Dielectric model

Fi. 4.30 A sigmoidal dielectric model smoothly varies the effective permittivity from SO to 1 as shown. 

Li) he so-called distance-dependent dielectric models. The simplest implementation of a dis-i.iiice-dependent dielectric is to make the relative permittivity proportional to the distance. Tine interaction energy between two charges qi and qj then becomes ... 

For reasons of space and because of their prime importance, we focus here on free energy calculations based on detailed molecular dynamics (MD) or Monte Carlo (MC) simulations. However, several other computational approaches exist to calculate free energies, including continuum dielectric models and integral equation methods [4,14]. 

Interpretation of the experimental study of quartz leads to the conclusions that below 6 GPa and greater than 2.5 GPa x-cut quartz responds as an approximation to the elastic-dielectric model, but that there are very signifi-... 

Equation (15) permits a straightforward analysis of dielectric continuum models of hydration that have become popular in recent decades. The dielectric model, also called the Bom approximation, for the hydration free energy of a spherical ion of radius R with a charge q at its center is... 

Tawa, G. J., and Pratt, L. R. (1994). Tests of dielectric model descriptions of chemical charge displacements in water. ACS Symposium Series 568, 60-70. 

Equation (2.39) leads to the prediction that AA should be proportional to p2. For a bulk solvent, this can be considered as a molecular equivalent of the well-known Onsager formula derived for the continuum dielectric model [12],... 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

Truhlar and coworkers [29, 232, 233] have proposed a quantum-mechanical-continuum dielectric model for aqueous solvation. This method is discussed by Truhlar and coworkers in Chapter I. 

Solvent effects can be incorporated into two kinds of solvation models, either those that consider each solvent molecule as an individual molecular species (explicit models), or those that deal with the averaged effect of the solvent molecules through use of a coarse-grained description of solvent (e.g., dielectric models, implicit solvent models, etc.). 

The upshot is that the Born theory of solvation fails because it regards the solvent as a continuous dielectric, whereas in fact solute ions (especially metal cations with z > 1) often interact in a specific manner with solvent molecules. In any event the molecular dielectric is obviously very lumpy on the scale of the ions themselves. The Born theory and other continuous dielectric models work reasonably well when metal ion solute species are treated as solvent complexes such as Cr(OH2)63+ rather than naked ions such as Cr3+, but the emerging approach to solvation phenomena is to simulate solvation dynamically at the molecular level using computer methods. 

Calculations based on the continuum dielectric model have been performed by the hydrated electron in the limit of zero cavity size (19). The general treatment is based on a variational calculation using hydrogenic type wave functions for the ground and the first excited states. This treatment is based on a Hartree Fock scheme, where the Coulomb and exchange interaction of the excess electron with the medium are replaced by the polarization energy of a continuous dielectric. The results obtained are summarized in Table V. The fair agreement obtained with... 

Figure 3 The screening factor F(co) of Cgo calculated on the basis of (a) the classical dielectric model of infinitesimally thin CgQ, Fclass(< ) [43] and (b) the quantum mechanical 5-potential model of infinitesimally thin CgQ, (a>) [38,39]. The calculated data for Fclass(o>) were obtained assuming...

The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),. .. However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations... 

The heterogeneous dielectric model presented is represented by a hemispherical cavity, C, having the radius R. The cavity is placed such that it is on the surface of Sm and embedded in S, and the volume is given by... 

The methodology that uses the dielectric model is instead the simpler and in principle the more suitable for the study of chemical reactions involving large molecular systems. In 1998, Amovilli et al [13] developed a computer code in which the solvent reaction field, including all the basic solute-solvent interactions, has been considered for Complete Active Space Self Consistent Field (CASSCF) calculations. 

The calculations presented in this chapter have been done using the MCSCF code of the package GAMESS [36] which contains also the more recent developments, made in our laboratory, on the solvent dielectric model for the study of solvation (see for example [37]). Our solvation model implemented in GAMESS allows a CASSCF calculation with the inclusion of all the basic solute-solvent interactions. 

---