  Finite-size corrections for charged defect supercell calculations typically
consist of image-charge and potential alignment corrections. A wide variety of
schemes for both corrections have been proposed for decades. Regarding the
image-charge correction, Freysoldt, Neugebauer, and Van de Walle (FNV) recently
proposed a novel method that enables us to accurately estimate the correction
energy a posteriori through alignment of the defect-induced potential to the
model charge potential [Freysoldt et al., Phys. Rev. Lett. 102, 016402 (2009)].
This method, however, still has two issues in practice. Firstly, it uses
planar-averaged potential for determining the potential offset, which cannot be
readily applied to relaxed system. Secondly, the long-range Coulomb interaction
is assumed to be screened by a macroscopic dielectric constant. This is valid
only for cubic systems and can bring forth huge errors for defects in
anisotropic materials. In this study, we use the atomic site electrostatic
potential as a potential marker instead of the planar-averaged potential, and
extend the FNV scheme by adopting the point charge model in an anisotropic
medium for estimating long-range interactions. We also revisit the conventional
potential alignment correction and show that it is fully included in the
image-charge correction and therefore unnecessary. In addition, we show that
the potential alignment corresponds to a part of first-order and full of
third-order image-charge correction; thus the third-order image-charge
contribution is absent after the potential alignment. Finally, a systematic
assessment of the accuracy of the extended FNV correction scheme is performed
for a wide range of material classes. The defect formation energies calculated
using around 100-atom supercells are successfully corrected even after atomic
relaxation within a few tenths of eV compared to those in the dilute limit.
