Spin correlation in Pr IV and Tm IV free ions

The coefficients \(c_{k}\) (k = 2, 4, 6) that pertain to spin-correlated matrix elements of the tensor operator \({{\varvec{U}}}^{{\varvec{(k)}}}\) have been evaluated by means of the differences \({{\varvec{U}}}^{{\varvec{(k)}}}\) (intermediate) \(-\) \({{\varvec{U}}}^{{\varvec{(k)}}}\) (LS) and the reduced matrix elements of the operator \({{\varvec{V}}}^{{\varvec{(1k)}}}\) . Only spin-allowed transitions have been considered from each ground level to the excited energy levels within the \(4\hbox {f}^{2}\) and \(4\hbox {f}^{12}\) configurations of the free ions Pr (3+) and Tm (3+), respectively. The values of the coefficients \(c_{k}\) thus found correspond in most cases by sign and order of magnitude to those determined in other sources as corrections to lanthanide (3+) crystal-field parameters.