Irreducible tensor operators definition

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

The choice of the phase and the normalization of the irreducible tensor operators is somewhat arbitrary [379]. Following [133], we will employ the following definition of matrix elements of tensor operators (other existing methods are discussed in Appendix D) ... 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

According to the foregoing discussion two-shell operators are expressed in terms of irreducible products of tensors W KkK lt, lj) and W Kklc li). Using a standard technique, we can represent the submatrix elements of these products in terms of relevant quantities defined for these tensors. We shall, therefore, confine ourselves to consideration of the submatrix element of tensor W KkK h,l2)- By its definition (17.58),... 

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