Reduced matrix elements tensor operators

Equations of this kind can also be derived for the special cases of reduced matrix elements of operators composed of irreducible tensors. Then, using the relation between CFP and the submatrix element of irreducible tensorial operators established in [105], we can obtain several algebraic expressions for two-electron CFP [92]. Unfortunately such algebraic expressions for CFP do not embrace all the required values even for the pN shell, which imposes constraints on their practical uses by preventing analytic summation of the matrix elements of operators of physical quantities. It has turned out, however, that there exist more general and effective methods to establish algebraic expressions for CFP, which do not feature the above-mentioned disadvantages. 

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

The reduced matrix element of the one-electron tensor operator for n-electrons collapses to... 

As for any one-electron operator, the (orbital) unit tensor operator has the following reduced matrix elements ... 

Thus any one-electron operator has reduced matrix elements equal to those of the unit tensor operator times its one-electron reduced matrix element (Z 1 1 l). Other results are as follows ... 

The reduced matrix elements of a double-tensor operator (formed as a scalar product of tensor operators of rank k and 1) become... 

The reduced matrix elements of a many-electron operator can be expressed through the unit tensor operator... 

Table 49 terms a Non-zero reduced matrix elements of the unit tensor operators between dn- ... 

The Wigner-Eckart theorem states that the matrix element of a tensor operator can be expressed through a more fundamental quantity - the reduced matrix element (which is free of projections of angular momenta) and a coupling coefficient... 

Having the reduced matrix element determined one can easily evaluate all the matrix elements of a tensor operator. 

The (orbital) unit tensor operator of rank k for one electron is introduced as having the reduced matrix element equal to unity, i.e. 

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... 

In this section, we follow the symmetry-adapted approach put forward by Acevedo et al. [10], and introduce the vibronic crystal coupling constants Av y(i, t), the tensor operators 0 (Txr i, t) and the general symmetry-adapted coefficients to give a master formula to evaluate the relevant reduced matrix elements as given below ... 

The disadvantages of the cogredient form of definition are connected with the presence of the factor (—1) in the normalizing coefficient 2Nk (D.32), which is a result of defining the reduced matrix element of the tensor operator (D.ll) into which this factor is introduced, in our opinion, without particular necessity. 

So far we know the selection rules for spin-orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for Cl wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step Cl wave functions are usually determined only for a single spin component, mostly Ms = S. The Ms quantum numbers determine the component of the spin tensor operator for which the spin matrix element (S selection rules dictated by the spatial part of the ME. 

The operators Op, and Op2 are used in the reduced matrix elements where, however, their tensor character without components fi has to be understood. Op, is a tensor of rank 1, Op2 a tensor of rank 0. 

For a tensor operator W4 2(/fi. ki ) which acts only on, say, part 1 of a coupled scheme, the reduced matrix element is given by... 

The matrix elements of the tensor operator Uq which are diagonal in the spin S can be rewritten using the Wigner-Eckart theorem with the 3-j symbols and the reduced matrix elements... 

The intensities of the spectral lines and the depolarization coefficients are functions of the reduced matrix elements of the polarizability tensor calculated by vibronic functions. In order to estimate the possibility of observation of the pure rotational Raman spectra under consideration, one has to consider in more detail the polarizability operator. Its components belonging to the line y of representation f can be presented in the form of a power series with respect to the displacement qriri active in the Jahn-Teller effect (the other components can be neglected as not active in the pure rotational Raman spectrum under consideration) ... 

Judd (29), in his classic paper of 1962, used the odd parity terms of the ligand field to accomplish this admixture. After applying second order perturbation theory and several simplifying assumptions, he showed that the electric dipole line strength between J-manifolds may be expressed as the sum of three terms, each being the product of an intensity parameter and a reduced matrix element of the tensor operator U of rank X. The electric dipole line strength, Se(j, can be written in the form... 

As a consequence of the Wigner-Eckart theorem the replacement theorem holds true a matrix element of any irreducible tensor operator can be expressed with the help of the matrix elements formed from the angular momenta and a ratio of the reduced matrix elements, hence... 

Case 4. When the first-rank tensor is an angular momentum operator (Tk = l, k = 0, Tt = 7(72), / = 1) then its reduced matrix elements become... 

The reduced matrix element of a vector operator (tensor operator of the first rank) is expressed in terms of the 67-symbol as follows... 

In these formulae the symmetry adaptation coefficients, 3/- and 6y-symbols occur along with the set of crystal field parameters Dq, Ds, Dt, Da and Dt. In addition, there are the reduced matrix elements of the n-electron unit tensor operators, evaluable with the help of the coefficients of fractional parentage (dn lvSL dnvSL) as follows... 

Here the reduced matrix element of the double tensor operator vSL V( 11 > v S L ) occurs it becomes expressed as follows (k = 1)... 

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