Potential from point charges

The success of such an approach relies on the ability to devise realistic potentials Fext(iO- One possibility is to distribute a set of solvent molecules at random and with random orientations in a finite, sufficiently large volume under a set of constraints. I.e., the density of the solvent molecules should be realistic, no two solvent molecules should come two close to each other, and no solvent molecule should come too close to the solute. Most often it is then assumed that the external potential Fext(f is of electrostatic nature and can be modeled as a superposition of those of the solvent molecules. These can, e.g., in turn be approximated as a superposition of potentials from point charges placed at the positions of the nuclei. Improvements will allow the point charges to be placed at other positions, and also higher-order multipoles can be included. These potentials are devised so that selected properties of the solvent are reproduced accurately. 

In their approach, Yamamoto and Kato let Fext(> be determined through a non-polarizable model for the solvent so that Fext(>0 can be written as a superposition of Coulomb potentials from point charges representing the solvent. They showed, subsequently, how electron-transfer processes could be treated within their approach, even in the case when quantum effects of light nuclei (most notably, protons) need to be included. Finally, they presented results for a specific electron-transfer process. 

Kliiner et al. [19] has analyzed the bimodal velocity distributions observed in NO desorption from NiO(0 01) shown in Fig. 24 by calculating a full ab initio potential energy surface (PES) for an excited state in addition to the PES for the ground state. Calculation of the electronically excited state uses a NiOj cluster embedded in a semi-infinite Madelung potential of point charges 2. The excited state relevant for laser-induced desorption is an NO -like intermediate, where one electron is transferred from the cluster to the NO molecule. 

Figure 4 Molecular electrostatic potential of water molecule, represented as a contour plot with intervals of 0.025 au. Red contours indicate regions of negative potential and blue represents positive, (a-b) Potential generated from full electron density, in and perpendicular to the molecular plane, respectively (c d) potential generated from point charges situated at three atomic positions (e-f) potential generated from point charges and dipoles situated at three atomic positions. (See color plate at end of chapter.)...

The Eq. (2.78) describes the dependence of the overpotential on the deposition time from point b to point c. The overpotential changes due to the change of the surface concentration of adatoms from Co,a at the equilibrium potential to some critical value Ccr.a at the critical overpotential, rj, at which the new phase is formed. Hence, the concentration of adatoms increases above the equilibrium concentration during the cathode reaction, meaning that at potentials from point b to point c there is some supersaturation. The concentration of adatoms increases to the extent to which the boundary of the equilibrium existence of adatoms and crystals has been assumed to enable the formation of crystal nuclei. On the other hand, the polarization curve can be expressed by the equation of the charge transfer reaction, modified in relation to the crystallization process, if diffusion and the reaction overpotential are negligible, as given by Klapka [48] ... 

ME, but of less mathematical rigour, is the singularity method (SM), which reconstructs the potential field by a superposition of singularity fields from point charges (Phillips 1995). Additionally, it is possible to approximate compact, isometric aggregates as porous spheres (PS), for which analytical expressions of the double layer structure are available (Ohshima 2008). Some of the mentioned methods assume a certain shape of the aggregates or primary particles, some of them are based on the Unearised PBE (BEM, ME, PS). Only a few have already been employed for particle clusters (CoeUio et al. 1996 Kwon et al. 1998 Schiefil et al. 2012). 

Let s find the potential at any arbitrary point (x, y) at distance r = (x -i-y2)i/2 from the origin in Figure 21.11(b). The vector r is at an angle 0 with respect to the x-axis. Use Equation (21.9) to define the potential at P from each of the two fixed point charges. The total electrostatic potential ip is the sum of the potentials from each charge of the dipole ... 

Suppose that you have a charge q at point A in water at a distance a from an oil/water interface (see Figure 21.16). The dielectric constant of water is Dw and the dielectric constant of oil is D . In general, we want to know the electrostatic potential at an arbitrary point P in water, at a distance r from A. At P, the electrostatic potential from the charge would be ipp = Cq/Dwr if there were no interface. But the presence of the dielectric interface perturbs the field at P. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

The electrostatic potential at a point r, 0(r), is defined as the work done to bring unit positive charge from infinity to the point. The electrostatic interaction energy between a point charge q located at r and the molecule equals The electrostatic potential has contributions from both the nuclei and from the electrons, unlike the electron density, which only reflects the electronic distribution. The electrostatic potential due to the M nuclei is ... 

Adsorption Forces. Coulomb s law allows calculations of the electrostatic potential resulting from a charge distribution, and of the potential energy of interaction between different charge distributions. Various elaborate computations are possible to calculate the potential energy of interaction between point charges, distributed charges, etc. See reference 2 for a detailed introduction. 

We have studied, by MD, pure water [22] and electrolyte solutions [23] in cylindrical model pores with pore diameters ranging from 0.8 to more than 4nm. In the nonpolar model pores the surface is a smooth cylinder, which interacts only weakly with water molecules and ions by a Lennard-Jones potential the polar pore surface contains additional point charges, which model the polar groups in functionalized polymer membranes. 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

Since a metal is immersed in a solution of an inactive electrolyte and no charge transfer across the interface is possible, the only phenomena occurring are the reorientation of solvent molecules at the metal surface and the redistribution of surface metal electrons.6,7 The potential drop thus consists only of dipolar contributions, so that Eq. (5) applies. Therefore the potential of zero charge is directly established at such an interface.3,8-10 Experimentally, difficulties may arise because of impurities and local microreactions,9 but this is irrelevant from the ideal point of view. 

Figure 5. Sketch of a work function-potential of zero charge plot. The line through the point of Hg has unit slope. The horizontal distance of Mi and M2 from the line measures AX in Eq. (28).

Figure 14. Plot of the potential of zero charge, Ea=0 (from Table 26), against the work function,

Figure 17. Plot of the potential of zero charge, EaJi, vs. the electron work function, . The point is the most probable value. Data for E0wq from Ref. 140 for Au (111) from Ref. 25 for Pt (111) from Ref. 867.

The potential of zero charge measures, on a relative scale, the electron work function of a metal in an electrochemical configuration, i.e., immersed in a solution rather than in a vacuum. Converted to an absolute value (UHV scale) and compared with the classic electron work function of the given metal, the difference between the two quantities tells us what occurs from the local structural point of view as the metal comes in contact with the solution. 

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