Continuum Poisson-Laplace equation

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. 

The contribution from electrons can be computed by using the wave function. We consider a vacuum hole (a cavity) in the dielectric continuum with an arbitrary shape. The solute molecule represented in the electric distribution pM is immersed and located in this cavity. The basic equation for dielectric continuum model is Poisson Laplace equation in the electromagnetics. The electrostatic field in the cavity and outside one can be obtained by solving the following equations. 

Summary of the continuum model. The method described in this section is the first application to calculating a solvated molecule s properties in a quantum chemical manner. Qualitative considerations seem to be reasonable in spite of their considerable simplicity. PCM can be regarded as a final achievement amongst the series of ab initio MO-dielectric continuum combined methods, since all the method is based on the Poisson-Laplace equation. However, it is important to pay much attention to the serious disadvantage in all these... 

Many continuum solvation methods prefer to attack the electrostatic problem by resorting to grid integration of the Poisson equation. The use of 3D grids makes it convenient to extend the model we have considered till now, characterized by a constant value of e, and by the corresponding Poisson and Laplace equations, i.e. 

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