The ionic positions of Gd³⁺ at orthorhombic sites in CaF₂ with a nearby charge-compensating M⁺ ion (M=Li, Na, K, Rb and Cs) are calculated from a reliable lattice relaxation model. The results are then employed in superposition model analyses of the corresponding spin parameters of the ground state 4f⁷ ⁸S_7/2 splitting of the Gd³⁺ ion available in the literature. The superposition model analyses indicate that the spin parameters can best be described in terms of some intrinsic parameters which follow a parabolic dependence on the ligand distances in addition to the conventional power-law form, and hence also assert the cancelling nature of various physical mechanisms contributing to the ground state splitting of Gd³⁺.