Irreducible tensor operators matrix elements

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. 

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 of the angular momenta... 

Of frequent interest is the need of evaluation of matrix elements for a tensor product of two irreducible tensor operators... 

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... 

Then, using the matrix elements for the complex irreducible tensor operators (see Ref. [11]) we arrive at the following expression for the matrix elements of the vibronic interaction ... 

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) ... 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

In order to determine whether orientation has taken place or not, it is convenient to expand the matrix elements /mm over irreducible tensor operators according to (5.17). As follows from Appendix D, reverse expansion (5.20) may be written conveniently in the following way ... 

The main reason for working with irreducible tensor operators stems from an important theorem, known as the Wigner-Eckart Theorem (WET)75,76 for matrix elements of tensor operators ... 

We work in the basis set 177,./, Q, I, F, Mp) where F= J+1 and i2 is the component of electronic angular momentum along the intemuclear axis. We shall ignore any possibility of i -degeneracy r refers to any other unspecified quantum numbers. Using the standard results for the matrix elements of the scalar product of two irreducible tensor operators, we obtain... 

In degenerate states the situation is essentially changed. In these cases in the decomposition of the product [P] into irreducible representations there are other representations in addition to the totally symmetric representation. As a result, there may be nonzero matrix elements for nonto-tally symmetric components of irreducible tensor operators of the polarizability and multipole moments. In particular, in the decomposition of [P] there may be representations contained in D with an / value less than lA, for which D a contains the totally symmetric representation. 

Throughout this paper semi-heavy type will be used for matrices and matrix elements, gothic type for irreducible tensor operators (6, 15) and GUI sans for the ligand field operator, for proj ection operators, and for rotation operators (11). 

The basic expressions for the matrix elements of the spin-orbit coupling operator have been derived by Griffith [48]. A double tensor operator is Xf[y where, for fixed y and varying M, X fy is an irreducible tensor operator with respect to spin variables. Similarly, for fixed S and varying y, Xf[y is an irreducible tensor operator with respect to space variables. A matrix element of a double tensor operator is now reduced as follows... 

Here T2(S S), T2(S V) and T2(V V) mean the second-rank irreducible tensor operators built up of the vector operators. Their reduced matrix elements are evaluated as follows... 

The triangular conditions for the 3y-symbols yield the restrictions M = M, M 1 and M 2, respectively, and closed formulae exist for all these 37-symbols [16]. The restriction to S = S 1 and S = S 2 originates in the triangular relationships in the corresponding 67-symbols involved in the reduced matrix elements of the second-rank irreducible tensor operators... 

Since is a sum of one-electron operators, matrix elements such as (21) and (23) may be evaluated directly after the terms have been expanded into microstates of the form of Eq. (19) and (20). Although the process is straightforward it is generally quite tedious. Much more elegant and powerful methods have been developed by Racah. These methods make full use of the Wigner-Eckart theorem to evaluate matrix elements of operators written in the form of irreducible tensors. Descriptions are to be found in Slater [53) and Judd [30). We shall apply these methods to evaluate (21) and (23). 

Here, as was already pointed out, it is noticed from the values for matrix elements of irreducible tensor operators [45-48] that the matrix elements of have notably large values only in the hypersensitive transitions among possible f-f transitions, whereas those of and are less directly related to the hypersensitive transitions [10]. Therefore, it can be interpreted that the characteristics of hypersensitivity are involved in the term of A = 2 in Eq. (7), namely xx- In fact, the magnitude relation X2 X4, x holds in LnX3 molecules in vapor phase [31, 32], and these oscillator strengths can be evaluated only by the term of A = 2 in Eq. (7). Comparison of the T2(dc) term for LnX3,... 

The exploitation of the community of the transformation properties of irreducible tensors and wave functions gives us the opportunity to deduce new relationships between the quantities considered, to further simplify the operators, already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements. Indeed, it is possible to show that the action of angular momentum operator Lf on the wave function, considered as irreducible tensor tp, may be represented in the form [86] ... 

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 ... 

The elements of the theory of angular momentum and irreducible tensors presented in this chapter make a minimal set of formulas necessary when calculating the matrix elements of the operators of physical quantities for many-electron atoms and ions. They are equally suitable for both non-relativistic and relativistic approximations. More details on this issue may be found in the monographs [3, 4, 9, 11, 12, 14, 17]. 

To conclude this chapter, let us present the main formulas for sums of unit tensors, necessary for evaluation of matrix elements df the energy operator. They will be necessary in Part 5. The matrix element of any irreducible tensorial operator may be written as follows ... 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

In this chapter we have found the relationship between the various operators in the second-quantization representation and irreducible tensors of the orbital and spin spaces of a shell of equivalent electrons. In subsequent chapters we shall be looking at the techniques of finding the matrix elements of these operators. 

In summary, the techniques described in this chapter allow us to derive expansions of the operators that correspond to physical quantities, in terms of irreducible tensors in the spaces of orbital, spin and quasispin momenta, and also to separate terms that can be expressed by operators whose eigenvalues have simple analytical forms. Since the operators of physical quantities also contain terms for which this separation is impossible, the following chapter will be devoted to the general technique of finding the matrix elements of quantities under consideration. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

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. 

The above recurrence relations allow the numerical values of CFP to be readily found and the availability of the above algebraic formulas for them makes it possible to establish similar algebraic expressions for other quantities in atomic theory, including the CFP with two detached electrons, the submatrix elements of irreducible tensors, and also the matrix elements of the operators of physical quantities [110]. 

As usual, the operators describing hyperfine interactions are to be expressed in terms of irreducible tensors. Then we are in a position to find formulas for their matrix elements. The corresponding operator, caused... 

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