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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Sign up to join this communityFirst, 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.