Hybrid functionals with fixed mixing parameter perform no better than PBE for fundamental band gaps of nanoscale materials

Hybrid functionals mixing the exact exchange with (semi)local functionals to reinstall the missing derivative discontinuity have been successfully employed to predict band gaps $({E}_{\mathrm{g}})$ in bulk semiconductors. Here we show that traditional hybrid functionals with fixed fractions of exact exchange do not perform significantly better than the most popular semilocal PBE-GGA functional for ${E}_{\mathrm{g}}$ of semiconductor nanostructures, since their band-gap corrections are essentially size independent. This is because they cannot respond properly to the variation in screening when size changes. They merely predict constant band-gap corrections to the PBE gaps in silicon nanowires (Si NWs) when wire diameter reduces, instead of the dramatic increase predicted by many-body $GW$ calculations. Moreover, these hybrid functionals generate almost identical wave functions compared with PBE for both bulk Si and Si NWs, whose overlaps with corresponding quasiparticle wave functions become much smaller than 1 for narrow NWs.