Principal axis system, orientation chemical shift

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 crystal structure of benzoic acid is monoclinic and contains four molecules per unit cell in the form of two magnetically inequivalent dimers with equal values of the chemical shift tensors but with their principal axis systems oriented... 

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. 

As mentioned above, the principal values of chemical shift tensor give information about three dimensional electronic state of a molecule. However, in order to understand behavior of the principal values, one should obtain information about the orientation of the principal axis system of a chemical shift tensor with respect to the molecular fixed frame. The orientations of the principal axis systems of the chemical shift tensors of L-alanine Cp -carbons in some peptides were calculated, whose L-alanine moieties have different main-chain dihedral-angles, (( >,v /H-57.40,-47.50)[aR-helix], (-138.8°,134.7°)[ pA-sheet], (-66.3°,-... 

The frequency contribution from a CS tensor also depends on the molecular orientation with respect to the external magnetic field. As illustrated in Figure 3(A), the principal axis system for a CS tensor is defined with the direction of the magnetic field in this text. A liquid sample of H20 is generally used for chemical shift referencing (set to be 0 ppm). Similar to the case of an 170 EFG tensor, a quantum chemical approach is useful for the spectral analysis. Since quantum chemical calculations yield the CS, o, the following conversion is required for making a direct comparison between theoretical and experimental values ... 

Atoms in molecules rarely possess spherically symmetric electron distributions due to the presence of chemical bonds or nonbonding rr-orbitals. The chemical shielding, therefore, depends on the orientation of the molecule with respect to the static magnetic field and the chemical shift is described by a second-rank tensor. The chemical-shift tensor is fully described by three principal values and three Euler angles that orient the principal axis system of the diagonalized chemical-shift tensor with respect to the molecular frame, fin, 522, and 633 (ppm) represent the three principal components of the shift tensor with the following rule 5ii 622 33. In... 

P indicates that the components are those for the principal axis system (PAS) of the tensor. The terms o(QpL(f)) are Wigner rotation matrix elements. They are functions of the set of Euler angles, Qpf (f), which relates the PAS of the chemical shift to the laboratory frame. Due to MAS, these angles are time dependent. A full treatment of the orientation dependence of the chemical shift requires the transformation between several different reference frames. 

As mentioned above, the principal values of the chemical shift tensor give information about the three-dimensional electronic state of a molecule. However, in order to understand the behavior of the principal values, one should obtain information about the orientation of the principal axis system of a chemical shift tensor with respect to the molecular fixed frame. Figure... 

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. 

The spectrum shown in Fig. 2C is typical of many seen for membrane systems. It is sometimes said that the observed Aphosphate group—the amplitude of angular excursion during motional averaging. This is unfortunately only partially true, for the shape and width of the pattern depend critically on the orientation of the axis of motional averaging with respect to the principal components of the chemical-shift tensor a,, G22> and < 33 > as has been pointed out forcefully by Thayer and Kohler (1981). Furthermore, it is often considered that the type of motional averaging, and the resultant powder spectrum just described, is... 

O - These values are related to the chemical shifts by adding or subtracting the shielding constant of the reference compound. The calculations thus not only allow extracting the values of the principal components but also provide the orientations of the individual components in the molecular system. This information is frequently used in solid-state studies to orient the experimentally determined principal values fin, fi22. and fi33 into the axis system of the molecule. 

We briefly discuss the effect of variation of the phosphodiester orientation on spectral linewidth. Figure 12 shows the crystal axis system (a,b,c) with the principal axis Zp of the tensor and the direction of the magnetic field, all of which are related by angles, ot, fi, 6, and x- According to Eq. (3), the observable chemical shift can be given by... 

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