Coefficient recoupling

One can construct wave functions by first coupling 1 and 2 and subsequently 3 or in any other order (Figure 5.1). The angular momentum wave functions corresponding to the two coupling schemes are related by a recoupling coefficient [Eq. (B.24)],... 

The recoupling coefficients have been calculated by Racah and are usually written in the form... 

Recoupling coefficients are important in computing matrix elements of operators. Consider, for example, the C operators defined, for triatomic molecules, in Eq. (4.68). For three bonds (tetratomic molecules) one has... 

When more than two representations of G z> G are coupled, there is more than one way to do the coupling. All of them are, however, equivalent, and one can go from one to the other by means of recoupling coefficients. In the case of three algebras one has genetically... 

The recoupling coefficients of SO(4) can also be written in terms of recoupling coefficients of SO(3). The explicit form (for recoupling of symmetric representations, which are the only ones needed in molecular physics) is... 

This recoupling coefficient is used in Chapter 5, where it is rewritten in a different form by making use of the W coefficients of Eq. (B.33). 

It can also be connected with Racah polynomials [25], although it cannot be reduced to the ordinary Racah s or 6-j coefficient which performs angular momentum recoupling. Indeed, like a Racah s recoupling coefficient it is orthogonal with respect to summation on two angular momentum quantum numbers (/ and X), but contains a... 

The recoupling coefficients just introduced are independent of the projections of angular momenta M (they appear as a scalar product and we know that the scalar product does not depend upon the coordinate system). The recoupling coefficients can be arranged into 6y-symbols (having a number of symmetry properties) through... 

These two states are interrelated through the recoupling coefficients... 

For a homonuclear system, SA = SB = Sc = S, (which is the most frequent case of mixed-valence systems), the 9/-symbol can be expressed through 6/-symbols and the recoupling coefficients become [9]... 

A detailed study of 0(4) transformation properties leads to an exact formula which is not explicitly shown. However, the important points are that (1) in the exact formula, the recoupling coefficient is written as... 

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