Full Hubbard-U corrections on electron-phonon coupling of strongly correlated materials via the finite displacement method

 The interplay between electron-electron and electron-phonon interactions is a central topic in condensed matter physics. Although the density functional theory plus Hubbard U correction (DFT+U) method is widely used to study electronic structures of strongly correlated materials, its extension to electron-phonon coupling (EPC) matrix and EPC calculations has received limited attention. Here, we implement an algorithm that integrates the DFT+U method and finite-displacement method for phonon and EPC matrix calculations. We obtain full Hubbard U corrections not only applied on electronic and phonon band structures, but more importantly also on EPC matrix. We demonstrate our algorithm in two prototypical correlated materials: infinite-layer nickelates La_1-xSr_xNiO₂ and ruthenium dioxide RuO₂. We find that i) while the Hubbard U corrections weakly increase the EPC of La_1-xSr_xNiO₂, its total EPC remains small and insufficient to account for the observed superconducting transition temperature T_c of about 10-30 K. Our result contrasts with the recent work (Phys. Rev. Lett. 133, 126401) showing that the full GW corrections yield an elevated EPC of La_1-xSr_xNiO₂ five times larger than its DFT value. ii) The Hubbard U corrections remove the imaginary phonon modes of RuO₂ when strained on TiO₂ substrate and substantially reduce its EPC. Our result alleviates the discrepancy between the previously obtained large theoretical EPC and the low experimental T_c. In both examples, we highlight that since correlation corrections (Hubbard U, GW, hybrid functional etc.) easily tune the shape of Fermi surface, accurate EPC calculations must be performed based on a realistic Fermi surface that is consistent with experimental observations.