# Lone pairs in MO theory?

I have learned that non-bonding MOs are lone pairs. However when considering the MO diagram for the $O_2$ molecule, I do not see any non bonding orbitals yet there are two lone pairs on each oxygen atom. Where do these come from?

Non-bonding MOs are usually lone pairs, but a lone pair does not necessarily mean there is a non-bonding MO.

You are trying to match what you learned from Lewis structures to the orbital picture. In general, that does not work (there may be exceptions).

Let's look at the MO diagram:

It is tempting to assign the bonding $\sigma_{2p}$ to the first (sigma) bond in $\rm O_2$. The two $\pi_{2p}$ bonding orbitals would then results in 2 $\pi$ bonds, hence we get a O-O triple bond. The further two electrons in the anti-bonding $\pi_{2p}^*$ orbitals could be assigned to one oxygen each, thus we have a biradical. The remaining two electron lone pairs might be assigned to the two $\sigma_{2s}$ orbitals. This is actually one of the two valid Lewis structures for $\rm O_2$. (The other possible Lewis structure has only a double bond and two lone pairs at each oxygen atom.)

The problem is that the lone pairs are ascribed to only one atom, but the MOs are delocalized over the molecule. Thus there is no direct relationship between them. A non-bonding orbital is usually just an atomic orbital, which you may thus ascribe to a lone pair. But that does not mean each lone pair leads to a non-bonding orbital.

It is however possible, to localize the molecular orbitals (by making appropriate linear combinations of MOs) in such a way you get atomic orbitals, which you might match again with the lone pairs. The thing here is, that there are many different sets of orbitals which are all valid descriptions of the bonding situations.

Also note, that MO diagrams are just a simplification as well.

Source of the MO Diagram

• As you started to mention, linear-combination of atomic orbitals (LCAO) with each molecular orbital being required to obey the rules of the full group ($\text{D}_\infty$) for $\text{O}_2$ using only the atomic orbitals designated in the subscripts of the MO. A better choice of orbitals would be using the natural bonding orbitals (NBO) derived from an accurate wavefunction. In either case, some should appear to be more bonding, while others are less so. – Alex Vaughn May 2 '17 at 17:23
• Actually, I was not talking about LCAO, but about linear combinations of molecular orbitals, which in turn are usually (but not necessarily) linear combination of atomic orbitals. I am not so familiar with NBOs, but I think, simply put, they are a variant of the localized orbitals I was referring to. – Feodoran May 2 '17 at 18:04