QM questions: MO theory orbital combination

[kayan]

1) When we learn about MO theory, we learn that there are different ways to combine atomic orbitals to find the molecular orbitals for a molecule. However, going back to the fundamental/physical meaning of orbitals, orbitals are the wavefunctions. Wavefunctions do not have any physical meaning (so I'm told). However, when you square the wavefunction, suddenly it represents the spatial probability of finding the electron.

Getting to my question: why is it that in MO theory, the AOs (no physical meaning) are added to construct the MOs instead of combining the SQUARE of the AOs (which have physical meaning) to construct the MOs? To see this from my perspective, think about the physicality of what we are doing: we are saying that if we have, say, 2 atoms close to each other, then a bond forms if the electrons are concentrated in between the nuclei. It would make more sense to me that you add the square of the AOs together since you want to add the regions of electron probability.

I thought about this question while thinking about the concept of orbital phase. If we added the square of the orbitals rather than the orbitals themselves, then there would be no phases since all negative wavefunctions would go positive. But the concept of phase is so central in MO theory. I'm sure this conclusion is wrong, but trying to reconcile it with a physical picture is making it hard to see why.

 

[dextercioby]

The linearity principle which allows you to take sums of quantum states is valid only for wavefunctions themselves, not for their squares. So a molecular orbital must still be an orbital, hence on an equal footing with the atomic orbitals which is made up from.

 

[kayan]

Thanks.
Why does the linearity principle prohibit taking linear combinations of the squared-orbitals? Also, is there a more physical explanation for this?

 

[dextercioby]

Because the whole mathematical machinery is based on vectors (of Hilbert spaces), not on reals (vectors' magnitudes). An atomic or a molecular orbital is a state and states are vectors in a Hilbert space by the most common axiomatization of Quantum Mechanics.

That's why the phase is important, because it's assigned to the wavefunction, not to its square.

 

[DrDu]

You should also remember that the MO's also aren't physical observable objects. Orbitals are also not general wavefunctions but only one-particle wavefunctions. We use them as basis states to approximate the whole many body wavefunction of a molecule. Even the latter one is not an observable quantity but we can calculate all relevant properties of the molecules from expressions quadratic in the wavefunction.