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My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ?

In other words, why & how the relative magnitude of the respective terms in the Hamiltonian, namely for LS & the central field (residual), results in the order the summing of the angular momenta (orbital & spin) ?

In my eyes, this is a fundamental question regarding the physical picture & the appropriate vector model.

Thanks, Bentzy.