What exactly are the frontier orbitals in diradical species like O2?

Question

I read about frontier orbitals, namely HOMO, LUMO. But what's exactly HOMO/LUMO? I know HOMO stands for highest occupied molecular orbital and LUMO stands for lowest unoccupied molecular orbital.

But I am not getting which orbital is to be tagged as HOMO, and which as LUMO. Consider the example of $\ce{O2}$, highest occupied MO is $\pi^*_\mathrm{2p}$ and lowest unoccupied MO is also $\pi^*_\mathrm{2p}$. Which orbital is HOMO/LUMO? Is $\pi_\mathrm{2p}$ the HOMO and $\pi^*_\mathrm{2p}$ the LUMO? If so, why? There is a similar case with $\ce{B2}$.

O2 is a bad example to consider. As for the bonding/nonbonding part: typically yes, but not always. — Ivan Neretin, Mar 04 '20 at 18:19
That's better. In this case I expect $\sigma2p$ for HOMO and $\pi2p$ for LUMO. Note that the LUMO is bonding. — Ivan Neretin, Mar 04 '20 at 18:51
@IvanNeretin is it $\sigma$2p or $\sigma$2s or $\sigma^\ast$2s? — Zenix, Mar 04 '20 at 18:54
OK, looks like I got it wrong. π2p sits below σ2p and hence is half-filled, which technically makes it both HOMO and LUMO. — Ivan Neretin, Mar 04 '20 at 18:57
It's important, that these labels only work within the framework of MO theory. There they are straight forward. The orbital highest in energy which is occupied is the HOMO and it does not matter how many fold the degeneracy is. The immediately following orbital would be the LUMO, which is unfortunately not always well defined. The case of bonding and antibonding is another story, when you venture out to more than two atoms one orbital can be both. Frontier Molecular Orbital Theory is a very helpful approximation, but it is quite crude. — Martin - マーチン, Mar 04 '20 at 19:07
For O2 and other radicals, it may be wise to use the term SOMO (singly-occupied molecular orbital), instead of HOMO/LUMO — Tan Yong Boon, Mar 05 '20 at 01:46
Truthfully, there are two questions here which are not very closely related and could stand as separate Q + A's. The first is how are the HOMO/LUMO defined in the case of a diradical like O2. The second is their bonding / antibonding character. — orthocresol, May 02 '20 at 16:50
Since the majority of what you have written pertains to the first question, I suggest removing the second and asking it separately. But I also would not be surprised if it has been already been asked and answered. You could do a quick search. — orthocresol, May 02 '20 at 17:38
@orthocresol I searched the second part, couldn't find one on SE, if you could tell the link. — Zenix, May 02 '20 at 18:53
at first glance: Bonding and Antibonding Why do bonding and anti-bonding states split in energy (mathematically)? Maybe How can two orbitals constructively and destructively interfere simultaneously? This: How can a molecular orbital with a nodal plane still be considered “bonding”? (!!) — Martin - マーチン, May 04 '20 at 13:25
@Martin - マーチン none of them answer particularly relation between HOMO/LUMO with BMO/ABMO. Also this post ain't getting reopening votes (2 till now, one of them is mine), what should I do? — Zenix, May 04 '20 at 21:43
That isn't really that surprising. The concept of bonding and anti-bonding orbitals is independent from the concept of HOMO and LUMO. It does - especially in Organic Chemistry - often get mixed up and non-bonding orbitals are introduced, too, but that's too much for the comments. I'll have to think about the Question, and we should continue conversations in [chat]. — Martin - マーチン, May 05 '20 at 08:59
@Martin - マーチン I am working on vector analysis right now, I would be grateful if you could spare some time from your busy schedule, tomorrow and (or in) succeeding days. Also, in which room will you be most comfortable? — Zenix, May 05 '20 at 11:44

What exactly are the frontier orbitals in diradical species like O2?

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