Recoupling transformation

A similar procedure can be applied to the wave functions (5.4), and one can write recoupling transformations... 

The definition of the invariant quantities, the 6-1 and the 9-1 S5nnbols may be inferred from a consideration of the orthogonal recoupling transformation between the different coupling schemes of irreducible products of the same degree ... 

The Q-l symbol is closely connected to the recoupling transformation between triple products. In order to obtain the expression, Eq. (22), one... 

In light of our previous discussion, a recoupling transformation coefficient must exist such that... 

As demonstrated by Griffin, Levitt, and coworkers in the late 1980s [21, 93], it is also possible to recouple homonuclear dipolar couplings through interference between isotropic chemical shifts and the rotor revolution. This phenomenon, called rotational resonance, occurs when the spinning frequency is adjusted to a submultiple of the isotropic chemical shift difference, i.e., ncor = ct> so o) °. To understand this experiment, the dipolar coupling Hamiltonian in (10) is transformed... 

Dipolar recoupling may also be accomplished using continuous rf irradiation as demonstrated in the heteronuclear and homonuclear case by the CP at MAS conditions (or for low-y heteronuclear spins called double-cross-polarization, DCP [103]) and HORROR (homonuclear rotary resonance) [26] experiments, respectively. These experiments may easily be described by transforming the description into the interaction frame of the rf irradiation using (14a) exploiting... 

The so-called HORROR experiment by Nielsen and coworkers [26] introduced continuous rf irradiation recoupling to homonuclear spin-pairs and initiated the later very widely used concept of /-encoded recoupling. Using a irreducible spherical approach as described above, the HORROR experiment (Fig. 2d) is readily described as starting out with the dipolar coupling Hamiltonian in (10) and x-phase rf irradiation in the form Hrf = ncor(Ix +SX), also here without initial constraint on n. The dipolar coupling Hamiltonian transforms into tilted frame (rotation n/2 around Iy + Sy)... 

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. 

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... 

The orbital transformation defined by Eq. (3.1) enforces both strong orthogonabty and perfect pairing. These constraints are not required of a general GVB wavefunc-tion [19], and much of our current work dispenses with these constraints. However, approximate GVB orbitals describe the essence of both covalent and recoupled pair bonds. 

We also note that the use of a DVR in one or two dimensions and a different basis (e.g. distributed Gaussians) for the remaining dimensions permit most of the advantages of the full DVR approach in terms of sequential diagonalization, truncation, and recoupling. " However the transformations (like 3.16) are less efficient since the function f will not be diagonal in a basis representation as it is in the DVR (e.g. the inner summation is over an additional index). We now turn to the applications of these techniques to several demanding problems in quantum dyiuunics. 

Eq. (2.32) shows that the components of o(ll kk ) transform under rotations like the components of the direct product 1 X 1. We can construct the irreducible tensor components by recoupling in the usual way[10,12] ... 

A review of the reactions and cationoid skeletal transformations in the series of polyfluoroarenes has been reported. The monoprotonated mechanism of the benzidine acid-catalysed rearrangement of hydrazobenzene has been compared with the diproto-nated mechanism monoprotonation results in concerted closed-shell sigmatropic shift, whereas diprotonation leads to a stepwise radical cation recoupling. ... 

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