Electronic Levels of 4f Elements

The degeneracy of a configuration in the zero order approximation is given by the product of 2(21i+ 1) for all outer electrons when no equivalent electrons are involved. For the Tf configuration which consist entirely of equivalent electrons, the degeneracies are given by the binomial coefficient which are 1, 14, 91, 364, 

3432 or N = 0-7. For the second half of the group, the degeneracies are the same as in the first half, in the opposite order in accordance with the principle that N 4f electrons are equivalent to 14 - N holes in the completed shell and a configuration with a certain number of holes has the same number and kind of states as a configuration with the same number of electrons. If the interaction between the electrons and between the orbital and spin angular moment are introduced by appropriate terms H etc. in the Hamiltonian, some of the 

Summed over all electron pairs, and the spin-orbit interaction by 

Where is the spin-orbit interaction constant for the i electron. For a Coulomb field with an effective nuclear charge Z , 

In the interaction between the states within one configuration is taken into consideration and, thus interaction between states of different configurations are left out, this is the approximation usually employed. It is reasonably good for the Tf configuration of trivalent rare earth ions because this configuration is isolated from all others with which it could interact. Because of symmetry considerations 

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