Dipole approximation for

We consider the expression of the lab frame photoelectron angular distribution for a randomly oriented molecular sample. The frozen core, electric dipole approximation for the differential cross-section for electron emission into a solid angle about a direction k can be written as... 

Nonvanishing Components of Second-Order Susceptibility Tensor for Second-Harmonic Generation in Electric-Dipole Approximation for Achiral and Chiral Isotropic (i.e. isotropic in the plane of the film) Films0... 

In this section, we discuss the work inquiring into the meaning of r/s, the distance between the two dipoles in Eqs. (12) and (13). The simplest approximation is to assume that r/s is equal to the internuclear distance between the nucleus in the ligand, the relaxation of which is being studied, and the metal ion. This amounts to the point-dipole approximation for both the nuclear and the electron spins. While such an approximation is perfectly... 

A compilation of data relevant for a discussion of the dipole approximation for photoionization processes in neon is given in Table 8.1. Since , should be small compared to reference, it can be seen that the approximation is well justified for excess photon energies of the order of the binding energy of the active nAelectron, and this statement also holds for other systems (see, however, [KJG95]). 

Discrete dipole approximation. For particles with complex shape and/or complex composition, presently the only viable method for calculating optical properties is the discrete dipole approximation (DDA). This decomposes a grain in a very big number of cubes that are ascribed the polarizability a according to the dielectric function of the dust material at the mid-point of a cube. The mutual polarization of the cubes by the external field and the induced dipoles of all other dipoles is calculated from a linear equations system and the absorption and scattering efficiencies are derived from this. The method is computationally demanding. The theoretical background and the application of the method are described in Draine (1988) and Draine Flatau (1994). 

W. H. Yang, G. C. Schatz, and R. P. Vanduyne. Discrete Dipole Approximation for Calculating Extinction and Raman Intensities for Small Particles with Arbitrary Shapes Journal of Chemical Physics, 1995, 103, 869-875. 

Yang, W. -H., Schatz, G. C., and Van Duyne, R. P. (1995). Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes. J. Chem. Phys. 103 869-875. 

We consider interaction of the rectilinear isolated single-wall CNT of the infinite length with the harmonic electromagnetic field of a frequency coq such that 2ncl( >o Rc (CNT radius). We assume that the field is incident normally on the CNT surface and the electric field vector is polarized along the CNT axis. We restrict our consideration to the dipole approximation for the n-electrons interaction with the electromagnetic field. Then the Von Neiman equation can be presented as follows [5]... 

For a carbonyl group in a Cjv symmetry environment, such as in formaldehyde, the dipole approximation for the n- r transition yields JWo f > 0 and Mo f = 0. Although the transition is magnetically strongly allowed and polarized along the CO axis, it is electrically forbidden. The absorption therefore is of very weak intensity and the rotational strength is equal to zero. Perturbations by vibrations or by an achiral solvent can affect and to such an extent that a small nonzero electric dipole transition moment results but again this produces only a very small absorption intensity and... 

The dinitroxide (20) is one of several nitroxide biradicals used to test the point-dipole approximation for electron-electron dipolar interactions/ ... 

Draine, B.T. and Flatau, P.J. (1994) Discrete-dipole approximation for scattering calculations. Journal of the Optical Society of America A, 11, 1491-1499. 

In the crystal containing molecules with a center of inversion and only in the dipole approximation for the intramolecular interaction, the parts of the total Hamiltonian H4 = H = He = 0, and at the same time the value T>f, is equal to zero. However, it is known that the value Df determining the so-called gas-condensed matter shift even exists in crystals with center of inversion (and thus due to the contribution of the nondipole intermolecular interaction) and can be of the order of 0.1 eV (in anthracene crystals the value of the gas-condensed matter shift is about 2000 cm-1). This indicates that even in centrosymmetric crystals the terms in the total Hamiltonian H4, Hs, He may be neglected only with some care. 

Davydov splitting for exciton resonance in anthracene, and for the first time obtained reasonable agreement with available experimental data. He used a dipole approximation for the intermolecular interaction and the only ingredients in his theory were the resonance frequencies and oscillator strength. In contrast to quantum theory described in this chapter the classical dipole theory does not take into account the contribution of the nondipole interaction, which are important in the majority of solids. It is clear that also multiexciton states including states with few quantum of excitations on the same molecule (what is forbidden for the two-level model) in classical harmonic oscillator theory contribute to the energy of excitons. However, in the framework of the classical theory it is impossible to develop the estimation of corrections which we discussed here. 

Results are given for k = 0 and several different directions of k. For purposes of comparison, the values calculated using the dipole approximation for all interactions with k perpendicular to the face (001) are also given in Table 3.5. Below, to give an impression of the state of art, we mention in some detail, for... 

If molecules of a crystal have no inversion center, the static dipole moments of a molecule in the ground and in the excited states are usually not equal to zero. In this case and in the dipole-dipole approximation for the expression (15.16) can be reduced to the interaction energy of dipoles Ps and Ps<... 

B. T. Draine and P. J. Flatau, The Discrete-Dipole Approximation for Scattering Calculations, Journal of Optical Society of America A, 11,1491-1499,1994. 

Estimation of the relative intensity of the autoionization process [27] has shown that the second-order process intensity constitutes 10-15% of the first-order process intensity. Based on the results obtained and the dipole approximation for electron transitions [28], the authors of [27] draw the conclusion that the SEFS structure is formed in the process of coherent scattering of a secondary electron emitted from the valence band as a result of excitation by an incident electron (the first-order process). By contrast, the intensity of autoionization, i.e., the second-order process, was estimated [29-31] with hydrogenlike wave functions. The autoionization intensity in the region of the existence of the SEFS spectrum was shown to be comparable to the intensity of the first-order processes. 

The intensity of the interference term in Eq. (23) is determined by the angular correlation function The use of the dipole approximation for the... 

Estimates of the Angular Correlation Functions of the Second-Order Process. The use of the dipole approximation for describing the core level ionization, i.e., the use of matrix elements [Eqs. (61), (66), (67)], leads to the dipole selection rules for the second-order processes too. Thus the angular correlation function of the final state [Eq. (33)] determined by T p) and T p)... 

Taking into account the scattering in the intermediate state in the second-order process gives rise to an interference term whose intensity is determined by the angular correlation function of the intermediate state [Eq. (32)]. With the obtained amplitudes of the core level ionization and core-hole annihilation and the amplitude of the second-order process, we make an estimate of the angular correlation function of the intermediate state. The dipole approximation for the amplitude of core level ionization defines the angular correlation function by the following ... 

In the Drude polarizable model, the only relevant adjustable parameter is the combination q /KD that corresponds to the atomic polarizability. In the limit of large Kd, the treatment of induced polarization based on Drude oscillators is formally equivalent to a point-dipole treatment such as used by AMOEBA. In practice, the magnitude of Kd is commonly chosen to achieve small displacements of Drude particles from their corresponding atomic positions, as required to remain close to the point-dipole approximation for the induced dipole associated with the atom-Drude pair [150] while preserving a stable integration of the equation of motion with a reasonable time step. For a fixed force constant Kd the atomic polarizability is determined by the amount of chaise assigned to the Drude particle. In the current implementation, the classical Drude model introduces atomic polarizabilities only to non-hydrogen atoms for practical considerations, as discussed below. However, this is adequate to accurately reproduce molecular polarizabilties, as seen in a number of published studies [127,142,146]. 

The distances of interest in pulsed ESR are usually sufficiently long to use the point dipole approximation for uncorrelated spins described in the section above. The magnitude of the coupling is then proportional to R where R is the electron-electron distance. By measuring the constant Du the distance R is accordingly obtained. 

The local structure of Cu(I)-NO adsorption complexes formed over Cu-L and Cu-ZSM-5 zeolites were studied by pulsed ENDOR and HYSCORE methods by Umamaheswari et al. [53]. The H ENDOR signals from residual distant protons were not detected in completely copper ion exchanged Cu-ZSM-5 zeolites. Such signals were, however, observed for the Cu-L zeolite, where the H form of the zeolite was 30% ion exchanged with Cu(II) ions and subsequently dehydrated to (auto)reduce Cu(II) to Cu(I). For both systems, very broad A1 ENDOR spectra were observed. The Al hf couplings were estimated using the point dipole approximation for the Cu(I)-NO center in Cu-L. The result shows that an aluminum framework atom is located in the third coordination sphere with respect to the NO molecule adsorbed on a Cu(I) cation site. 

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