Formula Racah

For algebraic evaluation of the 3j- and 6j-symbols the Racah formulae are useful (Table 58). Various vector coupling coefficients met so far are reviewed in Table 59. In point groups analogous coefficients occur (Table 60). 

For the direct evaluation of the 3/-symbol the Racah formula [8, 9] (equivalent to the Wigner formula [5, 6]) is helpful... 

For a proper addition of angular momenta certain vector coupling coefficients are required the Clebsch-Gordan coefficients, or alternatively the 3/-symbols, the 6/ -symbols and 9/-symbols, for the coupling of two, three and four angular momenta, respectively. The 3/-symbols can be calculated through the Racah formula the 6/ -symbols can be expressed in terms of the 3/-symbols and the 9/-symbols are evaluable with the help of either 3/-symbols or 6/ -symbols. For some special cases closed-form formulae also exist. 

There exist extensive tables of 3 - j symbols (Rotenberg et al. 1959) and computer subroutines for their calculation (Schulten and Gordon, 1976 Zare, 1988). The calculation is done using Racah s formula... 

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. 

However, this is not the case for the dN shell. For d3 there are two of the same 2D terms. This problem with the dN shell and, partially, the fN shell was solved by Racah in his paper [23] introducing the seniority quantum number. In accordance with formula (9.7) we can build the antisymmetric wave function of shell lN with the help of the CFP with one detached electron. However, we could also use the CFP with two detached electrons... 

For some special cases such formulas have long been known. Racah, for example, derived relationship (16.39) for the CFP detaching 0 electrons... 

As before [A] indicates orbital degeneracy Eq. (21) also involves W or Racah coefficients for spin and orbital coupling. Values for the spin coefficients may be calculated from formulae quoted by Brink and Satchler (17). Frequently, however, it may not be necessary to use this formula, since it may be clear from the selection rules that a given function (S5A5) of the ionised shell can only produce one allowed resultant state (S2 A 2), and in this case the intensity is entirely determined by the fractional parentage coefficient for the ionised shell ... 

When it is impossible to use real functions, the complex description is easily introduced since the reduced matrices and the 6-1 and 9-1 symbols are invariant in the two representations. It is further important that the reduced ligand-field parameters are the same, even though they have been defined on the basis of the real orbitals. In this connection it may be mentioned that Racah s formulae for the 3-j and 6-j symbols (4, 75), which are convenient for computer work, make it possible to generate ligand-field matrices by a rather simple algorithm. 

The values of the 3-1 symbols may be obtained from table 1 in terms of the 3-y S5unbols from published tables (75). For computer work the 3-j s5mibols may also be calculated by Racah s closed formula, Eq. (10.14) of (4), with the phase difference between F( ) and 3-y taken into account. 

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... 

Consider, for example, the D terms of f. Our basis consists of (Pf)D>, (Ff)D> and (Hf)D>. The roots of the 3 x 3 matrix of the Coulomb interaction yield the energies of and the two terms. We have already given ( D) in eq. (25). Its extraction leaves us with a quadratic equation for ( D), from which explicit expressions in terms of the Slater integrals can be obtained. Formulas for all the terms of f were given by Racah [1942b, eqs. (98)]. 

In spite of these difficulties, it is interesting to see how Racah s most vulnerable formula, eq. (82), which determines the lowest levels of the doubly ionized lanthanides, has stood the test of time. For N = 1 we get, from that equation,... 

Racah began his discussion of the second spectra by regretting that the absence of relevant data precluded the construction of formulas of the type represented by eqs. (80)-(82). Instead, he suggested that it might be useful (and much more convenient) to compare differences in energy between the lowest levels of opposite parity (the so-called system differences) in the second and third spectra. He noted, for example, that we can write... 

Electrostatic parameters and matrix elements for the configuration Racah defined the electrostatic parameters A, B, C for the configuration d through the following formulae ... 

The following relation holds between these eigenfunctions (formula 65 of Racah 1943) ... 

The relations between the parameters a, 0 and y defined by Rajnak and Wybourne (1963) (see formula (1.143 above), and those defined by Racah (1964), Trees (1964) and Z.B. Goldschmidt (1968a, 1968b) and also used in this chapter (formula 1.144 above) are ... 

The following general formula for the calculation of 3/ symbols is due to Racah ... 

With the abbreviation (Eqn [40]) Racah s formula for the 6/ coefficients reads ... 

The angular dependence of the coefficients C R, < a. < b) can be expressed in a closed form. The relevant formulae are obtained by asymptotic expansion of the polarization series truncated at some finite order. In practice such an asymptotic expansion is best performed by evaluating the polarization energies (as given by equations 9, 18, and 21) using the multipole expansion of the electrostatic potential l/ ri — r2. The latter expansion can be written in terms of either the Cartesian or the spherical tensors. The spherical formulation appears to be more popular because it leads much more easily to closed formulae and only this formulation will be considered in this article. Denoting by (r) the regular solid harmonic r Cim(0,), where Cim 0,) is the spherical harmonic in the Racah normalization and with the Condon and Shortley phase, one can write ... 

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