Wannier functions, minimal model and charge transfer in Pb$_9$CuP$_6$O$_{25}$

Recent preprints claimed that the copper doped lead apatite Pb$_9$CuP$_6$O$_{25}$ (LK99) might be a high-temperature superconductor because of its strong diamagnetism and transport properties. Motivated by the strongly correlated effects that can arise from a triangular lattice of Cu atoms with narrow bandwidth, we calculated the maximally projected Wannier functions from density functional theory simulations, and constructed a minimal two-orbital triangular model with Cu ($3d_{xz},3d_{yz}$) basis, and a four-orbital buckled honeycomb model with Cu ($3d_{xz},3d_{yz}$), O ($2p_x,2p_y$). Since the Coulomb interaction Ud is much larger than potential energy difference between Cu and O, charge transfer will occur for hole filling fraction $n_h > 1$. We further calculate the interaction parameters, and discuss the possible insulating state and corresponding spin exchange coupling.