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In addition to structure, we directly examine the density matrix of the 3$d$ orbitals produced by the DFT calculations and magnetization per Co atom (see Table~\ref{tbl:bacoso}). Non-integer occupancies are penalized since the interaction $U-J$ couples to $\sim \tr n_\sigma - \tr n_\sigma^2$, which is zero only for $n = 0$ or $1$. $1$ and indeed, structures where the density matrix are further away from integer occupation are more strongly penalized as $U-J$ increases. As expected, the experimental structure has nearly integer occupancies and thus remains low-energy while the energies of all other structures are increased.  [Gabi, Ran, I need some help here: do you see any clear correlations between the action of the U correction and the local structure / Co-Co distance / density matrix? It roughly looks like compounds that contain CoO$_2$S$_2$ tetrahedra behave like the ground state (in that their energies are parallel as a function of $U$. I'm having a hard time picking out other correlations. Ran has an idea regarding the Co-Co distance. Also, how is magnetization involved?]