Born repulsion expression

The Born repulsion expression stems from the Pauli principle, which results from the prohibition of electrons with the same spin occupying the same wavefunction, a situation that occurs when doubly occupied orbitals interact. Whereas the van der Waals interaction cannot be computed accurately from... 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

For the linear model cos cp = 0, sin (p = 1, and it will only be stable when d2E/d(p2 > 1 so that (1 — 8oc/r3) > 0 and (8oc/r3) < 1. Therefore, if we could allow the value 8a/r3 of a molecule to increase continuously, when it reaches unity the molecule would begin to have lower symmetry. If we still wish to correct the energy for the change of r, then we must differentiate the expression for E with respect to r and eliminate B. At this point, however, the exact calculation can be taken no further because in hydrogen compounds no normal Born repulsion exists. The calculation can be carried through successfully for all molecules not containing hydrogen ions. 

Several repulsive and attractive forces operate between colloidal species and determine their stability [12,13,15,26,152,194], In the simplest example of colloid stability, dispersed species would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The overlap causes a coulombic repulsive force acting against each surface, which will act in opposition to any attempt to decrease the separation distance (see Figure 5.2). One can express the coulombic repulsive force between plates as a potential energy of repulsion. There is another important repulsive force causing a strong repulsion at very small separation distances where the atomic electron clouds overlap, called Born repulsion. 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

In order to calculate the potential well, Born repulsion should be included in the calculations. Inslead of using an explicit expression for the Born forces, the interaction potential U is cut at a distance of about 4 A. Considering <5 of the order of 24 A, x-4A, >fp-l5-10 20 J and A — 7.5 10 20 J, one obtains that the optimum At is given by the expression... 

These results indicate that the Born-Mayer expression with four repulsive parameters... 

DFT quantum-chemical codes, the Born repulsive interaction is accurately computed. Expressions (4a) and (4b) are two-body interaction potentials commonly used in codes that predict energies or geometries based on empirical potentials. 

Combining equations 5.13 and 5.15 gives an expression for the lattice energy that is based on an electrostatic model and takes into account Coulombic attractions, Coulombic repulsions and Born repulsions between ions in the crystal lattice. Equation 5.16 is the Born-Lande equation. 

Eq. (16) gives the total attraction energy at the molecular level. It indicates that the attraction energy becomes more negative as the separation distance decreases. When the separation distance becomes so small, to the extent that the electron clouds of two units start to overlap, a repulsive force named as the Born repulsion energy is generated and can be expressed as ... 

The potential energy V of an ionic crystal can be expressed approximately as the sum of the Coulomb interaction and the Born repulsive potential ... 

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion tenns, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Born-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

In order to employ the Born treatment to explain the transformations of CsCl + CsBr solid solutions, we have used the revised van der Waals coefficients of CsBr proposed by Hajj.10 The repulsive parameters of CsBr can be evaluated as follows (i) In the expression for the lattice energy (eqn. (3)) we have assumed that p2 = apx as proposed by Rao and Rao.3 (ii) We calculate (8 >t + 662)CsBr by substituting the value of p, obtained from the compressibility relations.2... 

Besides this electrostatic attraction there is the repulsion which arises when the ions begin to touch each other. This repulsion originates in the general mutual repulsion of the electron clouds of atoms and ions, which always occurs when the clouds penetrate each other and the electrons do not form any atomic bond with a common electron pair (p. 147). The repulsion is difficult to calculate and so Born represented the repulsion energy by B rn, a function which, provided n is large, increases very rapidly with decreasing distance r, corresponding to almost hard spheres. B in this expression is a still undetermined factor of proportionality while the value of the exponent can be deduced from the compressibility n amounts to about 9. 

For application to nonmolecular solids, the bond description is similar but certain modifications are needed. First, the covalent energy must be multiplied by the equivalent number n of two electron covalent bonds per formula unit that must be broken for atomization. The evaluation of n will be discussed in detail presently. Second, the ionic energy must be evaluated as the potential energy over the entire crystal, corrected for the repulsions among adjacent electronic spheres. This is done by using the Born-Mayer equation for lattice energy, multiplying this expression by an empirical constant, a, which is 1 for the halides and less than 1 for the chal-cides, as follows ... 

Born suggested that this repulsive energy could be expressed by... 

Since crystals are not indefinitely compressible there is evidently a repulsion force which operates when the electron clouds of the ions begin to interpenetrate (without electron-sharing between the ions). This repulsion energy is not readily calculable, and Born represented it by 5/r , a function which increases very rapidly with decreasing distance r if n is large, that is, it corresponds to the ions being hard spheres. The expression for the lattice energy is now... 

Differentiation of this expression with respect to r, where the repulsive and attractive forces are in equilibrium, produces the Born-Lande potential for one pair of ions. For one mole of ions the expression must be multiplied by Avogadro s number (A ). 

The attractive Coulomb energy needs to be balanced against the contribution from the short-range repulsive forces that occur between ions when their closed shells overlap. There is no accurate simple expression for this repulsion. In the Born-Lande model it is assumed proportional to 1/r", where n is a constant that varies in the range 7-12 depending on the ions. The resulting expression for the lattice energy is... 

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