Quasispin for complex electronic configurations

2 Quasispin for complex electronic configurations tensors Tk and at II = 1, to the CFP with two detached electrons 

We shall now turn to the mathematical techniques that rely on the tensorial properties of the quasispin space of two subshells of equivalent electrons. They can be readily generalized to more complex configurations. 

The wave function of two subshells is generally constructed by vectorial coupling of the total angular momenta of each subshell 

Clearly, for operators corresponding to physical quantities we can derive expansions that enable us to make use of their tensorial properties in quasispin space. Specifically, operator (23.27) will be 

P( 1 - 2) is the operator of permutation of subscripts of the first and second subshells. 

There is another way of looking at the tensorial properties of operators and wave functions in the quasispin space of the entire two-shell configuration. If now we introduce the basis tensors for two subshells of equivalent electrons 

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Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

Coupling vectorially the quasispin operators of each shell Q = Qi +Q2 and assuming Mq = —J[ 2(/i + I2 + 1) — N, where N — iVj + N2, we generalize the quasispin concept to cover the case of complex electronic configurations. Then we can define the total quasispin quantum number for any configuration. For two shells of equivalent electrons we have, in such a case, the following wave function ... 

It has been shown earlier (see Chapters 15 and 16) that the technique relying on the tensorial properties of operators and wave functions in quasispin, orbital and spin spaces is an alternative but more convenient one than the method of higher-rank groups. It is more convenient not only for classification of states, but also for theoretical studies of interactions in equivalent electron configurations. The results of this chapter show that the above is true of more complex configurations as well. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

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