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

Abstract: As correlation strength has a key influence on the simulation of strongly correlated materials, many approaches have been proposed to obtain the parameter using first-principles calculations. However, the comparison of the different Coulomb strengths obtained using these approaches and an investigation of the mechanisms behind them are still needed. Taking lanthanide metals as an example, we research the factors that 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 both light and heavy lanthanides. To investigate these differences, 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, whereas LR shows a balance of the competition between the two factors. Additionally, the performance of these approaches is influenced by different starting points from PBE and PBE+$U$, especially for cRPA. Our results provide useful knowledge for understanding the $U_{\rm eff}$ of lanthanide materials, and similar analyses can also be used in the research of other correlation strength simulation approaches.