Internuclear vector

Finally, a given internuclear vector rjk takes a direction with respect to the 3 axis of the unit of structure defined by polar and azimuthal angles ( , r[). Because each unit possesses transverse isotropy, the second moment will involve functions of h, only, the functions of r being replaced by their average values. 

These moment studies have been performed on polymer systems such as polyethylene (or on penetrants in polymer systems) in which the interacting spins (protons or fluorines) reside on the same or on adjacent atoms. This allows essentially no freedom of variation in the internuclear vectors upon deformation of the network. The primary informational content therefore relates to independent segmental orientation distributions. By placing single spins on alternate segments, there should be much greater sensitivity to changes in the chain extension upon bulk deformation. 

Dipolar couplings are very powerful restraints for structure determination of biomolecules and the determination of protein-protein or protein-ligand interactions [16]. They depend on the orientation of an internuclear vector, and its distance and the angular dependence is given by the following formula ... 

Fig. 7.18 Schematic representation of cross-corre- two involved internuclear vectors, the differential lated relaxation of double and zero quantum co- relaxation affects the multiplet in the given way. herences. Depending on the relative angle of the...

The anisotropy of the overall tumbling will result in the dependence of spin-relaxation properties of a given 15N nucleus on the orientation of the NH-bond in the molecule. This orientational dependence is caused by differences in the apparent tumbling rates sensed by various internuclear vectors in an anisotropically tumbling molecule. Assume we have a molecule with the principal components of the overall rotational diffusion tensor Dx, Dy, and l)z (x, y, and z denote the principal axes of the diffusion tensor), and let Dx< Dy< Dz. 

The photoinduced absorbance anisotropy in a TPD experiment relaxes according to the same correlation function as in Eq. (4.16).(29) Effects of spatial variations in the excitation and probe beams, and chromophore concentration, have been treated and shown not to alter the final result.(29) NMR dipolar relaxation rates are expressed in terms of Fourier transforms of the correlation functions, 4ji< T2m[fi(0)] T2m[i2(f)]>> where fl(f) denotes the orientation of a particular internuclear vector. In view of Eq. (4.7), these correlation functions are independent of the index m, hence formally the same as in Eq. (4.16). For the analysis of NMR relaxation data, it is necessary also to evaluate Fourier transforms of the correlation functions. Methods to accomplish this in the case of deformable DNAs have been developed and applied to analyze a variety of data.(81 83)... 

As a point of departure we assume, within a conventional separation of nuclear and electronic motions, an effective Hamiltonian for the motion of two atomic nuclei and their associated electrons both along and perpendicular to the internuclear vector, directly applicable to a molecule of symmetry class for which magnetic effects are absent or negligible [25] ... 

Here T2 is the order of time in which the nucleus resides in a given spin state. If there are no restrictions on the directions available to the internuclear vector, then the time average can be replaced by a space average, with the result that... 

Hirshfeld (1964) pointed out that bond bending not only occurs in ring systems, but also results from steric repulsions between two atoms two bonds apart, referred to as 1-3 interactions. The effect is illustrated in Fig. 12.3. The atoms labeled A and A are displaced from the orbital axes, indicated by the broken lines, because of 1-3 repulsion. As a result, the bonds defined by the orbital axes are bent inwards relative to the internuclear vectors. When one of the substituents is a methyl group, as in methanol [Fig. 12.3(b)], the methyl-carbon-atom hybrid reorients such as to maximize overlap in the X—C bond. This results in noncolinearity of the X—C internuclear vector and the three-fold symmetry axis of the methyl group. Structural evidence for such bond bending in acyclic molecules is abundant. Similarly, in phenols such as p-nitrophenol (Hirshfeld... 

In this work we use an adiabatic electronic representation, and Jacobi nuclear coordinates are chosen r, the HE internuclear vector, and R, the vector joining the HE center-of-mass to the Li atom, in a body-fixed frame in which the three atoms lie on the a — body-fixed plane, with R being parallel to the body-fixed... 

Fig. 2. The angular coordinates used to describe the orientation of the magnetic field, B0, and an internuclear vector, r, relative to an arbitrarily chosen coordinate frame which is fixed to the molecule. Description in terms of (A) direction cosines and (B) polar angles.

The explicit expansion of the spherical harmonic functions describing the orientation of the z/th internuclear vector leads to the following expression,... 

From the expression of RDCs in the PAS of alignment (whether Eq. (12) or Eq. (13)), it is clear that a given RDC measurement, Djjes, does not correspond to a unique orientation of the internuclear vector. In fact the... 

Fig. 4. For axially symmetric alignment, a single measured residual dipolar coupling is consistent with orientations of the corresponding internuclear vector, r, which deviate from the principal (z) axis of the order tensor by an angle , thus describing the surface of a cone, and its inverse. If alignment departs from axial symmetry, then the cone of permissible orientations will require further description in terms of an azimuth angle and exhibit some distortion along the x and directions.

The applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

Fig. 7. The nature of information concerning the mean orientation and dynamics of an internuclear vector r, which can be obtained from RDC analysis. Upon diagonali-zation of the Cartesian dipolar interaction tensor R, described in the text, the mean vector orientation, r, will be described by the Euler angles a and /3. The eigenvalues will correspond to the axial and rhombic order parameters which describe the amplitude of motion. If the motion is asymmetric, as reflected in a nonzero rhombic order parameter, then the principal direction of asymmetry is described by the Euler angle y.

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