“…(iii) For simplicity, the D 2d approach instead of S 4 symmetry is adopted to describe the crystal-field interaction of (4) in this work. In fact, even though one takes the exact S 4 symmetry, the magnitudes of the imaginary parts of the rank-4 and rank-6 crystal-field parameters are very small, as shown in [21,32,33], and so their contributions to the coefficient C( 4 I 15/2 ;Γ γ (γ ) M J1 ) or C( 4 I 13/2 ;Γ γ (γ ) M J2 ) in Γ γ and the g factors are expected to be no more than 4%. (iv) Even in the approach of D 2d symmetry, the angular distortions for the polar angles θ j due to the size and charge mismatching substitution of Ca 2+ by Er 3+ are not considered in the calculations.…”