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DFT band gap problem is a well-known problem. LDA or GGA exchange-correlation functionals underestimate the band gap heavily. I saw some of the strategies to address this issue were:

  1. using hybrid functions such as PBE0 or HSE06.
  2. using meta-GGA level functional such as SCAN, rSCAN, r2SCAN, etc.
  3. using Koopman functional (KCW)
  4. post-processing the ground state DFT result with GW approximation, etc.

Now, I want to know if there exists a comparison of accuracy and/or efficiency among these strategies. Or could it be the case that it depends on the specific material?

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It's a nice question, it will be difficult to compare their accuracy and efficiency in a single plot, because the band gap problem is mainly divided into two parts: the self-interaction and the correlations, so it depends on the material. In all cases the GW approximation is the most accurate approximation, it describes the occupied states slightly like Hartree-Fock but has the advantage of describing the unoccupied states very accurately at the expense of computational time, here is a quantitative analysis. In general, all these approximations aim at reducing the computational time without degrading the accuracy.

The self-interaction error is a wrong description of occupied states by DFT, the correction needs to include a part of the Hartree-Fock exact exchange free of a self-interaction error to build a hybrid functional (PBE0, HSE06).

Hybrid functional are useless for strongly correlated systems, the problem is not the locality of the DFT exchange but the overall approximation limited to one electron. A standard DFT cannot describe the ground state, DFT+U is the best correction and the functional used is important. Localized orbitals in this case require more derivative of the density. The SCAN+U method is often more accurate than GGA+U or LDA+U, especially for ionic solids.

I don't know the correction of the Koopmans functional, but I guess it reduces the self-interaction error and brings back the Koopmans theorem.

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    $\begingroup$ Thank you for your answer. So, I guess it is safe to say that it depends on the material of interest. Since I am working with a strongly correlated system. It seems that a DFT+U and GW is the best choice in this case, similar to this work $\endgroup$
    – Abdul Muhaymin -Free Palestine
    Commented Aug 5, 2023 at 10:14
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    $\begingroup$ @AbdulMuhaymin Exactly and thanks for the link it looks like an interesting article. $\endgroup$
    – M06-2x
    Commented Aug 5, 2023 at 15:59
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    $\begingroup$ Your statement "the problem is... the overall approximation limited to one electron" is a bit misleading, it's only the approximation of the quasi-particle state by the KS state which is inaccurate. The ground state of these systems can be described perfectly by a single particle method, at least in principle, it just isn't a single "electron". $\endgroup$
    – Phil Hasnip
    Commented Aug 7, 2023 at 0:45
  • $\begingroup$ @PhilHasnip This statement academic and can be derived easily even if it not always intuitive. I didn't mention specifically KS state because HF (in the hybrid funct.) and post-HF are also inaccurate in this case. It will be very useful if you provide a reference. $\endgroup$
    – M06-2x
    Commented Aug 8, 2023 at 10:51
  • $\begingroup$ The Hohenberg-Kohn theorem means that the ground state properties are calculable exactly in principle, and the band-gap can be reformulated as a difference in ground state energies. Whether the DFT auxiliary system is one-, two-, or even no particles doesn't matter... in principle. Of course, in practice Kohn-Sham XC approximations do not provide accurate band-gap predictions, but it is not because of the one-particle approach. $\endgroup$
    – Phil Hasnip
    Commented Dec 18, 2023 at 0:39

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