Electrostatics, Poisson equation

To enable determination of Ci, electrostatic Poisson equation and, assuming that the solution is a viscous Newtonian fluid, by the Navier-Stokes momentum equation. The Poisson equation reads... 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

In the computational practice, the ASC density is discretized into a collection of point charges qk, spread on the cavity surface. The apparent charges are then determined by solving the electrostatic Poisson equation using a Boundary Element Method scheme (BEM) [1], Many BEM schemes have been proposed, being the most general one known as integral equation formalism (IEFPCM) [10]. 

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. 

An alternative to the use of finite differences or finite elements to discretize the differential operator is to use boundary element methods (BEM). " One of the most popular of these is the polarizable continuum model (PCM) developed originally by the Pisa group of Tomasi and co-workers. The main aspect of PCM is to reduce the electrostatic Poisson equation (1) into a boundary element problem with apparent charges (ASCs) on the solute cavity surface. 

The solvent reaction potential Vo in Eq,(1.2) is determined by solving the classical electrostatic Poisson equation which governs total electrostatic potential V = Vm + Vff (Vm is the electrostatic potential produced by the electronic and nuclear charge distribution of the solute). The Poisson problem has the form of a partial differential equations with domain in the whole three-dimensional space [2] ... 

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)). 

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 ... 

In reduced electrostatic units, the factor is eliminated and the Poisson equatic becomes ... 

The Poisson equation relates the electrostatic potential ([) to the charge density p. The Poisson equation is... 

This may be solved numerically or within some analytic approximation. The Poisson equation is used for obtaining the electrostatic properties of molecules. 

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 continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

IV. CLASSICAL CONTINUUM ELECTROSTATICS A. The Poisson Equation for Macroscopic Media... 

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 electrostatic approaches based on the Poisson equation have been used to address a wide variety of problems in biology. One particularly useful application is in the determination of the protonation state of titratable groups in proteins [46]. For... 

Solving the one-dimensional Poisson equation with the charge density profile pc z), the electrostatic potential drop near the interface can be calculated according to... 

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

Similar equations can be derived for the other ions. The charge density p x) is then obtained by adding the charge densities pertaining to all four kinds of ions. Both p x) and the electrostatic potential (f> x) are calculated by solving the Poisson equation self-consistently, and the particle distributions then follow by substituting (j)(x) into Eq. (25) and the respective equations for the other ions. 

The Poisson equation in electrostatic units for planar symmetry may be written... 

For small deviations from electroneutrality, the charge density at x is proportional to -(x)/kT9 where < is the difference of the electrostatic potential from its (constant) value when there is no charge density (the density of a species of charge z is proportional to 1 - zkT on linearizing the Boltzmann exponential). Then the Poisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation ... 

The Finite Difference Method (FD)168 169. This is a general method applicable for systems with arbitrary chosen local dielectric properties. In this method, the electrostatic potential (RF) is obtained by solving the discretized Poisson equation ... 

Resat, H. McCammon, J.A., Free energy simulations correcting for electrostatic cutoffs by use of the Poisson equation, J. Chem. Phys. 1996,104, 7645-7651. 

An alternative to the GB, COSMO, and Poisson electrostatic calculations is to model the solution to the Poisson equation in terms of pair potentials between solute atoms this procedure is based on the physical picture that the solvent screens the intra-solute Coulombic interactions of the solute, except for the critical descreening of one part of the solute from the solvent by another part of this solute. This descreening can be modeled in an average way to a certain level of accuracy by pairwise functions of atomic positions.18, M 65 One can obtain quite accurate solvation energies in this way, and it has recently been shown that this algorithm provides a satisfactory alternative to more expensive explicit-solvent simulations even for the demanding cases of 10-base-pair duplexes of DNA and RNA in water.66... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

E-A. Numerical or analytic solution of the classical electrostatic problem (e.g., Poisson equation) with homogeneous dielectric constant for solvent. 

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