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\emph{Global stability} -- We checked the thermodynamic stability of the proposed compounds against competing phases by selecting commonly known reactants and computing the formation enthalpies of the synthesis pathways as shown in Table~\ref{tbl:pathways}. We computed the total energies of formation $\Delta E = E_\text{products} - E_\text{reactants}$, and find that all differentials are positive, indicating the reactions target phases are unfavorable. However, it is known that many functional materials are metastable, protected from decay by large energetic barriers. The parent cuprate La$_2$CuO$_4$ is an example: as shown on the last line of Table~\ref{tbl:pathways}}, LCO is actually unstable by 28kJ/mol. We also examined the volume differentials $\Delta V = V_\text{products} - V_\text{reactants}$ with the knowledge that often high pressure synthesis allows otherwise unstable compounds to form. We indeed find that $\Delta V$ are overwhelming negative, meaning the application of high pressure may allow the formation of the target phases.
%% \begin{table}
%% \begin{tabular}{r|r|rl}
%% \hline
%% $\Delta E$ & $\Delta V$ & {c}{Synthesis pathway} & \\
%% \hline
%% \hline
%% 141 & -7.3 & La$_2$O$_2$S + CuS & $\rightarrow$ La$_2$CuO$_2$S$_2$\\
%% 223 & -3.4 & Y$_2$O$_2$S + CuS & $\rightarrow$ Y$_2$CuO$_2$S$_2$\\
%% 267 & -5.0 & Lu$_2$O$_2$S + CuS & $\rightarrow$ Lu$_2$CuO$_2$S$_2$\\
%% 356 & -3.0 & Sc$_2$O$_2$S + CuS & $\rightarrow$ Sc$_2$CuO$_2$S$_2$\\
%% 101 & -4.9 & La$_2$O$_2$S$_2$ + Cu & $\rightarrow$ La$_2$CuO$_2$S$_2$\\
%% \hline
%% 148 & -3.3 & La$_2$O$_3$ + CuS & $\rightarrow$ La$_2$CuO$_3$S \\
%% 454 & -0.7 & Sc$_2$O$_3$ + CuS & $\rightarrow$ Sc$_2$CuO$_3$S \\
%% 97 & -4.9 & La$_2$O$_2$S + CuO & $\rightarrow$ La$_2$CuO$_3$S \\
%% 269 & 2.8 & Sc$_2$O$_2$S + CuO & $\rightarrow$ Sc$_2$CuO$_3$S \\
%% \hline
%% 28 & -5.1 & La$_2$O$_3$ + CuO & $\rightarrow$ La$_2$CuO$_4$ \\
%% \hline
%% \end{tabular}
%% \caption{Synthesis pathways for various cuprate oxysulfides based on
%% substitution of sulfur for both (top block) or only one (middle block) of
%% the apical oxygens in $R_2$CuO$_4$. Energies in kJ/mol and volumes in
%% kJ/mol/GPa. Since the energies of formation ($\Delta E = E_\text{products}
%% - E_\text{reactants}$) are positive, none of these pathways appear
%% favorable at ambient conditions. However, high-pressure synthesis will
%% help stabilize these pathways, since the majority of volume differentials
%% ($\Delta V = V_\text{products} - V_\text{reactants}$) are negative. We
%% benchmark our method against the standard synthesis pathway for
%% La$_2$CuO$_4$, shown on the last line. Surprisingly, $\Delta E$ is
%% +28~kJ/mol, so either DFT systemmatically overestimates enthalpies (which
%% means the actual enthalpies for our hypothetical compounds are
%% \emph{smaller}, in our favor), or we must add a bi-directional uncertainty
%% of $\pm 30$~kJ/mol to the computed enthalpies. Additionally, positional
%% entropy of the apical $S$ in the half-substituted $R_2$CuO$_3$S compounds
%% should also assist in synthesis.}
%% \label{tbl:pathways}
%% \end{table}
\emph{Reexamination} -- In the intervening years, the maturation of materials databases allowed us to revisit the question of global stability. Various databases have computed and tabulated the convex hulls of binary, ternary and some quaternary systems, and provided tools for researchers to apply their framework to novel chemical systems. In the following, we describe our new understanding of the global stability of La$_2$CuO$_2$S$_2$ and La$_2$CuO$_3$S$_S$ against all known competing phases in the La-Cu-S-O chemical system. Since the Cu site contains significant correlations, we must address the effect of $U$ on the energies provided by density functional theory.
