Operator Racah

Reduced matrix element of Racah operator (rationalized spherical harmonics)... 

It is assumed that ligands bear the negative charges qx = - ezR (zr is a charge number). The inverse of the electron-ligand separation can be expanded into the basis of the Racah operator... 

Although analytic expressions for the potential constants exist, they are rarely calculated directly. The covalency degree, uncertainties of effective ligand charges and other conceptual drawbacks make such an approach problematic. The potential constants are more often taken as free parameters of the theory which enter the final formulae of electron spectroscopy, electron spin resonance and magnetochemistry. The potential constants in different representations of the crystal field potential obey simple proportionality relationships which can be found in special monographs [10-13]. For example, the potential expressed through the Racah operators... 

The remainder of may be cast into a similar form by an expansion of the Stevens operators 0 (i) belonging to the ith site into a well-ordered series of the operators at and a,. Lindg rd and Danielsen (1974) derived this expansion by matching corresponding matrix elements of Racah operators (linear combinations of Stevens operators) with the Bose operators. For example, O becomes... 

This term describes a shift in energy by Acim rn, for an orbital with quantum numbers I — 2, mi and that is proportional to the average orbital angular momentum (/z) for the TOj-spin subsystem and the so-called Racah parameters Bm, that in turn can be represented by the Coulomb integrals and The operator that corresponds to this energy shift is given by... 

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. 

Invariant operators were introduced by Casimir (1931) for SO(3). Racah (1950) generalized them to all orders. 

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. 

The seniority number (seniority for short), v, is a quantum number related to eigenvalues of the Racah seniority operator . The seniority matches the electron configuration ln in which the particular term first appears. We will see later that some matrix elements of the operators included in the Hamiltonian are diagonal in v so that their off-diagonal counterparts vanish. For some operators, however, there are crossing terms in v. 

This basis may be converted to the Fano-Racah convention [11] by multiplying all kets in the right-hand side of Eq. 10 by the same phase factor i. Adding spin results in a shell consisting of six possible states. These states may be created by operators... 

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 ... 

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 ... 

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]. 

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... 

A lemma by Racah (5), a consequence of Schur s lemma, provides a relationship between the coupling coefficients of a group and those of a sub-group. This relationship is here illustrated and standardized by introducing differential operator equivalents for the real standard bases of the three-dimensional rotation group Rs. 

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... 

Here, the elements of the first operator are (in Slater-Condon parameters) taken from Ref. [11) p. 53 and translated into Racah parameters... 

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