Direct dipolar interaction

The next step is to define the Hamiltonian for the system. For NMR it is easy to write down the Hamiltonian for a set of nuclei in a molecule that is tumbling rapidly. We know that each nucleus interacts with the magnetic field as we discussed in Chapter 2, that direct dipolar interactions average to zero and need not be considered, and that indirect spin coupling between any pair of nuclei can be handled as a scalar product. All of these factors are included in the Hamiltonian given. 

Although solid state 2H NMR techniques are also used widely in structural studies, the principal use of these techniques has been to obtain detailed information on reorientational motions in the solid state, and our discussion is focused on this aspect of 2H NMR. As discussed above, the quadrupole interaction is usually the dominant nuclear spin interaction in 2H NMR, and other nuclear spin interactions (e.g. dipole-dipole interaction, CSA and scalar J-coupling) are generally negligible in comparison. For 2H, the quadrupole interaction is typically about 150-250 kHz, whereas the direct dipolar interactions and CSA are typically about 10 kHz and 0.7 kHz (at 11.7 T) respectively. Since the EFG originates from... 

Unlike the high resolution easily achieved in solution state NMR, the resolution of a C NMR spectrum in the solid state is severely degraded by broadening due to the anisotropic direct dipolar interaction (which varies as the inverse... 

Through-space methods utilize the direct dipolar interaction. For and N, these interactions are removed by MAS. Hence, some form of recoupling is required. Recoupling does not simply restore the static Hamiltonian. Although the Hamiltonian may be of the same form, it will be smaller in scale and can have different angle dependencies or spin terms. For magnetization transfer, the choice of which recouphng scheme to use may depend upon all these factors. 

The anisotropy of the indirect (ind) spin-spin coupling of nuclei can lead to a term Dcalled a pseudodipolar interaction, which enters the spin Hamiltonian of Eq. (1) in the same way as the direct dipolar interactions of Eq. (2).4 Thus we may write... 

When directed dipolar interactions produced by polar groups or substituents are present in addition to the dispersive forces, then the variety of crystal structures increases. 

In this case the protons are non-equivalent, because of the neighboring dihydropyrrole ring. The protons thus give rise to an AB quartet. However, the observed splitting is the sum of the ordinary scalar spin-spin coupling and the direct dipolar interaction, and is therefore field dependent, showing the value -I9.92 Hz at 300 MHz and -I9.25 Hz at 600 MHz. The effective J is thus field dependent and is given by... 

Forgeron et al. measured the J( Xe, F) coupling constant in crystalline XeFa to be —5560+50 Hz whose sign was determined using the sign of the Xe — F direct dipolar interaction. No theoretical calculations have been published for J(Xe, F). 

Abraham et al., who add an extra term to Equation 19 to allow for direct dipolar interactions between solvent and solute (17). This term is small for solvents of low dielectric constant but becomes significant for those having b> 10. The theory, thus modified, seems to give fairly satisfactory descriptions of polar effects on conformational equilibria (18), free energies of transfer of ion-pairs (19), and activation free energies of reactions . 

The direct dipolar interaction, also known as direct dipole-dipole coupling, is a through-space interaction between the magnetic dipole moments of the two spins, I and S. Like the quadrupolar and the CS interaction, the dipolar interaction is anisotropic and depends on the orientation of the internuclear vector between two spins with respect to the direction of external magnetic field. The strength of the dipolar interaction increases... 

CP is mediated by the heteronuclear dipolar interaction between proton and the target spin. Since the direct dipolar interaction strongly depends on... 

In a nuclear magnetic resonance measurement in solution, the direct dipolar interaction Xo actually disappears because, due to rapid molecular motion, the interspin (internuclear) vectors are rapidly space-averaged within the time-scale of a measurement. Hence, the terms Xs and Xj are detectable as sharp lines or splittings in a high-resolution spectrum of H or and they can be related to detailed molecular structure or conformation of the substance investigated. In a solid, on the contrary, the directions of the internuclear vectors are stationary even if they are distributed randomly in space. Then, X gives a very wide linewidth to the zeroth-order absorption line and completely masks all lines due to Xs and Xj. 

Under typical solution NMR conditions, molecules are isotropically oriented, that is, they do not show a preferred orientation. As a consequence, interactions that depend on the orientation are averaged. One of these interactions is the direct dipolar interaction between pairs of nuclei that gives rise to dipolar couplings. Dipolar couplings depend on the inverse third power of the interatomic... 

In order to understand some of the essential features of the RR experiment, it is convenient to assume negligible chemical shielding anisotropy. Under these conditions, o) and e CS-perturbed Zeeman angular frequencies, are independent of time. In addition, it is convenient to use the spherical tensor notation to describe the direct dipolar interaction. Thus, the total Hamiltonian is... 

Figure 5. Schematic representation of three spins, A, B, and C interaeting via direct dipolar contacts (black arrows) and/or spin diffusion (dashed arrows), a) Trace of a 2D NOESY spectrum (right) and a corresponding trace from a 2D ROESY spectrum (left). Spins A and B are close in space, and spin diffusion mediates magnetization transfer between protons A and C, that are not close in space. The 2D ROESY experiment allows unambiguous discrimination between direct and spin diffusion effects, b) Protons A and C are close in space, and in addition magnetization is transferred between the two protons via spin diffusion. The effect is a cancellation of the 2D ROESY signal. A cancellation can also occur if indirect magnetization transfer involves more than one relay proton (proton B in this case). Therefore, a discrimination between spin diffusion and direct dipolar interaction is not possible in this case.

Figure 10. Parts of 2D trNOESY (a), 2D trROESY (b), and 2D QUIET-trNOESY (c) of a-Kdo-(2- 4)-a-Kdo-(2- 0)-allyl bound to mAb S25-2. The QUIET-trNOESY experiment was recorded with a 15 ms double-band selective Q3 inversion pulse (inversion of regions 4.10-3.60 ppm and 2.17-1.67 ppm). Peaks within the inverted regions show an opposite sign (bold lines, c) relative to the other cross peaks outside these regions. The mixing time was 250 ms for all experiments. A comparison of the spectra allows identification of spin-difiiision effects. Cross-peaks that are cancelled in the trROESY spectrum because spin diffusion and direct dipolar interactions take place at the same time (see discussion in the text) are marked with circles in the 2D trROESY spectrum (b). Reprinted with permission from Biochemistry [39).

The presence of neighboring magnetic nuclei alters the local field and therefore the energies of a nucleus. The direct dipolar interaction energy between any two magnetic moments pi and P2 separated by a vector r is... 

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