Coupling schemes for two-particle operator used in atomic calculations

General formulas for a two-electron operator for open-shell atoms in a coupled tensorial form are presented. Various representations of a two-electron operator in coupled tensorial form are studied. The consideration covers the cases when the two-electron operator represents atomic interactions or some effective interaction, for instance, arising from a perturbation theory. The second quantization technique is used. General formulas are derived in LS coupling. Two consistent approaches to make the coupled tensorial form for a two-particle operator are studied in detail.