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It is often said that the optical band gap, i.e. the first excitation energy, of a species is exactly equal to the difference between the (Kohn-Sham) HOMO and LUMO. This would mean that the state of the wavefunction corresponding to the density constructed by summing a set of filled K-S orbitals, except for the "original" HOMO, which is replaced by the "original" LUMO, is higher in energy (exactly by the optical band gap) than the ground state.

My question now follows- does this fact actually hold, assuming exact DFT functional- if it does, can it be generalised to (the cases where the orbitals being "swapped" before summation are not necessarily the HOMO/LUMO, just a filled K-S orbital and an unfilled K-S orbital), and/or to the cases where two filled orbitals are swapped with two unfilled orbitals(etc)?

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    $\begingroup$ I have never heard about that. The bands are for periodic systems whereas the HOMO/LUMO are for molecular systems. If you would like to compare (just for comparison), yes, the "band gap" is like the difference between LUMO/HOMO. $\endgroup$
    – Camps ♦
    Sep 19, 2022 at 12:10

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The optical band gap is not expected to equal the DFT band gap since excitonic effects are not included.

What you are referring to is the fundamental gap. The fundamental gap is related to electron addition and removal energies, while the optical gap is lower than the fundamental gap due to the attraction between electrons and holes.

Anyway, if we focus your question to the fundamental gap, then I’m not sure if what you’re saying holds either. You can read some of the introductory material in this paper , especially the section “fundamental gaps from KS theory.”

Briefly, they say that while the highest occupied orbital energy has a physical interpretation as the negative of the ionization potential, but there isn’t a corresponding expression for the lowest occupied molecular orbital that is rigorously true. They cite another paper which argues that the exact exchange correlation functional would underestimate the fundamental gap similarly to how local or semi local approximations underestimate it. You might also be interested in another paper that they cite which talks about related issues too.

You can, however, compute the fundamental gap of a molecule by computing the total energy of the +1 and -1 version of the molecule you are considering and this procedure is rigorous in KS DFT.

One method that has been proposed to calculate the fundamental gap from a difference of molecular orbital energies is described in this paper: a generalized Kohn-Sham system (which refers to one with Fock exchange included) can restore the derivative discontinuity that is missing in KS DFT.

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  • $\begingroup$ I have edited it to include more information about the optical gap and fundamental gap, and I think I also answered the question by showing that the premise underlying it isn’t quite right. $\endgroup$
    – AGS
    Sep 20, 2022 at 3:43
  • $\begingroup$ I was referring to EXCITATION, not ADDITION AND REMOVEMENT. $\endgroup$
    – Kanghun Kim
    Sep 20, 2022 at 18:06
  • $\begingroup$ Yes, I cover that in the beginning. There is no such rigorous guarantee for the fundamental gap, and there also isn’t one for excitation energies because the electron-hole interaction isn’t included. Are you trying to ask about things like XAS and core level excitation? If so, there are ways to calculate that, but there also won’t be a rigorous connection between such excitations and DFT band energy differences because the electron-hole interaction is not included and also because differences in eigenvalues from KS DFT don’t have a rigorous physical interpretation. $\endgroup$
    – AGS
    Sep 20, 2022 at 18:56
  • $\begingroup$ To clarify: you mention the optical gap, but the quantity that you refer to (the difference in energy between the highest occupied and lowest unoccupied orbitals) should actually be compared to the fundamental gap because of the neglect of electron-hole interactions. In any case, whether you are interested in the optical or fundamental gap, the KS DFT band gap should not be rigorously interpreted as giving either of those. It is, however, often somewhat close in some systems. Is your question “what are accurate ways to calculate a XAS spectrum (or some related quantity)?” $\endgroup$
    – AGS
    Sep 20, 2022 at 19:01
  • $\begingroup$ All papers linked are between paywalls my institution does not seem to be affiliated with though <_> $\endgroup$
    – Kanghun Kim
    Sep 21, 2022 at 6:42

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