# In orbital hybridisation, what determines the number of orbitals that hybridise to be at the same energy level?

For example, in phosphorus pentachloride five orbitals hybridize, but why not 6, as in the hydrated Al ion? I have read that the reason six orbitals hybridize is that a maximum of six oxygens can stably fit around it. Why is it not the same case for phosphorus? Secondly, why does water $sp^3$ hybridize even though it has enough lone pairs in its p orbital to bond with 2 oxygens. Surely it would take more energy to pull up the lower $s$ orbital to bond than to allow the $p$'s, already elevated to that energy, to do the bonding?

The difficulty in talking about hybridization lies in the fact that it tends to confuse atomic and molecular orbitals, or (equivalently?) one- and many-electron wave functions.

Mixing atomic orbitals generally only becomes possible when we do a mathematical approach which treats more than one electron at once. That, however, is mathematically rather complex, and the resulting wave functions are molecular orbitals, which are entirely different from atomic orbitals. Molecular orbitals are constructed as combinations of atomic orbitals, which means that it is possible to e.g. add s and p wave functions together to obtain something that looks like sp hybrids. But, and this is important, there are actually many different possibly ways for such combinations, and none of them should really be considered physical reality; and furthermore, this "mixing together" has to follow very strict mathematical rules which in the end affects the entire set of molecular orbitals at once. Describing molecular orbitals in terms of hybrid orbitals (or something similar) is possible if one does it thoroughly, in terms of a full and correct mathematical treatment. Still, even when applying the concept in this way, it is somewhat arbitrary, since it represents only one of many possible solutions (and depending on the problem, not even the best one).

In the simple and qualitative terms often introduced in chemistry (especially organic chemistry), it is a very crude concept which should be regarded with great caution and especially not as something that would be some sort of reality. Hybridization is a mathematical procedure for many-electron wave functions; nothing more, nothing less.

In more technical terms: The role of one- and many-electron wave functions in atoms is crucial here. Atomic orbitals are assumed to be hydrogen-like wave functions; in the one-electron atom, the orbitals are $\ell$-degenerate, so linear combinations of wave functions for different orbitals would mathematically still be solutions to the Schrödinger equation. In many-electron atoms, the degeneracy breaks down, so combining different orbital wave functions would be invalid, since they belong to different eigenvalues. We do, however, employ a many-electron Hamiltonian such as the Fock operator in this case, and unless I am mistaken, the electron-electron interactions lift this restriction. So, by some unitary transformation of the many-electron wave function of a single atom, I guess one might arrive at something like hybrid orbitals. Then again, this does not reflect physical reality.

This is only partly an answer to your question. I think teaching hybridization in depth is a bad pedagogical choice, so I am not the right person to explain the inner workings of this theory.

Don't spend too much time try to get the details of the how's and why's of hybridization… There are arguments over how useful it is as a concept in theoretical chemistry, and it certainly has severe limitations. For example, the picture derived from hybridization for the water molecule is simply wrong: it predicts two equivalent lone pairs on the oxygen atom, and they are not.

Just think in terms of molecular orbitals!