Complex atoms, angular momenta terms

In more complex atoms there may be a strong coupling between the motion of different electrons. The states are usually described in terms of the total orbital angular momentum L and the total spin angular momentum... 

A wavefunction, ip, is a solution to the Schrodinger equation. For atoms, wavefunctions describe the energy and probabihty of location of the electrons in any region around the proton nucleus. The simplest wavefunctions are found for the hydrogen atom. Each of the solutions contains three integer terms called quantum numbers. They are n, the principal quantum number, I, the orbital angular momentum quantum number and mi, the magnetic quantum number. These simplest wavefunctions do not include the electron spin quantum number, m, which is introduced in more complete descriptions of atoms. Quantum numbers define the state of a system. More complex wavefunctions arise when many-electron atoms or molecules are considered. 

In this case, then, the orbital degeneracy can be described by introducing into the spin Hamiltonian the operator for the unquenched part of the total orbital angular momentum. The term y4L S2 is the analogue of the commonly adopted free-atom form fL S, which is appropriate in a strictly central-field situation. Again, however, the effective Hamiltonian corresponds to a simple model, all the complexities of the real system... 

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