Coupling recoupling coefficients

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)],... 

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 hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

In the simplest setup, the two strong field components may be set identical to Ci = Cs = C. The relatively large CIX or CSX term averages isotropic and anisotropic chemical shift effects as well as the heteronuclear dipolar coupling interaction between 15N or 13C and H. The difference of - or the sum of - the B coefficients selects the form of the recoupled heteronuclear dipole-dipole coupling interaction, as expressed in terms of the effective Hamiltonian in the interaction frame of the rf irradiation... 

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