The Electrostatic Continuum

The model used for ion solvation energies since the work of Bernal and Fowler16 has considered simple electrostatic interactions between the ion and the permanent and induced charges on solvent molecules, plus a term for estimating the work involved in importing the ionic charge 

The Electrostatic Gibbs Energy in a Continuous Polar Medium 

The local internal field FA is produced by an external field EA. To account for the Onsager cavity101 in the medium (see below), the ratio of Fa to Ea is qA for the sake of generality. By definition, Aa/Ea = eA, the microscopic or local static dielectric constant in element dV, which may be a function of the (external or internal) field. Since Ep a + Ec = Fa - Aa, the total electrostatic energy in the volume element dV is given by  

Since Aa = ze/a2 for both a point charge and for a charge uniformly distributed over a surface, and dV is 47ia2da, where a is the ion-dipole center distance, the electrostatic energy due to the presence of an ion in the surrounding dielectric continuum of dielectric constant e0 may be approximated by the integral  

Since Bernal and Fowler,16 the charging radius r0 in the Born equation has been put equal to the Inner Sphere radius, or approximately the ion to water molecule center distance plus 1.4 A. At least for 1+ ions, this gives a fairly good approximation to the Gibbs energy of interaction of the ion with the outer Dielectric Continuum if aT and s are constant throughout the medium. High-valency ions are discussed in Section IV. 

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While a partly classical inner-sphere contribution to the activation energy in cases where the electrostatic continuum approximation applies has been assumed by Marcus for a redox-type (i.e., non-atom-transfer) process, no successful efforts have been made to describe the instantaneous radiationless electron transfer mechanisms for heavy nuclei (see, however. Ref. 78). Overall, it seems reasonable to conclude that the instantaneous transfer theory is an unlikely hypothesis for combined atom and electron transfer. A more reasonable approach is dependent on the fact that the lifetime of an activated molecule (i.e. the proton tightly bound to its neighboring water molecules, e.g., in 11304" ) is very long compared with the transition time for one single-electron passage. 

Calculations of A<, from the electrostatic continuum model and from quantum mechanical models have been reviewed and compared/ The distinction between electrostatic displacement D and field E is emphasized. Values of A are compared for different physical models of the reacting molecules, e.g., conducting spheres (the model usually considered in previous literature) and cavities of various dimensions. In the electrostatic model a formula for A has been given, which applies to any system which has a symmetrical binuclear structure, and from which Marcus two-sphere " and Cannon s ellipsoidal " models can be deduced as special cases. 

For the evaluation of there are several formulas based on the shape and size of the solute and on different parameters of the solvent surface tension [43], surface tension with macroscopic correction [44], isothermal compressibility [45] and geometrical parameters of the solvent molecules [46]. The first three techniques follow the same philosophy. They do not rely on a detailed description of the discrete solvent and make use of experimental bulk parameters of the solvent, in analogy with the dielectric constant of the electrostatic continuum model. As an example of this kind of approach we present Sinanoglu s formula [44]... 

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]... 

It is often the case that the solvent acts as a bulk medium, which affects the solute mainly by its dielectric properties. Therefore, as in the case of electrostatic shielding presented above, explicitly defined solvent molecules do not have to be present. In fact, the bulk can be considered as perturbing the molecule in the gas phase , leading to so-called continuum solvent models [14, 15]. To represent the electrostatic contribution to the free energy of solvation, the generalized Bom (GB) method is widely used. Wilhin the GB equation, AG equals the difference between and the vacuum Coulomb energy (Eq. (38)) ... 

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 ... 

The Poisson equation describes the electrostatic interaction between an arbitrary charge density p(r) and a continuum dielectric. It states that the electrostatic potential ([) is related to the charge density and the dielectric permitivity z by... 

The conductor-like screening model (COSMO) is a continuum method designed to be fast and robust. This method uses a simpler, more approximate equation for the electrostatic interaction between the solvent and solute. Line the SMx methods, it is based on a solvent accessible surface. Because of this, COSMO calculations require less CPU time than PCM calculations and are less likely to fail to converge. COSMO can be used with a variety of semiempirical, ah initio, and DFT methods. There is also some loss of accuracy as a result of this approximation. 

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]. 

The continuum model, in which solvent is regarded as a continuum dielectric, has been used to study solvent effects for a long time [2,3]. Because the electrostatic interaction in a polar system dominates over other forces such as van der Waals interactions, solvation energies can be approximated by a reaction field due to polarization of the dielectric continuum as solvent. Other contributions such as dispersion interactions, which must be explicitly considered for nonpolar solvent systems, have usually been treated with empirical quantity such as macroscopic surface tension of solvent. 

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. 

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. 

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... 

The Polarizable Continuum Model (PCM) employs a van der Waals surface type cavity, a detailed description of the electrostatic potential, and parameterizes the cavity/ dispersion contributions based on the surface area. The COnductor-like Screening... 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

The electrostatic contributions, identified as /t x born and modeled on the basis of a dielectric continuum, are typically a substantial part of the... 

To exploit the concept of PMF to represent solvent in free energy calculations, practical approximations must be constructed. A common approach is to treat the two components Z H/"P(X) and Z lYelec(X) separately. Approximations for the nonpolar term are usually derived from geometric considerations, as in scaled particle theory, for example [62], The electrostatic contribution is usually derived from continuum electrostatics. We consider these two contributions in turn. 

Continuum electrostatics approximations in which the solvent is represented as a featureless dielectric medium are an increasingly popular approach for the electrostatic... 

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