The expressions for various possible mechanisms under the point-multipole approximation which contribute to the zero-field splitting parameter ${}_{}{}^6{}_{}{}^{}S$ in the spin Hamiltonian $D$ for a paramagnetic $H_S=D[3\mathrm{}{S}_z^2-S(S+1\mathrm{}\left)\right]$ ion in a trigonal symmetry have been derived. Important differences have been noticed in this axial symmetry in contrast with the results in the tetragonal symmetry considered previously. The applications of the expressions to two hypothetical cases and the real case of Cd${\mathrm{Cl}}_2$:${\mathrm{Mn}}^{2+}$ reveal that both the spin-spin and spin-orbit interactions are important. In the case of Cd${\mathrm{Cl}}_2$:${\mathrm{Mn}}^{2+}$, the spin-orbit contribution ${(-10.43\mathrm{}\times 10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$) cancels a large part of the spin-spin contribution (+15.35×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$), giving the value $D=+4.92\mathrm{}\times {10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$. This compares favorably with the results (at room temperature) of < +5×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ for $D$ from experiments on single crystals and ∼ + 1×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ from experiments on powdered samples. The latter approximate result from powdered samples has been deduced by reducing the uncertainty in $D$ using the graph illustrating the experimental variation of $D$ with temperature. For ${\mathrm{CdCl}}_2$:${\mathrm{Mn}}^{2+}$ it has also been pointed out that the value of $D$ varies from -2.11×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ to + 10.49×${10}^{-4\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$ if one considers the existing uncertainty in the crystal structure parameter obtained from x-ray diffraction. In the present study, the presence of higher multipoles because of the ions, and the distortion of the lattice surrounding the magnetic ion, have not been taken into account.

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