A generalization of Clebsch-Gordan Coefficients and Wigner
3*j*-Symbol and Wigner 6*j*-Symbol which arises in the coupling of four angular momenta and can
be written in terms of the Wigner 3*j*-Symbol and Wigner 6*j*-Symbol.
Let tensor operators and act, respectively, on subsystems 1 and 2. Then the reduced matrix element
of the product
of these two irreducible operators in the coupled representation is given
in terms of the reduced matrix elements of the individual operators in the uncoupled representation by

(1) |

Shore and Menzel (1968) give the explicit formulas

(2) | |

(3) | |

(4) |

**References**

Gordy, W. and Cook, R. L. *Microwave Molecular Spectra, 3rd ed.* New York: Wiley, pp. 807-809, 1984.

Messiah, A. ```' Symbols.'' Appendix C.III in *Quantum Mechanics, Vol. 2.* Amsterdam, Netherlands: North-Holland, pp. 567-569
and 1066-1068, 1962.

Shore, B. W. and Menzel, D. H. *Principles of Atomic Spectra.* New York: Wiley, pp. 279-284, 1968.

© 1996-9

1999-05-26