  Magnetism of transition metal (TM) oxides is usually described in terms of
the Heisenberg model, with orientation-independent interactions between the
spins. However, the applicability of such a model is not fully justified for TM
oxides because spin polarization of oxygen is usually ignored. In the
conventional model based on the Anderson principle, oxygen effects are
considered as a property of the TM ion and only TM interactions are relevant.
Here, we perform a systematic comparison between two approaches for spin
polarization on oxygen in typical TM oxides. To this end, we calculate the
exchange interactions in NiO, MnO, and hematite (Fe2O3) for different magnetic
configurations using the magnetic force theorem. We consider the full spin
Hamiltonian including oxygen sites, and also derive an effective model where
the spin polarization on oxygen renormalizes the exchange interactions between
TM sites. Surprisingly, the exchange interactions in NiO depend on the magnetic
state if spin polarization on oxygen is neglected, resulting in non-Heisenberg
behavior. In contrast, the inclusion of spin polarization in NiO makes the
Heisenberg model more applicable. Just the opposite, MnO behaves as a
Heisenberg magnet when oxygen spin polarization is neglected, but shows strong
non-Heisenberg effects when spin polarization on oxygen is included. In
hematite, both models result in non-Heisenberg behavior. General applicability
of the magnetic force theorem as well as the Heisenberg model to TM oxides is
discussed.
