part 1

Our calculations solves the single-ion Hamiltonian:

where:

B_n^m -denote CEF parameters whereas is applied external field.
μ_B– Bohr magneton,
g_e=2.002324– Lande factor for spin ,
λ_s-o– Spin-orbit constant.In description of the CEF interactions the equivalent operator technique developed by Eliott and Steves in 1953 has been applied. See also work of Hutchings from 1964. The Stevens operators are, in fact, factors of the multipolar charge moment.

Hamiltonian providing energies (eigenvalues) of the electronic states with their eigenfunctions. It diagonalizes (2L+1) (2S+1)-for 3d-ions, or (2J+1)-for f-ions, (H_3d or H_4f ). In case of H_3d first term is the CEF interactions written with the Stevens operators.

Stevens operators for |J,J_z > representation:

Number of Stevens parameters in Hamiltonian for different symmetries:

Transformation of Stevens operators on rotation of the z Axis into the perpendicular plane x,y so that z' makes an angle α with x, and y' is parallel to z':

The diagonalization of the CEF Hamiltonian provide the discrete spectrum of energy levels E_i together with the eigenvectors Γ_i(|LSL_zS_z>).

More about the CEF calculation techniques you can find, for instance, in: Operator methods in ligand field theory , H. Watanabe, Prentice-Hall, Inc. Engelwood Cliffs, New Jersey (1966)