Tensorial operators

Energy, which is the observable quantity associated with the Hamiltonian operator, is a pure number or, more precisely, a scalar quantity. Therefore, the Hamiltonian must be a scalar operator. In this section, the prescriptions for constructing a scalar operator from combinations of more complicated operators, such as vector angular momenta, and for evaluating matrix elements of these composite scalar operators are reviewed briefly. 

Some operators, such as the interelectronic electrostatic interaction e2/ nj, are obviously scalar quantities. Others are scalar products of two tensorial operators. A tensor of rank zero is a scalar. A tensor of rank one is a vector. There are several ways of combining two vector operators the scalar product 

The general expression for the scalar product of two tensorial operators of rank k is 

The power of the Wigner-Eckart theorem (Messiah, 1960, p. 489 Edmonds, 1974, p. 75) is that it relates one nonzero matrix element to another, thereby vastly reducing the number of integrals that must either be explicitly evaluated or treated as a variable parameter in a least-squares fit to spectral data. For example, consider S k a tensor operator of rank k that acts exclusively on spin variables. The Wigner-Eckart theorem requires 

If a matrix element is evaluated for a particular value of E and E by another method (see Section 3.4.1), for example for the maximum value of E = 5 and 

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While finding the numerical values of any physical quantity one has to express the operator under consideration in terms of irreducible tensors. In the case of Racah algebra this means that we have to express any physical operator in terms of tensors which transform themselves like spherical functions Y. On the other hand, the wave functions (to be more exact, their spin-angular parts) may be considered as irreducible tensorial operators, as well. Having this in mind, we can apply to them all operations we carry out with tensors. As was already mentioned in the Introduction (formula (4)), spherical functions (harmonics) are defined in the standard phase system. 

Thus, utilizing the concept of irreducible tensorial sets, one is in a position to develop a new method of calculating matrix (submatrix) elements, alternative to the standard way described in many papers [9-11, 14, 18, 21-23]. Indeed, the submatrix element of the irreducible tensorial operator can be expressed in terms of a zero-rank double tensorial (scalar) product of the corresponding operators (for simplicity we omit additional quantum numbers a, a1) ... 

Expressions of this kind can be found also for complex tensorial products of tensorial operators. Thus, we see that, indeed, we arrive at a new... 

A fundamental role is played in theoretical atomic spectroscopy by the Wigner-Eckart theorem, the utilization of which allows one to find the dependence of any matrix element of an arbitrary irreducible tensorial operator on projection parameters,... 

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]... 

On the other hand, the use of the representations of orthogonal group R21+1 in theoretical atomic spectroscopy gives additional information on the symmetry properties of a shell of equivalent electrons, allowing one to establish new relationships between the matrix elements of tensorial operators, including the operators, corresponding to physical quantities. 

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 ... 

As was mentioned in the previous paragraph, the Wigner-Eckart theorem (5.15) is fairly general, it is equally applicable for both approaches considered, for tensorial operators, acting in various spaces (see, for example, Chapters 15,17 and 18, concerning quasispin and isospin in the theory of an atom). 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

Apart from phase system (13.42) some of the tensorial operators (e.g. unit tensors and their sums (5.27), (5.28)) are defined in a pseudostandard phase system (see Introduction, Eq. (5)), for which... 

Ranks of such tensorial operators are not confined in parentheses, for the notation itself is used to indicate the phase system. Equation (13.43) does not hold in general, for the same reasons as (13.42). 

For another unit tensorial operator Vkl (5.28) in the second-quantization representation we have... 

If, according to [14], we introduce unit tensorial operator w kK, the submatrix element of which is... 

The number of tensorial operators with different possible values of ranks k,K and appropriate values of the projections is predetermined by the number of possible projections of the second-quantization operators that enter into the tensorial product. This number is equal to (4/ + 2)2. 

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)... 

Using (16.31)-(16.34), we can establish from (16.56) the following relations between the submatrix elements of irreducible tensorial operators Uk and Vkl [92] ... 

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. 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

Let us proceed to the two-electron operators. The corresponding procedure is essentially based on the exploitation of the tensors, composed of unit tensorial operators (see Chapter 5). Let us consider first the electrostatic interaction (19.6). Using (5.42) we find (notice that further on in this... 

For the fN shell we have to take into consideration terms containing expressions of the kind (5.36) and (5.37). As was shown in [127, 129], parameters a and / account for the superposition of all configurations which differ from ground lN by two electrons. If the admixed configurations differ from the principal one by the excitation of one electron, then we have to introduce one extra parameter T, the coefficient of which will be described by the matrix element of the tensorial operator of the type [Uk x Uk x Uk ]°, for which, unfortunately, there is no known simple algebraic expression. This correction is important only for N > 3. 

Here n ensures the normalization condition of the wave function. The use of the concept of irreducible tensorial operators in this approach [220] opens up the possibility of exploiting such a method for complex electronic configurations however, so far it has been applied only to light atoms and ions. 

The quantitative characteristic of the alignment created is given, as already stated, by multipole moments of even rank. A more rigorous treatment of the expansion of the quantum mechanical density matrix over irreducible tensorial operators will be performed later, in Chapter 5 and in Appendix D. As an example we will write the zero, second and fourth rank polarization moments and [Pg.62]

Note that this is similar to the Hamiltonian describing the dipolar interaction of two nuclear magnetic moments, presented in equation (8.9), but is opposite in sign. Using the results derived in the first part of appendix 8.1, we see that the dipolar Hamiltonian (8.227) may be written as a cartesian tensorial operator ... 

Table 6. The syrntnetrical matrices of the unit tensorial operators 2) and 23 with respect to the d basis (only elements on and above the diagonal are given). The unit tensorial operator 23, whose matrix elements are the 3 l symbols, is defined by its reduced matrix [Ref. (15) p. 243] so that in this case...

It is often of interest to perform an expansion of the type of Eq. (65). Hobson has developed a projection operator which extracts (projects) the harmonic hi out of the homogenous polynomial /j. This operator S is an irreducible tensorial operator of the degree zero for the three-dimensional rotation group and has the form [Ref. [18) p. 127)]... 

We have seen how the operator S is able to project hi out of fi. There is a general alternative manner of obtaining hi which belongs to the irreducible tensorial set of the degree I of the three-dimensional rotation group. This is another irreducible tensorial operator of the degree zero, the operator for the resolution of the identity within I space [Ref. (77) p. 383] which may be written... 

Eq. (84) is a general form of the set U of unit tensorial operators, whose reduced matrices are... 

The 3-1 symbols represented as matrix elements of the general unit tensorial operator 33 For l s, p, and d the entries of... 

By use of this relation one can express any tensorial operator desired in terms of the operators [( i x As an example, we use appendix 2... 

The ligand-field operator is a spatial operator and is a special case of a one-electron operator. Such operators are conveniently treated in the formalism developed by Racah (14). He has shown that the matrix elements of a one-electron irreducible tensorial operator... 

The matrix element is given also by the Wigner-Eckart theorem, Eqs. (27) or (28b), and therefore one concentrates upon the evaluation of the reduced matrix. This is conveniently calculated in terms of the unit tensorial operator (13) of the degree k, which in view of Eq. (33) is defined through the reduced matrix... 

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