Quadrupole approximation

Raab [30] has shown the inadequacies of this procedure, which, in the case of dynamic fields, cannot be extended beyond the quadrupole terms and may lead to inconsistencies with the interaction Hamiltonian (7). Accordingly, we retain here definition (34) within the limits of the quadrupole approximation. To higher order only the set of nontraceless moments (22)-(27) can be properly used. 

Within the quadrupole approximation, the terms entering the first-order Hamiltonian (15) are... 

In the case of monochromatic radiation and within the quadrupole approximation, Eqn. (65) can be recast in the form... 

Within the quadrupole approximation, the magnetic field is irrotational and the Hermitian operator representing the Lorentz magnetic force on the electrons is... 

To some purposes, one can define new molecular tensors that are independent of the origin [40]. At any rate, it can be easily proven that the induced moments (69)-(72) and fields (73)-(74) are, order by order, independent of the origin chosen for the multipole expansion, provided that all the terms of the same order of magnitude are retained. Thus, within the quadrupole approximation, both the magnetic field and the electric field gradient must be taken... 

Thus, in the approximation of the quadrupole electrostatic interaction described by f Ft, the centers of gravity of the pairs, when drawn as a function of h, lie on two parabolas, one for each value of j, the minima of both parabolas occur at h = —i these are drawn in fig. 1.39 with Fo= 124238 cm , = 643cm and Fz = 0.325 cm , and with the experimental pairs for comparison. Both the observed values for the centers of gravity of the pairs, and their calculated values in the quadrupole approximation, are listed in table 1.44. It can be seen... 

Next, we will not consider terms higher than the second, i.e. the dipole term. This is in accordance with the electric quadrupole approximation, which is based on the fact that the effects of electric quadrupole and magnetic dipole terms are of the same order of magnitude and smaller than the electric dipole terms. They should therefore be treated together, as we do here, while second-order magnetic moments should be treated together with third electric moments, which we have not discussed either. The remaining (dipole) term... 

Within the quadrupole approximation, the rate of energy loss from a source is therefore given by... 

I is the bond length. The experimental quadrupole moment is consistent with a charge, q, of approximately 0.5e. In fact, a better representation of the electrostatic potential around the nitrogen molecule is obtained using the five-charge model shown in Figure 4.20. 

The charge distribution of the molecule can be represented either as atom centred charges or as a multipole expansion. For a neutral molecule, the lowest-order approximation considers only the dipole moment. This may be quite a poor approximation, and fails completely for symmetric molecules which do not have a dipole moment. For obtaining converged results it is often necessarily to extend the expansion up to order 6 or more, i.e. including dipole, quadrupole, octupole, etc. moments. 

A comparison of MP2/6-31G structural parameters of 1,2-oxazole 19 (isox-azole) and 1,3-oxazole 20 with microwave data is provided by Kassimi et al. (Scheme 16) [96JPC8752]. The general agreement is excellent. The same authors investigated dipole moments, quadrupole moments, octopole moments, and dipole polarizabilities of 19 and 20 together with several oxadiazoles and oxatriazoles [96JPC8752, 99JPC(A) 10009]. For the mean polarizability of these species, they found the approximative formula... 

While mathematically attractive, this force law is of limited interest physically it represents only the interaction between permanent quadrupoles, and even this with neglect of angles of orientation. However, although the details of the dependence of viscosity upon temperature are affected by the force law used, the general form of the hydrodynamic equation in the Navier-Stokes approximation is not affected. 

For example, octahedral quadrupole splitting observed for the cis-octahedral analogs 7,8). More recently, temperature-dependent Mossbauer measurements have been used in conjunction with Raman spectroscopy to determine molecular weights 453) and lattice rigidity 460) of various organotin compounds. 

Microwave spectra obtained from PHgD and PHDa in a magnetic field of about 25 kG showed Zeeman effects, from which molecular g values were calculated. They were 20 times smaller than those for ammonia. The molecular quadrupole moments of phosphine and ammonia were approximately the same. Magnetic susceptibilities and molecular quadrupole moments were also compared. 

Foyt et al. [137] interpreted the quadrupole-splitting parameters of low-spin ruthenium(II) complexes in terms of a crystal field model in the strong-field approximation with the configuration treated as an equivalent one-electron problem. They have shown that, starting from pure octahedral symmetry with zero quadrupole splitting, A q increases as the ratio of the axial distortion to the spin-orbit coupling increases. 

A triple-quadrupole mass spectrometer with an electrospray interface is recommended for achieving the best sensitivity and selectivity in the quantitative determination of sulfonylurea herbicides. Ion trap mass spectrometers may also be used, but reduced sensitivity may be observed, in addition to more severe matrix suppression due to the increased need for sample concentration or to the space charge effect. Also, we have observed that two parent to daughter transitions cannot be obtained for some of the sulfonylurea compounds when ion traps are used in the MS/MS mode. Most electrospray LC/MS and LC/MS/MS analyses of sulfonylureas have been done in the positive ion mode with acidic HPLC mobile phases. The formation of (M - - H)+ ions in solution and in the gas phase under these conditions is favorable, and fragmentation or formation of undesirable adducts can easily be minimized. Owing to the acid-base nature of these molecules, negative ionization can also be used, with the formation of (M - H) ions at mobile phase pH values of approximately 5-7, but the sensitivity is often reduced as compared with the positive ion mode. 

We assume that, on formation of B- XY, a fraction 5j (i = intermolecular) of an electronic charge is transferred from the electron donor atom of Z of the Lewis base B to the npz orbital of X and that similarly a fraction 5p (p = polarisation) of an electronic charge is transferred from npz of X to n pz of Y, where z is the XY internuclear axis and n and n are the valence-shell principal quantum numbers of X and Y. Within the approximations of the Townes-Dailey model [187], the nuclear quadrupole coupling constants at X and Y in the hypothetical equilibrium state of B- -XY can be shown [178] to be given by ... 

Each of the two beamlines are focussed by a single electromagnetic quadrupole doublet, and a beam size of about 1 pm2 is achieved with a beam current of approximately 50 pA. Electrostatic deflectors are mounted to permit beam sweeping on the sample both to produce images and to limit heating effects. 

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