Rotationally invariant ab initio evaluation of Coulomb and exchange parameters for $DFT+U$ calculations

Conventional density functional theory (DFT) fails for strongly correlated electron systems due to large intra-atomic self-interaction errors. The $DFT+U$ method provides a means of overcoming these errors through the use of a parametrized potential that employs an exact treatment of quantum mechanical exchange interactions. The parameters that enter into this potential correspond to the spherically averaged intra-atomic Coulomb $(U)$ and exchange $(J)$ interactions. Recently, we developed an ab initio approach for evaluating these parameters on the basis of unrestricted Hartree–Fock (UHF) theory, which has the advantage of being free of self-interaction errors and does not require experimental input [Mosey and Carter, Phys. Rev. B 76, 155123 (2007)]. In this work, we build on that method to develop a more robust and convenient ab initio approach for evaluating $U$ and $J$⁠. The new technique employs a relationship between $U$ and $J$ and the Coulomb and exchange integrals evaluated using the entire set of UHF molecular orbitals (MOs) for the system. Employing the entire set of UHF MOs renders the method rotationally invariant and eliminates the difficulty in selecting unambiguously the MOs that correspond to localized states. These aspects overcome two significant deficiencies of our earlier method. The new technique is used to evaluate $U$ and $J$ for $Cr2O3$⁠, FeO, and $Fe2O3$⁠. The resulting values of $U-J$ are close to empirical estimates of this quantity for each of these materials and are also similar to results of constrained DFT calculations. $DFT+U$ calculations using the ab initio parameters yield results that are in good agreement with experiment. As such, this method offers a means of performing accurate and fully predictive $DFT+U$ calculations of strongly correlated electron materials.