Maximally localized Wannier functions in LaMnO$_3$ within PBE$+U$, hybrid functionals, and GW: an efficient route to construct ab-initio tight-binding parameters for $e_g$ perovskites

Using the newly developed VASP2WANNIER90 interface we have constructed maximally localized Wannier functions [1] (MLWFs) for the $e_g$ states of the prototypical Jahn-Teller magnetic perovskite LaMnO$_3$ at different levels of approximation for the exchange-correlation kernel. These include conventional density functional theory (DFT) with and without additional on-site Hubbard $U$ term, hybrid-DFT, and single shot GW. By suitably mapping the MLWFs onto an effective $e_g$ tight-binding (TB) Hamiltonian [2,3] we have computed a complete set of TB parameters providing the band dispersion in excellent agreement with the underlying {\em ab initio} and MLWF bands. The method-dependent changes of the TB parameters and their interplay with the electron-electron interaction term are discussed and interpreted, outlining a guidance for more elaborate treatments of correlation effects in effective Hamiltonian-based approaches. \newline [1]\,I.\,Souza,\,N.\,Marzari,\,and\,D.\,Vanderbilt,\,Phys.Rev.B\,65,\,035109\,(2001).\newline [2]\,R.\,Kov\'a\v{c}ik\,and\,C.\,Ederer,\,Phys.Rev.B\,81,\,245108\,(2010).\newline [3]\,R.\,Kov\'a\v{c}ik\,and\,C.\,Ederer,\,Phys.Rev.B\,84,\,075118\,(2011).