Dipolar approximation

The options couple and weak denote a scalar coupling interaction. Couple is adequate for a strongly coupled second order spin system and weak for a coupling that can be described by an AX approximation. Dipolar describes a dipolar coupling, only the energy conserving part of the dipolar Hamiltonian are implemented. The qpolar option is reserved for the coupling interaction of a quadrupolar nucleus. Before a nucleus can appear in the interaction statement it must be defined in the nucleus statement. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

To a first approximation, the energy (Up) that arises during the interaction between two dipolar molecules is given by. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

Sinee ir, a, and (3 are approximately normalized seales, the coeffieients s, a. and b are measures of the relative weights of the dipolarity/polarizability, HBD ability, and HBA ability of the solvent. Equation (8-77) has been extended to take aeeount of the eavity effect and certain anomalies, as we will see later in this seetion. 

Of course these requirements cannot be fulfilled simultaneously. For example, a low vapor pressure of the liquid electrolyte is obtained only by using more viscous dipolar aprotic solvents such as propylene carbonate, but high solvent viscosity generally entails a low conductivity. Nevertheless, a large number of useful solvents and electrolytes is available, allowing a sufficiently good approximation to an ideal electrolyte. 

Mohanty et al. were the first to introduce pendent r-butyl groups in die polymer backbones. The resulting material was quite soluble in aprotic dipolar solvents.83 The PEEK precursors were prepared under a mild reaction condition at 170°C. The polymer precursor can be converted to PEEK in die presence of Lewis acid catalyst A1C13 via a retro Friedel-Crafts alkylation. Approximately 50% of die rerr-butyl substitutes were removed due to die insolubility of the product in die solvent used. Later, Risse et al. showed diat complete cleavage of f< rf-butyl substitutes could be achieved using a strong Lewis acid CF3SO3H as both die catalyst and the reaction medium (Scheme 6.15).84... 

The extent of the agreement of the theoretical calculations with the experiments is somewhat unexpected since MSA is an approximate theory and the underlying model is rough. In particular, water is not a system of dipolar hard spheres.281 However, the good agreement is an indication of the utility of recent advances in the application of statistical mechanics to the study of the electric dipole layer at metal electrodes. 

This simple theoryis based on the expectation that, to a reasonable degree of approximation, proton-proton, dipolar contributions to the measured spin-lattice relaxation-rate are pairwise additive and decrease as a simple sixth power of the interproton distance. The simplified version of the dipole-dipole mechanism is summarized in the following two equations for spin i coupled intramolecularly with a group of spins j... 

In this Section we want to present one of the fingerprints of noble-metal cluster formation, that is the development of a well-defined absorption band in the visible or near UV spectrum which is called the surface plasma resonance (SPR) absorption. SPR is typical of s-type metals like noble and alkali metals and it is due to a collective excitation of the delocalized conduction electrons confined within the cluster volume [15]. The theory developed by G. Mie in 1908 [22], for spherical non-interacting nanoparticles of radius R embedded in a non-absorbing medium with dielectric constant s i (i.e. with a refractive index n = Sm ) gives the extinction cross-section a(o),R) in the dipolar approximation as ... 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

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