Graphical approach to the spin-orbit interaction

We derive a new form for the matrix element of the two-particle mutual spin-orbit interaction in the Breit-Pauli approximation. The complexity of standard tensor-operator expansion techniques is obviated by suitably decomposing the mutual spin-orbit interaction into products of spin and orbital angular-momentum operators, whose matrix elements in the $l{m}_{i}s{m}_s$ scheme are obtained in a straightforward manner. Graphical representations are given for both the one- and two-particle spin-orbit interactions, thereby permitting the analytical evaluation by graphical techniques of the spin-orbit interaction between arbitrary configurations in any coupling scheme. The graphical procedure for evaluating these matrix elements is demonstrated for $\mathrm{LS}$-coupled configurations having two nonequivalent electrons outside closed shells. Various properties of the two-particle mutual spin-orbit operator are easily illustrated using its graphical representation.