  Effects of the structural distortion associated with the $\rm OsO_6$
octahedral rotation and tilting on the electronic band structure and magnetic
anisotropy energy for the $5d^3$ compound NaOsO$_3$ are investigated using the
density functional theory (DFT) and within a three-orbital model. Comparison of
the essential features of the DFT band structures with the three-orbital model
for both the undistorted and distorted structures provides insight into the
orbital and directional asymmetry in the electron hopping terms resulting from
the structural distortion. The orbital mixing terms obtained in the transformed
hopping Hamiltonian resulting from the octahedral rotations are shown to
account for the fine features in the DFT band structure. Staggered
magnetization and the magnetic character of states near the Fermi energy
indicate weak coupling behavior.
