Capability of the free-ion eigenstates for crystal-field splitting
  Any electron eigenstate |\Psi>of the paramagnetic ion open-shell is characterized by the three independent multipole asphericities A_{k}=<\Psi || C^(k)}||\Psi> for k=2,4 and 6 related to the second moments of the relevant crystal-field splittings by \sigma_{k}^{2}=<[1/(2J+1)>]A_{k}^{2}S_{k}^{2}, where S_{k}^{2}=<[1/(2k+1)>]\sum_{q}|B_{kq}|^{2}. The A_{k} can serve as a reliable measure of the state |\Psi>capability for the splitting produced by the k-rank component of the crystal-field Hamiltonian and allow one to verify any fitted crystal-field parameter set directly comparing the calculated and the experimental second moments of the relevant crystal-field splittings. We present the multipole characteristics A_{k} for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4f^{N} ions applying in the calculations the advanced eigenfunctions of the free lanthanide ions obtained based on the M. Reid f-shell programs. Such amended asphericities are compared with those achieved for the simplified Russell-Saunders states. Next, they are classified with respect to the absolute or relative weight of the A_{k} in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the A_{k} variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm^{3+}, Er^{3+}, Nd^{3+}, and Pr^{3+} ions. The detailed mechanisms of such A_{k} changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained. 
Jacek Mulak and Maciej Mulak
physics.chem-ph cond-mat.mtrl-sci