Theoretical unification of hybrid-DFT and DFT plus U methods for the treatment of localized orbitals

Hybrid functionals serve as a powerful practical tool in different fields of computational physics and quantum chemistry. On the other hand, their applicability for the case of correlated d and f orbitals is still questionable and needs more considerations. In this article we formulate the on-site occupation dependent exchange correlation energy and effective potential of hybrid functionals for localized states and connect them to the on-site correction term of the DFT+ U method. The resultant formula indicates that the screening of the onsite electron repulsion is governed by the ratio of the exact exchange in hybrid functionals. Our derivation provides a theoretical justification for adding a DFT+ U-like on-site potential in hybrid-DFT calculations to resolve issues caused by overscreening of localized states. The resulting scheme, hybrid DFT+ V-w, is tested for chromium impurity in wurtzite AlN and vanadium impurity in 4H-SiC, which are paradigm examples of systems with different degrees of localization between host and impurity orbitals.