Pseudopotentials, an overlooked source and remedy of DFT errors

First-principles calculations rely heavily on pseudopotentials, yet their impact on accuracy is hardly addressed. In this work, we show that most pseudopotentials to date introduce errors, which manifest themselves as errors of atomic energy levels, leading to a $de facto$ deviation from the Hohenberg-Kohn theorem. We consider the atomic-level adjusted pseudopotentials, whose interplay with exchange-correlation functional provides a pragmatic correction that balances accuracy and efficiency. We benchmark our theory with bandgap calculation for 54 semiconductors containing monovalent Cu. The results, compared to those from conventional studies, not only remove all erroneous metal predictions for 11 compounds, but also reduce the mean relative error from 80\% to 20\%. Overall accuracy even exceeds those of standard hybrid functionals and GW methods.