Relativistic approach and quasispin for one subshell

As we have already seen in Chapters 11 and 12, the realization of one or another coupling scheme in the many-electron atom is determined by the relation between the spin-orbit and non-spherical parts of electrostatic interactions. As the ionization degree of an atom increases, the coupling scheme changes gradually from LS to jj coupling. The latter, for highly ionized atoms, occurs even within the shell of equivalent electrons (see Chapter 31). 

With jj coupling, the spin-angular part of the one-electron wave function (2.15) is obtained by vectorial coupling of the orbital and spin-angular momenta of the electron. Then the total angular momenta of individual electrons are added up. In this approach a shell of equivalent electrons is split into two subshells with j = l 1/2. The shell structure of electronic configurations in jj coupling becomes more complex, but is compensated for by a reduction in the number of electrons in individual subshells. 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

Consequently, with second quantization, the approach using Hamiltonian (2.1)-(2.7) and relativistic wave functions (2.15) differs from the approach using Hamiltonian (1.16)—(1.22) and the non-relativistic wave 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2 

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