Imaginary Bonding Interactions Usually in chemistry, we deal with bonding interactions. That is, if I have the diatomic A-A molecule or A-B molecule, there's a favorable interaction (i.e., a bond) and a prototypical MO diagram like this:



As I'd say in a lecture, one orbital goes down in energy, one orbital goes up in energy. I can use Hückel theory to define the energies of the bonding and anti-bonding interactions:

$$e_{ab} = e_a + \frac{H_{ab}^2}{e_a - e_b}$$
$$e^*_{ab} = e_b + \frac{H_{ab}^2}{e_b - e_a}$$

Now, consider if $H_{ab}$ is imaginary.. we'd use the same equations to get a very unusual MO diagram:



How would you describe such an interaction? It's clearly not the traditional favorable bonding. It's not a non-bonding interaction. And it's not the traditional anti-bonding case like $\ce{He2}$, because if I put 2 electrons into this system, the anti-bonding orbital isn't filled.

I ask the question because in a survey we've performed using DFT, we find many such combinations.

I'm leaning towards calling it an imaginary bond, but if there are known discussions, I'd be curious to read references.
The short answer is that I need to come up with better wording to describe the interaction and that Hückel is surprisingly bad for these systems.

I'll give a slightly longer answer for now, and add a reference as we submit the paper.

First, an example system:



So "A" is a strongly electron-deficient dinitrothiophene ring (an electron acceptor) and "B" is a strongly electron-rich diaminothiophene ring. The figure in the question is a subset of the entire MO diagram, focusing on the HOMOs of the monomers and the HOMO and HOMO-1 of the dimer.

Why do I think this is a strange system? Well, the particle-in-a-box and Hückel models are usually pretty good at explaining conjugated organic molecules. If you make a longer conjugated system, the HOMO should go up in energy, and the HOMO-LUMO gap goes down (delocalization).



The alternative, localization, would have the orbitals remain at the same energy level:



Both of those "limits" can be explained in the Hückel model by changing the $H_{ab}$ or $\beta$ parameter (depending on how you write the equation, the symbol is different). In the localized case, $H_{ab} = 0$.

What's strange about these strong donor-acceptor cases is that as I said, it's not really delocalized, or localized.

Instead, as one comment correctly deduced, the DFT calculation (B3LYP/6-31G*) shows a large degree of charge transfer, but not necessarily localization. The dipole moment computed to be >7 Debye.

If one attempts to use Hückel to analyze these results, you must infer that $H_{ab}$ is imaginary. This indicates an "open system" because electrons are indeed transferred - from monomer "B" to monomer "A" creating the large dipole moment.

My overly-simplified question was probably a bit confusing, but:


It's interesting that no simple model captures charge transfer unless you allow Hückel to have imaginary (non-physical) $H_{ab}$ parameters. (NB, my colleagues aren't too happy with this concept, suggesting it simply indicates Hückel is a poor choice.)
It's surprising that in prototypical conjugated systems, I can't (yet) find a discussion about this alternative to delocalization and localization. We find it commonly as the difference in energy level increases.