  We propose a self-consistent site-dependent Hubbard $U$ approach for DFT+$U$
calculations of defects in complex transition-metal oxides, using Hubbard
parameters computed via linear-response theory. The formation of a defect
locally perturbs the chemical environment of Hubbard sites in its vicinity,
resulting in different Hubbard $U$ parameters for different sites. Using oxygen
vacancies in SrMnO$_3$ as a model system, we investigate the dependence of $U$
on the chemical environment and study its influence on the structural,
electronic, and magnetic properties of defective bulk and strained thin-film
structures. Our results show that a self-consistent $U$ improves the
description of stoichiometric bulk SrMnO$_3$ with respect to GGA or GGA+$U$
calculations using an empirical $U$. For defective systems, $U$ changes as a
function of the distance of the Hubbard site from the defect, its oxidation
state and the magnetic phase of the bulk structure. Taking into account this
dependence, in turn, affects the computed defect formation energies and the
predicted strain- and/or defect-induced magnetic phase transitions, especially
when occupied localized states appear in the band gap of the material upon
defect creation.
