deletions | additions       diff --git a/bacoso3.tex b/bacoso3.tex  index 7b800d1..1c57ea6 100644  --- a/bacoso3.tex  +++ b/bacoso3.tex 
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In addition to structure, we directly examine Finally,  the density matrix degree of itineracy of a material is linked to its structure. Shown in Fig.~\ref{fig:bacoso-structs} are representative structures for each group. The ligands  of the3$d$ 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 $n = 0$ either oxygen  or $1$ sulfur,  and indeed, structures where the density matrix are further away form either a dumbell, trigonal planar, square planar, tetrahedral, or square pyramidal cage. The atomic distance  frominteger 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 central Co atom to its surrounding ligands is largely fixed --  the U correction and large variation lies in how  the local structure / Co-Co distance / density matrix? It roughly looks like compounds that contain CoO$_2$S$_2$ tetrahedra behave like cages are connected. Crystal structures where  the ground state (in that their energies cages  are parallel as a function of $U$. I'm having a hard time picking out other correlations. Ran has connected by  an idea regarding the Co-Co distance. Also, how is magnetization involved?] edge lead to smaller cobalt-cobalt distances and more itinerant behavior. Structures with large cobalt-cobalt distances are more localized.