Continuum constant

Turning now to the case of a flat continuum (constant 8), we can readily construct the case of two overlapping resonances, in which case the phase shift has a resonant part... 

Also, the approach provides the potential for constant-stress molecular dynamics [40,45] with realistic retracting forces acting on the atomistic periodic box. This would provide an advantage over the current situation, where constant-stress molecular dynamics is weakened by the fact that the device, which maintains constant stress, (a) acts directly and without time delay on each atom, and (b) requires an arbitrary wall mass. The question of efficiency for atomistic-continuum constant-stress molecular dynamics must be answered first, of course. 

P(E) corresponds to the number of occupied states per unit energy range. Only under conditions of an unstructured continuum, constant probability and constant transport and escape, would a photoelectron spectrum be a direct measure of the density of states, I)(E). 

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

In the reaction field method, the space surrounding a dipolar molecule is divided into two regions (i) a cavity, within which electrostatic interactions are sunnned explicitly, and (ii) a surrounding medium, which is assumed to act like a smooth continuum, and is assigned a dielectric constant e. Ideally, this quantity will be... 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

Another way of calculating the electrostatic component of solvation uses the Poisson-Boltzmann equations [22, 23]. This formalism, which is also frequently applied to biological macromolecules, treats the solvent as a high-dielectric continuum, whereas the solute is considered as an array of point charges in a constant, low-dielectric medium. Changes of the potential within a medium with the dielectric constant e can be related to the charge density p according to the Poisson equation (Eq. (41)). 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

The final class of methods that we shall consider for calculating the electrostatic compone of the solvation free energy are based upon the Poisson or the Poisson-Boltzmann equatior Ihese methods have been particularly useful for investigating the electrostatic properties biological macromolecules such as proteins and DNA. The solute is treated as a body of co stant low dielectric (usually between 2 and 4), and the solvent is modelled as a continuum high dielectric. The Poisson equation relates the variation in the potential (f> within a mediu of uniform dielectric constant e to the charge density p ... 

Another common approach is to do a calculation with the solvent included in some approximate manner. The simplest way to do this is to include the solvent as a continuum with a given dielectric constant. There are quite a few variations on this technique, only the most popular of which are included in the following sections. 

Reactions. Supercritical fluids are attractive as media for chemical reactions. Solvent properties such as solvent strength, viscosity, diffusivity, and dielectric constant may be adjusted over the continuum of gas-like to Hquid-like densities by varying pressure and temperature. Subsequently, these changes can be used to affect reaction conditions. A review encompassing the majority of studies and apphcations of reactions in supercritical fluids is available (96). 

Simple considerations show that the membrane potential cannot be treated with computer simulations, and continuum electrostatic methods may constimte the only practical approach to address such questions. The capacitance of a typical lipid membrane is on the order of 1 j.F/cm-, which corresponds to a thickness of approximately 25 A and a dielectric constant of 2 for the hydrophobic core of a bilayer. In the presence of a membrane potential the bulk solution remains electrically neutral and a small charge imbalance is distributed in the neighborhood of the interfaces. The membrane potential arises from... 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

Another variant that may mrn out to be the method of choice performs the alchemical free energy simulation with a spherical model surrounded by continuum solvent, neglecting portions of the macromolecule that lie outside the spherical region. The reaction field due to the outer continuum is easily included, because the model is spherical. Additional steps are used to change the dielectric constant of that portion of the macromolecule that lies in the outer region from its usual low value to the bulk solvent value (before the alchemical simulation) and back to its usual low value (after the alchemical simulation) the free energy for these steps can be obtained from continuum electrostatics [58]. 

Solvent effects on chemical equilibria and reactions have been an important issue in physical organic chemistry. Several empirical relationships have been proposed to characterize systematically the various types of properties in protic and aprotic solvents. One of the simplest models is the continuum reaction field characterized by the dielectric constant, e, of the solvent, which is still widely used. Taft and coworkers [30] presented more sophisticated solvent parameters that can take solute-solvent hydrogen bonding and polarity into account. Although this parameter has been successfully applied to rationalize experimentally observed solvent effects, it seems still far from satisfactory to interpret solvent effects on the basis of microscopic infomation of the solute-solvent interaction and solvation free energy. 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

Some authors plot log k or AG against 1/e rather than against the Kirkwood function. Since 1/e is nearly linearly related to (e — 1)/(2e + 1), within the assumptions of a theory in which the solvent is treated as a continuum this substitution of variable is not serious. Another approach is to interpret the solvent dependence of the Hammett reaction constant p on a dielectric constant function. ... 

The quantitative theory of ionic reactions, within the limitations of a continuum model of the solvent, is based on the Bom equation for the electrostatic free energy of transfer of an ion from a medium of e = 1 to the solvent of dielectric constant... 

One femily of models for systems in non-aqueous solution are referred to as Self-Consistent Reaction Field (SCRF) methods. These methods all model the solvent as a continuum of uniform dielectric constant e the reaction field. The solute is placed into a cavity within the solvent. SCRF approachs differ in how they define the cavity and the reaction field. Several are illustrated below. 

Now, we should ask ourselves about the properties of water in this continuum of behavior mapped with temperature and pressure coordinates. First, let us look at temperature influence. The viscosity of the liquid water and its dielectric constant both drop when the temperature is raised (19). The balance between hydrogen bonding and other interactions changes. The diffusion rates increase with temperature. These dependencies on temperature provide uS with an opportunity to tune the solvation properties of the liquid and change the relative solubilities of dissolved solutes without invoking a chemical composition change on the water. 

---