Controlling Projection-Space Artifacts in DFT+U via Projection-Consistent U_(eff)

Density functional theory augmented with a Hubbard correction (DFT+U) is widely used to treat localized electronic states, but its predictions are often sensitive to the choice of the local projection space defining the correlated subspace. This sensitivity poses a practical challenge for computational reproducibility, particularly when projection parameters vary across codes, basis sets, or materials. In this work, we systematically investigate how the effective on-site Coulomb interaction $U_{\mathrm{eff}}$, determined \textit{ab initio} using constrained density functional theory, depends on the size of the local projection space in all-electron APW+lo calculations. Using rutile and anatase TiO$_2$ and $\beta$-MnO$_2$ as representative test cases, we show that applying a single fixed $U_{\mathrm{eff}}$ across different projection choices introduces artificial projection-driven errors in total energies, including spurious magnetic ordering transitions and unphysical sensitivity of phase stability. These artifacts are eliminated when $U_{\mathrm{eff}}$ is determined in an internally consistent manner for each projection space, yielding projection-consistent DFT+U predictions for lattice parameters, phase energetics, and magnetic ground states. By analyzing total-energy trends alongside the spatial characteristics of the localized $d$ orbitals, we demonstrate that the systematic reduction of $U_{\mathrm{eff}}$ with increasing projection size originates from orbital relaxation and enhanced electronic screening associated with orbital spatial extension. These results provide a physically motivated framework for controlling projection-space artifacts in DFT+U calculations and for obtaining energetically robust predictions across diverse correlated materials and computational setups.