Title: Comparative study of first-principles approaches for effective Coulomb interaction strength $U_{\rm eff}$ between localized $f$-electrons: lanthanide metals as example

Abstract: Since the correlation strength has key influence on the simulation of strongly correlated materials, many approaches have been proposed to obtain the parameters using first-principles calculation. However, the comparison of the different Coulomb strength from these approaches and investigation of the mechanism behind are still needed. Taking lanthanide metals as example, we research the factors which affect the effective Coulomb interaction strength $U_{\rm eff}$ by local screened Coulomb correction (LSCC), linear response (LR) and constrained random-phase approximation (cRPA) in VASP. The $U^{\rm LSCC}_{\rm eff}$ value increases from 4.75 eV to 7.78 eV, $U^{\rm LR}_{\rm eff}$ is almost stable at about 6.0 eV (except for Eu, Er and Lu), and $U^{\rm cRPA}_{\rm eff}$ shows a two-stage decreasing trend in light lanthanides and heavy lanthanides. To investigate these difference, we established a scheme to analyze coexistence and competition between the orbital localization and the screening effect. We find that LSCC and cRPA are dominated by the orbital localization and the screening effect respectively, while LR shows a balance of the competition between the two factors. Additionally, the performance of these approaches are influenced by different starting points from PBE and PBE+$U$, especially for cRPA. All these results give useful knowledge for the $U_{\rm eff}$ of lanthanide materials, and similar analysis can also be used in the research of other correlation strength simulation approaches.