  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 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 Mn_1 (spin-up) site, and the four O 2s/2p orbitals at
each of the O_1 and O_2 sites. Maximal localization identifies uniquely four
Wannier functions for each O, which are trigonally-distorted sp^3-like
orbitals. They display a weak covalent bonding between O 2s/2p states and
minority-spin d states of 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 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 which
diagonalize the r^2 operator, so that they display the D_3d symmetry of the
Mn_1 site.
