Dynamic polarizability tensor

The energy-dependent — r r polarization propagator is thus the dynamic polarizability tensor, i.e. the induced electric dipole moment in units of an external electric field. 

The response to frequency-dependent external fields may be obtained from Hartree-Fock response theory, yielding dynamical polarizabilities and hyperpolarizabilities. The identification of excitation energies as the poles of the dynamical polarizability tensor may be invoked to calculate excitation energies as well as one-photon and two-photon transition moments from the time development of the ground state [40-42]. 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

Lastly, we consider the diffusive contribution to the signal. Since this portion of the signal arises from molecular reorientation, it should be completely depolarized unless these diffusive reorientational dynamics also have a significant DID component. The orientational decay will be made up of exponential components, the number of which depends on the molecular symmetry and the relationship between the principal axes of the diffusion and polarizability tensors of the molecules (8). If these tensors share no axes, the orientational decay will be composed of a sum of five exponentials. If the tensors share one axis, the decay will be composed of three exponentials. If the tensors share all three axes, the decay will be composed of two exponentials. If the molecule is further a symmetric top, then reorientation about the axis of symmetry cannot be observed, and the decay will be composed of a single exponential. In principle, considerably more information is available when the principal axes of the diffusion and polarizability tensors are not shared however, in practice it is virtually impossible to find a unique fit to the sum of five exponentials, some of which may have very small amplitudes. In the remainder of this chapter we will therefore concentrate on symmetric-top liquids. 

It is possible to express the second-order induction energy in terms of the multipole moments of any small molecules involved and their static polarizability tensors " - but further simplification is difficult for a pair of polyatomic subunits. A similar analysis permits the dispersion to be placed within the context of dynamic polarizabilities. In the case of a pair... 

The polarizability of an atom or molecule describes the response of the electron cloud to an external field. The atomic or molecular energy shift KW due to an external electric field E is proportional to i for external fields that are weak compared to the internal electric fields between the nucleus and electron cloud. The electric dipole polarizability a is the constant of proportionality defined by KW = -0(i /2. The induced electric dipole moment is aE. Hyperpolarizabilities, coefficients of higher powers of , are less often required. Technically, the polarizability is a tensor quantity but for spherically symmetric charge distributions reduces to a single number. In any case, an average polarizability is usually adequate in calculations. Frequency-dependent or dynamic polarizabilities are needed for electric fields that vary in time, except for frequencies that are much lower than electron orbital frequencies, where static polarizabilities suffice. 

The dynamic dipole polarizability tensor a can be expressed through eq. (1.55) when both A and B operators are the electric dipole moment... 

In this section, we shall summarize a study on dynamical polarizability and hyper polarizability tensors a, P, and 7 of urea in vacuo and in water we have published on the Journal of Molecular Structure (Theodxem). Urea is a compound that has been thoroughly investigated both experimentally and... 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

In the same way, we have calculated the static and dynamic polarizabilities for the MNA clusters described in the previous section. For the dynamic part, fKB =. 6 eV which corresponds to a wavelength of 1064 nm. The results for the longitudinal component of the polarizability tensor (ofaa) are reported in Tables VI and VII for ID arrays, and in Tables VIII and IX for 2D and 3D clusters. Tables X, XI, XII, and Xin list the results for the other diagonal t sor components in 2D and 3D clusta. ... 

It is seen from Fig. 3.61 that the SEIRA spectrum is identical to the one obtained by IRRAS. Only the synunetric (ai) modes (1352 and 1413 cm for CO2 and NO2 groups, respectively) appear, while the antisynunetric ((>1) modes (1528 and 1592 cm", respectively) are practically absent in both spectra. This can be understood assuming that p-NBA is adsorbed at the Ag surface as the p-nitrobenzoate ion with its C2 axis normal to the metal surface, as sketched in Fig. 3.62, provided charge transfer does not occur [390]. If this is the case, the dynamic dipole moment of the symmetric and antisymmetric CO2 and NO2 stretches is directed perpendicular and parallel to the metal surface, respectively. From this observation, Osawa and Yoshii [361] concluded that the surface selection rule (SSR) for metal surfaces is also valid for SEIRA, which was confirmed by Zhang et al. [367], In fact, analysis of the polarizability tensor of a molecule adsorbed on a metal particle [392, 394] has confirmed the dominance of the component, where the z-axis is normal to the surface at the adsorption site. Greenler et al. [395] have performed both classical and quantum-mechanical calculations for the interaction of the electromagnetic field with particles of varying sizes and arrived at the conclusion that the SSR should only be applied to... 

With optical techniques, vibrational dynamics are probed on spatial scales much greater than molecular sizes, or unit cell dimensions in crystals, commonly encountered. The scale is directly related to the wavelength of the incident radiation (in the range fl om 1 to 10(X) p,m in the infrared or about 0.5 p,m for Raman). Oscillators at very short distances, compared to the wavelength, are excited exclusively in phase. For molecules, only overall variations of the dipole moment or polarizability tensor can be probed. In crystals, only a very thin slice of reciprocal space about the centte of the BriUouin zone (k 0) can be probed. This corresponds to in-phase vibrations of a virtuaUy infinite number of unit cells. With optical techniques, band intensities are largely determined by symmetry-related selection rules, although these rales hold only in the harmonic approximation. 

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