Potential energy of electrostatic interaction

According to Coulomb s law, the potential energy of electrostatic interaction U between two point charges, q and q2 in vacuum, is given by... 

On insertion of the field (34) into equation (55) one obtains the general expression for the potential energy of electrostatic interaction between two electric multipolar systems ... 

With the general definition of the electric multipole (40) and n-th order field (53), the potential energies of electrostatic interaction between electric multipoles and fields take the general form ... 

Energy of Induced Multipole Interaction.—Interaction between a Multipolar System and External Fields. We calculated above the potential energy of electrostatic interaction between a multipole system and external fields, or thefirst-orderenergyduetothefirstpowerofthefield. Besides that energy, which took account only of the reoriratation of permanent multipoles, we have to take into consideration contributions due to the drcum-stance that an external electric field induces higher-order multipole moments given by the expressions (72) and (79). 

Work done and potential energy of electrostatic interactions... 

Hence to obtain an expression for the potential energy of electrostatic interactions between the two charges requires that an expression for the work done in bringing the two charges from a... 

The distribution of the ions in the ionic atmosphere of the y -ion is governed by the total potential energy of electrostatic interactions at all positions, r, from the central reference y -ion. For any ion of type, i, in the ionic atmosphere of the y -ion, this potential energy is given by Zie j/j. Here is the charge on the ion of type i, and x]/j is the potential at this ion due to the central y -ion and all the other ions of the ionic atmosphere, but it does not include a contribution from the ion which is at position, r. 

The potential energy of electrostatic interaction between ft and the reaction field is ft Er. The corresponding quantum-mechanical operator in atomic units is... 

Table 10.5 Approximate expressions for the potential energy of electrostatic interactions, Mr, as a function of closest distance of approach, H. In the table, z is the counter-ion change number (e.g. with NaCI as electrolyte, z is either I or -1, depending on whether Na or Ct is the counter-ion to the surface). The "ionic valency" is the same as the "charge number" (both terms include the sign of the charge). More expressions are available in Israelachvili (2011), e.g. for electric double layer interaction between two cylinders or a cylinder and a flat surface...

Interaction of incident electrons with the electrostatic potential (ESP) gives the possibility to reconstmct the potential from transmission electron diffraction (ED) experiments. ESP and the electron density determine all physical properties of crystals (e.g. energy of electrostatic interaction, characteristics of the electrostatic field in crystals, dipole, quadmple and other momentum of nuclear, diamagnetic susceptibility. 

The last term in the formula (1-196) describes electrostatic and Van der Waals interactions between atoms. In the Amber force field the Van der Waals interactions are approximated by the Lennard-Jones potential with appropriate Atj and force field parameters parametrized for monoatomic systems, i.e. i = j. Mixing rules are applied to obtain parameters for pairs of different atom types. Cornell et al.300 determined the parameters of various Lenard-Jones potentials by extensive Monte Carlo simulations for a number of simple liquids containing all necessary atom types in order to reproduce densities and enthalpies of vaporization of these liquids. Finally, the energy of electrostatic interactions between non-bonded atoms is calculated using a simple classical Coulomb potential with the partial atomic charges qt and q, obtained, e.g. by fitting them to reproduce the electrostatic potential around the molecule. 

Energy of Electrostatic Interaction Between Two Multipole Systems.— Consider two arbitrary electric charge systems qt and with multipoles and mutually distant by r (Figmre 5). The potential energy of their interaction can be determined from equation (54) on repladng therein the -th order field by the field F " of order n produced by the... 

One of the main functions of the solvent is simply to reduce the forces of interaction between ions and thereby reduce the electrostatic potential energy of this interaction. Physically this corresponds to allowing the ions to exist as ions. This is a bulk effect. 

The potential due to the ionic atmosphere at the surface of the ion, i.e. at a distance a/2 from the centre of the central reference ion. Non-ideality in electrolyte solutions is a result of electrostatic interactions obeying Coulomb s Law. The potential energy of such interactions is given in terms of ... 

To elucidate these forces, one best begins with the interaction of two permanent dipoles Pi and p2 at a distance r from each another. Compare also Fig. 2.1a. The potential energy of the interaction of the two dipoles pi and py is found from electrostatics to be... 

The total potential energy of two spherical particles is equal to the sum of electrostatic potential energy and the potential energy of molecular interaction between the particles ... 

An expression for the force of electrostatic repulsion between two charged crossed hemicylindrical surfaces is given in Equation (3.5) and in fact this turns out to be equivalent to interaction between a sphere and a flat plate. Integration of this expression leads directly to the potential energy of electrostatic repulsion, namely. 

The molecular theory of Van der Meer and Vertogen is based on a specific molecular model that is not in contradiction with experiment. At the same time Barbero and Durand [89] have shown that the molecular tilt is an intrinsic property of any layered quadrupolar structure. This idea has been used by Poniwierski and Sluckin in their model [84] that presents a rather general mechanism for the stabilization of the smectic C phase. It is interesting to note that the mathematical form of the interaction potential in the Poniwierski-Sluckin theory is similar to the potential (Eq. 85). The energy of electrostatic interaction between two axial quadrupoles, employed in [84], can be written as... 

From one force held to the next, the balance of energy terms may be different. For example, one force held might use a strong van der Waals potential and no electrostatic interaction, while another force held uses a weaker van der Waals potential plus a charge term. Even when the same terms are present, different charge-assignment algorithms yield systematic differences in results and the van der Waals term may be different to account for this. 

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. 

In a solution of a solute in a solvent there can exist noncovalent intermolecular interactions of solvent-solvent, solvent-solute, and solute—solute pairs. The noncovalent attractive forces are of three types, namely, electrostatic, induction, and dispersion forces. We speak of forces, but physical theories make use of intermolecular energies. Let V(r) be the potential energy of interaction of two particles and F(r) be the force of interaction, where r is the interparticle distance of separation. Then these quantities are related by... 

The ions in solution are subject to two types of forces those of interaction with the solvent (solvation) and those of electrostatic interaction with other ions. The interionic forces decrease as the solution is made more dilute and the mean distance between the ions increases in highly dilute solutions their contribution is small. However, solvation occurs even in highly dilute solutions, since each ion is always surrounded by solvent molecules. This implies that the solvation energy, which to a first approximation is independent of concentration, is included in the standard chemical potential and has no influence on the activity. 

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