Racah coefficient

Racah coefficients, multidegenerate nonlinear coupling, higher order coupling, 243... 

The symbol W is called a Racah coefficient. Quite often, instead of the Racah coefficient W, one uses a Wigner 6 - j symbol... 

If a molecule contains two or more open-shells, it is necessary to consider the coupling that already exists between the different open-shells in the molecule. The probability of ionization is usually expressed in terms of Racah coefficients and fractional parentage coefficients. 

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 ... 

L. C. Biedenharn, Tables of the Racah Coefficients, Oak Ridge National Labatory, Physics Division, ORNL-1098, 1952. 

The energies of the various terms are expressed either by Racah coefficients A, B and C, or by the Slater-Condon parameters Fo, F2 and F4 to which the former set are related by... 

In this example, there are nine uncoupled states and, of course, nine coupled states. The coefficients that have been worked out are linear expansion coefficients, and so they can be arranged in a 9 x 9 matrix. This is the transformation (matrix) from the uncoupled basis to the coupled basis. This matrix and its inverse contain the coefficients for making up one kind of basis function from the other set. These coefficients are used in many problems in quantum physics, and they are often called vector coupling coefficients. Since they can be worked out once and for all for specific choices of /j and there are many tabulations of the values, and many alternate procedures have been devised for finding them. They are sometimes scaled by different factors for convenience in different circumstances. Names given to vector coupling coefficients under various prescriptions include Clebsch-Gordon coefficients, Racah coefficients, 3 -j symbols, and 6- j symbols. 

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