Quadrupoles and dipoles

Note that two distinct types of interactions (ion-quadrupole and dipole-dipole) contribute to an overall R 3 dependence, and the number of distinct multipole types having similar R n dependences continues to increase with increasing n. For uncharged systems, the dipole-dipole interaction (5.23d) is expected to dominate, with an angular term that favors parallel alignment of the two dipoles. 

Since more than one rf interval has been measured, it is, in principle, possible to determine both the quadrupole and dipole polarizabilities from the microwave resonance data alone. The procedure is identical to that used earlier for alkali atoms,15 except that the factors kd and kq must be introduced. If we define... 

In the region of negligible overlap, the three-body interactions can be calculated, just like the two-body ones, from imaginary frequency polarizabilities. This has been done, for example, by Doran78 for interaction of triplets of noble-gas atoms. He gives the coefficients for dipole-dipole-dipole, dipole-dipole-quadrupole, dipole-quadru-pole-quadrupole, quadrupole-quadrupole-quadrupole, and dipole-dipolc-octupole terms. 

Vibrational product state distributions have been obtained for reactions studied in crossed molecular beams using the technique of beam electric resonance spectroscopy [109]. This method uses the focusing action of electric quadrupole and dipole fields to measure the radio frequency Stark spectrum of the reaction products, which must possess a dipole moment. This has restricted this technique to reactions producing alkali halides. 

Curran et al. [ 16] conclude their analysis of the 1A NF spectrum by comparing the magnitudes of the magnetic hyperfine, quadrupole and dipole moment parameters with predictions from self-consistent field calculations due to O Hare and Wahl [19]. The agreement is satisfactory, and one supposes that contemporary calculations would be in even better agreement with experiment. One of the main purposes of the measurements and spectral analysis described in this section is to provide accurate benchmarks for theoretical calculations, and also physical insight into the nature of the molecular bonding. 

Most separated local field experiments concentrate on the CSA. In speetra of halfinteger nuclei with / > 1/2, the CSA is often much less important than the second-order quadrupole interaction. Static or MAS spectra display a typical quadrupole powder pattern from which the quadrupole coupling constant and the asymmetry parameter can be determined. These quadrupole parameters can then be used to determine the principal components of the electric field gradient tensor, which are related to the local environment of the quadrupole nucleus. Knowledge of the local geometry is extremely valuable, and correlation of the quadrupole and dipole information of OH groups provides useful insights into structure. 

Experimental measurements of Aie BE can also be used to determine quadrupole moments for dipolar systems. The values obtained in this way refer to an origin which is denoted as the effective quadrupole center (EQC) [135]. The EQC is defined as Aie point at which the combinaAon of dipole-quadrupole and dipole-magnetic dipole polarizabilities multiplied with the dipole moment given in Eq. (27)... 

The symmetry of linear molecules imposes less constraints on the multipole polarizability tensors than does tetrahedral symmetry, so the forms of the respective tensors for linear molecules are usually more highly complicated than for tetrahedral ones. This certainly complicates interpretation of light scattering results but has much less influence on complications in running our program. By means of the same methods as in the case of tetrahedral molecules we calculate the spherical irreducible components of the dipole-dipole, dipole-quadrupole, and dipole-octopole polarizability tensors as dipole-dipole... 

Fig. 7 Middle panel — ferrous (left) and ferric (right) pre-edge features in the Fe K-edge including the relative contribution of (quadrupole and dipole) to the intensity at the bottom of each spectrum, upper Oh and lower Td. The states responsible for the splitting are indicated on the Tanabe-Sugano diagrams d7 left final state of Fe(II), d6 right final state of Fe(III). Altered from [15], with permission, Copyright American Chemical Society, 1997...

If the existence of the stationary signal at time >Ti is not determined by the interactions between quadrupole and dipole-dipole reservoirs but by the spin-lattice relaxation, we can neglect processes in which the spins absorb the quanta of the dipole-dipole interaction modulated by the RF field and can omit corresponding terms in the total Flamiltonian of the spin system. Such reduced total Hamiltonian coincides with the effective Hamiltonian Hgff in Equation (37). 

For purely quadrupolar and dipole-quadrupole systems, (3.11) and (3.12) must be solved numerically. This is briefly discussed in Section III.B.3. We note that for purely quadrupolar fluids (p = 0), only the four projections (000), (220), (222), and (224) occur in the MSA solution. MSA results for dipole, quadrupole, and dipole-quadrupole fluids have been compared with MC calculations, and in general the agreement is very poor. Several examples of this are given in Section III.D. 

This situation can be compared [153-155] to N2O on graphite, a molecule which has slightly larger quadrupole and dipole moments and a solid phase which is structurally very similar to the bulk a-CO solid. There is a residual entropy of about In 2 at low temperatures, and thus the bulk N2O solid does not undergo a head-tail ordering transition [153, 155]. Also no heat capacity anomaly was detected for N2O on graphite [153, 155] down to 3-4 K, which could be rationalized [153] in terms of the striking difference in the dynamical behavior of the N2O and a-CO solids [251]. 

The ratio between dipole-quadrupole and dipole-dipole transition probabilities is given in Ref. 66) as... 

G. N. Patey and J. P. Valleau, Fluids of spheres containing quadrupoles and dipoles A study using perturbation theory and Monte Carlo computations, /. Chem. Phys. 64,170 (1976). 

Using the perturbation theory proposed by Larsen et al. Karakatsani and Economou have extended the PC-SAFT equation to account for dipole-dipole, dipole-quadrupole, quadrupole-quadrupole and dipole-induced dipole interactions. The exact second- and third-order perturbation terms in the work of Larsen are however rather complex and so a simplified version of the two terms was also proposed based on the work of Nezbeda and co-workers. While simplifying the equation, in an effort to generate a more usable, engineering-type, approach, the simplification of the dipolar term in the model introduces an additional pure-component model parameter. We refer to these as the PC-SAFT-KE and truncated PC-SAFT-KE (denoted PC-PSAFT and tPC-PSAFT by the authors respectively) approaches. In both KE approaches, the multipoles are assumed to be uniformly distributed over all segments in the molecule. The truncated and full PC-SAFT-KE approaches have been applied to study a wide range of polar fluids and their mixtures, " with the truncated equation found to be as accurate as the full PC-SAFT-KE model, and the inclusion of polar interactions found to improve the theoretical predictions in most cases. ... 

G. Maroulis, Dipole-quadrupole and dipole-octopole polarizability for CH4 and CF4. 

G. Maroulis, Dipole-quadrupole and dipole-octupole polarizability for CH4 and CF4. J. Chem. Phys. 105(18), 8467-8468 (1996)... 

Maroulis and Hohm have calculated the dipole-dipole, dipole-quadrupole, and dipole-octopole polarizabilities of Ge(CH3)4 using the FF procedure at the HF, MP2, and B3LYP levels of approximation. From comparison with their data deduced from collision-induced light-scattering experiments, they have concluded that reliable dipole-dipole polarizability values are obtained, that good agreement is achieved for the dipole-octopole polarizability but that there remains a discrepancy for the dipole-quadrupole polarizability (Table 3). 

Maergoiz et al. [28-30] performed classical trajectory calculations for ion-dipole, ion-quadrupole and dipole-dipole collisions, deriving results for capture rate coefficients and expressing them in terms of two reduced parameters, the reduced temperature, d, closely related to the Tr parameter in the work of Chesnavich and co-workers, and the Massey parameter, which is equal to the ratio of the coUisional timescale to the rotational period of the neutral. 3> 1 corresponds to the adiabatic limit (see below). They gave parameterized expressions for k /ki, similar to those given by Su and Chesnavich [26], but extending the range of validity. 

Though the calculated values are rough indications, we can see from the table that the same effect that caused the van der Waols attraction in noble gases, also for the simple molecules yields a contribution which is already of the same order of magnitude as the total attraction. The averaged charge distributions in the molecule (quadrupole and dipole moment respectively) have not even been taken into account. 

The quadrupole and octopole moments, the dipole polarizability, the first and second dipole hyperpolarizabilities, and the dipole-quadrupole and dipole-dipole-quadrupole polarizabilities were calculated using MPPT(4) theory [8]. 

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