The recent experimental claim of room-temperature ambient-pressure superconductivity in a Cu-doped lead- apatite (LK-99) has ignited substantial research interest in both experimental and theoretical domains. Previous density functional theory (DFT) calculations with the inclusion of an on-site Hubbard interaction $U$ consistently predict the presence of flat bands crossing the Fermi level. This is in contrast to DFT plus dynamical mean-field theory (DMFT) calculations, which reveal the Mott insulating behavior for the stoichiometric ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ compound. Nevertheless, the existing calculations are all based on the parent $P{6}_3/m\text{−}{\mathrm{Pb}}_{10}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ structure, which is argued to be not the ground-state structure. Here, we revisit the electronic structure of ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ with the energetically more favorable $P\overline{3}$ structure, fully taking into account electronic symmetry breaking. We examine all possible configurations for Cu substituting the Pb sites. Our results show that the doped Cu atoms exhibit a preference for substituting the Pb2 sites rather than the Pb1 sites. In both cases, the calculated substitutional formation energies are large, indicating the difficulty in incorporating Cu at the Pb sites. We find that most of the structures with Cu at the Pb2 site tend to be insulating, while the structures with both Cu atoms at the Pb1 sites (except one configuration) are predicted to be metallic by $\mathrm{DFT}+U$ calculations. However, when accounting for the electronic symmetry breaking, some Cu-doped configurations previously predicted to be metallic (including the structure studied in previous $\mathrm{DFT}+U$ calculations) become insulating. Our work highlights the importance of symmetry breaking in obtaining a correct electronic state for ${\mathrm{Pb}}_9\mathrm{Cu}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$, thereby reconciling previous $\mathrm{DFT}+U$ and $\mathrm{DFT}+\mathrm{DMFT}$ calculations.

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• Electronic structure
• Crystal structures
• Density functional theory
• First-principles calculations