Zero-quantum relaxation

Rotational resonance of inhomogeneously broadened systems was studied by Heller et al.ul With nonlinear least-squares fitting, the distance, the inhomogeneous broadening, the zero-quantum relaxation time, and their respective errors, can be obtained. For short distances, all three parameters can... 

Proton-driven spin diffusion (see also Appendix A) is the classical spin-diffusion experiment for low abundant spins. The line width of the one- and zero-quantum lines of the S-spins are mainly determined by the heteronuclear dipolar couplings while the homonuclear I-spin dipolar coupling makes the broadening of the levels homogeneous. Suter and Ernst [12] calculated an approximate value for the zero-quantum relaxation time... 

In addition to this possibility of determining isotropic /-coupling constants, the -echo concept has been extended to address two further aspects of general interest in the context of 2 experiments experimental determination of zero-quantum relaxation characteristics, and excitation of doublequantum coherence through dipolar coupling. ... 

The dipolar relaxation mechanism is unique because the dipolar coupling interaction contains two spin terms involving mutual spin flips - zero quantum t J. J. t and double quantum J J < j j transitions. All other relaxation processes are limited to single spin interchange. While the decoupler is on, the spin population differences between the levels irradiated are equalized as the rate of energy input from the decoupler greatly exceeds the outflow by relaxation. In dipolar coupled systems, the availability of double and zero quantum relaxation pathways (cross-relaxation) produces non-Boltzmann spin populations in the energy levels of the observed nucleus. These perturbed populations are measured as the nuclear Overhauser effect (nOe) effect rj). The relationship is... 

Reff = observed dipolar coupling constant t = time T20 = spin term in the spherical tensor representation of the dipolar Hamiltonian = zero-quantum relaxation time constant U = propagator = magne-togyric ratio of spin / A/ = anisotropy of the indirect spin-spin interaction 0 = angle between the applied field and the internuclear vector A = dephasing parameter /Uq = permeability of free space Vj. = rotor frequency in Hz 1/, = isotropic resonant frequen-... 

Overall, this treatment predicts that the scalar term for rotational diffusion is independent of double quantum relaxation (W2) and depends only on zero quantum relaxation. The coupling factor can assume values between 0.5 and —1.0 for pure dipolar and pure scalar relaxation, respectively. Moreover, the curves in Fig. 4 clearly show the expected field dependence with low values for the coupling factor, and therefore low enhancements for high magnetic fields. 

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

Figure 1.44 Transitions between various energy levels of an AX spin system. A, and Aj represent the single-quantum relaxations of nucleus A, while Xi and Xj represent the single-quantum relaxations of nucleus X. W2 and are double- and zero-quantum transitions, respectively.

Since the equilibrium state has been disturbed, the system tries to restore equilibrium. For this it can use as the predominant relaxation pathways the double-quantum process (in fast-tumbling, smaller molecules), leading to a positive nOe, or the zero-quantum process 1% (in slower-tumbling macromolecules), leading to a negative nOe. 

Is it possible to predict the predominant mode of relaxation (zero-quantum or double-quantum) by observing the sign of nOe (negative or positive) ... 

The positive nOe observed in small molecules in nonviscous solution is mainly due to double-quantum relaxation, whereas the negative nOe observed for macromolecules in viscous solution is due to the predominance of the zero-quantum 1% cross-relaxation pathway. 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

The cross-correlated relaxation rate observed for double- or zero-quantum coherence involving A1 or A2 and B1 or B2 for two dipolar interactions therefore takes the following form ... 

To illustrate how cross-correlated relaxation can be used to measure the angle between two bond vectors, we will use the example of the generation of double and zero quantum coherence between spins A1 and B1 and call the angle between the Ax-A2 and B1-B2 vectors 8 (Fig. 7.18). 

Fig. 7.18 Schematic representation of cross-corre- two involved internuclear vectors, the differential lated relaxation of double and zero quantum co- relaxation affects the multiplet in the given way. herences. Depending on the relative angle of the...

The method relies on the measurement of cross-correlated relaxation rates in a constant time period such that the cross-correlated relaxation rate evolves during a fixed time r. In order to resolve the cross-correlated relaxation rate, however, the couplings need to evolve during an evolution time, e.g. tt. The first pulse sequence published for the measurement of the cross-correlated relaxation rate between the HNn and the Ca j,Ha i vector relied on an HN(CO)CA experiment, in which the Ca chemical shift evolution period was replaced by evolution of 15N,13C double and zero quantum coherences (Fig. 7.20). 

Therein, cross-correlated relaxation T qHj c h °f the double and zero quantum coherence (DQ/ZQ) 4HizCixCjj generated at time point a creates the DQ/ZQ operator 4HjzCjJCiy. In the second part of the experiment, the operator 4HJZCjxQy is transferred via a 90° y-pulse applied to 13C nuclei to give rise to a cross peak at an(i... 

As an example of the measurement of cross-correlated relaxation between CSA and dipolar couplings, we choose the J-resolved constant time experiment [30] (Fig. 7.26 a) that measures the cross-correlated relaxation of 1H,13C-dipolar coupling and 31P-chemical shift anisotropy to determine the phosphodiester backbone angles a and in RNA. Since 31P is not bound to NMR-active nuclei, NOE information for the backbone of RNA is sparse, and vicinal scalar coupling constants cannot be exploited. The cross-correlated relaxation rates can be obtained from the relative scaling (shown schematically in Fig. 7.19d) of the two submultiplet intensities derived from an H-coupled constant time spectrum of 13C,31P double- and zero-quantum coherence [DQC (double-quantum coherence) and ZQC (zero-quantum coherence), respectively]. These traces are shown in Fig. 7.26c. The desired cross-correlated relaxation rate can be extracted from the intensities of the cross peaks according to ... 

SQ TOCSY TROSY ZQ ZQC single quantum total correlation spectroscopy transverse relaxation-optimized spectroscopy zero quantum zero-quantum coherence... 

The longitudinal cross-relaxation rate (see Eq. (13)) originates solely from the terms in the dipolar Hamiltonian involving both spins, namely those terms corresponding to zero-quantum and double-quantum transitions so that... 

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