Principal-axis system

The electric field gradient is again a tensor interaction that, in its principal axis system (PAS), is described by the tluee components F Kand V, where indicates that the axes are not necessarily coincident with the laboratory axes defined by the magnetic field. Although the tensor is completely defined by these components it is conventional to recast these into the electric field gradient eq = the largest component,... 

Rotational constants G = A, B or C are inversely proportional to principal moments of inertia Ia through the expressions G = h/Sn2Ia, where a refers to one of the three principal inertia axis directions a, b or c. The Ia are related to the coordinates of the atoms i in the principal axis system via the... 

Figure 2 Principal axis system of a shielding tensor assumed to be of axial symmetry.

Figure 5 Definition of the polar angles in the principal axis system of the diffusion-rotation tensor.

Figure 17 The a,a,2,6-tetrachlorotoluene. Small letters refer to the rotation-diffusion principal axis system.

Because of the non-coincidence of the g- and -matrix principal axes, the various parallel features correspond to different orientations of the magnetic field in the g-matrix principal axis system. These orientations are given in Table 4.16. 

In the coordinate system that diagonalizes g, the related D-tensor is also diagonal. Expanding the fine structure term in the principal axis system, we have ... 

The coupling tensor Rlm in the laboratory frame is time dependent due to the motions of spin-bearing molecules. It can be expressed in terms of the rotational transformation of the corresponding irreducible components pln in the principal axis system (PAS) to the laboratory frame by... 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

Fig. 3. The principal axis system of the order tensor corresponds to the specific positioning of a set of coordinate axes relative to the molecule, as indicated by the Sxx, Syy and Szz labelled axes. The rotation which diagonalizes the order tensor also carries the initial molecular coordinate frame, specified by the x, and z labelled axes, into the PAS of the order tensor. This figure was prepared using the program Module.193...

Fig. 9. Comparison of the principal axis systems of alignment for the TAR RNA in bacteriophage Pfl as (A) experimentally determined and (B) predicted using the program SSIA.100...

Fig. 10. Measurement of RDCs in a second, different alignment medium can significantly reduce the possible corresponding internuclear vector orientations. Permissible orientations will correspond to the intersection of cones, which are placed according to the respective principal axis systems of ordering.

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

Let us summarize the results. Using the principal-axis system, we defined the nuclear mass-weighted Cartesian coordinates qt by (6.1) and (6.3) ... 

Solution. Adopting a principal axis system in which k is parallel to the c-axis and i and j lie in the basal plane, the diffusivity tensor will have the form given in Eq. 4.66 and the flux will be given by... 

The first is Coov> which has a principal axis about which any rotation is a symmetry operation, plus an infinity of vertical planes containing the principal axis. Systems that in addition have inversion symmetry belong to D h-... 

In the principal axis system of Vy, in which the tensor is diagonal, the quadrupolar Hamiltonian for a single spin is... 

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