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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}). The Non-integer occupancies are penalized since the interaction  $U-J$ term penalizes non-integer occupancies. couples to $\sim \tr n_\sigma - \tr n_\sigma^2$, which is zero only for $n = 0$ or $1$.  We can rationalize this behavior by [need to look at the data here — maybe it has to [Gabi, Ran, I need some help here:  do with you see any clear correlations between  the local ligand environment action  of the Co atom, U correction  andmaybe we won’t be able to rationalize it.]  Roughly,  the correction depends on local structure / Co-Co distance / density matrix? It roughly looks like compounds that contain CoO$_2$S$_2$ tetrahedra behave like  the occupation 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 3d orbitals to which U is applied, namely the energy difference between two states with occupations n1 and n2 is roughly $∆E \sim U(n1-1/2)(n2-n1)$. [Gabi, Ran, this Co-Co distance. Also, how  is just my hypothesis. Ran, could you check this from the data?] magnetization involved?]