Continuum electrostatic approximation

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

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

Approximate analytical theories of solvation dynamics are typically based on the linear response approximation and additional statistical mechanics or continuum electrostatic approximations to Cy(r). The continuum electrostatic approximation requires the frequency-dependent solvent dielectric response For example, the Debye model, for which e(a>) = + (cq - )/(l +... 

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

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... 

In Section III we described an approximation to the nonpolar free energy contribution based on the concept of the solvent-accessible surface area (SASA) [see Eq. (15)]. In the SASA/PB implicit solvent model, the nonpolar free energy contribution is complemented by a macroscopic continuum electrostatic calculation based on the PB equation, thus yielding an approximation to the total free energy, AVP = A different implicit... 

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. 

It should be emphasized that this description of solvation as a purely electrostatic process is greatly over-simplified. Short-range interactions exist as well, and the physical exclusion of the solvent from the space occupied by the solute must have its own dynamics. Still, for solvation of ions and dipolar molecules in polar solvents electrostatic solvent-solute and solvent-solvent interactions dominate, and disregarding short-range effects turns out to be a reasonable approximation. Of main concern should be the use of continuum electrostatics to describe a local molecular process and the fact that the tool chosen is a linear response theory. We will come to these points later. 

Over the recent years implicit solvent models have undergone a transition to relatively mature methodology that is now widely employed in molecular dynamics simulations and related applications. Most popular are implicit solvent models based on a decomposition of the solvation free energy into electrostatic and nonpolar components. The electrostatic free energy is typically obtained according to a continuum electrostatics model that is described by Poisson theory or by the more approximate but much more efficient Generalized Born formalism. 

The discussion of continuum electrostatics in Section 11.2.1 was limited to solution of Poisson s equation, which can be achieved exactly (for classical solutes) or to a good approximation (for QM solutes) using PCMs. In biomolecular applications, however, the objective is usually solution of the Poisson-Boltzmann equation [4, 33]. For low concentrations of dissolved ions, the latter is often replaced by the linearized Poisson-Boltzmann equation (LPBE),... 

The electrostatic contribution can be obtained by solving the Poisson equation in the continuum dielectric approximation. Although this approximation has not been systematically tested for interfacial systems, its recent applications to bulk solutions proved to be highly successful [45,46]. The conventional continuum, dielectric model can be considered as an implementation of second-order perturbation theory [47]. The first-order term is assumed to vanish and all terms beyond the second order are neglected. It is not clear, however, how well these approximations hold near an interface. In particular, interfacial solvent molecules have preferred orientations due to the interfacial excess electric field. They will, therefore, not be randomly oriented around the cavity volume of the solute - a requisite for the first-order term to vanish. Furthermore, it has... 

A qualitative theoretical explanation of this model based on ionic surface charge densities or continuum electrostatics is possible, but quantitative interpretations are difficult, even more so when molecular ions like SCN" are studied, which cannot be approximated by spheres with an isotropic surface charge density. 

Marcus s model assumes the validity of a linear response approximation and that a continuum electrostatic description of the interface is suitable for the purpose of calculating the activation free energy. Furthermore, to obtain expressions for the rate constant, the interface is assumed to be either a mathematically sharp plane or a broad homogeneous phase. Unfortunately, an insufficient... 

Let denote the potential obtained from a standard PB calculation assuming a constant bulk dielectric coefficient and < )var( ) denote that obtained by assuming a variable dielectric coefficient (i.e., Eqs. [385]-[388]). Now let Vb z) be the MC potential (a sum of Lennard-Jones and electrostatic terms) that a cation in the system would experience in a bulk dielectric continuum. The approximate effect of including a variable dielectric coefficient in the Monte Carlo simulation can be found by replacing the bulk MC potential Vb z) with the PB-corrected potential VB(r)- -< )var(r) — < )B(r). Of course the resulting ion distribution will not be self-consistent with the assumed (PB-derived) dielectric coefficient map, but it will be a noticeable improvement on the original MC distribution using a bulk dielectric. 

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. 

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. 

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)). 

We can exploit the new results for packing contributions to reconsider the outer shell contribution in Eq. (33). For ionic solutes, the outer shell term would represent the Born contribution because it describes a hard ion stripped of any inner shell ligands. A Born model based on a picture of a dielectric continuum solvent is reasonable (see Section III,B, and Fig. 9, color insert). With that motivation, we first separate the outer shell term into an initial packing contribution and an approximate electrostatic contribution as... 

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