Double quantum relaxation

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. 

Each of these numbers is multiplied by 1/r6, reflecting the distance dependence of the dipole-dipole interaction. Now we see that double-quantum relaxation does in fact dominate the dipole-dipole relaxation of small molecules, and our cartoon model of relaxation exclusively by the DQ pathway during the mixing time is not that far off. Likewise, the assumption that only ZQ relaxation occurs for large molecules (see exercise above) is also qualitatively correct. 

Which agrees with the result obtained using the irreducible tensor formalism.34 The triple-quantum filtration pulse sequence, applied to a spin- quadrupolar nucleus, produces almost exactly the same result as does the double-quantum filtration sequence, except for two differences. The double-quantum relaxation rate is faster than the triple-quantum relaxation rate, and the triple-quantum FID is a factor of 1.5 larger than the double-quantum FID.34 The triplequantum filter will thus require less than half the number of acquisitions to equal the signal-to-noise of the double-quantum sequence. 

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. 

A single-quantum transition involves one spin only, whereas the zero- and doublequantum transitions involve two spins at the same time. The zero- and double-quantum transitions give rise to cross-relaxation pathways, which provide an efficient mechanism for dipole-dipole relaxation. 

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) ... 

Fig. 9.10 (A) Definition of the dihedral angle d between the spin-pairs ij and k,l. (B) Cross-correlated relaxation can be detected by differences in intensities of the multiplet components of double quantum coherences...

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. 

Finally, the double quantum (DQ) spin spin relaxation time T2d can be determined using the pulse sequence 90° — x — 45° — 90° —t — 45°.51 The first three pulses create the DQ coherence, and the last read pulse converts the DQ to a single quantum coherence for detection. 

Artifactual peaks are even more dangerous than noise since they may not always be immediately recognized and may lead to erroneous assignments. An important source of artifacts is instability in the steady-state condition, e.g. if the relaxation delay is set too short. A commonly encountered example is presented by peaks which occur at the double quantum frequencies in DQF-COSY spectra. For detailed treatments of aspects of noise and artifacts see [4, 5]. 

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 ... 

Moreover, we note that recently in reconstructing relaxation times via the time-temperature superposition principle using double quantum nuclear magnetic resonance (DQ-NMR) the and power laws were invoked without giving the spatial information of NSE [75]. 

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... 

Figure 2 The four-level diagram for a system of two interacting spins, in this case an electron (S) and nucleus with a positive gyromagnetic ratio (/). The intrinsic electron and nuclear spin relaxation are given by p and w°, respectively, and the dipolar and/or scalar interactions between the electron and nuclear spin are represented by p, w0, w, and w2. The transition w0 is known as the zero-quantum transition, while w, is the singlequantum transition and w2 is the double-quantum transition. Nuclear and electronic relaxation through mechanisms other than scalar or dipolar coupling are denoted with w° — 1/Tio and p — 1/Tie, where Ti0 and T1e are the longitudinal relaxation times of the nucleus and electron in the absence of any coupling between them. Since much stronger relaxation mechanisms are available to the electron spin, the assumption p>p can be safely made. Adapted with permission from Ref. [24].

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