Tensors Racah

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. 

Defining the unit double tensor (Racah, 1942b)... 

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 basic texts on tensor products are de Shalit and Talmi (1963) and Fano and Racah (1959). 

The factor C k of B(R) is often called the /th component of the Racah spherical tensor of rank k the three tensor components of rank 1 may be considered unit basis vectors spanning the (spherical) space. 

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. 

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 ... 

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]. 

The constant A = 3.96 cm-1 has been obtained from the free-molecule zero-field splitting (Mizushima, 1975) and C is a Racah spherical harmonic with 1 = 2. The tensor that describes the interaction between the magnetic dipole moments ge Sp, where ge equals 2.0023 and fxB is the Bohr magneton, can be written immediately as... 

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... 

In spite of this, there does exist a general theoretical method for dealing with just this situation of the coupling of three (or more) angular momenta. It is the irreducible tensor method of Racah (29 and ITigwer (JO). 

The application of irreducible tensorial sets to quantum mechanical problems provides a powerful tool for the calculation of observables. This was recognized in the early forties by G. Racah (72—74) who developed the tensor method and used it for an extension of the theory of atomic spectra. 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

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. 

As stated earlier on several occasions, the algebraic method should not be viewed on a mere mimicking of other well-established approaches to solving molecular spectroscopy problems. However, one could have just such an impression if the problems are limited to very simple cases that can be addressed equally well by traditional methods and do not carry the embarrassing burden of Lie algebras or Racah s tensor calculus. Nonetheless, every introductory article must start with simple examples and only then proceed to more complex ones. Sections III.C.2 and IV.B reveal the algebraic approach as capable of providing reliable and alternative solutions to nontrivial questions. Here alternative means in a faster way and with fewer arbitrary parameters. In this section we basically... 

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). 

We turn now to a first consideration of expressions for the orbital and spin matrix elements which arise when there is no restriction placed on the magnitude of the scattering vector. The matrix elements can be expressed in terms of Racah tensors, as mentioned before. 

Our definition of spherical harmonics Yq k) and Clebsch-Gordan coefficients are in accord with the standard references. The quantity A(K, K ) is the product of radial integrals, and various factors which arise from the vector coupling coefficients of n equivalent electrons. The latter component of A(K, K ) is essentially embodied in the Racah tensor. Let... 

If a reduced matrix element between many-electron states is to be expressed in terms of those of single-electron states, we have to introduce the concept of fractional parentage coefficients pioneered by Racah. Collecting the general parts of a reduced matrix element between states with n equivalent electrons, we are led to the definition of a Racah tensor and, in this case, the fractional parentage coefficients will appear explicitly only in this definition ... 

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