  The spins of the low-spin Ir4+ (S = 1/2, d5) ions at the octahedral sites of
the oxides Sr3NiIrO6, Sr2IrO4 and Na2IrO3 exhibit preferred orientations with
respect to their IrO6 octahedra. We evaluated the magnetic anisotropies of
these S = 1/2 ions on the basis of DFT calculations including spin-orbit
coupling (SOC), and probed their origin by performing perturbation theory
analyses with SOC as perturbation within the LS coupling scheme. The observed
spin orientations of Sr3NiIrO6 and Sr2IrO4 are correctly predicted by DFT
calculations, and are accounted for by the perturbation theory analysis. As for
the spin orientation of Na2IrO3, both experimental studies and DFT calculations
have not been unequivocal. Our analysis reveals that the Ir4+ spin orientation
of Na2IrO3 should have nonzero components along the c- and a-axes directions.
The spin orientations determined by DFT calculations are sensitive to the
accuracy of the crystal structures employed, which is explained by perturbation
theory analyses when interactions between adjacent Ir4+ ions are taken into
consideration. There are indications implying that the 5d electrons of Na2IrO3
are less strongly localized compared with those of Sr3NiIrO6 and Sr2IrO4. This
implication was confirmed by showing that the Madelung potentials of the Ir4+
ions are less negative in Na2IrO3 than in Sr3NiIrO6, Sr2IrO4. Most
transition-metal S = 1/2 ions do have magnetic anisotropies because the SOC
induces interactions among their crystal-field split d-states, and the
associated mixing of the states modifies only the orbital parts of the states.
This finding cannot be mimicked by a spin Hamiltonian because this model
Hamiltonian lacks the orbital degree of freedom, thereby leading to the
spin-half syndrome. The spin-orbital entanglement for the 5d spin-half ions
Ir4+ is not as strong as has been assumed lately.
