Interaction energy of two shells in LS coupling

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. 

As was mentioned in previous chapters, the wave function of an atom must be antisymmetric with respect to the transposition of the coordinates (sets of quantum numbers) of each pair of electrons. The antisymmetric wave function of a shell of equivalent electrons may be constructed with the help of the CFP (formulas (9.7) and (9.8) for LS and jj couplings, respectively). Antisymmetrization of the atomic wave function with respect to electrons belonging to different shells may be ensured with the help of generalized CFP [14] or by imposing certain phase conditions on the 

For the special case of two equivalent electrons formula (20.1) turns into (9.1). 

Matrix elements of the operators of the interaction energy between two shells of equivalent electrons may be expressed, with the aid of the CFP, in terms of the corresponding two-electron quantities. Substituting in such formula the explicit expression for the two-electron matrix element, after a number of mathematical manipulations and using the definition of submatrix elements of operators composed of unit tensors, we get convenient expressions for the matrix elements in the case of two shells of equivalent electrons. The corresponding details may be found in [14], here we present only final results. 

Let us start with the one-electron operators. Their intershell matrix elements are small or vanish. Therefore, the total matrix element of such an operator is simply equal to the sum of corresponding quantities describing that interaction within each shell. Thus, for operators T and P (1.15) we have  

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In this case we talk about jj coupling. If the weaker electrostatic interaction is then applied, the individual jj s couple to a resultant J. The resulting energy-level diagram is quite different compared with the case of LS coupling (i.e., there is no interval rule). In the right-hand part of Fig.2.9 the jj structure for two equivalent p electrons is shown. In the outer shells... 

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