The multi-electron Hamiltonian including the relativistic spin-orbit coupling term (but excluding other relativistic corrections) is

where and are given by eq240 and eq261a respectively.We shall now show that and commute with and therefore with . Assuming ,

Expanding the RHS of the second equality of the above equation and using eq99, eq100, eq101, eq165, eq166 and eq167, . In general,

Similarly,

Furthermore, . Using eq243, . Combining this with eq284,

Next,

Using the identity and substituting eq284 and eq285 in the above equation, . In general,

Using eq244, eq245, . So,

Therefore, and commute with . We have also shown in an earlier article that commutes with , and . This implies that we can select a common complete set of eigenfunctions for , , and . . Unlike the case of the hydrogen atom, and no longer commute with for a multi-electron atom (see eq182 and eq182a). Despite that, the quantum numbers , and are used to characterise the states of elements in the first three periods of the periodic table, because spin-orbit coupling is weak for these atoms.

Similar to hydrogen, the eigenvalue equation of for a multi-electron atom is solved by treating as a perturbation if spin-orbit coupling is weak, with the unperturbed part of the eigenvalue equation solved using the Hartree-Fock method.

###### Question

Evaluate using the first order perturbation theory.

###### Answer

For a multi-electron system, . So, , which when substituted in and using eq133, eq160 and eq205a, gives

where .

is evaluated empirically to be a constant for a given term.