Repulsive potential between

Because of coulomb repulsions, the core repulsive potential between ions of like sign may be assumed to be negligibly small in the most significant configurations (see Section II-A). Consequently, in these configurations, the interaction potential between ions of like sign is a function only of the distance between the ions and is independent of the type of ions. The cation-anion pair potentials may be written as... 

Here h i), z =l or 2, stands for the core Hamiltonian for electron i moving in the field of nuclei a and b separated by distance R, and g(l,2) is the repulsion potential between two electrons away from each other by distance rl2. 

Moliere central potential — an analytical expression to approximate the repulsive potential between two atoms at short range. The universal pairwise potential requires only two parameters — the atomic numbers of the interacting atoms. 

There exist several linkages of the SBUs which should be avoided, for example, the short separation of two metal atoms (M Mj), one ligand atom and one bridging atom (Lj Bj), and L Mj in SBUs unlinked. To prevent these undesirable linkages, other potential functions have to be considered, including the repulsive potential between Mj Mj pairs, the attractive potential between L, - - - Mj pairs, and the repulsive potential between Lj Bj pairs. A repulsive potential between Mr Mj pairs prevents SBUs from overlapping with each other. The distance between the Mr Mj pair is limited to 3.4 A for D4R. The Lennard-Jones potential parameters used in assembling the D4R are provided in Table 7.2. 

Because dispersion and repulsion potentials between two partners fall off so rapidly with distance the binding energy depends very much upon the shape of the surface. For example, a molecule bound at the centre of a hemispherical cavity of radius Zq (Fig. 13a) is subject to a dispersion potential 4 times as great as for a molecule located on a smooth surface [31]. Inside a long, narrow capillary, terminated by a hemisphere at whose centre the molecule is located (Fig. 13b) the attractive potential is roughly 7 times that of a plane surface [31]. Inside a topographic depression a bound molecule is in close contact with a larger number of atoms than on a smooth surface. 

A repulsive potential between the solute and the cavity walls is added to the reaction field(s). This repulsion is due to exchange interaction between the solute and the solvent molecules, which is modeled as... 

Frenkel defects in general do not play an important role in molecular crystals. The asymmetric shape of the molecules and the steeply increasing repulsive potential between molecules at short intermolecular distances make the occurrence of interstitial molecules in molecular crystals thermodynamically improbable. The... 

Assuming constant surface potential, the repulsion potential between two spherical particles is given by. 

It was later realized that the tight binding approximation may be directly deduced as a rigorous approximation to the density functional theory, is the opinion of the author of [34]. The other possibUities in principle are three center terms and an overlap of orbitals on distant lattice sites. One computes the hopping integral types of Vssa, Vssx,, V pjr, and a pairwise repulsive potential between atoms. 

The influence of the concentration of the electrolyte in the solution upon the repulsive potential between the two plates is entirely expressed by the quantity x. As x is proportional to nK the value of f (u, z), as read from the table, must bo... 

Fig. 35. The repulsive potential between two spherical particles, when the exact expression for high potentials is applied. Dotted lines = repulsive potential according to the approximated aG...

This approach is not limited to the study of solutions. It has been extended to the case of a molecule interacting with the surface of a solid [24]. In this case, the problem is even simpler provided that the solid is a dielectric since the cavity docs not need to be created and one only has to introduce a repulsive potential between the molecule and the solid. It can be extended to the quantum chemical study of a fragment of macromolecule in electrostatic interaction with the rest of the system. This is particularly interesting to study the chemical processes in the active site of a biological macromolecule like an enzyme 25. Ilierefore, several fields of chemistry are already concerned with this methodology. 

We define the potential of mean force AF s) as the free energy difference when the center of the particle is located at s -)- i and when it is at the center of the cell. (Hence, s is the separation between the wall and the surface of the sphere.) We present AF(s) for three different radii in Fig. 6. The theory predicts a positive, repulsive potential between the particle and the wall. The intensity of this repulsion increases with the radius of the particle the simulations predict a global repulsive force, with the occurrence of oscillations and local extrema. The presence of these minima and maxima is due to the formation of smectic-like modulation of the LC density by the confining wall. (See Fig. 7)... 

For practical applications in polymer dispersions (ie also during heterophase polymerizations) the effect of charge stabilization due to permanent charges, either by ionic stabilizers, comonomers, or initiator fragments, is much more important. The repulsive potential between two equally charged particles (yR,es) separated at a center-to-center distance of (dpp + D) is given by equation 35, where Q is the charge on each particle and a is the permittivity of the continuous phase. 

FIGURE 4.72 Types of coordination by vertices, edges and polyhedra faces, with the distance between the coordination cations ranging in 1 0.58 0.38 proportion, thus properly increasing the repulsive potential between them after Heyes (1999). 

The structure factor Ss r(q) mirrors repulsive interaction between the stars in good and 6 solvents. Star polymers in dilute solution can be assimilated to soft colloidal particles. Interaction between star polymers in dilute solutions has a character of soft repulsion that arises at separation d 2Rs,ar between the star centers. The binary repulsive potential between the stars was evaluated by Witten and... 

Unfortunately there is as yet no known way to obtain the repulsion energy from properties of the separate molecules. An attempt has been made to characterise the repulsive surface of a molecule by performing IMPT calculations between the molecule and a suitable test particle, such as a helium atom. Because the helium atom has only one molecular orbital and is spherically symmetrical, such calculations can be done much more easily than calculations involving two ordinary molecules. From the data for the repulsion between molecule A and the test particle, and between B and the test particle, it may be possible to construct a repulsive potential between A and B. Some limited progress has been made with this idea. An alternative approach has been based on the suggestion that the repulsion energy is closely correlated with the overlap between the molecular wavefunctions, but this seems likely to be more useful as a guide to the form of analytic models than as a direct route to accurate potential functions. 

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