  Finding an accurate ab initio approach for calculating the electronic
properties of transition metal oxides has been a problem for several decades.
In this paper, we investigate the electronic structure of the transition metal
monoxides MnO, CoO, and NiO in their undistorted rock-salt structure within a
fully iterated quasiparticle self-consistent GW (QPscGW) scheme. We study the
convergence of the QPscGW method, i.e., how the quasiparticle energy
eigenvalues and wavefunctions converge as a function of the QPscGW iterations,
and we compare the converged outputs obtained from different starting
wavefunctions. We find that the convergence is slow and that a one-shot
G$_0$W$_0$ calculation does not significantly improve the initial eigenvalues
and states. It is important to notice that in some cases the "path" to
convergence may go through energy band reordering which cannot be captured by
the simple initial unperturbed Hamiltonian. When we reach a fully iterated
solution, the converged density of states, band-gaps and magnetic moments of
these oxides are found to be only weakly dependent on the choice of the
starting wavefunctions and in reasonably good agreement with the experiment.
Finally, this approach provides a clear picture of the interplay between the
various orbitals near the Fermi level of these simple transition metal
monoxides. The results of these accurate {\it ab initio} calculations can
provide input for models aiming at describing the low energy physics in these
materials.
