Before turning to HCSCO, we benchmarked the methodology on HgBa\({}_{2}\)CuO\({}_{4}\) itself. We find that at fixed oxygen stoichiometry, HBCO lies 74meV/atom above the hull. Upon modeling the doped compound using via a \(3\times 3\) supercell calculation (11% doping), we find HBCO to be even more unstable at 130meV/atom above the hull. As noted above in our work with LSCO, metastable structures theoretically predicted to be \(\sim\)100meV/atom above the hull do indeed exist. Thus, we propose that \(\sim\)100meV/atom uncertainty is a reasonable confidence interval.

With that caveat in mind, we find that HCSCO lies 170meV/atom above the hull at fixed oxygen stoichiometry, and 240meV/atom for fixed oxygen chemical potential, which means the compound is likely unstable, but not out of the realm of possibility for successful synthesis.

Electronic structure – In order to evaluate the prospects of superconductivity, we examine the orbitally-resolved density of states (Fig. \ref{fig:hcsco-bands}). There is indeed a single band crossing the Fermi level Hg(CaS)\({}_{2}\)CuO\({}_{2}\), similar to the cuprates. Additionally, we find the charge-transfer energy of HCSCO to be 1.94eV, identical to that of the other Hg-based cuprates, so we expect similar superconducting transition temperatures. The jury is still out on this material as it has not been successfully synthesized.