Dipolar interaction relaxation

From SCRP spectra one can always identify the sign of the exchange or dipolar interaction by direct exammation of the phase of the polarization. Often it is possible to quantify the absolute magnitude of D or J by computer simulation. The shape of SCRP spectra are very sensitive to dynamics, so temperature and viscosity dependencies are infonnative when knowledge of relaxation rates of competition between RPM and SCRP mechanisms is desired. Much use of SCRP theory has been made in the field of photosynthesis, where stnicture/fiinction relationships in reaction centres have been connected to their spin physics in considerable detail [, Mj. 

Thus, identification of all pairwise, interproton relaxation-contribution terms, py (in s ), for a molecule by factorization from the experimentally measured / , values can provide a unique method for calculating interproton distances, which are readily related to molecular structure and conformation. When the concept of pairwise additivity of the relaxation contributions seems to break down, as with a complex molecule having many interconnecting, relaxation pathways, there are reliable separation techniques, such as deuterium substitution in key positions, and a combination of nonselective and selective relaxation-rates, that may be used to distinguish between pairwise, dipolar interactions. Moreover, with the development of the Fourier-transform technique, and the availability of highly sophisticated, n.m.r. spectrometers, it has become possible to measure, routinely, nonselective and selective relaxation-rates of any resonance that can be clearly resolved in a n.m.r. spectrum. 

Here, is the magnetization of spin i at thermal equilibrium, p,j is the direct, dipole-dipole relaxation between spins i and j, a-y is the crossrelaxation between spins i and j, and pf is the direct relaxation of spin i due to other relaxation mechanisms, including intermolecular dipolar interactions and paramagnetic relaxation by dissolved oxygen. Under experimental conditions so chosen that dipolar interactions constitute the dominant relaxation-mechanism, and intermolecular interactions have been minimized by sufficient dilution and degassing of the sample, the quantity pf in Eq. 3b becomes much smaller than the direct, intramolecular, dipolar interactions, that is. 

Eq. 16 is an extremely useful criterion for examining the extent of dipolar interaction in a multispin system, and gives the relaxation method a major advantage over the n.O.e. method. The equivalent quantitative test for the n.O.e. experiment requires all but the receptor nucleus to be saturated and this is not readily performed in practice. 

For a rigidly held, three-spin system, or when existing internal motion is very slow compared to the overall molecular tumbling, all relaxation methods appear to be adequate for structure determination, provided that the following assumptions are valid (a) relaxation occurs mainly through intramolecular, dipolar interactions between protons (b) the motion is isotropic and (c) differences in the relaxation rates between lines of a multiplet are negligibly small, that is, spins are weakly coupled. This simple case is demonstrated in Table V, which gives the calculated interproton distances for the bicycloheptanol derivative (52) of which H-1, -2, and -3 represent a typical example of a weakly coupled, isolated three-spin... 

While the rate of change of dipolar interaction depends on t its magnitude depends only on the internuclear distance and is independent of t,. Thus the dipole-dipole relaxation depends on the molecular correlation time T the internuclear distance r, and the gyromagnetic ratios of the two nuclei, y and js -... 

Nuclear spins can be considered as dipoles that interact with each other via dipolar couplings. While this interaction leads to strongly broadened lines in soUd-state NMR spectroscopy, it is averaged out in isotropic solution due to the fast tumbUng of the solute molecules. In Uquid-state NMR spectroscopy, the dipolar interaction can only be observed indirectly by relaxation processes, where they represent the main source of longitudinal and transverse relaxation. 

Tessari, M., Vis, H., Boelens, R., Kaptein, R., Vuister, G. W. Quantitative measurement of relaxation interference effects between Hn CSA and H- N dipolar interaction correlation with secondary structure. J. Am. Chem. Soc. 1997, 119, 8985-8990. 

The relaxation enhancement displayed for canthaxanthin, 7 -apo-7 -(4-carboxyphenyl)-P-carotene and BI was analyzed to provide interspin distances. The dipolar interactions and the distances can be determined according to procedures described elsewhere (Budker et al. 1995, Rakowsky et al. 1995, Eaton and Eaton 2000, Rao et al. 2000) and based on simulations of the paramagnetic metal ion contribution, WAA. 

To measure distances in the wider temperature range, this procedure was modified. Relaxation of the carotenoid occurs through several different mechanisms including the dipolar-dipolar interaction. Assuming that kAA is the rate constant of the dipolar-dipolar interaction and K=(k,l + k2 + k3 +. ..) is the sum of the rate constants of all other relaxation pathways, we can extract kAA from the following equation ... 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

The coupling term, traditionally denoted by cr B (which has however nothing to do with the screening coefficient of Section 2.2), is the so-called cross-relaxation rate and is a relaxation parameters which depends exclusively on the dipolar interaction between nuclei A and B, contrary to auto-relaxation rates which are compounds of several contributions. For instance, if A is a carbon-13, the auto-relaxation rate can always be written as... 

This is the beauty of this quantity which provides specifically a direct geometrical information (1 /r% ) provided that the dynamical part of Equation (16) can be inferred from appropriate experimental determinations. This cross-relaxation rate, first discovered by Overhau-ser in 1953 about proton-electron dipolar interactions,8 led to the so-called NOE in the case of nucleus-nucleus dipolar interactions, and has found tremendous applications in NMR.2 As a matter of fact, this review is purposely limited to the determination of proton-carbon-13 cross-relaxation rates in small or medium-size molecules and to their interpretation. 

It can be noticed that the maximum NOE factor (2 when A is a carbon-13 and B a proton) is reached under extreme narrowing (see Section 6) conditions and if RA arises exclusively from the A-B dipolar interaction. On the other hand, the cross-relaxation rate gab is easily deduced from the NOE factor and from the A specific relaxation rate... 

Finally, it can be noted that there also exist dipolar-dipolar crosscorrelation rates which involve two different dipolar interactions. These quantities may play a role, for instance, in the carbon-13 longitudinal relaxation of a CH2 grouping.11,12 Due to the complexity of the relevant theory and to their marginal effect under proton decoupling conditions, they will be disregarded in the following. 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

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