It is noteworthy that the use of the atomic approach does not introduce significant additional errors to the H MF SO results. Next, we study the errors affecting the various approximations to the spin-orbit Hamiltonian in the calculation of the energy gap between ground and first excited multiplets. While due to lack of dynamical correlation in our approach these calculations would be of limited use to model experimental data, we are only interested here to estimate the typical error affecting spin-orbit Hamiltonians for the calculation of such energy gaps, as a term of comparison with that estimated for the ground multiplet magnetic excitations that are the focus of this study. The results are represented in Figure (see Table in SI). Here, the bare H (1) SO fails completely, with a general overestimation of the crystal field gaps larger than +70% for all the ions but Gd(III). Differently, the P 7/2 levels for Gd(III) are underestimated (∆E = 35631, 35687, 35728, 35822 cm -1 by H