Definition of the electrostatic potential

The factor 47t 0 occurs when the quantities are expressed as SI units e0 is the permittivity of free space, equal to 8.854 187 82-10 12 C2 N 1 m-2. The factor 47ie0 ( = 1.112 626 5-10 10 C2 N 1 m 2) disappears when either atomic or cgs units are used, and we shall omit it in the expressions given. However, its inadvertent omission in numerical calculations will lead to meaningless results. 

The difference between the electrostatic potential at two points is equal to the work required to bring a unit charge from one point to the other. The choice of zero potential is arbitrary, but the potential is commonly defined as zero when the particles are at infinite distance. Thus, the electrostatic potential at a point is the work required to bring a unit of charge from infinity to that point. 

For a continuous charge distribution, the potential is obtained by integration over the space containing the distribution. At a point defined by r, the potential is given by 

For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain 

Since the first contribution is positive, while the second is negative, the sign of the potential at a point depends on whether the nuclei or the electrons dominate at that particular point. An electrophilic reagent approaching this distribution will be attracted by the electrons, but repelled by the nuclei. 

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Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ij)a of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By just outside we mean a position very close to the surface, but so fax away that the image interaction with the phase can be ignored in practice, that means a distance of about 10 5 — 10 3 cm from the surface. Obviously, the outer potential i/ a U a measurable quantity. 

Electrochemistry deals with charged particles that have both electrical and chemical properties. Since electrochemical interfaces are usually referred as electrified interfaces, it is clear that potential differences, charge densities, dipole moments, and electric currents occur at these interfaces. The electrical properties of systems containing charged species are very important for understanding how they behave at interfaces. Therefore, it is important to have a precise definition of the electrostatic potential of a phase [1-6]. Note that what really matters in electrochemical systems is not the value of the potential but its difference at a given interface, although it is illustrative to discuss its main properties. 

Inserting equation (2) into the definition of the electrostatic potential ... 

Fig. 1. Notations adopted in the definition of the electrostatic potential in a point P(r) generated by the electronic and nuclear charges of a molecule...

Ihe electrostatic interaction energy ot the pomt-like unit charge (probe) m position r with molecule A is described by the following formula (the definition of the electrostatic potential produced by molecule A, Fig. 14.14a) ... 

The results of electrostatic potential calculations can be used to predict initial attack positions of protons (or other ions) during a reaction. You can use the Contour Plot dialog box to request a plot of the contour map of the electrostatic potential of a molecular system after you done a semi-empirical or ab initio calculation. By definition, the electrostatic potential is calculated using the following expression ... 

To sum up, there are two potentials which are definite V, the electrostatic potential in empty, or nearly empty, space just outside the phase and fa the thermodynamic electrochemical potential of a charged component i. 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

The definition of the Madelung potential follows from the expression of the electrostatic energy of a lattice of point charges ... 

The relation in Eq. (19) is simply a definition of an electrostatic potential, due to some distribution of Coulomb charges. 

With this definition of the electric moments we have already achieved our first goal, because we have removed or at least hidden the integration over the charge distribution in the Taylor expansion of the electrostatic potential... 

To quantify the value of the electrostatic potential acting on the atoms of a rigidly cut three-dimensional lattice, one has to extend the definition of the Madelung constant. In an ionic representation of a binary compound, the electrostatic potential which is exerted on an ion of type i, located in the nth plane - indexed from the vacuum side - may be written in the following way ... 

The outer electrical potential of a phase is the electrostatic potential given by the excess charge of the phase. Thus, if a unit electric charge is brought infinitely slowly from infinity to the surface of the conductor to a distance that is negligible compared with the dimensions of the conductor considered (for a conductor with dimensions of the order of centimetres, this distance equals about 10 4cm), work is done that, by definition, equals the outer electric potential ip. 

The approach to the mathematical definition of the interface model is very simple. For every layer in the interface, the charge is defined once as a function of chemical parameters and once as a function of electrostatic parameters. The functions for charge are set equal to each other and solved for the unknown electrochemical potentials. Mathematical techniques for solving the equations have been worked out and described in detail (9). 

Since /(r) is the electrostatic potential energy per unit charge, the gradient of this parameter with distance must be equal to the force acting on a unit charge - which is the definition of the electric field. Hence it follows that... 

Representation of the density n(r) [or, effectively, the electrostatic potential — 0(r)] near any one of the sinks as an expansion in the monopole and dipole contribution only [as in eqn. (230c)] is generally, unsatisfactory. This is precisely the region where the higher multipole moments make their greatest contribution. However, the situation can be improved considerably. Felderhof and Deutch [25] suggested that the physical size of the sinks and dipoles be reduced from R to effectively zero, but that the magnitude of all the monopoles and dipoles, p/, are maintained, by the definition... 

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