Quadrupolar potential

Fig. 1.17 The quadrupole mass analyzer is formed by four circular or hyperbolic rods placed in parallel. O Quadrupolar potential.

We have presented a eomparative review of the mean-field theory of different types of eoupled rotors. We have eonsidered planar and linear rotors in dipolar and quadrupolar potentials. 

The component of the DC quadrupolar potential in the z-axis direction is described by the following equation. 

The atomic multipole expansion of the BI electrostatic potential is extremely useful, when the long-range, purely point-multipolar part of the potential yields an important contribution. This is the case in crystals, where the multipolar sums (up to the quadrupolar potential) are conditionally convergent lattice sums. Special techniques, like Ewald summation method [130, 131] and its generalizations [132] are needed to handle properly these infinite sums. Recently we applied the multipolar BI method, coupled with Ewald summation for the evaluation of electrostatic potentials and fields in zeolite cavities [133] and for the prediction of the IR frequency sequence of the different acidic sites in H-faujasite [134]. 

Poniewierski and Stecki [245] combined hard Onsager spherocylinders with an attractive Maier-Saupe type quadrupolar potential. The Lenard-Jones interaction potential often used in isotropic phases with a repulsive term proportional to and an attractive part proportional to r" depends only on the intermolecular distances r, and is therefore not suited to the description of anisotropic fluids. A modification of this isotropic potential that accounts for the relative orientations of anisotropic particles has been proposed by Berne and Pechukas... 

In Eq. (8-7), which is Coulomb s law, the charges are to be aeeompanied with their signs. Because of the high-order reciprocal dependence on distance in Eqs. (8-11) and (8-12), these quadrupolar interactions are usually negligible. For uncharged polar molecules the dipole-dipole interaction of Eq. (8-10), which has the dependence, is the most important contributor to the electrostatie potential energy. 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

A common feature of many mesogenic molecules is the presence of polar substituents and aromatic cores [3]. The electrostatic interactions between such groups can be incorporated into a molecular potential with the addition of dipolar and quadrupolar terms, respectively. Rather than represent these permanent electrostatic interactions by using a model in which a charge distribution is scattered over the surface of the molecule, it is very common to use one (or more) point multipoles [2,29]. Thus for an electrostatic Gay-Berne model, the pair potential is given by the sum... 

The potential of mean torque, parameterized by the local interaction tensors Xa and Xc (assume to be cylindrical) for the aromatic core and the C C segment, respectively, can be mapped out at each temperature by fitting the observed quadrupolar splittings in the mesophase. Furthermore, the order matrix of an average conformer of the molecule can also be calculated.17 Now pn is needed to describe the internal dynamics. 

A quadrupole mass analyzer is made of four hyperbolic or circular rods placed in parallel with identical diagonal distances from each other. The rods are electrically connected in diagonal. In addition to an alternating radiofrequency (RE) potential (V), a positive direct current (DC) potential (U) is applied on one pair of rods while a negative potential is applied to the other pair (Fig. 1.17). The ion trajectory is affected in x and y directions by the total electric field composed by a quadrupolar alternating field and a constant field. Because there is only a two-dimensional quadrupole field the ions, accelerated after ionization, maintain their velocity along the z axis. 

These, and similar data for other systems, demonstrate the tremendous potential that the MICR technique has for the qualitative elucidation of potential energy surfaces of relatively complex organic reactions. Once implementation of the quadrupolar excitation technique has been effected to relax ions to the cell center, the technique will become even more powerful, in that the determination of highly accurate unimolecular decomposition lifetimes of chemically activated intermediates will also become possible. No other technique offers such a powerful array of capabilities for the study of unimolecular dissociation mechanisms and rates. 

Numerous X-ray investigations have unravelled the solid state structure of contact and solvent-separated ion pairs. It was therefore considered to be of interest to evaluate also the potential of solid state NMR as a tool for the investigation of this structural problem. In addition to the study of chemical shifts discussed above (Section II.B), the quadrupole coupling constant of the nuclide Li, x( Li), was expected to be an ideal sensor for the bonding situation around the lithium cation because, due to its dependence on the electric field gradient, the quadrupolar interaction for this spin-3/2 nucleus is strongly influenced by local symmetry, as exemplified in Section II.C.3. This is also shown with some model calculations in Section ILF. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

Plenio and Diodone have also reported fluorocrown ethers (Table 5), which exhibit chemical shift response upon binding Ca + [314]. Of course, calcium could potentially be analyzed directly by " Ca NMR, however, its natural abundance is <0.2%, its sensitivity is <1% that of H, and being quadrupolar, it is liable to extensive line broadening [59]. Thus, the application of F NMR with appropriately designed reporter molecules gives insight into cytosolic [Ca +]. 

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