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

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