Understanding and correcting the spurious interactions in charged supercells

The supercell technique is widely spread for the simulation of charged point defects. Charged defects in a supercell are unfortunately subjected to spurious image interactions, which are usually handled by introducing two correcting terms: a Madelung-type correction that accounts for the electrostatic interactions of repeated charges in a compensating background and a potential alignment term that refers the charged supercell to the electron reservoir. We demonstrate that the Madelung correction already brings a large potential shift that slowly converges as $1/L$ with increasing supercell sizes. We hence define a potential alignment devoid of any double counting. We finally propose a simple evaluation for the nearest-neighbor interaction that removes the remaining spurious hybridization of the defect wave functions between images. The application of these three corrections together drastically speeds up the convergence with respect to supercell size for all defects that are not too shallow.