Hubbard U parameters for transition metals from first principles

Using the linear response-based constrained local density approximation (cLDA) approach we systematically computed the Hubbard $U$ parameters for series of $3d, 4d,$ and $5d$ transition metals. We compare the results with estimations by the constrained random phase approximation (cRPA) method and discuss the performance of the self-consistent density functional theory + $U$ ($\mathrm{DFT}+U$) method for prediction of lattice parameters, work functions, $d$-bandwidths and $d$-band centers. Interestingly, we found that blindly applied the standard, fully localized limit (FLL) version of the $\mathrm{DFT}+U$ approach heavily overestimates the positions of $d$-band centers with respect to the Fermi level, but much better agreement with experiment is obtained when applying a more realistic, Wannier-type representation of $d$ orbitals for projection of $d$ states occupancies. We present another, independent estimate of the Hubbard $U$ parameter based on the comparison of Hartree-Fock and DFT eigenvalues, and positions of $d$-band centers. The so-derived estimates are surprisingly well consistent with the ones derived from the above-mentioned first principles approaches, and allow for validation of cRPA or cLDA results for the disputed cases, including Cu, Ag, and Au for which large $U$ parameters are obtained from the cLDA method.