Electric quadrupole approximation

Results obtained from the alkali iodides on the isomer shift, the NMR chemical shift and its pressure dependence, and dynamic quadrupole coupling are compared. These results are discussed in terms of shielding by the 5p electrons and of Lbwdins technique of symmetrical orthogonalization which takes into account the distortion of the free ion functions by overlap. The recoilless fractions for all the alkali iodides are approximately constant at 80°K. Recent results include hybridization effects inferred from the isomer shifts of the iodates and the periodates, magnetic and electric quadrupole hyperfine splittings, and results obtained from molecular iodine and other iodine compounds. The properties of the 57.6-k.e.v. transition of 1 and the 27.7-k.e.v. transition of 1 are compared. 

The above-mentioned nonlinear optical effects can be described by the perturbation of the electromagnetic held intensity under the electric dipole approximation. Actually, this approximation is broken in optical near-helds. Hence, a perturbation effect of multipole such as electric quadrupole or magnetic dipole should also be considered, although such a higher-order effect is normally negligible. Indeed, electric quadrupole contributions can be comparable with electric dipole contributions... 

Since the d orbitals have quadrupolar character, this distortion must be modelled using a point electric quadrupole with the shape of the d i orbital oriented with a negative pole along the z axis and the positive pole in the x-y equatorial plane (Fig. 8.7(a)). No fluxes have been calculated for Jahn-Teller distorted cations, either with or without the quadrupolar contribution but, as in the other cases, the experimental bond valences are expected to provide a good approximation to the bond fluxes. 

The predominant interaction for a 2H spin system is the quadrupolar interaction, which couples the electric quadrupole moment of the 2H nucleus to its electronic surrounding. This interaction is a second-rank tensor Hq which lies approximately along the C-2H bond in organic molecules. Thus, in practice, 2H nuclei may be considered to be isolated. It shows that the 2H NMR formalism is similar to that of an isolated proton pair [8] ... 

Short of the ab initio calculations, there are several semi-empirical approaches to the calculation and interpretation of electric quadrupole coupling constants. These were developed originally by Townes and Dailey [47, 48] and are well documented in the book by Gordy and Cook [49], They are based on the linear combination of atomic orbitals approximation for molecular orbitals, mentioned earlier in equation (7.266) and described in more detail in chapter 6 ... 

Quadrupole potential. In a second approximation, equation (45) yields, with equations (35) and (40), the potential due to an electric quadrupole in the form ... 

The NMR lineshape of solids is determined by the quadrupolar interaction, which can be described by two parameters the quadrupole frequency, o Q, and the asymmetry parameter, t/ (19,20). The parameter q)q is determined by the electric quadrupole moment of the deuteron and the zz component of the electric field gradient at the deuteron site. For deuterons bonded to carbon atoms, the asymmetry parameter is approximately zero and the z axis is along the C—D bond. In this case, the dependence of the resonance frequency, m, from the orientation of the molecule with respect to the magnetic field applied is given by a relation similar to Eq. (18) (19). 

Here p( ) is the electric dipole (El) operator for molecule c located at position R , Qij(c) is the corresponding electric quadrupole (E2) operator, and m( c) is the magnetic dipole (Ml) operator. The diamagnetization does not contribute to this order of approximation. We also recognize in Eqs. (12) and (13) the microscopic transverse displacement electric field, dx, whose quantum operator form will be discussed in the next section. Explicit expressions for the components of the... 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

All Stable chlorine, bromine and iodine nuclei possess an electric quadrupole moment and in non-symmetric environment their NMR spectra are usually broad and cannot be studied by ordinary high resolution techniques. Notable exceptions are halide ions in aqueous solution where the NMR signals become reasonably narrow (approximate line widths at half height are = 8 Hz for Cl and 5 Hz for Cl... 

The interaction between groups is thus approximated by the electrostatic interaction between point dipoles. The values of the dipoles can be either the transition moments or the permanent moments of the groups. However, for the electric dipole forbidden transitions, i,oa is zero and in this approximation we would need to use the next term involving the electric quadrupole transition... 

The second-rank tensor P,y (m) depends on the velocity dipole operator, while Mif(co) depends on the velocity dipole operator and on the magnetic dipole operator and finally T (co) on the velocity dipole operator and the velocity form of the electric quadrupole operator, respectively. Their mathematical expressions are reported and described in detail in Chapter 2. Once more, like we did for TPA, invoking the BO approximation and integrating over the electronic coordinates, the TPCD intensity between vibronic states can be written in terms of elements of electronic transition tensors Pej/it, rj, co), Me,f x, rj, co), and T yr(x, rf, co) between the vibrational states and Z5(/)) associated with the initial and final electronic states 0,) and 0/), respectively. 

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