# Hybridisation and the Schrödinger equation

I am slightly confused about hybridisation and how it relates to molecular and atomic orbitals, despite having pored through many sources online. I was hoping someone could verify whether my current understanding is correct, in particular regarding what hybridisation actually is/does because I have not read this explicitly, but am assuming it is the case from what I have read so far:

• The Schrödinger equation can be solved to give the atomic orbitals (at least, some simplification involving the effective nuclear charge can be used to find the outer (and inner?) atomic orbitals.
• In hybridisation, we first consider the shape of a molecule and then we consider each atom seperately and how we could combine the atomic orbitals so that the geometry about each atom is correct. Now what I am particularly unsure about is: Are the hybridised orbitals entirely a conceptual construct, or are they mathematical solutions to the Schrödinger equation? I was thinking that maybe the hybrid atomic orbitals that are created are a linear combination of the atomic orbitals, and thus this linear combination also solved the Schrödinger equation and can exist but would have a different energy to the individual atomic orbitals? I am finding it difficult to think how a linear combination of the atomic orbitals could produce something with a completely different shape though (although I suppose the orbitals are orientated and essentially vectorial, so I think it could work). Perhaps the hybrid atomic orbital form could also be, say, the product of the atomic orbital forms? Although then I do not think this would solve the Schrödinger equation anymore. But I am not sure. And anyway, all of these ideas assume that hybridisation has a mathematical basis in the Schrödinger equation, which I am not sure about at all!
• Then molecular orbital theory takes the orbitals of each atom (I think theoretically it takes every orbital in every atom?) in the molecule and combines them to form a molecular orbital for the whole molecule - no longer simply an overlap between two atoms. This certainly also solves the Schrödinger equation. From what I understand, theoretically the calculation to form MOs should involve only the AOs of each atom, but to simplify the situation sometimes hybrid atomic orbitals are used in MO construction? If so, surely the hybrid atomic orbitals must be a linear combination of the atomic orbitals that solve the Schrödinger equation, so that this can be used in molecular obrital theory which is based in the solution to the Schrödinger equation? Also, where in the calulcation are the coefficients for each atomic orbital arising? Is it from the boundary conditions including things like molecule geometry, or perhaps that we know the energy of a molecule and when we put this particular energy in to the Schrödinger equation it gives us the coefficients?

I apologise for the long post and I realise there seem to be many questions within here, however they are all linked and about the mathematical underpinnings of hybridisation/valence bond theory and molecular orbitals, so I thought they belonged in one post.

• When you say hybridization, are you referring to LCAO (linear combination of atomic orbitals)? Or are you referring to the hybridization taught in organic chemistry classes (where they only deal with the valence orbitals)? May 19, 2017 at 19:42
• @QuantumAMERICCINO I think the latter (sp/sp2/sp3 mixing etc)- i.e. the hybridisation where you want it to match up with the shape of the molecule
– Meep
May 19, 2017 at 20:31
• An important point is that the total electron density described by the hybrid orbitals is the same as the density of the unhybridized orbitals. We're just carving that space up differently in terms of names like drawing different boundaries on a map. The land isn't changed by the change in names. Feb 12, 2019 at 2:15