  The most recent discovery, the Cu-doped lead apatite LK-99, is a proposed
room-temperature superconductor operating under ambient pressure. However, this
discovery has brought a slew of conflicting results from different scientific
groups. While some observed the absence of electrical resistance, others could
not confirm any signs of superconductivity in LK-99. Here, we investigate the
structural and electronic properties of LK-99 and its antecedent compounds
through quantum mechanics (QM) and QM-based molecular dynamics (QM-MD)
simulations. Our study elucidates the insulating nature of base compounds,
Pb$_{10}$(PO$_4$)$_6$O and Pb$_{10}$(PO$_4$)$_6$(OH)$_2$, spotlighting their
large band gaps. Notably, Cu doping in LK-99 disrupts its symmetry, yielding a
distorted ground-state crystal structure with a triclinic P1 symmetry and
CuO$_4$ square coordination. Such alterations predispose LK-99 to exhibit
semiconducting behaviors, characterized by a flat band above the Fermi energy,
arising from Cu-3d and O-2p orbitals. In addition, the S doping sustains the
triclinic P1 symmetry but leads to a significantly reduced band gap, with a
band emerging primarily from Cu-3d and S-3p orbitals. These findings are
important in understanding LK-99's structural and electronic properties and
provide a strategic compass for the development of high-T$_C$ superconductors.
