20 - Interaction energy of two shells in LS coupling

Summary

While studying the energy spectra or other spectral quantities of atoms and ions having complex electronic configurations, one ought to consider the expressions for the matrix elements of the operators both within each shell of equivalent electrons and between each pair of these shells. For example, in order to find the energy spectrum of the ground configuration 1s²2s² of the beryllium atom, we have to calculate the interaction energy in each shell ls² and 2s² as well as between them. The last case will be discussed in this chapter. If there are more shells, then according to the two-particle character of interelectronic interactions, we have to account for this interaction between all possible pairs of shells.

Let us notice that momenta of each shell may be coupled into total momenta by various coupling schemes. Therefore, here, as in the case of two non-equivalent electrons, coupling schemes (11.2)–(11.5) are possible, only instead of one-electronic momenta there will be the total momenta of separate shells. To indicate this we shall use the notation LS, LK, JK and JJ. Some peculiarities of their usage were discussed in Chapters 11 and 12 and will be additionally considered in Chapter 30. Therefore, here we shall restrict ourselves to the case of LS coupling for non-relativistic and JJ (or jj) coupling for relativistic wave functions. We shall not indicate explicitly the parity of the configuration, consisting of several shells, because it is simply equal to the sum of parities of all shells.