Similar errors are shown by both the atomic H AMF SO and H ABP SO approximations, where µ falls in the range ±1% for all the ions, with the exception of Gd(III). For the latter, the exclusion of the 2e-SOC multi-center integrals in the computation leads to a quite large underestimation of the energy gaps (∆E = 0.2, 0.5, and 0.8 cm -1 , µ ≈ -12%). Within the atomic approximation, Ho(III), Er(III), and Tm(III) show the largest 3σ values (3σ ≈ 2.0%, 3.0% and 1.5%, respectively), while 3σ < 1.2% for all the other metals.