Interplay of covalency and correlations in the edge shared spin 1/2 $A_{3}T_{2}$O$_{4}$ chain compounds ($A$ = Na, K; $T$ = Cu, Ni)

Na$_{3}$Cu$_{2}$O$_{4}$, K$_{3}$Cu$_{2}$O$_{4}$ and K$_{3}$Ni$_{2}$O$_{4}$ belong to a new class of quasi-1D insulating cuprates which feature strongly buckled, one-dimensional $^{1}_{\infty}$CuO$_{2}$ ribbon-like chains consisting of edge-sharing CuO$_{4}$ plaquettes. Structural analysis of the metal-oxygen bond lengths and thermodynamic measurements[1,2,3] imply that these systems are intrinsically charge ordered ($\ldots$ (Ni/Cu)$^{2+}$-(Ni/Cu)$^{3+}$-(Ni/Cu)$^{2+}$-(Ni/Cu)$^{3+}\ldots$) and show dominant antiferromagnetic interactions. No electronic structure analysis of these systems exist to date. Using density functional theory based calculations (LDA, Wannier functions, LDA+$U$), we analyze the microscopic origin of the magentic interactions in these systems. The main interaction along the chains are the second neighbor superexchanges. Nonetheless, a careful analysis of the first neighbor interaction between the magnetic (Cu$^{2+}$/Ni$^{3+}$) cation and the non-magnetic cation (Cu$^{3+}$/Ni$^{2+}$) is necessary. We report on the interplay of covalency, crystal field splitting and correlations in these systems. [1] Z. Anorg. Allg.Chem. vol. 462, 92 (1980). [2] J. Solid State Chem. vol. 178, 3708 (2005). [3] Z. Anorg. Allg. Chem. vol. 637, 1101 (2011).