NOEs in a two-spin system

One should note here that despite providing an important relaxation mechanism, dipolar couplings do not usually produce observable splittings in solution state NMR spectra. This is because, although the couplings have a finite value at any instant in time, they are averaged precisely to zero on the NMR timescale by the rapid isotropic tumbling of a molecule. 

F ure 8.5. The direct, through-space interaction between two near spin-V4 nuclei (the dipolar interaction). This fluctuates as the molecule tumbles in solution and can provide a time-dependent field capable of inducing spin transitions. 

Qualitatively then, one can predict from the previous arguments that molecules that tumble rapidly in solution are likely to favour the higher energy W2 process and hence exhibit positive NOEs whilst those that tumble slowly will favoiur the Wp process and thus display negative NOEs indeed, this is what is observed in practice. 

Quantitative expressions for the relaxation rates in a dipolar coupled two-spin system have been derived, thus  

When a molecule tumbles so rapidly in solution such that 1, all terms in these expressions containing w become negligible and the rates simplify to  

Consider a system comprising only two homonuclear spin- /2 nuclei, I and S, that exist in a rigid molecule which tumbles isotropically in solution, that is, it has no preferred axis about which it rotates. The two nuclei do not 

The power available within a molecular system to induce transitions by virtue of its molecular tumbling is referred to as the spectral density J(co) (Section 2.5) and this provides a measure of how the relaxation rates Wq, W] and W2 vary as a function of tumbling rates. This is illustrated schematically in Fig. 8.6 for three different correlation times. An alternative description of the spectral density is that it represents the probability of finding a fluctuating magnetic component at any given frequency as a result of the motion and as such the area under each of the curves of Fig. 8.6 must then be equal. Thus, for a molecule with a short tc (rapid tumbling) there exists an almost 

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Figure 8.8. The variation in the maximum theoretical homonuclear steady-state NOE in a two-spin system as a function of molecular tumbling rates (defined by the dimensionless parameter cOoTc). The region of fast motion is the extreme narrowing limit and that of slow motion is the spin-diffusion limit.

E8-5. Why is it that in a two spin system the size of transient NOE enhancements depends on Rj, Rs and steady state experiment the enhancement only depends onR1 and [Pg.205]

Having taken the trouble to see how the relaxation rates in a two-spin system depend on molecular motion, we are now in a position to predict the behaviour of the NOE itself as a function of this motion and of intemuclear separation. Taking the rale constant Eq. (8.4) and substituting these into that for the NOE Eq. ((8.2)) produces the curve presented in Fig. 8.8 for the theoretical variation of the homonuclear NOE as a function of molecular tumbling rates as defined by (where u>o is the spectrometer observation frequency, approximately equal to uii and ujs). Note this is for a two-spin system, which relaxes solely by the dipole-dipole mechanism and as such represents the theoretically maximum possible NOE. The curve has three distinct regions in it, which we shall loosely refer to as the fast, intermediate and slow motion regimes. For those molecules that tumble rapidly in solution (short those in the extreme narrowing limit), the NOE has a... 

Equations [l]-[3] allow us to calculate how the NOE behaves in a two-spin system. As explained in more detail below, there are two main ways of observing NOEs. In the first, one spin S is perturbed, for example, by a selective 180° pulse almost all two-dimensional (2D) methods work in a similar way. This results in the buildup of an NOE at adjacent spins I (Figure 1), with an initial rate given simply by... 

Our discussion of the NOE was based on a two-spin system with solely mutual dipolar relaxation. When there are more than two spins in magnetically... 

The NOE was explained for a two-spin system (7 = 1/2) by Solomon in 1955 (see Ref. 4). Figure 13 shows that such a system has four possible energy levels (= spin states) connected by six possible transitions. 

The previous section considered the NOE for the hypothetical case of a two-spin system in which the spins relax exclusively via mutual dipole-dipole relaxation. In progressing to consider more realistic multispin systems two key issues will be addressed how the presence of other spins affects the magnitudes of steady-state NOEs and how these reintroduce distance dependence to the NOE. These considerations lead to the conclusion that steady-state NOE measurements must be used in a comparative way to provide structural data, and that they do not generally provide estimates of intemuclear distances per se. 

The previous section considered the NOE for the hypothetical case of a two-spin system in which the spins relax exclusively via mutual dipole-dipole relaxation. In progressing to consider more realistic multispin systems, two key issues will be... 

In an isolated two-spin system, the NOE (or, more accurately, the slope of its buildup) depends simply on where d is the distance between two protons. The difficulties in the interpretation of the NOE originate in deviations from this simple distance dependence of the NOE buildup (due to spin diffusion caused by other nearby protons, and internal dynamics) and from possible ambiguities in its assignment to a specific proton pair. Mofec-ufar modeling methods to deaf with these difficulties are discussed further below. 

The murrastifolines exhibited characteristic carbazole UV spectra. Their H-NMR spectra showed signals for aromatic methoxy ((5 3.57-4.11) and methyl groups ( 2.52-2.56). In the aromatic region, signals of a four-spin system and two singlets for H-4 (6 7.56-7.63) and H-2 (S 6.85-7.00) were observed. The H-NMR spectra, NOE studies, and mass fragments at m/z 210 or 211 confirmed murrayafoline A (7) as the common structural unit for all murrastifolines (Schemes 2.42-2.44). 

The terms pc and py correspond to 1/Tic and 1/Tih, respectively, and CTCH is the cross-relaxation rate. It should be stressed that the simplicity of the above equation is a consequence of the rareness of the I spins and of the dominant strength of the dipolar interaction between directly bonded nuclei. The situation for homonuclear proton spin systems is often more complicated, since the protons usually constitute a much larger spin system, and a separation into distinct two-spin systems may be not valid in this case. The broadband irradiation of the protons yields, in a steady state, Mhz = 0 and M z = Mj (1 rj). The factor 1 + 77 is called, as introduced above, the nuclear Overhauser enhancement factor. The NOE factor is related in a simple way to the equilibrium magnetizations of the I- and S-spins (which are proportional to the magnetogyric ratios 71 and 7s), the cross-relaxation rate and the relaxation rate of the I-spin ... 

It is intuitive that the NOE will be larger the larger the cross relaxation, and smaller the larger p/. It may be appropriate here to discuss in a more quantitative way the response of a dipole-coupled two spin system to selective and non-selective experiments, and the relationship between the results of these experiments and the quantity of interest, p/. The qualitative behavior has been anticipated in Section 3.13. As already described (and see also Section 9.2), when measuring T of a signal one can excite (for instance, invert) all the signals... 

Heteronuclear incoherent magnetization transfer is the transfer of longitudinal magnetization. It can proceed in the laboratory frame and in the rotating frame. The nuclear Overhauser effect (NOE) [Nogl] is a manifestation of polarization tranter in the laboratory frame. In the extreme narrowing limit saturation of dipolar relaxation of the I doublet of a heteronuclear IS two-spin- system leads to an enhancement of the S-spin polarization by a factor... 

In a particular two-spin system, S relaxes quickly and I relaxes slowly. Which experiment would you choose in order to measure the NOE enhancement between these two spins Include in your answer an explanation of which spin you would irradiate. 

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