Effective on-site Coulomb interaction and electron configurations in transition-metal complexes from constraint density functional theory

Effective on-site Coulomb interactions ($U_{\mathrm{eff}})$ and electron configurations in the localized $d$ and $f$ orbitals of metal complexes in transition-metal oxides and organometallic molecules, play a key role in the first-principles search for the true ground-state. However, wide ranges of values in the $U_{\mathrm{eff}}$ parameter of a material, even in the same ionic state, are often reported. Here, we revisit this issue from constraint density functional theory (DFT) by using the full-potential linearized augmented plane wave method. The $U_{\mathrm{eff}}$ parameters for prototypical transition-metal oxides, TMO (TM$=$Mn, Fe, Co, Ni), were calculated by the second derivative of the total energy functional with respect to the $d$ occupation numbers inside the muffin-tin (MT) spheres as a function of the sphere radius. We find that the calculated $U_{\mathrm{eff}}$ values depend significantly on the MT radius, with a variation of more than 3 eV when the MT radius changes from 2.0 to 2.7 a.u., but importantly an identical valence band structure can be produced in all the cases, with an approximate scaling of $U_{\mathrm{eff}}$. This indicates that a simple transferability of the $U_{\mathrm{eff}}$ value among different calculation methods is not allowed. We further extend the constraint DFT to treat various electron configurations of the localized $d$-orbitals in organometallic molecules, TMCp$_{\mathrm{2}}$ (TM$=$Cr, Mn, Fe, Co, Ni), and find that the calculated $U_{\mathrm{eff}}$ values can reproduce the experimentally determined ground-state electron configurations.