A shell of equivalent electrons

Non-relativistic and relativistic cases of a shell of equivalent electrons 

In previous chapters we considered the wave functions and matrix elements of some operators without specifying their explicit expressions. Now it is time to discuss this question in more detail. Having in mind that our goal is to consider as generally as possible the methods of theoretical studies of many-electron systems, covering, at least in principle, any atom or ion of the Periodical Table, we have to be able to describe the main features of the structure of electronic shells of atoms. In this chapter we restrict ourselves to a shell of equivalent electrons in non-relativistic and relativistic cases. 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

N electrons with the same values of quantum numbers n,7 (LS coupling) or tijljji (jj coupling) are called equivalent. The corresponding configurations will be denoted as nlN (a shell) or nljN (a subshell). A number of permitted states of a shell of equivalent electrons are restricted by the Pauli exclusion principle, which requires antisymmetry of the wave function with respect to permutation of the coordinates of the electrons. 

The wave function for the particular case of two equivalent electrons may be constructed, using vectorial coupling of the angular momenta and antisymmetrization procedure. For LS and jj coupling, it will look as 

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On the other hand, the use of the representations of orthogonal group R21+1 in theoretical atomic spectroscopy gives additional information on the symmetry properties of a shell of equivalent electrons, allowing one to establish new relationships between the matrix elements of tensorial operators, including the operators, corresponding to physical quantities. 

Here i indicates the coordinates, on which the operator acts. Symbol = is used to emphasize that the equality is valid in the sense of matrix elements for a shell of equivalent electrons. When using (5.41), the spin-angular part of the scalar product of the operators, acting on spatial coordinates... 

The existence of two coupling schemes for a shell of equivalent electrons is conditioned by the relative values of intra-atomic interactions. If the non-spherical part of electronic Coulomb interactions prevails over the spin-orbit, then LS coupling takes place, otherwise the jj coupling is valid. As we shall see later on, for the overwhelming majority of atoms and ions, including fairly highly ionized ones, LS coupling is valid in a shell of equivalent electrons, that is why we shall pay the main attention to it. 

A set of pairs of quantum numbers n,7, with the indicated number of electrons having these quantum numbers, is called an electronic configuration of the atom (ion). Thus, we have already discussed the cases of two non-equivalent electrons and a shell of equivalent electrons. If there is more than one electron with the same nf, then the configuration may look like this ... 

As was already mentioned, due to the Pauli exclusion principle, which states that no two electrons can have the same wave functions, a wave function of an atom must be antisymmetric upon interchange of any two electron coordinates. For a shell of equivalent electrons this requirement is satisfied with the help of the usual coefficients of fractional parentage. However, for non-equivalent electrons the antisymmetrization procedure is different. If we have N non-equivalent electrons, then a wave function that is antisymmetric upon interchange of any two electron coordinates can be formed by taking the following linear combination of products of one-electron functions [16] ... 

In Chapter 9 we discussed the classification of the terms and energy levels of a shell of equivalent electrons using the LS coupling scheme. Here we shall consider the case of two non-equivalent electrons. As we shall see later on, generalization of the results for two non-equivalent electrons to the case of two or more shells of equivalent electrons is straightforward. 

We remind the reader that jj coupling inside a shell of equivalent electrons requires the use of relativistic wave functions, whereas for LS... 

Unit tensors are especially important for group-theoretical methods of studying the lN configuration. We can express the infinitesimal operators of the groups [10, 24, 98], the parameters of irreducible representations of which are applied to achieve an additional classification of states of a shell of equivalent electrons, in terms of them. 

Group-theoretical methods of classification of the states of a shell of equivalent electrons. Casimir operators... 

In this chapter we have found the relationship between the various operators in the second-quantization representation and irreducible tensors of the orbital and spin spaces of a shell of equivalent electrons. In subsequent chapters we shall be looking at the techniques of finding the matrix elements of these operators. 

The traditional description of the wave function of a shell of equivalent electrons was presented in Chapter 9. Here we shall utilize the second-quantization method for this purpose. In fact, the one-electron wave function is... 

The operator of total quasispin angular momentum of the shell can be obtained by the vectorial coupling of quasispin momenta of all the pairing states. For a shell of equivalent electrons, instead of (15.35) we have... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

But this phase factor can be selected so that the signs of the CFP for almost filled shells are the same as in the quasispin method. It is worth recalling here that finite transformations generated by quasispin operators define the passage to quasiparticles. In much the same way, in the quasispin space of a shell of equivalent electrons the unitary transformations... 

In Chapter 14 we have already discussed the group-theoretical method of classification of the states of a shell of equivalent electrons. Remembering that second-quantization operators in isospin space have an additional degree of freedom, we can approach the classification of states in isospin basis in exactly the same way. 

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