Uniform dielectric media

Since electrostatic effects dominate the thermodynamic cycle as shown in Figure 10-2, major development efforts have focused on the calculation of electrostatic energy for transferring the neutral and charged forms of the ionizable group from water with dielectric constant of about 80 to the protein with a low dielectric constant (see later discussions). This led to the development of continuum based models, where water and protein are described as uniform dielectric media, and enter into the linearized Poisson-Boltzmann (PB) electrostatic equation,... 

When a plane wave undergoes total internal reflection at a planar interface between two semi-infinite, uniform dielectric media, as in Fig. 10-2, an evanescent field is set up in the rarer medium and the reflected wave differs in... 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

To understand solvation dynamics, it is necessary to recall some aspects of dielectric relaxation in the framework of the simple continuum model, which treats the solvent as a uniform dielectric medium with exponential dielectric... 

The physical significance of these variables is apparent when they are evaluated in the Onsager cavity description of solvation, which treats the solute as a sphere (which we will assume here is unpolarizable) of radius a. The solvent is modeled as a uniform dielectric medium with a static dielectric constant s and an optical dielectric constant op. The following relationships apply in the Onsager cavity description... 

A major area of theoretical interest has been on solvent effects, and several techniques have been applied to the calculation of NLO properties. " The most common (and simplest) method is the reaction field model, where the solute molecule is in a cavity of solvent, which is treated as a uniform dielectric medium. Cavity approaches are problematic. How do you pick the cavity size How do you pick the cavity shape How do you model stronger, specific interactions (such as hydrogen bonding) The work of Willetts and Rice " illustrated the inability of reaction field models to adequately treat solvent effects even though they tried both spherical and ellipsoidal cavities. Mikkelsen et al. attempted to provide specific interactions with their solvent model by explicitly including solvent molecules inside the cavity. These and related issues need to be addressed further if computational chemists are to develop truly useful procedures capable of including solvent effects in NLO calculations. Recent work by Cammi, Tomasi, and co-workers " has attempted to address these issues within the polarized continuum model (PCM) and have included studies of frequency-dependent hyperpolarizabilities. 

Let us consider an electrical potential at a point in a space where electric charges are distributed as point charges in a uniform dielectric medium. The electrical potential at a point in the space is expressed by ... 

The earliest attempt to predict the free energy of a water cluster containing an ion originates from the classical drop model of a small cluster, Eq. (40). Wilson attempted to calculate the free energy of formation of a liquid drop of radius r on an ion of radius ri and charge e. He treated the problem by considering the ion as embedded in a uniform dielectric medium and obtained... 

This accurate analysis is accompanied by a rigorous foniial elaboration of the model to which I shall return later. A few years before Onsager, Bell [7] presented a model consisting of a dipole, fi, within a spherical cavity immersed in a continuous uniform dielectric medium. This model is quite similar to the Onsager one, but is summarily treated and with some errors now it is completely forgotten, and is only quoted in extensive reviews of the subject. 

The solvent is modeled as a uniform dielectric medium. The dynamics of the ith particle (either a labeled bead or a labeled ion) is taken as... 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as... 

The continuum model of solvation has evolved from these beginnings. The solvent is treated as a continuous polarizable medium, usually assumed to be homogeneous and isotropic, with a uniform dielectric constant e.11-16 The solute molecule creates and occupies a cavity within this medium. The free energy of solvation is usually considered to be composed of three primary components ... 

The notion of homogeneity is not absolute all substances are inhomogeneous upon sufficiently close inspection. Thus, the description of the interaction of an electromagnetic wave with any medium by means of a spatially uniform dielectric function is ultimately statistical, and its validity requires that the constituents—whatever their nature—be small compared with the wavelength. It is for this reason that the optical properties of media usually considered to be homogeneous—pure liquids, for example—are adequately described to first approximation by a dielectric function. There is no sharp distinction between such molecular media and those composed of small particles each of which contains sufficiently many molecules that they can be individually assigned a bulk dielectric function we may consider the particles to be giant molecules with polarizabilities determined by their composition and shape. 

A number of theoretical models for solvation dynamics that go beyond the simple Debye Onsager model have recently been developed. The simplest is an extension of Onsager model to include solvents with a non-Debye like (dielectric continuum and the probe can be represented by a spherical cavity. Newer theories allow for nonspherical probes [46], a nonuniform dielectric medium [45], a structured solvent represented by the mean spherical approximation [38-43], and other approaches (see below). Some of these are discussed in this section. Attempts are made where possible to emphasize the comparison between theory and experiment. 

In this derivation it is assumed that the field and displacement are uniform throughout the volume of the condenser, and that the field is zero outside the condenser. This is true for the isotropic dielectric medium with the edge effects neglected. If this were not true, then the work done on differential volumes would have to be considered and the total work would be obtained by integration over the volume of the condenser. 

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

The treatment of polarization based on the assumption that water has a uniform dielectric constant involves a fundamental difficulty. Indeed, a uniform, continuous distribution of dipoles on a planar surface does not generate a field in a homogeneous medium and hence is not able to polarize the water. If the dipoles are distributed in the sites of a 2D planar square lattice, the field is... 

Consider a macroscopic metal electrode Its conduction electrons all tend to travel at the outer surface of the electrode, and they will induce positively charged cations to move close to the electrode surface the "electrons inside electrode cations" system is called the Helmholtz32 double layer [19]. If this layer of cations were at a fixed distance d from a flat electrode and if the medium had a uniform dielectric constant e, then a voltage-independent capacitance of this double layer would be (ee0/d) experimentally, these assumptions are invalid, and too simplistic. In fact, gegenions will also... 

For an uniform isotropic dielectric medium, the vectors D, E, P have the same direction, and the susceptibility is coordinate-independent therefore... 

In a hypothetical, isotropic, structureless, uniform medium of dielectric constant D, the quantities F r)y t/(r), and E r) are all reduced by the factor 1 /D. For water at 25 C with D = 78.5, for the preceding cases we find F(r) = 1.2 X 10 dynes, U(r) = 0.83 Kcal/mole, and E r) = 7.2 X 10 volts/cm, somewhat reduced but still large. In considerations of solvent power it is the ratio U r)/kT which indicates the extent to which the dielectric medium is important (relative to thermal motion) in shielding charges from each other. 

Consider a q-dot as a two-level system near a metal nanoparticle in a uniform transparent dielectric medium, as shown in the inset in Fig. 1. Both particles are... 

A comparison of anion solvation by methanol, a protic solvent, and dimethylformamide, a dipolar aprotic solvent, is instructive. The electrostatic contribution, d/i , to the Gibbs free energy of solvation per mole of an ion is sometimes estimated quite successfully (Stokes, 1964) from the Bom model, in which a charged sphere of radius r is transferred from vacuum to a medium of uniform dielectric constant, c. The Bom equation (17) suggests that an anion should be similarly solvated in methanol and in DMF, because these solvents have effectively the same dielectric constant (33-36). The Born equation makes no allowance for chemical interactions, such as hydrogen-bonding and mutual... 

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