Relaxivity contributions

In the presence of a field H, rotating at the precessional frequency the nuclear system can absorb energy, following which nuclear relaxation occurs. Thus, the equation of motion must include both the precessional and the relaxation contributions ... 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

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. 

In proton-relaxation experiments, R, values are used extensively, whereas 7, values are more frequently reported for C relaxation measurements. Although there is no special merit in this preference for C 7, values, the pairwise additivity of relaxation contributions in proton-relaxation experiments is more clearly apparent for the relaxation rates. 

Under these conditions, and neglecting external relaxation-interactions according to Eq. 4, the pairwise relaxation-contributions, py, are evaluated from Eqs. 10-12, provided that an equal number of experimentally determined values are available. For a weakly coupled, three-spin system,... 

Nevertheless, the overall structural problem can be solved from combined n.O.e. and single-selective relaxation-measurements through the evaluation of individual cross-relaxation terms, (Ty. According to Noggle and Shirmer, the n.O.e. value is a function of the cross-relaxation between spins i and j and the relaxation contributions of the neighboring protons to spin i, that is. 

Some simplified assumptions for extracting interproton relaxation-contributions from a single set of nonselective relaxation-rates have been made for a tetrachlorotetra-0-mesyl-ga/ac(o-sucrose derivative (43). By con-... 

However, the relaxation contributions obtained from Eq. 22 were not satisfactorily compared with those obtained from specific, deuterium-substitution experiments and single- and double-selective relaxation-rates. Moreover, the errors estimated for the triple-pulse experiments were very much larger than those observed for the other techniques. This point will be discussed next. 

Ratio of Interproton, Dipolar Relaxation Contributions, pij/pik, for a Molecule Which Is Tumbling Isotropically. (Reproduced, with permission, from Ref. 5.)... 

InterprotoD Distances (A) for 44, Calculated from the Specific, Interproton Dipole-Dipole Relaxation Contributions, and Assnming that —1-8... 

Figure 4. Plot of the difference between experimental G (uaT) and its relaxational contribution versus the reduced frequency for six representative networks.

In most cases, the orbital relaxation contribution is negligible and the Fukui function and the FMO reactivity indicators give the same results. For example, the Fukui functions and the FMO densities both predict that electrophilic attack on propylene occurs on the double bond (Figure 18.1) and that nucleophilic attack on BF3 occurs at the Boron center (Figure 18.2). The rare cases where orbital relaxation effects are nonnegligible are precisely the cases where the Fukui functions should be preferred over the FMO reactivity indicators [19-22], In short, while FMO theory is based on orbitals from an independent electron approximation like Hartree-Fock or Kohn-Sham, the Fukui function is based on the true many-electron density. 

It is noteworthy that the neutron work in the merging region, which demonstrated the statistical independence of a- and j8-relaxations, also opened a new approach for a better understanding of results from dielectric spectroscopy on polymers. For the dielectric response such an approach was in fact proposed by G. Wilhams a long time ago [200] and only recently has been quantitatively tested [133,201-203]. As for the density fluctuations that are seen by the neutrons, it is assumed that the polarization is partially relaxed via local motions, which conform to the jS-relaxation. While the dipoles are participating in these motions, they are surrounded by temporary local environments. The decaying from these local environments is what we call the a-process. This causes the subsequent total relaxation of the polarization. Note that as the atoms in the density fluctuations, all dipoles participate at the same time in both relaxation processes. An important success of this attempt was its application to PB dielectric results [133] allowing the isolation of the a-relaxation contribution from that of the j0-processes in the dielectric response. Only in this way could the universality of the a-process be proven for dielectric results - the deduced temperature dependence of the timescale for the a-relaxation follows that observed for the structural relaxation (dynamic structure factor at Q ax) and also for the timescale associated with the viscosity (see Fig. 4.8). This feature remains masked if one identifies the main peak of the dielectric susceptibility with the a-relaxation. 

As an example of behavior of a typical Gd-complex and Gd-macromolecule we discuss here the NMRD profiles of a derivative of Gd-DTPA with a built-in sulfonamide (SA) and the profile of its adduct with carbonic anhydrase (see Fig. 37) 100). Other systems are described in Chapter 4. The profile of Gd-DTPA-SA contains one dispersion only, centered at about 10 MHz, and can be easily fit as the sum of the relaxation contributions from two inner-sphere water protons and from diffusing water molecules. Both the reorientational time and the field dependent electron relaxation time contribute to the proton correlation time. The fit performed with the SBM theory, without... 

However, certain assumptions must be made in order to derive the QSC. First, it is often assumed that the relaxation of Li is totally dominated by the quadrupole mechanism, i.e. T = r . If the relaxation of Li is measured on the same sample, relaxation contributions from other mechanisms can be eliminated . is related to QSC by equation 11 ... 

Since the d orbitals are not allowed to relax in a one electron ECP it may appear that a third prerequisite is that the frozen d approximation should be valid, i.e. the relaxation of the d orbitals should not influence the bonding appreciably. In reality the effect of d-shell relaxation on various metal cluster properties is appreciable e.g. the d-shell relaxation effect on chemisorption energy of oxygen on a Nis cluster is about 40 kcal/mol[23]. However, a small d orbital relaxation is not a necessary prerequisite for the development of a one electron ECP provided that the relaxation is not dominated by covalency effects. The covalent contribution to the bonding of an oxygen atom to a Cus cluster (where the d-shell relaxation contribution to the binding energy is 17 kcal/mol) is only a few kcal/mol[24]. 

Fig. 3.13. Field dependence of the Curie spin relaxation contributions to R u and R2M (arbitrary scale) zr = 2 x 10 9 s.

In principle, there may also be a Curie relaxation contribution of the contact-type whenever there is chemical exchange or intramolecular rotation to modulate the coupling. The contribution to R2M would then be... 

The presence of contact relaxation indicates that a given moiety is covalently bound to a paramagnetic metal ion and provides an estimate of the absolute value of A (Eqs. (3.26) and (3.27)). Sometimes the contact coupling constant can be evaluated by chemical shift measurements, and it is therefore possible to predict whether the contact relaxation contributions to R m, Rim or both, are negligible or sizable. 

Unlike hyperfine isotropic shifts, which often contain pseudocontact and contact contributions of the same order of magnitude, relaxation rates can often be recognized to be dominated by only one of the possible contributions. In addition, whereas contact and pseudocontact shifts may happen to have different signs, thereby making their separation more uncertain, relaxation contributions are obviously always positive and additive. 

Some qualitative guidelines can be given to make an a priori estimate of the relative weight of dipolar, contact, and Curie relaxation contributions. Consider first the fast motion limit where Rim = Rim and none of the frequency-dependent terms is dispersed. The equations take the simple form already noted ... 

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