# Are there any cases where LDA/GGA overestimates band gaps? If so what is the reason?

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?

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.
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.