Dipole approximation polarizability

We have described our most recent efforts to calculate vibrational line shapes for liquid water and its isotopic variants under ambient conditions, as well as to calculate ultrafast observables capable of shedding light on spectral diffusion dynamics, and we have endeavored to interpret line shapes and spectral diffusion in terms of hydrogen bonding in the liquid. Our approach uses conventional classical effective two-body simulation potentials, coupled with more sophisticated quantum chemistry-based techniques for obtaining transition frequencies, transition dipoles and polarizabilities, and intramolecular and intermolecular couplings. In addition, we have used the recently developed time-averaging approximation to calculate Raman and IR line shapes for H20 (which involves... 

Abstract Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. 

The discontinuity of the interface leads to two contributions to the second order nonlinear polarizability, the electric dipole effect due to the structural discontinuity and the quadrupole type contribution arising from the large electric field gradient at the surface. Under the electric dipole approximation, the nonlinear susceptibility of the centrosymmetric bulk medium 2 is zero. If the higher order magnetic dipole... 

By taking as a reference the calculation in vacuo, the presence of the solvent introduces several complications. In fact, besides the direct effect of the solvent on the solute electronic distribution (which implies changes in the solute properties, i.e. dipole moment, polarizability and higher order responses), it should be taken into account that indirect solvent effects exist, i.e. the solvent reaction field perturbs the molecular potential energy surface (PES). This implies that the molecular geometry of the solute (the PES minima) and vibrational frequencies (the PES curvature around minima in the harmonic approximation) are affected by the presence of a solvating environment. Also, the dynamics of the solvent molecules around the solute (the so-called nonequilibrium effect ) has to be... 

The dielectric constant is a macroscopic property of the material and arises from collective effects where each part of the ensemble contributes. In terms of a set of molecules it is necessary to consider the microscopic properties such as the polarizability and the dipole moment. A single molecule can be modeled as a distribution of charges in space or as the spatial distribution of a polarization field. This polarization field can be expanded in its moments, which results in the multipole expansion with dipolar, quadrupolar, octopolar and so on terms. In most cases the expansion can be truncated to the first term, which is known as the dipole approximation. Since the dipole moment is an observable, it can be described mathematically as an operator. The dipole moment operator can describe transitions between states (as the transition dipole moment operator and, as such, is important in spectroscopy) or within a state where it represents the associated dipole moment. This operator describes the interaction between a molecule and its environment and, as a result, our understanding of energy transfer. 

In quantum mechanics the definition of molecular polarizabilities is given through time-dependent perturbation theory in the electric dipole approximation. These expressions are usually given in terms of sums of transition matrix elements over energy denominators involving the full electronic structure of the molecule [42]. 

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). 

Draine, B.T., and Goodman, J.J. (1993) Beyond clausius-mossotti wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophysical J., 405 685-697. 

By using simple scaling laws for the polarizability and first hyperpolarizability as a function of the molecular size, Hurst and Munn [93] addressed, within the point dipole approximation, the relationship between nd molecular elongation and found that very large can be obtained for compact molecules, provided that the ratio between the polarizability and the molecular volume is large. 

One step better for determining accurate nonlinear susceptibilities was made by using high-level ab initio (hyper)polarizabilities [100,101], and this made possible the calculation of nonlinear susceptibilities without recourse to experimental information except for the crystal structure. For the nonpolar benzene crystal, the differences in the x tensor between the submolecule treatment and the point-dipole approximation were small [100]. However, the differences were larger for some components of the tensor of the urea crystal [101], which is polar. The effects of the surroundings, due mostly to the permanent molecular... 

The problem of metallic particles like gold and silver particles is similar to the previous case except that the material is now highly polarizable. Hence, the polarization sheet is excited by the local field which cannot be taken as the incoming field only. It must be taken as the superposition of the incoming field and the polarization field. This problem is rather difficult in general and several theories have been proposed in the past [35-40]. For arbitrary shapes, one may directly use a numerical approach like the discrete dipole approximation (DDA) for instance. It has however been solved analytically for spherical particles by G. Mie and H. Chew et al. in... 

To begin, from Eqs. (41) and (47) the explicit result for the nonlinear polarizability athat mediates an m-photon process may be written, in the electric dipole approximation, as follows ... 

It is instructive to take as a first example the general expression for molecular polarizability, the response tensor that formally mediates elastic light scattering in the electric dipole approximation. The result is obtained by application of Eq. (74) with m = 2 (one photon is annihilated and another of the same frequency is created). Here there are only two time orderings, or state-sequence pathways, as illustrated in Figs. 5 and 6, respectively. Each generates a term whose numerator is a product of transition dipole moment components. For... 

Aside the SOS oaches, the (hyper)polarizabilities can be evaluated by differentiating die energy (Eo) or the pole moment with respect to the external electric fields. Since, in the electric dipole approximation, the field-dependent dipole moment is related to the energy through the following expression ... 

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]. 

There are basically two numerical approaches to obtain approximate solutions to the Schrodinger equation variational and perturbational. In calculations, we usually apply variational methods, while perturbational methods are often applied to estimate some small physical effects. The result is that most concepts (practically all the ones we know of) characterizing the reaction of a molecule to an external field come from the perturbational approach. This leads to such quantities (see Chapter 12) as dipole moment, polarizability, and hyperpolarizability. The computational role of perturbational theories may, in this ctmtext, be seen as being of the second otder. 

The non-additive DIS contributions are described in terms of the first dipole dynamic polarizability or by the corresponding approximate expression we have introduced for the dimeric case. 

The most fundamental starting point for any theoretical approach is the quantum mechanical partition function PF), and the fundamental connection between the partition function and the corresponding thermodynamic potential. Once we have a PF, either exact or approximate, we can derive all the thermodynamic quantities by using standard relationships. Statistical mechanics is a general and very powerful tool to connect between microscopic properties of atoms and molecules, such as mass, dipole moment, polarizability, and intermolecular interaction energy, on the one hand, and macroscopic properties of the bulk matter, such as the energy, entropy, heat capacity, and compressibility, on the other. 

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