# Can a MO energy computed from a HF or DFT calculation for one molecule be directly compared to that of another molecule?

This was inspired by a recent question How is the zero energy defined for molecular orbitals? which made me pause to think if I understood clearly or not about the zero of energy of an MO.

For example, I know you cannot compare the total electronic energy of one molecule to another (as computed by any conventional HF or DFT code for example) since the zero is different - since they have different numbers/types of nuclei.

But is the same true of a MO energy, for example HOMO energy? That post I linked above had an answer describing how the zero of MO energy is an infinitely diffuse orbital of a single electron, which I take to mean irrespective of how many nuclei the system has or where they are. Consequently - is it possible to compare for example the HOMO energy of one molecule to the HOMO energy of some other molecule of different size, constituents, etc? Whether that comparison is useful is another question, of course!

(Energy differences like HOMO-LUMO gaps can be compared safely, that at least seems intuitive to me)

## 1 Answer

Yes, orbital energies can be compared amongst different molecules.

Suppose you have two molecules, A and B, and you want to compare their HOMO levels. Now consider the supermolecule consisting of A and B separated by an infinite distance, and suppose that there is no electron transfer between the molecules*. Then the HOMO levels of A and B are both occupied orbital levels of the supermolecule, although only one of them remains the HOMO, while the other becomes a more deeply buried occupied orbital**. Since you already know that the different orbitals of the same system can be compared, this proves that orbitals of different molecules can be compared. This reasoning works equally well for all orbitals.

Another way to understand this is to observe that the HOMO level is related to ionization potentials, via the Koopmans' and Janak's theorem for HF and DFT, respectively. Therefore the difference between the HOMO levels of molecules represents the difference of ionization potentials, which is of course a meaningful quantity. This argument does not work so well for LUMOs, and even less so for other orbitals.

*Even if the electron transfer is energetically favorable, we can still converge to the state where the electron transfer has not taken place, and the state remains meaningful as an excited state of the system. This is reminiscent to the DeltaSCF method for charge transfer states.

**When the HOMOs of A and B are degenerate, they can mix up to an arbitrary phase in the supermolecule. But in this case we can always choose the phase to be 0 or $$\pi$$, so that the supermolecule orbitals remain localized on the respective molecules.

• They can only be compare if obtained with the same level of theory. You cannot compare a result from HF with DFT.
– Camps
Nov 25, 2022 at 12:32