Principal elements chemical shift tensors

The isotropic chemical shift is the average value of the diagonal elements of the chemical shift tensor. Advances in solid state NMR spectroscopy allow one to determine the orientation dependence, or anisotropy, of the chemical shift interaction. It is now possible to determine the principal elements of a chemical shift powder pattern conveniently, and the orientation of the principal axes with more effort. Hence, instead of settling for just the average value of the chemical shift powder pattern, one can now aim for values of the three principal elements and the corresponding orientations in a molecular axis system. 

TABLE 1 The values of the principal elements of the chemical shift tensor for selected O-phosphorylated amino acids obtained from the analysis of the intensities of the spinning sidebands 31P CP/MAS spectra... 

To illustrate the power of PISA wheels and dipolar waves to determine the structure of helical peptides and proteins in uniaxiaUy oriented lipid bilayers. Fig. 6a-c show SIMPSON/SIMMOL-simulated PISEMA spectra of an ideal 18-residue a-helix with a tilt angle of 10°-30° relative to Bq. In these simulations, we have tried to mimic experimental conditions by including a random distribution of the principal components of the chemical shift tensor and the dipolar coupling. The chemical shift distribution is 6 ppm for each principal element and has been established as follows we obtained — 85000 N isotropic chemical shifts reported to the BioMagResBank and selected only the — 31000 located in helical secondary stractures to have a data set independent on secondary chemical shifts. The standard deviation on the N chemical shifts for these resonances was — 6 ppm. With the lack of other statistically reliable experimental methods to establish such results for the individual principal elements of the N CSA tensor, we assumed the above variation of 6 ppm for all three principal elements. The variation of the H- N dipolar coupling was estimated by investigating the structures for a small number of a-helical membrane proteins for which the structures were established by liquid-state NMR spectroscopy. These showed standard deviations... 

FIGURE 41. Solid-state NMR spectra of the C-12 retinal resonance in ihodopsin (a) obtained by taking the difference between C-12 rhodopsin and unlabeled rhodopsin spectra. Spectra were obtained at two spinning speeds, 2.36 kHz (left) and 2.84 kHz (right), and are compared with simulations (b and c) based on different chemical shift tensors. The simulations in (b) are for a chemical shift tensor having the principal tensor elements of the aU-trans PSB chloride salt (on = 58, <722 = 133, <733 = 212), while the simulations in (c) use the shift tensor values (on = 41, <722 = 149, <733 = 209) obtained for C12 rhodopsin by MoUevanger and coworkers . Reprinted with permission from Reference 55. Copyright (1990) American Chemical Society... 

Duijvestijn et al. [15] determined the orientation of the chemical shift tensor of rran -polyacetylene, which is the same sample used by Yannoni and Clark [14] by means of a CP technique in combination with dynamic nuclear polarization. They established that the tensor element S33 has its principal axis perpendicular to the molecular plane, and the Sn axis makes an angle of 43 5° with respect to the single bond, and 78 6° with respect to the double bond. Nakai et al. [16] have determined the orientation of the C chemical shift tensor of a s-polyacetylene by the 2D chemical shift/C—H... 

Recently, Jarrett et al. [74] characterized the magnitude and orientation of the chemical shift tensor of the nuclei in model compounds for poly(im-ide)s. The principal values of the chemical shift tensors were evaluated from the nonspinning spectra of four model compounds. The shift tensors span 120 ppm with the 822 element showing the greatest sensitivity to changes in structure. The orientation of the shift tensor was determined by using doubly-labelled materials. Experiments of this type make possible the sophisticated multidimensional NMR experiments for determining order and dynamics in polymers, as described by Schmidt-Rohr and Spiess [3]. 

Simulations [69, 70] of the spinning sideband patterns of P spectra obtained at 80.9 MHz with a low MAS speed (1-2 kHz) (not shown here) were carried out on all the polysiloxane-immobilized phosphine samples [71] to provide values of the principal elements of the chemical shift tensor, which often are valuable in elucidating chemical structure. Two model compounds, diphenyl-isopropylphosphine oxide and l,6-bis(diphenylphosphino)hexane, were used to provide CSA tensor elements that would be approximately representative for the polysiloxane-immobilized phosphine oxide and phosphine moieties, respectively. The experimental P spectra were then simulated on the basis of the principle tensor elements obtained on the two model compounds 117.1, 86.8 and -93.7 ppm for diphenyl-isopropylphosphine oxide and 7.3, -29.8 and -40.9 ppm for l,6-bis(diphenylphosphino)hexane. The simulations showed that the P chemical shift anisotropies of both phosphine and phosphine oxide moieties are qualitatively similar within each of these two categories for all the samples examined, implying very similar... 

Reconstruction of Chemical Shift Anisotrc s. The principal values of the chemical shift tensors for the carbonyl, protonated aromatic, and non protonated aromatic carbons were determined by a Herzfeld-Berger analysis (24) of the intensities of the sidebands from magic angle spinning (14). Herzfeld and Berger have defined two parameters, p and p, which are related to the chemical shift tensor elements en, 0-22, and <733 by ... 

The principal elements of the chemical shift tensors for the carbonyl and aromatic carbons (Table III) have been used to produce the calculated spectra shown in Figure 5(d)-(0< The calculated spectra are in good agreement with the experimental spectra (Figures 5,(a)-(c). 

The chemical shift anisotropy is usually described in a principal axis system, which is usually not the molecular axis system. In the principal axis system, the chemical shift tensor is diagonal. The elements of this tensor contribute to the NMR spectrum via these two equations ... 

In solid-state NMR, a very important concept is that the resonance frequency of a given nucleus within a particular crystallite depends on the orientation of the crystallite [3—5]. Considering the example of the CSA of a nucleus in a carboxyl group, Fig. 9.1 illustrates how the resonance frequency varies for three particular orientations of the molecule with respect to the static magnetic field, Bq. At this point, we note that the orientation dependence of the CSA, dipolar, and first-order quadrupo-lar interactions can all be represented by what are referred to as second-rank tensors. This simply means that the interaction can be described mathematically in Cartesian space by a 3 X 3 matrix (this is to be compared with scalar and vector quantities, which are actually zero- and first- rank tensors, and are specified by a single element and a 3 X 1 matrix, respectively). For such a second-rank tensor, there exists a principal axes system (PAS) in which only the diagonal elements of the matrix are non-zero. Indeed, the orientations illustrated in Fig. 9.1 correspond to the orientation of the three principal axes of the chemical shift tensor with respect to the axis defined by Bq. 

Chemical shift tensor In NMR, the chemical shift anisotropy is described by a second-rank tensor (a 3 x 3 matrix). Can generally be expressed in a coordinate frame where all off-diagonal elements vanish. In this principal axis system, the chemical shift tensor is fully described by the three diagonal elements—the principal components, 5n, 822, and 533—and the three eigenvectors or Euler angles describing the orientation of the principal axes with respect to an arbitrary frame. 

The tensor elements of the chemical shift also need to be defined and in the literature there have been a number of approaches adopted to define parameters describing the tensor. Naturally the three principal components of the tensor and the spatial orientation of these components on the molecular frame will define the tensor. However, papers often do not define the individual tensor components and the parameters adopted by each paper to define the tensor need to be carefully considered. A recent suggestion (Mason 1993) for standardising the definitions is to have the three elements in the PAS designated Sn, 822 and 833, with 8n corresponding to the highest frequency. Then the isotropic value is still the average of the sum of these three terms with the span defined as... 

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