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It is well known that LDA/GGA often underestimates band gaps. So I wonder if there are any cases where LDA/GGA overestimates the band gap?

In these cases, what is the reason?

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First, it's worth noting that DFT with both LDA or GGA underestimate band gap and not overestimate it usually. In extreme cases, it predicts that your compound is metal (i.e. no band gap at all)!

The reason is that there is a discontinuity in the derivative of energy with respect to number of electrons. Both LDA and GGA suffers from this problem. One example is band gap of $\text{La}_{2}\text{CuO}_{4}$ a high temperature superconductor with band gap of 2 eV but LDA, PBE, and PW91 predict it as a metal! ref: Perry et. al.

More reads here:

Update: If you are looking for a case that DFT with GGA or LDA overestimate the band gap and if you consider PBE (Perdew–Burke-Ernzerhof) exchange-correlation as a part of GGA, yes there are couple of examples that band gap is overestimated for compounds of $\text{PbSe}$ and $\text{Bi}_{2}\text{Se}_{3}$ because there is a strong spin-orbit coupling in these compounds with small band gap4.

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    $\begingroup$ It seems that the question acknowledges that it's "well known" that LDA/GGA under-estimate band gaps, and they are asking for examples where LDA/GGA over-estimates the band gap. Do you concur? $\endgroup$
    – Nike Dattani
    May 3, 2020 at 1:28
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    $\begingroup$ Thank you for the update. It would also be interesting if there's any examples without spin-orbit coupling, since regular DFT isn't meant for dealing with spin-orbit coupling. $\endgroup$
    – Nike Dattani
    May 3, 2020 at 1:46
  • $\begingroup$ @Alone Programmer, I'm sorry that the question is not directly related to the topic, but I was very interested in your statement. You wrote that the band gap of LaCuO is about 2 eV. However, how can this satisfy the well-known formula wikimedia.org/api/rest_v1/media/math/render/svg/… relating the band gap to the critical temperature? In the best case, the critical temperature is 40K. $\endgroup$
    – Disciple
    Jun 1, 2020 at 22:01

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