Two shells of equivalent electrons

In many physical problems we come across excited configurations consisting of several open shells or at least one electron above the open shell. Therefore, we have to be able to transform wave functions and matrix elements from one coupling scheme to the other for such complex configurations. If K denotes the configuration, and m, fifi stand for the quantum numbers of two different coupling schemes, then for the corresponding wave functions formulas of the kind (12.1), (12.2) hold, whereas the matrix element of some scalar operator D transforms as  

Formula (12.19) for the particular case of the energy operator and of the transformation from the LS coupling to another one, in the general case of two shells of equivalent electrons, inside which there is LS coupling and their total momenta are LiSi and L2S2, respectively, will be of the form (for simplicity we skip the designations of the shells themselves and of the additional quantum numbers, distinguishing the terms of the shell with the same L,S,)  

In a similar way the remaining transformation matrices from JK to LK, and from JJ to LK and JK coupling schemes may be found 

Formulas (12.21)—(12.26) are fairly simple, the corresponding transformation matrices are expressed only in terms of 6j- and -coefficients, which can be easily calculated with the help of algebraic formulas (see [11]), tables of their numerical values or using standard computer codes. These formulas can be easily generalized to cover the case of a larger number of open shells. 

Finally in this section let us discuss the case when the transformation 

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In the group-theoretical treatment of mixed configurations, we may, in analogy with the case of one shell, introduce the basis tensors for two shells of equivalent electrons [101]... 

Let us take a closer look at the concept of isospin referring to two shells of equivalent electrons with the same orbital quantum numbers. Our reasoning will partially follow the work [123]. 

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. 

Electrostatic interaction energy operator Q in (1.15) or in irreducible form (19.6) may formally be written for the case of two shells of equivalent electrons as... 

In these formulas the energy difference AE is measured in atomic units, transition probabilities are obtained in seconds and the submatrix element in the cases of one or two shells of equivalent electrons for LS coupling may be taken from (27.3) or (27.4). When calculating using intermediate coupling one has to bear in mind that the appropriate wave functions are of the form (11.10). 

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 ... 

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