Dielectric cavity model

G. Karlstrom, Proton transport in water modeled by a quantum chemical dielectric cavity model, J. Phys. Chem., 92 (1988) 1318. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

We have given some highlights of a theory which combines the familiar multistate VB picture of a molecular system with a dielectric continuum model for the solvent which accounts for the solute s boundary effects — due to the presence of a van der Waals cavity which displays the solute s shape — and includes a quantum model for the electronic solvent polarization. 

The dynamics of carbon-halogen bond reductive cleavage in alkyl halides was studied by MP3 ab initio calculations, using pseudopotentials for the halogens and semidiffuse functions for the heavy atoms [104], The effect of solvent was treated by means of the ellipsoidal cavity dielectric continuum model. Both a concerted (i.e., a one-step) and a stepwise mechanism (in which an anion radical is formed at first) were... 

The physical meaning of the relationship described in the previous subsection becomes apparent when we consider the popular special case of the Onsager cavity model that arises if we assume that the solvent s dielectric properties are well described by a Debye form. 

These results allow a test of the Onsager cavity model for a uniform dielectric continuum solvent with a dielectric response that is well modeled by Eq. (24). Our group recently tested this model for methanol. In this case, both high frequency (co) data (see Barthel et al. [Ill]) and short time resolution C(t) data [32] exist. 

The unique properties of dilute metal-ammonia solutions depend not upon the nature of the metal species, but upon the solvated electron common to all these solutions. Thus, the electron-in-a-cavity model (17, 19, 21) seems best suited to describe the species present in these solutions since the model is independent of the type of cation present. Jortner and his associates (15, 16) have extended this model by assuming that the cavity arises from polarization of the medium by the electron. The energy levels of the bound electrons are obtained by using a potential function containing the static and optical dielectric constants of the bulk medium as parameters. Using one-parameter hydrogen-like wave functions for the first two bound states of the electron, the total energy of the ith state is expressed as... 

The results of the SCRF models depend strongly on the radius Rx used for the definition of the spherical interface between the solute and the solvent. Unfortunately, the dielectric theory does not provide an answer for the question of which value is appropriate for this radius. Owing to the implicit assumption of the dielectric continuum models that the electron density of the solute should be essentially inside the cavity, any value of Rx below a typical van der Waals (vdW) radius would not be meaningful. On the other hand, at least at the distance of the first solvent shell, i.e., typically at two vdW radii, we should be in the dielectric continuum region. However, there is no clear rationale for the right value between these two limits other than empirical comparison of the results with experimental data. Among others, the choice of spherical cavities which correspond to the liquid molar-solute volume has proved to be successful. 

Within the dielectric continuum model, the electrostatic interactions between a probe and the surrounding molecules are described in terms of the interaction between the charges contained in the molecular cavity, and the electrostatic potential these changes experience, as a result of the polarization of the environment (the so-called reaction field). A simple expression is obtained for the case of an electric dipole, /a0, homogeneously distributed within a spherical cavity of radius a embedded in an anisotropic medium [10-12], by generalizing the Onsager model [13]. For the dipole parallel (perpendicular) to the director, the reaction field is parallel (perpendicular) to the dipole, and can be calculated as [10] ... 

At the end of the 1960s, Barriol gave up the mono-molecular cavity model in exchange for a multi-molecular one (but for few molecules only), and constructed a purely statistical theory of the dielectric constant, that could take into account fluctuations in time. [42] And what is interesting for our purpose is that, with his statistical theory, he found out the same results as those derived from the preceding model, thanks to hypotheses that throw light on the conditions of validity of the relations deduced from these calculations . [43]... 

Different solvation methods can be obtained depending on the way the (Vs(r p)) xj tern1 is calculated. So, for instance, in dielectric continuum models ( Vs(r p)) x is a function of the solvent dielectric constant and of the geometric parameters that define the molecular cavity where the solute molecule is placed. In ASEP/MD, the information necessary to calculate Vs(r, p))[Xj is obtained from molecular dynamics calculations. In this way (Vs(r p))[Xj incorporates information about the microscopic structure of the solvent around the solute, furthermore, specific solute-solvent interactions can be properly accounted for. For computational convenience, the potential Vs(r p)) X is discretized and represented by a set of point charges, that simulate the electrostatic potential generated by the solvent distribution. The set of charges, is obtained in three steps [26] ... 

Dielectric hydration models serve as primitive theories against which more detailed molecular descriptions can be considered. Of particular interest are temperature and pressure variations of the hydration free energies, and this is specifically true also of hydrated polymer electrolyte membranes. The temperature and pressure variations of the free energies implied by dielectric models have been less well tested than the free energies close to standard conditions. Those temperature and pressure derivatives would give critical tests of this model (Pratt and Rempe, 1999 Tawa and Pratt, 1994). But we don t pursue those tests here because the straightforward evaluation of temperature and pressure derivatives should involve temperature and pressure variation of the assumed cavity radii about which we have little direct information (Pratt and Rempe, 1999 Tawa and Pratt, 1994). 

Another model, in which the reactant is represented by a dielectric cavity with point charge in its center, has been forwarded by German und Kuznetsov [127]. It is beyond the scope of the present review to discuss all the above-mentioned improvements in detail. 

Theoretical formulations of reorganization in the course of electron-transfer processes have undergone a number of advances in recent years. The relative importance of various solvent contributions (including translational as well as orientational response, and inductive and dispersion as well as elecrostatic interactions) can depend strongly on the polarity (i.e., dipolar, higher multipolar, or nonpolar) as well as other molecular features of the solvent [21, 47-49]. Molecular-level perspectives on solvent response are of great utility in helping to parameterize effective cavity models (e.g., in conjunction with conventional [50] or spatially nonlocal [47] dielectric models). Additivity relationships traditionally assumed to pertain to sol-... 

The determination of local field factors is a difficult topic, even for such simplified models as the dielectric continuum models, and for this reason we will not go into further detail about the determination of local field factors in this chapter, referring instead to Chapter 4. We note, however, that for the case of a spherical cavity, Onsager demonstrated that the local field is related to the macroscopic field through [47]. 

Other modifications of the original Marcus model have been suggested [27]. Many reactants are not spherical in shape and are better approximated as ellipsoids. In this case a much more complex expression for the effective distance R is obtained which depends on the length of the two axes which describe the shape of the ellipsoid. Another improvement in the model is to describe each reactant as a dielectric cavity with fixed charges located within it. In this case, the calculation of Gxo requires a description of the charge distribution within the reactants and an estimate of the local permittivity in the dielectric cavity. 

Recently, a new category of methods, the cavity model, has been proposed to account for the solvent effect. Molecules or supermolecules are embedded in a cavity surrounded by a dielectric continuum, the solvent being represented by its static dielectric constant. The molecules (supermolecules) polarize the continuum. As a consequence this creates an electrostatic potential in the cavity. This reaction potential interacts with the molecules (supermolecules). This effect can be taken into account through an interaction operator. The usual SCF scheme is modified into a SCRF (self consistent reaction field) scheme, and similar modifications can be implemented beyond the SCF level. Several studies based on this category of methods have been published on protonated hydrates. They account for the solvent effect on the filling of the first solvation shell (53, 69), the charges (69, 76) and the energy barrier to proton transfer (53, 76). 

The chemical shifts of polar molecules are frequently found to be solvent dependent. Becconsall and Hampson have studied the solvent effects on the shifts of methyl iodide and acetonitrile. The results obtained from dilution studies in various solvents may be explained as arising from a reaction field around the solute molecules. The spherical cavity model due to Onsager was used to describe this effect, and this model was completely consistent with the experimental data when a modified value for the dielectric constant, s, of the particular solvent was used. 

To account for the polarization effects, the enzyme surrounding to a first approximation can he considered as a homogenous polarizable medium, which can he modeled using some dielectric cavity techniques. In addition, to model the steric effects that the enzyme surrounding imposes on the active site, it has been shown to be very useful to simply fix atoms at the edge of the active site model. The combination of continuum solvation and the coordinate-locking scheme represents a quite simple but yet powerful way to account for the parts of the enzyme that are not included in the model. 

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