  Applying the correlated electronic structure method based on density
functional theory plus the Hubbard $U$ interaction, we have investigated the
tetragonal scheelite structure Mott insulator KOsO$_4$, whose $e_g^1$
configuration should be affected only slightly by spin-orbit couping (SOC). The
method reproduces the observed antiferromagnetic Mott insulating state,
populating the Os $d_{z^2}$ majority orbital. The quarter-filled $e_g$ manifold
is characterized by a symmetry breaking due to the tetragonal structure, and
the Os ion shows a crystal field splitting $\Delta_{cf}$ = 1.7 eV from the
$t_{2g}$ complex, which is relatively small considering the high formal
oxidation state Os$^{7+}$. The small magnetocrystalline anisotropy before
including correlation (i.e., in the metallic state) is increased by more than
an order of magnitude in the Mott-insulating state, a result of a strong
interplay between large SOC and a strong correlation. In contrast to
conventional wisdom that the $e_g$ complex will not support orbital magnetism,
we find that for the easy axis [100] direction the substantial Os orbital
moment $M_L\approx-0.2 \mu_B$ compensates half of the Os spin moment $M_S$ =
0.4$\mu_B$. The origin of the orbital moment is analyzed and understood in
terms of additional spin-orbital lowering of symmetry, and beyond that due to
structural distortion, for magnetization along [100]. Further interpretation is
assisted by analysis of the spin density and the Wannier function with SOC
included.
