Spectroscopy spherical tensor operators

The use of a spherical basis for the representation of tensor components facilitates the application of (tensor) operator techniques to the calculation of matrix elements in atomic spectroscopy. 

Unit tensors play, together with spherical functions, a very important role in theoretical atomic spectroscopy, particularly when dealing with the many-electron aspect of this problem. Unit tensorial operator uk is defined via its one-electron submatrix element [22]... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

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