Racah tensor operators

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

Here y m are the spherical harmonic functions Q m = yj47r/(2k + 1) y, m is the Racah tensor operator = rk Ykm is the irreducible tensor operator Pk m (not to be confused with the Legendre polynomials) are unnormalised homogeneous polynomials of Cartesian coordinates proportional to the function rk Ykm + Yk m) Ok are referred to as equivalent operators which are constructed of only the angular momentum operators. 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

The desired coupled basis will be performed by the methods given by Racah and Wigner. Making use of the Wigner-Racah formalism and of the Wigner-Eckart theorem and observing some rules for the matrix elements of the products of tensor operators, we obtain for the matrix elements of the quadrupole interaction operator/Z ... 

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

The reduced matrix element on the right-hand side of (54) consists of singleelectron reduced matrix elements belonging to the specific operators, and the reduced matrix element of the corresponding Racah tensor. The latter, clearly, is independent of the nature of Uy and u. ... 

The decade ended with the publication of the two-volume work of Slater (1960) on atomic structure. A cautious treatment is given of tensor operators and fractional parentage. Expansions in determinantal product states are still resorted to, and there is no discussion of Racah s use of groups. A little later, the writer attempted to remedy that deficiency (Judd 1963). His prefatory assurance that the needs of the experimentalist were borne in mind was sourly commented on, in a private conversation, by Edien. The recollection that the book of Condon and Shortley (1935) gave similar problems to Russell (its dedicatee), provided some solace. 

These ideas can be readily extended to include the lanthanide configurations 4f or, for that matter, the general configuration 1 Racah credited Bacher and Goudsmit (1934) for realizing that two-electron excitations from / can be represented by two-electron operators within to second order in perturbation theory. To show how this can come about in detail, Racah introduced the single-electron tensor operators zffH ) that convert a state of to one of As such,... 

Although the use of any one of these two methods completely solves the problem of calculating the matrix elements of Hi for configurations, several short-cuts can be achieved corresponding to various special cases. These also are due to Racah (1943, 1949). He defined unit tensor-operators such that... 

C Racah s tensor operator of order / and component q [57Edm],... 

Racah s spherical tensor operator of order I and component q, 9 and are the polar coordinates of the intemuclear axis... 

These are the numbers that evidently show the role of the J-O Theory in the field with undefined limits due to the broad applications of its achievements. In order to understand why this theory is so important, its physical background must be presented. It is possible to conclude briefly that the J-O Theory is a simple application of the outstanding beauty of tri-positive lanthanide ions, and in particular their unusual electronic structure. Its features are defined in the language of Racah algebra applied for the concept of effective tensor operators. The simplicity and clarity of this approach, including the well-known Judd-Ofelt parametrization scheme of the/-spectra based on (10.1), when successfully applied to very complex systems makes one wonder how is it possible that this tool works so well in fact this query is its power. 

From a physical point of view this relativistic model is also based on the perturbation approach, and at the second order, similarly as in the case of the standard J-O Theory, the crystal field potential plays the role of a mechanism that forces the electric dipole/ t—>f transitions. The only difference is that now the transition amplitude is in effectively relativistic form, as determined by the double tensor operator, but still of one particle nature. Furthermore, the same partitioning of space as in non-relativistic approach is valid here. The same requirements about the parity of the excited configurations are expected to be satisfied. As a final step of derivation of the effective operators, the coupling of double inter-shell tensor operators has to be performed. This procedure is based on the same rules of Racah algebra as presented in the case of the standard J-O theory. However, the coupling of the inter-shell double tensor operators consists of two steps, for spin and orbital parts separately. Thus, the rules presented in equations (10.15) and (10.16) have to be applied twice for orbital and spin momenta couplings, resulting in two 3j— and two 6j— coefficients. 

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. 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

During the last two decades a number of new versions of the Racah algebra or its improvements have been suggested [27]. So, the exploitation of the community of transformation properties of irreducible tensors and wave functions allows one to adopt the notion of irreducible tensorial sets, to deduce new relationships between the quantities considered, to simplify further on the operators already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements, as an alternative to the standard Racah way. It is based on the utilization of tensorial products of the irreducible operators and wave functions, also considered as irreducible tensors. 

This relationship expresses the adjoint character of the annihilation operators. The effect of the time reversal operator on these tensors in the Fano-Racah phase convention is given by ... 

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