitals is an infinite set of atomic orbitals.

A finite set of atomic orbitals will be a less accurate approximation of the molecular orbitals.

So if you represent a low-lying molecular orbital as only a linear combination of core atomic orbitals, for example the lowest energy sigma MO of NN as only linear combination of the two 1s-1s orbitals, there will be some error associated with this approximation. And if you do not include the 1s orbitals in the representation of the the higher molecular orbitals there will be some error associated with this approximation. However, in the cases that I know of the error is small.

For example, in the following early molecular orbital SCF LCAO papers, the 1s orbital are linearly combined (mixed, hybridized) with the other atomic orbitals:

SCF LCAO MO Study of Li2 J. Chem. Phys. 27, 369 (1957)

An SCF LCAO MO Study of N2 J. Chem. Phys. 23, 569 (1955)

Electronic Population Analysis on LCAO–MO Molecular Wave Functions. I J. Chem. Phys. 23, 1833 (1955)

As the nitrogen paper summarizes "As was expected, the inner-outer-shell mixing is extremely small".

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