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:

enter image description here

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.

  • 1
    $\begingroup$ If you put two electrons into this system, together they will be better off than $(e_a+e_b)/2$, which means this is still a bonding interaction. $\endgroup$ Jul 25, 2016 at 18:50
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    $\begingroup$ I guess one aspect is whether the matrix representation of Hamiltonian is Hermitian. If it is, $H_{ab} = H_{ba}^* $. Solving the secular equation will lead to $H_{ab} H_{ab}^*$, which is positive-semi definite (different than than $H_{ab}^2$). On the other hand, if such Hamiltonian is used to describe open-system, it is in-general non-Hermitian, the imaginary bond will be possible $\endgroup$
    – user26143
    Jul 25, 2016 at 20:15
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    $\begingroup$ IIRC, in Hückel theory the Hamiltonian matrix is assumed to be real from the get go: $H_{ab} = H_{ba} = \beta$ and the MO energies formulas are valid only within this assumption. Thus, this formulas simply could not be used if $H_{ab}$ is not necessarily real. One have to apply the variational principle for this different case to get the MO energies formulas first. $\endgroup$
    – Wildcat
    Jul 25, 2016 at 20:26
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    $\begingroup$ I'm no professional and I could be completely wrong about this, but my first thought on seeing that diagram is that both electrons would end up on the left side and any interaction would be purely ionic. $\endgroup$
    – f''
    Aug 18, 2016 at 14:16
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    $\begingroup$ Just want to add that in accordance with the IUPAC definition of the bonding MO the above mentioned case is, in fact, "traditional favorable bonding". I quote from the definition: "Generally, the energy level of a bonding MO lies lower than the average of the valence orbitals of the atoms constituting the molecule." Note that bonding MO is not even required to lie lower than the valence orbitals, just lower than the average of them. $\endgroup$
    – Wildcat
    Aug 20, 2016 at 9:44

1 Answer 1


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:

enter image description here

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).

enter image description here

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

enter image description here

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.

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