Polarizability dielectric sphere

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

The generalization of the work, by Kirkwood and by Frohlich, on the relation between the mean-square fluctuation in dipole moment, , of a dielectric sphere and its zero-firequency polarizability, o4,j,(0), namely ... 

The above expressions were derived for the polarizabilities of molecules in free space or in a dilute gas (mostly air). However, we often encounter molecules interacting in a liquid solvent medium, which reduces the interaction pair potential by around e, or more the extent of this reduction depends on several factors. First of all, the intrinsic polarizability and dipole moment of an isolated gas molecule may be different when it is itself in the liquid state, or alternatively when dissolved in a solvent medium. This is because of the difference in interaction strength and also the separation distance between molecules. Thus, the polarizability values are best determined by experiment. Second, a dissolved molecule can only move by displacing an equal volume of solvent from its path. If the molecule has the same polarizability as the solvent molecules, that is if no electric held is reflected by the molecule, it is invisible in the solvent medium and does not experience any induction force. Thus, the polarizability of the molecule, a, must represent the excess or effective polarizability of a molecule over that of the solvent. Landau and Lifshitz applied a continuum approach and modeled a molecule, i, as a dielectric sphere of radius, ah having... 

To extend this relationship to the molecular domain, suppose that the dielectric sphere contains Ns molecules, each of which has polarizability o. Then,... 

These effects have been illustrated by calculations for simple idealized cases. There appear to be quite pronounced effects due to the particle and they must be taken into account in any application of inelastic scattering for quantitative determinations of the amount of a specific molecule in ian aerosol. Model calculations of inelastic scattering by isotropically polarizable electric dipoles uniformly distributed within an otherwise nonabsorbing dielectric sphere exhibit a variation of more than two orders of magnitude in the inelastically scattered intensity per active molecule as the particle size is varied. 

The term i Pcisphico) is named radiative damping [18] of the sphere polarizability. This imaginary term takes directly into account that the dipole induced in the (metallic or dielectric) sphere by the external incident field, re-irradiates the energy in the free-space. As this scattering process occurs in different directions, then, the net effect is that the incident field is attenuated in the propagation direction as it will be for a pure absorption process. 

Figure 6.28 Sketch showing the direction of the electric dipole moment p induced in a dielectric sphere with dielectric constant inside a dielectric fluid having dielectric constant Cj by the inhomogeneous electrical field E. (a) The particle is more polarizable than the fluid, that is, > e,. (b) The particle is less polarizable than the fluid, that is, < e,. (c) and (d) The effective charges and directions of p and dip corresponding to (a) and (b), respectively...

Solvation effects have been incorporated into the calculations of anionic proton transfer potentials in a number of ways. The simplest is the microsolvation model where a few solvent molecules are included to form a supermolecular system that is directly characterized by quantum mechanical calculations. This has the advantage of high accuracy, but is limited to small systems. Moreover, one must assume that a limited number of solvent molecules can adequately model a tme solution. A more realistic approach is to explicitly describe the inner solvation shell with quantum calculations and then treat the outer solvation sphere and bulk solvent as a continuum (infinite polarizable dielectric medium). In this way, the specific interactions can be treated by high-level calculations, but the effect of the bulk solvent and its dielectric is not neglected. An ej tension of this approach is to characterize the reaction partners by quantum mechanics and then treat the solvent with a molecular mechanics approach (hybrid quantum mechanics/molecular mechanics QM/MM). The low-cost of the molecular mechanics treatment allows the solvent to be involved in molecular dynamics simulations and consequently free energies can be calculated. In more recent work, solvent also has been treated with a frozen or constrained density functional theory approach. ... 

To answer this question, let us first consider a neutral molecule that is usually said to be polar if it possesses a dipole moment (the term dipolar would be more appropriate)1 . In solution, the solute-solvent interactions result not only from the permanent dipole moments of solute or solvent molecules, but also from their polarizabilities. Let us recall that the polarizability a of a spherical molecule is defined by means of the dipole m = E induced by an external electric field E in its own direction. Figure 7.1 shows the four major dielectric interactions (dipole-dipole, solute dipole-solvent polarizability, solute polarizability-solvent dipole, polarizability-polarizability). Analytical expressions of the corresponding energy terms can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres (Suppan, 1990). These four non-specific dielectric in-... 

On the other hand, electrostatic models regard the ligands or the whole crystal as polarizable units and thereby lead to weaker Coulomb and spin-orbit interactions. In a dielectric screening model (DSM) from Morrison et al. (1967) the f element is placed within an empty sphere with radius Rs which is embedded into an infinite medium with dielectric constant e. This leads to a reduction AFk of the Slater parameters (Newman, 1973) ... 

Rather than take a limit of large separations between relatively small spheres a of incremental polarizability a (it ), we can think of interactions within dilute suspensions or solutions. At relatively large separations, the shape and the microscopic details of an effectively small speck become unimportant. The only feature that is of interest is that the dilute specks ever so slightly change the dielectric and ionic response of the suspension compared with that of the pure medium. When the suspension of spheres is vanishingly dilute, esusp is simply proportional to their number density N multiplied by a(/ ) / susp — m (/ ) + Naa(i ) [see Fig. L1.42(a)]. 

The dilute limit emerges when a /z3 dilute limit, assume that the dielectric response of dense suspension follows the Lorentz-Lorenz or Clausius-Mossotti relation6 e = [(1 + 2Na/3)/(l - Na/3)] (This is the next approximate form when the number density N is too high to allow the linear relation e = 1 + Na.) Below what density N will this e be effectively linear in polarizability Expand... 

Inside the sphere where the interactions take place, the use of statistical mechanics is required. To represent a dielectric with dielectric constant , consisting of polarizable molecules with a permanent dipole moment, Frohlich [6] introduced a continuum with dielectric constant s X, in which point dipoles with a moment id are embedded. In this model, id has the same nonelectrostatic interactions with the other point dipoles as the molecule had, while the polarizability of the molecules can be imagined to be smeared out to form a continuum with dielectric constant 00 [7]. 

Where a is the polarizability, e is the frequency dependent dielectric function [4J], and V the volume of the dipole. The radius of each sphere is calculated using a/R=1.612 42, where a is the spacing between the particles, 40 nm... 

B. J. Alder, H. L. Strauss, and J. J. Weis. Dielectric properties of fluids composed of spheres of constant polarizability. J. Chem. Phys., 62 2328-2334 (1975). 

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