  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 calculations, which reveal the Mott insulating behavior for the
stoichiometric Pb$_{9}$Cu(PO$_4$)$_6$O compound. Nevertheless, the existing
calculations are all based on the $P6_3/m$ structure, which is argued to be not
the ground-state structure. Here, we revisit the electronic structure of
Pb$_{9}$Cu(PO$_4$)$_6$O with the energetically more favorable $P\bar{3}$
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
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 structures with Cu at the Pb2 site tend to be
insulating, while the structures with both two Cu atoms at the Pb1 sites
(except one configuration) are predicted to be metallic by 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 DFT+$U$ calculations) become insulating. Our work
highlights the importance of symmetry breaking in obtaining correct electronic
state for Pb$_{9}$Cu(PO$_4$)$_6$O, thereby reconciling previous DFT+$U$ and
DFT+DMFT calculations.
