Three-dimensional dipolar system

To evaluate the third contribution to in Eq. (F.56), we realize from Eq. (6.17a) that 

In writing the first term on the right side of Eq. (F.68) we introduced the projection of the total dipole moment M [see Eq. (6.13)] onto the 7-axis, namely 

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F.2 Three-dimensional dipolar system F.2.1 Self-energy... 

However, in all the papers mentioned above the authors analyzed only three-dimensional (3D) systems, while a two-dimensional (2D) case is also experimentally observed surfaces of various absorbers, heterogeneous catalysts, photocatalysts, etc. In [137], Fel dman and Lacelle examined the quenched disorder average of nonequilibrium magnetization, i.e., a free induction decay G(t) and its relative fluctuations for dipolar coupled homonuclear spins in dilute substitutionally disordered lattices. The studies of NMR free induction decays and their relative fluctuations revealed that the functional form of the disorder average (G(t))c depends on the space-filling dimentionality D of the lattice. Explicit evaluations of these averages for dilute spin networks with D = 1, 2, 3 were presented in [137] ... 

The critical exponents a, j8, y, 5, are expressed in the perturbation expansion in terms of the parameter e=d -d, and the expansion coefficients depend upon an order parameter dimensionality. In the diluted systems d = 6, in tricritical points and in three-dimensional dipolar-coupled Ising ferromagnets d = 3. Among lanthanide compounds uniaxial and... 

The high-resolution and the high scaling factor of 2D PISEMA, for the first time, enabled the use of the dipolar dimension to resolve resonances from non-selectively or uniformly labeled proteins. Three-Dimensional experiments were used to enhance the resolution of resonances from uniformly N-labeled peptides and proteins embedded in lipid bilayers. This was successfully demonstrated on aligned samples containing uniformly N-labeled membrane-associated peptides and proteins. Two-dimensional PISEMA spectra of some of these systems showed limited resolution due to a small frequency dispersion of resonances from a-helices oriented on the surface of the bilayer in both N chemical shift and H- N dipolar coupling dimensions. However, when an additional H chemical shift dimension was invoked, the 3D H chemical shift/ H- N dipolar coupling/ N chemical shift spectra of these systems considerably increased the resolution of peaks. ... 

Regarding the use of Eq. (6.26) in practice we note that the same comments made earlier apply here as well [see discussion after Eq. (6.17)]. A detailed discussion of optimal choices for the Ewald parameters a and for dipolar systems can bo found in Rc fs. 243 and 244. Finally, readers who are interested in performing MD simulations of dipolar fluids are referred to Appendix F.2.2 where we present explicit expressions for forces and torques associated with the three-dimensional Ewald sum [see Eq. (6.26)]. Moreover, explicit expressions for various components of the stress tensor can be found in Appendix F.2.3. 

Thus, for both the ionic and the dipolar systems, the actual use of the rigorously derived Ewald summation for slab systems loads to a substantial increase in computer time. One way of dealing with this problem would be to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable interpolation scheme. Another strategy is to employ approximate methods such as the one presented in the subsequent Section 6.3.2. 

The entire system is then placed into a medium of infinite dielectric constant that prevents formation of surface charges at the outer boundaries. Thus, we can employ the conventional three-dimensional Ewald sums [see Eqs. (6.15) and (6.26) for the Coulombic and dipolar case, respectively] with tin-foil boundary conditions (i.e., r/ — c ). 

For both situations, we calculate the (dimensionless) dipolar energy per particle, a Ux)/fj, N, via the slab-adapted three-dimensional Ewald sum [see Eq. (6.44)] and with the rigorous Ewald method for dipolar slab systems... 

Figure 6.2 Dimensionless energy per particle for dipolar crystalline (fee) slabs as a function of the number of lattice layers, assuming perfect order along the ar-axis (a) and along the z-axis (b). Included are results from direct summation (O), the rigorous Ewald sum for slab systems (A) (sec Appendix F.3.1.2], and the slab-adapted three-dimensional Ewald sum (x) [see Eq. (6.44)j. Part (b) additionally includes results from the latter method when the correction term [see Eq. (6.43)) is neglected ( ).

For dipolar particles, all for( e contributions within the slab-adapted three-dimensional Ewald sum coincide with those for truly three-dimensional systems discussed in Appendix F.2.2. This result arises because the correction term to the total dipolar energy [see Eqs. (6.44) and (6.43)] is independent of particle positions. There is, however, a contribution to the total torque that we need to consider separately. The total torque can be cast as... 

The principle purpose of correlation experiments is to establish a one-to-one mapping from the signal to its source i.e. to the particular atomic nucleus in the molecule. This assignment task involves identification of the members in the coupling network, referred to as the spin system. In addition, correlation experiments, as such or with modifications, are suitable for measurements of scalar and dipolar couplings. Correlation in the two dimensions is the most natural dimensionality because the spin-spin interactions are pair wise. Three-dimensional or experiments of higher dimensionality are constructed from concatenated two-dimensional experiments. Homonuclear three-dimensional experiments, such as TOCSY-NOESY, are not considered here because in many cases the multidimensional heteronuclear experiments are superior. 

A combination of circular dichroism, sodium dodecyl sulfate-polyacrylamide gel electrophoresis, chemical crosslinking, and analytical ultracentrifugation studies showed that both the apo- and metallated derivatives of H21(31-mer) form two-stranded a-helical coiled coils in aqueous solution. Further characterization of these derivatives by EPR spin-label experiments helped to determine its three-dimensional backbone structure. In these studies, a Cys-21 mutant of the 31-mer coiled coil, H21/C21(31-mer), was prepared and labeled with a thiol-specific nltroxide spin label (MTSL = l-oxyl-2,2,5,5-tetramethyl-A -pyrroline-3-methyl-methanethiosulfonate) at position 21 of the peptide sequence which is the site of metal substitution in the ET heterodimer. Comparison of the low-temperature, dipolar-broadened spectrum of the spin-labeled dimer with those of magnetically dilute peptide samples yielded a backbone-to-backbone distance that was nearly identical to that of the GCN4 homodimer. Based on these results, computer modeling studies provided an estimate of the metal-to-metal distance in the ET heterodimer of m-m > 25 A. The electron-transfo properties of this system are now being studied by a combination of laser flash-quench and pulse radiolysis techniques. 

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