Racah’s formula

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... 

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. 

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

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. 

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,... 

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