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 LSD+U schemes. The thirteen uppermost occupied spin-up bands correspond in a pure ionic scheme to the five Mn $3d$ orbitals at the Mn₁ (spin-up) site and the four O $2s/2p$ orbitals at each of the O₁ and O₂ sites. Maximal localization identifies uniquely four Wannier functions for each O, which are trigonally distorted sp³-like orbitals. They display a weak covalent bonding between O $2s/2p$ states and minority-spin d states of Mn₂, 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 Mn₁ 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 r² operator, so that they display the D3d symmetry of the Mn₁ site.