  Recently, Cu-substituted lead apatite LK-99 was reported to have
room-temperature ambient-pressure superconductivity. Here we utilize density
functional theory (DFT) total energy and harmonic phonon calculations to
investigate the thermodynamic and dynamic stability of two lead phosphate
apatites in their pure and Cu-substituted structures. Though
Pb$_{10}$(PO$_4$)$_6$O and Pb$_{10}$(PO$_4$)$_6$(OH)$_2$ are found to be
thermodynamically stable (i.e., on the T=0K ground state convex hull), their
Cu-substituted counterparts are above the convex hull. Harmonic phonon
calculations reveal dynamic instabilities in all four of these structures.
Oxygen vacancy formation energies demonstrate that the addition of Cu dopant
substituting for Pb increases the likelihood of the formation of oxygen
vacancies on the anion site. We propose a new possible phase in this system,
Pb$_8$Cu$_2$(PO$_4$)$_6$, where two monovalent Cu atoms are substituted for two
Pb(1) atoms and the anion oxygen is removed. We also propose several reaction
pathways for Pb$_9$Cu(PO$_4$)$_6$O and Pb$_8$Cu$_2$(PO$_4$)$_6$, and found that
both of these two structures are likely to be synthesized under a 1:1 ratio of
reactants Pb$_2$SO$_5$ and Cu$_3$P. Our work provides a thorough foundation for
the thermodynamic and dynamic stabilities of LK-99 related compounds and we
propose several possible novel synthesis reaction pathways and a new predicted
structure for future studies.
