Superposition of configurations in quasispin space

The operators of orbital, spin and quasispin angular momenta of the two-shell configuration are expressed in terms of sums of one-shell triple tensors (15.52)  

The wave function of the two-shell configuration (17.42) corresponds to the representation of uncoupled quasispin momenta of individual shells. The eigenfunction of the square of the operator of total quasispin and its z-projection can be written as follows in the scheme of the vectorial coupling of momenta in quasispin space  

It will represent a two-shell multi-configuration wave function in which the coefficients of expansion in configurations are the conventional Clebsch-Gordan coefficients. 

The concept of the vectorial coupling of quasispin momenta was first applied to the nucleus to study the short-range pairing nucleonic interaction [117]. For interactions of that type the quasispin of the system is a sufficiently good quantum number. In atoms there is no such interaction - the electrons are acted upon by electrostatic repulsion forces, for which the quasispin quantum number is not conserved. Therefore, in general, the Hamiltonian matrix defined in the basis of wave functions (17.56) is essentially non-diagonal. 

In certain special cases the approximate symmetries in atoms are sufficiently well explained using the quasispin formalism. In particular, the quasispin technique can be utilized to describe fairly accurately configuration mixing for doubly excited states of the two-electron atom. In the quasispin basis the energy matrix of the electrostatic interaction operator of such configurations is nearly diagonal, and hence the quantum number of total quasispin Q is approximately good . 

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Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

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