Comparative analysis of methods for calculating Hubbard parameters using cRPA

In this study, we present a systematic comparison of various approaches within the constrained random-phase approximation (cRPA) for calculating the Coulomb interaction parameter $U$. While defining the correlated space is straightforward for disentangled bands, the situation is more complex for entangled bands, where different projection schemes from hybridized bands to the target space can yield varying sizes of interaction parameters. We systematically evaluated different methods for calculating the polarizability functions within the correlated space. Furthermore, we analyze how different definitions of the correlated space, often constructed through Wannierization from Kohn-Sham orbitals, defines the orbital localization and play a crucial role in determining the interaction parameter. To illustrate these effects, we consider two sets of representative correlated d-orbital oxides: LiMO$_{2}$ (M = V-Ni) as examples of isolated d-electron systems and SrMO$_{3}$ (M = Mn, Fe, and Co) as cases of entangled d-electron systems. Through this systematic comparison, we provide a detailed analysis of different cRPA methodologies for computing the Hubbard parameters.