A shell of equivalent electrons lN

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

In theoretical atomic spectroscopy usually a phase system for the wave functions is chosen which ensures real values of the CFP. In this case the transformation matrices will acquire only real values, too. Let us notice that the transformation matrix in (12.12), according to (12.4), is reciprocal to that in (12.11). Due to the orthonormality of the sets of wave functions, these matrices obey the orthonormality conditions  

Let us present the formula for these transformation matrices, convenient for practical utilization (the details of its deduction may be found in [18, 94]). It has the form of the following recurrence relation  

In (12.15) we have the two usual transformation matrices (see Chapter 6), describing the change of the coupling of three momenta, and two more for four momenta. Expressing them in terms of 6j- and 97-coefficients according to formulas (6.42)-(6.44), we get the final form of the transfer- 

using this equation, we are able in a recurrent way, starting with the two-electron configuration, to find the numerical values of this transformation matrix for all shells of equivalent electrons we are interested in. The following two-electron transformation matrices serve as the initial ones  

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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. 

Another possibility to account for correlation effects in a shell of equivalent electrons is to use the so-called extended method of calculation. In a conventional single-configuration approach, all electrons of shell lN are described by one and the same radial orbital. In an extended method each electron in shell lN has different radial functions, angular parts remaining... 

The efficient way of constructing the wave function of the states of equivalent electrons permitted by the Pauli exclusion principle is by utilization of the methods of the coefficients of fractional parentage (CFP). The antisymmetric wave function xp(lNolLSMlMs) of a shell nlN is constructed in a recurrent way starting with the antisymmetric wave function of N— 1 electrons xp(lN lociLiSiMLlMsl). Let us construct the following wave function of coupled momenta ... 

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