Maximally localized Wannier functions in antiferromagnetic MnO within the FLAPW formalism

We have calculated the maximally localized Wannier functions of MnO in its antiferromagnetic (AFM) rhombohedral unit cell, which contains two formula units. Electron Bloch functions are obtained with the linearized-augmented-plane-wave method within both the local-spin density (LSD) and the $\mathrm{LSD}+U$ schemes. The thirteen uppermost occupied spin-up bands correspond in a pure ionic scheme to the five Mn $3d$ orbitals at the ${\mathrm{Mn}}_1$ (spin-up) site and the four O $2s/2p$ orbitals at each of the ${\mathrm{O}}_1$ and ${\mathrm{O}}_2$ sites. Maximal localization identifies uniquely four Wannier functions for each O, which are trigonally distorted ${\mathrm{sp}}^3$-like orbitals. They display a weak covalent bonding between O $2s/2p$ states and minority-spin d states of ${\mathrm{Mn}}_2,$ which is absent in a fully ionic picture. This bonding is the fingerprint of the interaction responsible for the AFM ordering, and its strength depends on the one-electron scheme being used. The five Mn Wannier functions are centered on the ${\mathrm{Mn}}_1$ site, and are atomic orbitals modified by the crystal field. They are not uniquely defined by the criterion of maximal localization and we choose them as the linear combinations that diagonalize the $r^2$ operator, so that they display the $D_{3d}$ symmetry of the ${\mathrm{Mn}}_1$ site.