Quantum Coupling

K y,/2ir)5f s> /,y, nonsecular terms can be neglected, hence only zero-quantum coupling terms need to be considered. The tensor elements c J = sin and c J = — sin correspond to single-quantum operators and can be neglected. The tensor elements c j = cos djj and = 1 represent a mixture of zero-quantum and doublequantum terms, and only the zero-quantum term + c yliyljy)... 

In general, the zero-quantum coupling tensor between two spins i and i has the form... 

The following four characteristic zero-quantum coupling tensors between any pair of spins i and j constitute idealized limiting cases for experimentally relevant Hartmann-Hahn experiments. These characteristic zero-quantum coupling tensors are characterized by effective coupling constants which are related to the actual coupling constants /,y by the scaling factors 5, ... 

Planar zero-quantum coupling tensors are characteristic for most heteronuclear Hartmann-Hahn experiments (see Section XI). Here the effective coupling constant between two heteronuclei i and j is scaled by s, <1/2 [see Eq. (115)]. Planar /-coupling tensors (with... 

The condition of Eq. (179) is fulfilled if the matrix elements of both operators A and B are either purely real or purely imaginary. For example, the effective Hamiltonians for characteristic zero-quantum coupling topologies with coupling tensor types /, P, L, or O (see Section V.B) are symmetric in the usual product basis. Because the matrix representations of the operators /, and /y are either real-valued (a =x or z) or purely imaginary (a = y) in this basis, the condition of Eq. (179) if fulfilled. In a system consisting of N spins 1/2, the operators and / have the same norm Tr /, = Tr /y = 2 and... 

Hence, in the idealized zero-quantum coupling topologies that are characteristic for most Hartmann-Hahn-type experiments, the magnetization-transfer functions between two single spins 1/2 are independent of the direction of the transfer (Griesinger et al., 1987a). 

Independent of the experimental implementation, idealized Hartmann-Hahn transfer functions can be calculated for characteristic zero-quantum coupling topologies (see Section V.B). Except for simple two-spin systems, transfer functions are markedly different for different zero-quantum coupling tensor types (see Fig. 10). This difference results from different commutator sequences that occur in the Magnus expansion of the density operator zero-quantum coupling topologies are shown schematically in Fig. 11. 

Under the idealized zero-quantum coupling topologies (see Section V.B), the transfer of magnetization between two spins 1 /2 that are part of an arbitrary coupling network is identical in both directions (see Section VI). This symmetry with respect to the direction of the transfer is related to the symmetry of homonuclear, two-dimensional Hartmann-Hahn spectra with respect to the diagonal (Griesinger et al., 1987a). In Hartmann-Hahn experiments, the properties of the multiple-pulse sequence can induce additional symmetry constraints (Ernst et al., 1991). 

In the reduced-dimensionality space the dynamics is treated exactly, e.g.. by the quantum coupled-channel approach. The remaining degrees of freedom are described in one of several approximate ways which will be reviewed below. The advantage of this approach is that it is feasible for systems of arbitrary complexity. In addition, it enables one to calculate cross sections, rate constants, etc. that are implicitly averaged over those degrees of freedom not explicitly treated dynamically, thus enabling a direct comparison to experiments which in most cases are not fully state-resolved. The degrees of freedom which are neither state-resolved experimentally nor treated dynamically will often be the same because they are usually the low-frequency motions such as rotation which are widely populated initially and finally in a collision. In the next section we shall review the elements of this theory for reactive systems with particular emphasis on resonances. 

Han, M., Vestal, C.R. and Zhang, Z.J. (2004) Quantum couplings and magnetic properties of CoCrxFe2-x04 (0 < x < 1) spinel ferrite nanoparticles synthesized with reverse micelle method. /. Phys. Chem. B, 108, 583-597. 

ABSTRACT. Recent work on quantum coupled oscillators and the collinear dynamics of three bodies, as models for unimolecular and bimolecular reactions, is reviewed with special reference to the role of resonances. The approach, semiclassical in spirit, exploits the approximate separability of the radius of hyperspherical formulations and allows to localize the breakdown of adiabaticity at "ridges in the potential", where transitions between modes occur. 

I Quantum Coupling The self-consistent determination of the interaction energy at a QM level is the hallmark of the quantum coupling approach. The spatial partition of the entire system is shown in Figure 8.2. In specific, the total energy can be expressed as ... 

The original QC formulation assumes that the total energy can be written as a sum over individual atom energies. This condition is not satisfied by quantum mechanical models. To address this limitation, in the present QCDFT approach, the nonlocal region is treated by either the mechanical or the quantum coupling QM/MM approaches [61, 62]. Here, for simplicity, we only discuss the mechanical coupling QCDFT in which the nonlocal... 

The next step is the calculation of the phases dispersions for the transmitted waves when quantum coupled (through the photonic transfer) with those diffracted, as follows (Biagini, 1990 Birau Putz, 2000) ... 

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