Theoretical energy-level schemes are derived to first order of perturbation theory and compared with those empirically derived for both a crystal of CaF₂: Sm³⁺ and one of CdF₂:Sm³⁺ in a crystal field of tetragonal symmetry. The resulting deviation of the energies between the theoretical and empirical schemes is within 40 cm^-1 for ten of the twelve crystal-field energy levels of the ground ⁶H multiplet states terminating the observed fluorescence lines. The theoretical schemes are derived for a unique set of values for the five crystal-field parameters of the tetragonal field. This is substantial additional support for an assignment of tetragonal symmetry for both of these crystals. Tables are given of the resulting approximate values of the crystal-field parameters, level energies, and level eigenfunctions.

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