Isolated spin pair approximation

Nilges et al. used a very complete analytical treatment, but showed that the calculations would be tractable if a cutoff of 4 A were used, apparently without introducing too great an approximation.67 Applying the method to squash trypsin inhibitor, they found that atomic shifts of only 1 A greatly improved agreement with experimental data and that the previous isolated spin pair approximation had caused a slight systematic contraction of the structures. 

The buildup of direct NOEs is approximately linear at short mixing times. In this regime (the isolated spin-pair approximation (ISPA)),118 it is assumed that the intensity of the observed crosspeak is directly proportional to r 6. The proportionality can be estimated by measuring the intensities (7ref) of NOEs between protons that... 

These investigators employed the CHARMm force field with the carbohydrate parameters of Ha et al. jn j s MD simulations in vacuo. They note particularly, as have others, >135 that the use of the isolated spin pair approximation (ISPA), which is often used to convert NOE intensities into interproton distances, can be extremely inaccurate. Within ISPA, the assumption is made that the NOE intensity NOEfj) between two protons arises only from spin relaxation between the two protons. This approximation neglects the effects of spin diffusion and internal motion. Taking for calibration a known interproton distance (r f) and its associated NOE (NO f), the ISPA distance two protons i and j) may be derived from Eq. [14].i35... 

The nOe data may be obtained by either one- or two-dimensional methods. These through-space effects may be used to estimate intemuclear distances. However, nOe cross-peak volumes are inversely proportional to the sixth power of the distance between the correlated protons only in the case of rigid spherical molecules whose tumbling is isotropic. The simplest method that has been proposed for the interpretation of nOe-derived distance data is the isolated spin pair approximation. The matrix treatment is more rigorous, although rel5dng also on several approximations. A set of equations describing the cross-relaxation pathways of all the protons in the molecule is cast into a matrix form and solved. 

Figure 31 presents experimental H DQ MAS spinning sideband patterns for the aromatic protons in (a) the crystalline and (b) the LC phases of a-deuterated HBC—C12.22 The MAS frequency was 35 and 10 kHz in (a) and (b), respectively, with two rotor periods being used for excitation/reconversion in both cases, such that rrcpi equals 57 and 200 /us in the two cases. The dotted lines represent best fit spectra simulated using the analytical time-domain expression for an isolated spin pair in eq 6. As noted in section VIIB, the aromatic protons exist as well isolated pairs of bay protons, and, thus, an analysis based on the spin-pair approximation is appropriate here. As is evident from the insets on the right of Figure 31, the DQ MAS spinning sideband patterns are very sensitive to the product of the D and rrcpi. The best-fit spectra for the solid and LC phases then correspond to DI(Zji)s equal to 15.0 0.9 and 6.0 0.5 kHz, respectively. 

The predominant interaction for a 2H spin system is the quadrupolar interaction, which couples the electric quadrupole moment of the 2H nucleus to its electronic surrounding. This interaction is a second-rank tensor Hq which lies approximately along the C-2H bond in organic molecules. Thus, in practice, 2H nuclei may be considered to be isolated. It shows that the 2H NMR formalism is similar to that of an isolated proton pair [8] ... 

Most of protein structural information from NMR is obtained in the form of nuclear Overhauser effects or NOEs between pairs of protons that are less than 6 A apart through space. An NOE between a spin pair carries distance information, but only short distances are observed because NOEs have an inverse sixth power dependence on distance. However, the distance cannot be uniquely determined given a measured NOE intensity without making some assumption about the environment of the spin pair and the motion of the vector between them. The simplest model for obtaining a distance from cross peak intensities in a nuclear Overhauser effect spectrum (NOESY) is the isolated ri d spin pair (RRNN - rigid rotor nearest neither) approximation (Jardetzky and Roberts, 1981). In this approximation the observed cross peak intensity, which is proportional to the cross relaxation rate, is related to a sini e intemudear distance, r. 

For simplicity, in equation (3c) S oc is assumed to be mainly along the director axis n, and n is aligned parallel tol. For Btoc Bo, the total 7, is equal to 7,2, whereas in the opposite case B,oc Bq one has 7, = 7,o- Evidently, the finite local field contribution complicates the control of the angle adjustment without an exact knowledge of, oc, and because of equation (3c) the inclinations of 90° become impossible by external field switches. Furthermore, the spectral densities for 7,0 are not discussed in the literature to the same extent as for 7,2, nor does there exist a critical experimental examination of the validity of the basic expression equation (3a). Approximate predictions about with the Redfield formalism give, for the completely isolated, i.e., uncoupled proton spin-pair (/ = 1) and high spin-temperature approach ... 

Having demonstrated the achievement of high-resolution sohd state NMR capability, the authors describe experiments that combine the high-resolution aspect of MAS NMR with methods that retain the structure and/or dynamic information inherent in the anisotropic interactions. Rotational-echo double resonance (REDOR) allows the determination of D between isolated heteronuclear spin pairs. D is related simply and without approximation to intemuclear separation. Hence, REDOR makes possible the unambiguous direct determination of intemuclear distance between the labeled spin pair, independent of pair orientation, i. e., in amorphous and /or microaystaUine solids, and extends our abihty to quantitatively explore complex materials. It is also possible to extract intemuclear distance from homonuclear dipolar coupled spin pairs, and these experiments are also reviewed. 

Let ns consider a linear chain of N statistical segments, fixed at its extremities. An average orientation is indnced along the chain hy these constrains. To estimate the effect of this average orientation on NMR parameters, the simplest pictnre is to consider that each segment carries an isolated pairs of spins, nsnally protons (two-spin approximation). Within this framework, the time evolntion of the transverse magnetization (t) for a spin pair attached to a chain may he written in the very simple form... 

In all the cases discussed above, only the response of the spin system composed of dipolar coupled spin-1/2 pairs was considered. Proton DQ-NMR spectroscopy on elastomers—static and spinning at the magic angle—proved that the consideration of isolated spin-1/2 pairs is a crude approximation (68). The dipolar couplings between the protons belonging to various fimctional groups are shown to be active in two-dimensional DQ-MAS spectra for high excitation/reconversion times (68,73). 

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