Apparent surface charge distribution

The solute-solvent potential term VJ represents the electrostatic interaction between the solute nuclei and electrons and the apparent surface charge distribution polarized medium. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into discrete elements, called tesserae, and by substituting the apparent charge a by a collection of point charges qk, each one placed at the center of a tessera sk. The point charges can be obtained as ... 

This is a special case of the previous problem with more than one boundary. It is particularly interesting in the study of association/dissociation processes and for the evaluation of two-body effective potentials for molecules separated by the solvent. The ASC method is the simplest to be used. In fact, in all many-boundary problems, one has to replace the definition of an apparent surface charge distribution a given in equation (14) by a set of charge distributions Oij, such that ... 

The reaction potential VR is therefore a single-layer potential. In order to calculate the apparent surface charge (ASC) distribution a, one makes use on the one hand of the relations... 

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. 

Over the last years, the basic concepts embedded within the SCRF formalism have undergone some significant improvements, and there are several commonly used variants on this idea. To exemplify the different methods and how their results differ, one recent work from this group [52] considered the sensitivity of results to the particular variant chosen. Due to its dependence upon only the dipole moment of the solute, the older approach is referred to herein as the dipole variant. The dipole method is also crude in the sense that the solute is placed in a spherical cavity within the solute medium, not a very realistic shape in most cases. The polarizable continuum method (PCM) [53,54,55] embeds the solute in a cavity that more accurately mimics the shape of the molecule, created by a series of overlapping spheres. The reaction field is represented by an apparent surface charge approach. The standard PCM approach utilizes an integral equation formulation (IEF) [56,57], A variant of this method is the conductor-polarized continuum model (CPCM) [58] wherein the apparent charges distributed on the cavity surface are such that the total electrostatic potential cancels on the surface. The self-consistent isodensity PCM procedure [59] determines the cavity self-consistently from an isodensity surface. The UAHF (United Atom model for Hartree-Fock/6-31 G ) definition [60] was used for the construction of the solute cavity. 

In the PCM procedure Vei is expressed in terms of an apparent charge distribution a(s) which is spread on the cavity surface (Apparent Surface Charge method, ASC) ... 

The apparent surface charge (ASC) approach appears to be a quite versatile method to calculate the reaction potential > (r), using either a quantum or a classical description of the solute molecule. According to classical electrostatics, the reaction potential can be described at any point in space in terms of an apparent charge distribution, a, spread on the cavity surface. Calling o-(s) the apparent charge per unit area, at a point s of the cavity surface E, one may write... 

Equation (11.7) can be used to eliminate the exterior derivative of (p from Eq. (11.6). Then, given some initial approximation for rp (perhaps just tpf, which is known once the solute s wave function has been computed), one could compute the surface charge, and thus the reaction-field potential, without the need to perform any calculations outside of the solute cavity. For a QM solute, this procedure must then be iterated to self-consistency. The original PCM of Miertus, Scrocco, and Tomasi [60, 61] used precisely this approach this model is now known as D-PCM. It is less desirable than more modern PC Ms, owing to the need to compute the normal electric field, which may be subject to increased numerical noise relative to later formulations that involve only electrostatic potentials [77]. Perhaps more significantly, the formulation of this model has conflated the apparent and actual surface charge distributions, and corresponds to a neglect of volume polarization [13]. 

With respect to other QM continuum models, the PCM method represents of the interaction operator Vi l ) (i.e. of the solvent reaction potential Va) in terms of an apparent surface charge (ASC) charge distribution a spread on the boundary F of the cavity (C) hosting the solute M. 

These methods combine a QM representation of solute with a classical continuum description of the solvent [18-23]. The methodology is equivalent to that of classical continuum methods, except that a) the solute charge distribution is allowed to relax by the solvent reaction field, and b) the solute-solvent interaction is computed at the QM level. Most QM continuum methods work within the multipole or apparent surface charge approaches, even though other formalisms are also available [18-23]. The solvent reaction field is introduced into the solute Hamiltonian by means of a perturbation operator (R in equation 22) that couples the solvent reaction field to the solute charge distribution. At this point, it is worth noting that equation 22 is not lineal, since T and R are mutually dependent. This means that a self-consistent process in which both the wavefunction and the reaction field are treated simultaneously is required to solve equation 22. This is the reason why these methods are typically known as self-consistent reaction field (SCRF) methods. 

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