  An effective spin model for Mott insulators is determined by the symmetries
involved among magnetic sites, electron fillings, and their interactions. Such
a spin Hamiltonian offers insight to mechanisms of magnetic orders and magnetic
anisotropy beyond the Heisenberg model. For a spin moment S bigger than 1/2,
single-ion anisotropy is in principle allowed. However, for $d^3$ Mott
insulators with large cubic crystal field splitting, the single-ion anisotropy
is absent within the LS coupling, despite S = 3/2 local moment. On the other
hand, preferred magnetic moment directions in $d^3$ materials have been
reported, which calls for a further theoretical investigation. Here we derive
the single-ion anisotropy interaction using the strong-coupling perturbation
theory. The cubic crystal field splitting including $e_g$ orbitals, trigonal
distortions, Hund's coupling, and spin-orbit coupling beyond the LS scheme are
taken into account. For compressed distortion, the spin-orbit coupling at
magnetic sites can favor either the easy-axis or the easy-plane while that of
anions leads to easy-axis anisotropy. We apply the theory on $\rm{CrX}_3$ with
X = Cl and I, and show the dependence of the single-ion anisotropy on the
strength of the spin-orbit couplings of both magnetic and anion sites.
Significance of the single-ion anisotropy in ideal two-dimensional magnets is
also discussed.
