Dipole polarizable sphere

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-... 

If solvent (or environment) relaxation is complete, equations for the dipole-dipole interaction solvatochromic shifts can be derived within the simple model of spherical-centered dipoles in isotropically polarizable spheres and within the assumption of equal dipole moments in Franck-Condon and relaxed states. The solvatochromic shifts (expressed in wavenumbers) are then given by Eqs (7.3) and (7.4) for absorption and emission, respectively ... 

In [43], the carbon cage was represented as a classical infinitesimally thin spherical conducting sphere. Methods of classical physics were used to determine the dipole polarizability ad co) of the sphere. Accordingly,... 

One can also deduce a simple expression for the dipole polarizability of the hydrogenic system confined to a sphere. For this, one observes that for large values of R, the polarizability tends to the hydrogenic value... 

For R 0, the polarizability will tend to that of a particle in a sphere. For deducing the dipole polarizability in this case, we use the expression in Equation (3.9). The equation for the first order perturbation in the wave function for the ground state can be written as... 

Such specifics of interaction between dipoles of H O, cations and anions mostly determine the structure of water solution. The simplest idea of it is provided by the statistical theory of diluted solutions of strong electrolytes proposed by Peter Joseph Debye (1884-1966) and Erich Armand Hiickel (1896-1980) in 1923. Under this theory ions are treated as rigid non-polarizable spheres separated by a uniform medium with high value of the dielectric constant. At that, structure of the solution is function of distances dipoles H O and ions. Depending on it, it is customary to distinguish molecular and supramolecular structure. Molecular structure is determined by a direct effect of ions on the orientation and mobility of water dipoles and is manifested first of all by the formation of hydrates. Supramolecular structure is caused by undisturbed interaction of H O molecules between each other (Figure 1.2). 

In the case of isotropic polarizabilities and uniaxial permanent dipoles the electric fields which actually work on the molecules are given by Eqs. (3.18) and (3.34). The internal field of polar polarizable spheres is given by... 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

Thus, the applied field induces a dipole moment proportional to the field. The ease with which the sphere is polarized may be specified by the polarizability a defined by... 

It is instructive to develop the solution for scattering by a small sphere of radius, a X. In such a limit the sphere is represented as a point dipole, and to determine its polarizability, the interaction of the sphere with the electric field is modeled as shown in Figure 4.5. The restriction that the sphere is much smaller that the wavelength of light suggests that to a first approximation, the electric field, at an instant in time, appears to the sphere as a uniform field. We must solve the following time-independent Maxwell s equations [1],... 

To identify the polarizability of the sphere, it is left to compare this solution to the solution of the electric field in the presence of a point dipole. To solve this problem, the simple model of a point dipole displayed in Figure 4.6 is used. Here two opposite point charges are separated by a distance / about the origin. The charge distribution is then... 

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. 

Polarizability goes as the cube of radius The electrostatic potential set up by a dipole of moment /xclipoie has the form (/dipole A2) cos (), where () is the angle between the dipole direction and the line to the position where the dipole potential is being sensed.5 For example, a metallic sphere of radius a placed in a constant external electric field Eo... 

Suppose that a tetrahedral molecule such as CCI4 is irradiated by plane polarized light (E ). Then, the induced dipole (Section 1.7) also oscillates in the same yz-plane. If the molecule is performing the totally symmetric vibration, the polarizability ellipsoid is always sphere-like namely, the molecule is polarized equally in every direction. Under such a circumstance, I (Iy) = 0 since the oscillating dipole emitting the radiation is confined to the xz-plane. Thus, pp = 0. Such a vibration is called polarized (abbreviated as p). In liquids and solutions, molecules take random orientations. Yet this conclusion holds since the polarizability ellipsoid is spherical throughout the totally symmetric vibration. 

One other aspect of nonprimitive electric double layer theories which is particularly relevant to the inner Stern region are the models for the water molecule and the ions. The simplest models for a water molecule and an ion are a hard-sphere point dipole and point charge, respectively. A more realistic model of the hard-sphere water molecule would include quadrupoles and octupoles and also polarizability. However the hard-sphere property is best avoided and replaced, for example, by a Lennard-Jones potential. An alternative to a multipolar water model are three point charge sites associated with the atoms within the water molecule. 

Over the last years, the basic concepts embedded within the SCRF formalism have undergone some significant improvements, and there are several commonly used variants on this idea. To exemplify the different methods and how their results differ, one recent work from this group [52] considered the sensitivity of results to the particular variant chosen. Due to its dependence upon only the dipole moment of the solute, the older approach is referred to herein as the dipole variant. The dipole method is also crude in the sense that the solute is placed in a spherical cavity within the solute medium, not a very realistic shape in most cases. The polarizable continuum method (PCM) [53,54,55] embeds the solute in a cavity that more accurately mimics the shape of the molecule, created by a series of overlapping spheres. The reaction field is represented by an apparent surface charge approach. The standard PCM approach utilizes an integral equation formulation (IEF) [56,57], A variant of this method is the conductor-polarized continuum model (CPCM) [58] wherein the apparent charges distributed on the cavity surface are such that the total electrostatic potential cancels on the surface. The self-consistent isodensity PCM procedure [59] determines the cavity self-consistently from an isodensity surface. The UAHF (United Atom model for Hartree-Fock/6-31 G ) definition [60] was used for the construction of the solute cavity. 

Onsager 0 equations (Section 5.10), one can extract the dipole moments of polar molecules and the polarizability of any solute molecule. One needs a capacitance cell whose electrodes are as close to each other as practical (for higher capacitances) and reasonable solubilities. If the shape of the solute is very different from the sphere used in the Debye model, then the ellipsoidal cavity has been treated theoretically [13] and applied to hypsochromism [14]. 

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]. 

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