Wannier functions and exchange integrals: The example of Li Cu 2 O 2

Starting from a single band Hubbard model in the Wannier function basis, we revisit the problem of the ligand contribution to exchange and derive explicit formulas for the exchange integrals in metal oxide compounds in terms of atomic parameters that can be calculated with constrained LDA and $\mathrm{LDA}+\mathrm{U}$. The analysis is applied to the investigation of the isotropic exchange interactions of $\mathrm{Li}{\mathrm{Cu}}_{2}{\mathrm{O}}_{2}$, a compound where the Cu-O-Cu angle of the dominant exchange path is close to 90\ifmmode^\circ\else\textdegree\fi{}. Our results show that the magnetic moments are localized in Wannier orbitals which have strong contribution from oxygen atomic orbitals, leading to exchange integrals that considerably differ from the estimates based on kinetic exchange only. Using $\mathrm{LSDA}+\mathrm{U}$ approach, we also perform a direct ab initio determination of the exchange integrals $\mathrm{Li}{\mathrm{Cu}}_{2}{\mathrm{O}}_{2}$. The results agree well with those obtained from the Wannier function approach, a clear indication that this modelization captures the essential physics of exchange. A comparison with experimental results is also included, with the conclusion that a very precise determination of the Wannier function is crucial to reach quantitative estimates.