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For multi-electron atoms for which the Hamiltonian (including the spin-orbit coupling) reads

$$H=\sum_i T_i-Z\sum_i V_i+\sum_i V_i^{\text{s.o.}}+\sum_{i>j}V_{ij}$$

The $T_i$ are the kinetic energy of the individual electrons, $V_i$ are the nuclear Coulomb attraction, $V_i^{\text{s.o.}}$ is the spin-orbit coupling, and $V_{ij}$ are interelectron Coulomb repulsion.

In http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_atomic.pdf (page 103), using Wigner-Eckert, they are able to diagonalize the spin-orbit in terms of the total orbital and spin angular momenta of all the electrons.

Although it says it is beyond the scope of the course, I would still like to know how to do it. I am pretty familiar with the Wigner-Eckert theorem in its basic form (that the matrix elements of a tensor-operator can be written in terms of a C.G. coefficient times a reduced matrix element). Thanks.

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