Structureless dielectric continuum

The energy spacings between levels associated with solvent dipole reorientations are small, 1-10 cm-1. Since the spacings are well below kBT at room temperature ( 200 cm-1), the contribution of the solvent to the energy 6f activation for electron transfer can be treated classically. The results of classical treatments, where the solvent is modelled as a structureless dielectric continuum, will be discussed in later sections. 

An expression has been derived by Marcus34 and Hush35 for A0 assuming the solvent to be a structureless dielectric continuum characterized by the macroscopic dielectric constants Dop and Ds. D0p and Ds are the optical and static dielectric constants, respectively, and Dop = n2 where n is the index of refraction in the visible spectral region. In the limit that the reactants can be treated as two non-interpenetrating spheres, AQ is given by equation (23). 

The charge is assumed to be uniformly distributed on the surface of the sphere. Such an ion is transferred from vacuum, with a relative dielectric permittivity equal to 1, to the solvent, which is considered to be a structureless dielectric continuum characterized by the static dielectric permittivity, This transfer may be divided into two processes the transfer of a noncharged sphere from vacuum to continuum and the charging (to ne) of the transferred sphere. 

Since Eq. (28) was obtained under assumptions similar to those used by Born, the calculation of AGq suffers from the same limitations as the Born solvation model. The dielectric continuum model is valid for electron transfer in a structureless dielectric medium with a reactant approximated by a hard conducting sphere. It is obeyed when the specific solute-solvent interactions are negligible. 

Water plays an important role in the chemistry and physics of bulk solutions and interfaces, including electrochemistry and macromolecules in solution. Usually the water is treated as a structureless, dielectric continuum, such as in the Debye-Hiickle approximation for electrolytes, the Gony-Chapman-Stern " (GCS) approximation for the electrical double layer and the DLVO approximation for colloids. Properties sensitive to the molecular nature of water cannot be determined by these theories. 

In continuum models, the solvent is considered as a homogeneous medium (structureless dielectric continuum), which is polarized by the solute molecule immersed in a solvent cavity. In this model, A a,b can be written as a sum of least three terms (Zielinski et al., 1978 Sinanoglu, 1968) ... 

The solvent reorganisation energy in eq. 4.32 is the work required to transform the medium around the reactants from its configuration in R° to that in P°. Regarding the solvent as structureless dielectric continuum, these solvent configurations differ because the equihbrium state of polarisation of the medium around the reactants, D and A and the products, and A , with their necessarily different charges, are different. Provided that... 

The classical Marcus expression, eq. 4.31, assumes the internal modes of R and P to behave classically and is primarily aimed at adiabatic reactions with /c i = 1, offering no method of evaluating /Qi < 1 for nonadiabatie transfer between more distant or weakly coupled donors and acceptors. This requires nonclassical approaehes, to which we now turn. Semiclassical approaches, dealt with in this section, incorporate the quantisation of the internal modes of D and A but retain the classical treatment of the surrounding medium as a structureless dielectric continuum... 

An expression has been derived by Marcus and Husy for Aq assuming the solvent to be a structureless dielectric continuum characterized by the macroscopic dielectric constants Z)o and... 

The first step to statistically correct the Gouy-Ghapman theory for the diffuse double layer used a restricted primitive electrolyte model. This model considers ions to be charged hard balls of identical radii in a structureless dielectric continuum with constant dielectric permittivity. There are three main approaches to creating a statistical theory with this model. The... 

Dielectric continuum models such as the Bom model consider the solvent to be a structureless continuum of relative permittivity s. The Gibbs energy of solvation of an ion, AsoivG, is calculated by the difference of the charging process in a vacuum (s = 1) and in the solvent (e) ... 

Xhe Debye-Hiickel (DH) theory (Debye and Huckel, 1923) gives a simple expression for the excess volume in the framework of the primitive model, which consider the systems as ions immersed in a continuum, structureless solvent of dielectric constant, ,... 

One of the most fundamental problems in electrochemical surface science is distribution of the electric potential and the particles at the interface. The classical model which prevailed until about 1980 treated the electrode surface as a perfect and structureless conductor and did not take into account the surface electronic structure. The electrolyte was considered as an ensemble of hard, point ions immersed in a dielectric continuum. This approximation neglected the fine structure of the solvent molecules and the solute as well as their discrete interactions. In recent years, much progress has been made in providing a more realistic model of the solid-liquid electrochemical interface by applying quantum mechanical theories to model the metal... 

In these methods the medium is represented by a structureless dielectric continuum. This is reasonable because the largest effect frequently arises from electrostatic fields emanating from neighbouring ions or dipoles. The method may also be applicable for a series of related reactions when specific interactions are of the same... 

In the simplest model of solvation, the solvent is treated as a structureless and continuous medium of dielectric constant e. In 1920, Born developed the earliest polarizable continuum model. He treated the ion as a point charge q located in the center of a hollow sphere with radius R. The hollow charged sphere is embedded in a classical dielectric continuum having a relative dielectric constant s. The electrostatic contribution to the free energy, evaluated in Section 11.1.1.1, is given by... 

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. 

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)). 

In this section, we shall focus on the use of CMs to study molecules at the interface between a solid and a fluid (gas or liquid). In particular, we reserve the term continuum models to approaches that consider both the solid and the fluid as structureless continuum bodies characterized by their dielectric response, and treat the molecule at some microscopic level. 

By contrast, the description given by a continuum description does not require any knowledge of the solvent configuration around the solute as a structureless continuum dielectric is introduced instead. The response of such a dielectric to the presence of the solute is determined by its macroscopic properties (namely the dielectric constant and the refractive index) and thus it will be implicitly averaged. Contrary to what happens in a QM/MM approach, here a single calculation on a given solute contained within the continuum dielectric will be sufficient to get the correct picture of the solvated system. 

Solvent effects can significantly influence the function and reactivity of organic molecules.1 Because of the complexity and size of the molecular system, it presents a great challenge in theoretical chemistry to accurately calculate the rates for complex reactions in solution. Although continuum solvation models that treat the solvent as a structureless medium with a characteristic dielectric constant have been successfully used for studying solvent effects,2,3 these methods do not provide detailed information on specific intermolecular interactions. An alternative approach is to use statistical mechanical Monte Carlo and molecular dynamics simulation to model solute-solvent interactions explicitly.4 8 In this article, we review a combined quantum mechanical and molecular mechanical (QM/MM) method that couples molecular orbital and valence bond theories, called the MOVB method, to determine the free energy reaction profiles, or potentials of mean force (PMF), for chemical reactions in solution. We apply the combined QM-MOVB/MM method to... 

The Polarizable Continuum Model (PCM)[18] describes the solvent as a structureless continuum, characterized by its dielectric permittivity e, in which a molecular-shaped empty cavity hosts the solute fully described by its QM charge distribution. The dielectric medium polarized by the solute charge distribution acts as source of a reaction field which in turn polarizes back the solute. The effects of the mutual polarization is evaluated by solving, in a self-consistent way, an electrostatic Poisson equation, with the proper boundary conditions at the cavity surface, coupled to a QM Schrodinger equation for the solute. 

In principle all kind of interactions are contained In (3.6.11. In the present section we shall consider a solid-liquid interface although the treatment is also valid for liquid-liquid Interfaces. Solid and liquid su e taken as primittve, l.e. as structureless continuums with dielectric permittivities and = , respectively. In this model the surface is hard, planar and uniformly charged. Considering the surface charge a° as discrete would mean a further improvement. The... 

This is precisely what is meant when the medium is described as a structureless dielectric or continuum. In particular, when discussing the role of the solvent in electrolyte solutions it is often described as a continuous medium, or a structureless medium. Most of the theoretical discussions of electrolyte solutions formulate the theoretical equations in terms of factors which involve the macroscopic quantity, Use of this bulk quantity in the equations implicitly means a description in terms of the solvent being a strucmreless dielectric, with no microscopic or molecular structure. 

The QM/continuum models are characterized by a representation of the environment as a structureless dielectric, solely characterized by its macroscopic dielectric constant, e, which determines the environment polarization as a response to the presence of the In the most widespread of these... 

The simplest simulated system is a Stockmayer fluid structureless particles characterized by dipole-dipole and Lennard-Jones interactions, moving in a box (size L) with periodic botmdary conditions. The results described below were obtained using 400 such particles and in addition a solute atom A which can become an ion of charge q embedded in this solvent. The long range nature of the electrostatic interactions is handled within the effective dielectric environment seheme. In this approach the simulated system is taken to be surrounded by a continuum dielectric environment whose dielectric constant e is to be chosen self consistently with that eomputed from the simulation. Accordingly, the electrostatic potential between any two partieles is supplemented by the image interaction associated with a spherical dielectric boundary of radius (taken equal to L/2) placed so that one of these... 

Generally, methods for calculating can be represented by two main categories implicit or explicit solvent models [38, 47-58]. The main difference between these two categories is the representation of the solvent strueture around the solute. Implicit Continuum Solvent Models (ICSMs) treat the solvent around the solvated molecule as a structureless polarizable medium characterized by a dielectric constant, e [49, 59,60]. In turn, in explicit solvent models (ESMs) both solute and solvent molecules in the solute-solvent systems are described at the atomistic level. There are two... 

Although not rigorously correct, the approximation of water as a structureless homogeneous continuum dielectric medium is used by many simulative methodologies. Both Brownian dynamics (see the section entitled Implicit Solvation Brownian Dynamics) and electrodiffusive approaches (see the section on Flux-Based Simulation) include the water in the electrostatic picture as a continuous dielectric background with polarizability appropriately tuned... 

The theories of van der Waals and double-layer forces are both continuum theories wherein the intervening solvent is characterized solely by its bulk properties such as refractive index, dielectric constant, and density. When a liquid is confined within a restricted space, it ceases to behave as a structureless continuum. At small surface separations, the van der Waals force between two surfaces is no longer a smoothly varying attraction instead, there arises an additional solvation force that generally oscillates between attraction and repulsion with distance, with a periodicity equal to some mean dimension of the liquid molecules. 

As shown above the size of the explicit water simulations can be rather large, even for a medium sized protein as in the case of the sea raven antifreeze protein (113 amino acid residues and 5391 water). Simulations of that size can require a large amount of computer memory and disk space. If one is interested in the stability of a particular antifreeze protein or in general any protein and not concerned with the protein-solvent interactions, then an alternative method is available. In this case the simulation of a protein in which the explicit waters are represent by a structureless continuum. In this continuum picture the solvent is represented by a dielectric constant. This replacement of the explicit solvent model by a continuum is due to Bom and was initially used to calculate the solvation free energy of ions. For complex systems like proteins one uses the Poisson-Boltzmann equation to solve the continuum electrostatic problem. In... 

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