Effective crystalline electric field potential in a j-j coupling scheme

We propose an effective model on the basis of a j-j coupling scheme to describe local f-electron states for realistic values of Coulomb interaction U and spin-orbit coupling A, for future development of microscopic theory of magnetism and superconductivity in f(n)-electron systems, where n is the number of local f electrons. The effective model is systematically constructed by including the effect of a crystalline electric field (CEF) potential in the perturbation expansion in terms of 1/lambda. In this paper, we collect all the terms up to the first order of 1/lambda. Solving the effective model, we show the results of the CEF states for each case of n = 2-5 with O-h symmetry in comparison with those of the Stevens Hamiltonian for the weak CEF. In particular, we carefully discuss the CEF energy levels in an intermediate coupling region with lambda/U in the order of 0.1 corresponding to actual f-electron materials between the LS and j-j coupling schemes. Note that the relevant energy scale of U is the Hund's rule interaction. It is found that the CEF energy levels in the intermediate coupling region can be quantitatively reproduced by our modified j-j coupling scheme, when we correctly take into account the corrections in the order of 1/lambda in addition to the CEF terms and Coulomb interactions which remain in the limit of lambda = infinity. As an application of the modified j-j coupling scheme, we discuss the CEF energy levels of filled skutterudites with T-h symmetry.