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$\Delta E$ & $\Delta V$ & {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}
