Apparent surface charge

Given a cavity segmentation by m segments i, of sizes s, and centered at positions fj, the dielectric surface polarization charge densities, at, and the corresponding apparent surface charges, qt = Sjffj, can be calculated from the exact dielectric boundary... 

The efficient construction of proper and sufficiently accurate segmentations of a molecular-shaped cavity is an important technical aspect of apparent surface charge models, because it has a strong influence on both the accuracy and speed of the calculations. Before going into details, some common features will be discussed. 

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... 

Let us remark incidentally that the van der Waals, solvent-accessible and solvent-excluded molecular surfaces commonly used in apparent surface charge calculations, can be discretized without resorting to a polyhedral approximation. Indeed, these surfaces are made of pieces of spheres and tori and it is therefore possible to mesh and compute integrals on the molecular surfaces since analytical local maps are available [19],... 

For practical calculations, the integral over Y has to be discretized, which introduces an additional numerical error. An alternative consists in applying the Galerkin approximation to system (1.38), which is equivalent to Equation (1.37). The discretized apparent surface charge [cr] is obtained by solving successively the linear systems... 

The apparent surface charge a involved in the above expression satisfies the integral equation... 

A more sophisticated description of the solvent is achieved using an Apparent Surface Charge (ASC) [1,3] placed on the surface of a cavity containing the solute. This cavity, usually of molecular shape, is dug into a polarizable continuum medium and the proper electrostatic problem is solved on the cavity boundary, taking into account the mutual polarization of the solute and solvent. The Polarizable Continuum Model (PCM) [1,3,7] belongs to this class of ASC implicit solvent models. 

The case of the dielectric version of PCM (DPCM) is more complicated than CPCM, as the system of equations which must be solved to compute the apparent surface charges is [3] ... 

The bottleneck of a calculation in solution is the evaluation of the polarization which, in the case of PCM, corresponds to the evaluation of the apparent surface charges. In particular, the bottleneck is represented by the evaluation of the products between the integral matrices of the electrostatic potential (matrix S in Equation (1.8.6)) or of the normal component of the electric field (matrix D in Equation (1.92)) and the apparent charges vector q. Thus the criterion we use to compare the standard and the simultaneous approach is based on the number of matrix products (Sq or D q) necessary in the whole optimization process. We also remind the reader that the dimension of the matrices is equal to the square of the number of the surface elements. 

The authors would like to thank Prof. Berny Schlegel for his contribution in the discussion that led to the idea of the area-weighted, apparent surface charges. Also we would like to thank Prof. Benedetta Mennucci for her continuing interest and her encouragement. 

In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute-solvent electrostatic interactions can be described in terms of a solvent reaction potential, Va expressed as the electrostatic interaction between an apparent surface charge (ASC) density a on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density cr by a collection of point charges qk, placed at the centre of each element sk. We thus obtain ... 

Most of the quantum chemical calculations of the nuclear shielding constants have involved two classes of solvation models, which belong to the second group of models (n), namely, the continuum group (i) the apparent surface charge technique (ASC) in formulation C-PCM and IEF-PCM, and (ii) models based on a multipolar expansion of the reaction filed (MPE). The PCM formalism with its representation of the solvent field through an ASC approach is more flexible as far as the cavity shape is concerned, which permits solvent effects to be taken into account in a more accurate manner. 

The vector q contains the apparent surface charges, and the superscript ONIOM indicates that it is equilibrated with the integrated density. In this scheme, the cavity is built around the entire cluster, which is then used in all three ONIOM subcalculations. However, the density o-ONIOM is only avaqai-qe after ap three ONIOM subcalculations have been completed. The wavefunctions and apparent surface charges can therefore no longer be optimized simultaneously, and we need to resort to an iterative scheme in which the calculation of the wavefunction and charges are alternated. This increases the computational cost significantly, and we explored approximations to ONIOM-PCM/A that are computationally more feasible. 

In ONIOM-PCM/B, we assume that the reaction field obtained in the low level calculation on the full system is a good approximation of that in ONIOM-PCM/A. We equilibrate the apparent surface charges with the low level calculation on the real... 

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. 

Fig. 2.4 The dependence of s on separation between C hho2 and C hlr.m molecules in LHCII is explained by the formation of a common cavity (a) schematic depiction of the interaction between two transition dipoles (arrows) immersed within cavities in a polarizable medium, (b) the apparent surface charges of the donor spread on the acceptor microenvironment, and (c) the spread of the surface charge over the cavities changes the magnitude and functional form of the electronic coupling...

An immediate consequence of the onset of spontaneous polarization in a body is the appearance of an apparent surface charge density and an accompanying depolarizing field ED as shown in Fig. 2.43(a). The energy associated with the... 

Therefore, the short-range interactions due to ion hydration do not modify qualitatively the double layer interaction, which remains repulsive at distances larger than the range of the interfacial interaction of the ions. The magnitude of the repulsion is, however, modified, since the relation between the true surface charge density a and the "apparent surface charge density a depends on AW, and hence on the specificity of the ions. 

Moving now to QM/continuum approaches, we shall limit our exposition to the so-called apparent surface charges (ASC) version of such approaches, and in particular to the family known with the acronym PCM (polarizable continuum model) [11], In this family of methods, the reaction potential Vcont defined in Eq. (1-2) has a form completely equivalent to the Hel part of the Z/qm/mm operator defined in Eq. (1-4), namely ... 

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