deletions | additions       diff --git a/bacoso2.tex b/bacoso2.tex  index d4810e1..cb5134f 100644  --- a/bacoso2.tex  +++ b/bacoso2.tex 
...
To understand the behavior of the energies as a function of $U$, we first roughly grouped the structures by the behavior of the $E$ vs. $U-J$ curve, which we have indicated by colors in Fig.~\ref{fig:reordering}. The largest group has a slope of roughly $∆E \sim 0.3U$. A second subset has energies that are relatively constant $(∆E \sim const)$. The third group, which appeared to have the lowest energies in the U = 0 run, rapidly increases in energy with $∆E \sim 0.7U$.  In order the to  rationalize this behavior, we begin by examining the structural motifs associated with each group. Shown in Fig.~\ref{fig:bacoso-structs} are representative structures for each group. The ligands density matrix  of the 3$d$ orbitals produced by the DFT calculations and magnetization per  Co atom atom. The Coulomb $U$ in the LDA+U formalism penalizes non-integer occupancies by coupling to $tr(n_\sigma - n_\sigma^2)$, where $n_\sigma$ is the density matrix of the 3$d$ orbitals. For the case of diagonal density matrices, this term indeed vanishes when the orbital occupancies  are either oxygen 0  or sulfur, 1,  and form either a dumbell, trigonal planar, square planar, tetrahedral, or square pyramidal cage. Additionally, the Co-Co distance greatly changes depending on is maximal at $1/2$. We have tabulated  the connectivity diagonal elements  of $\n_\sigma$ in Table~\ref{tbl:bacoso}, and we find structures with itinerant electronic behavior, i.e. non-integer density matrices, to be more strongly penalized as $U$ increases. The experimental structure has nearly integer occupancies and thus remains low-energy while  the ligand cages. energies of all other structures increase.  [Ran, do you have ideas as to how magnetization is involved?]