Energy repulsion

The repulsion energy, rep, is modeled as proportional to the intermolecular overlap. The total overlap integral between the charge densities of two molecules A and B is calculated by numerical integration over the original uncondensed densities  

The attractive potential arising from dispersion forces between two isolated molecules may be written in the form 

Neglecting the higher order contributions to the dispersion energy and combining the inverse sixth-power attraction with the inverse twelfth-power repulsion leads to the familiar Lennard-Jones potential function  

The interaction between two different molecules are generally represented by the arithmetic mean of o and the geometric mean of c  

Setting di /dr = 0 in Eq. (2.3) gives, for the equilibrium separation between centers of an isolated pair of molecules, = 2 a. Thus, the excluded volume (the van der Waals co-volume b) may be seen to be four times the actual volume of the molecules  

To estimate the dispersion-repulsion forces for larger molecules for which Lennard-Jones force constants are not available it is customary to retain the general form of Eq. (2.3) but to replace the attractive and repulsive constants (4 o and 4 o ) by the semiempirical constants A and B  

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Using the conditions of the Langmuir approximation for the double-layer repulsion, calculate for what size particles in water at 25°C the double-layer repulsion energy should equal kT if the particles are 40 A apart. 

Figure Al.5.2 First-order Coulomb (O) and exchange-repulsion ( ) energies for Fle-FIe. Based on data from Komasa and Thakkar [70].

The exchange-repulsion energy is approximately proportional to the overlap of the charge densities of the interacting molecules [71, 72 and 73]... 

The necessity to calculate the electrostatic contribution to both the ion-electrode attraction and the ion-ion repulsion energies, bearing in mind that there are at least two dielectric ftmction discontinuities hr the simple double-layer model above. 

Figure 2-117. Dependence of the van der Waals energy on the distance between two non-con-nected atom nuclei. With decreasing atoiTiic distance the energy between the two atoms becomes attraction, going through a minimum at the van der Waals distance. Then, upon a further decrease in the distance, a rapid increase in repulsion energy is observed.

The iotal energy of a system is equal to the sum of the electronic energy and the Coulombic nuclear repulsion energy ... 

Figure 4-15 A van der Waals Potential Energy Function. The Energy minimum is shallow and the interatomic repulsion energy is steep near the van der Waals radius.

The reason the Schroedinger equation for molecules cannot be separated appears in the last term, involving a sum of repulsive energies between electrons. To... 

To see how and under what conditions stability is enhanced or diminished, we need to consider the symmetry of the orbital (9-32), Flectrons in the antisymmetric orbital r r have a 7ero probability of occurring at the node in u where U] = rj. Electron mutual avoidance of the node due to spin correlation reduces the total energy of the system because it reduces electron repulsion energy due to charge... 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

When two or more molecular species involved in a separation are both adsorbed, selectivity effects become important because of interaction between the 2eobte and the adsorbate molecule. These interaction energies include dispersion and short-range repulsion energies (( ) and ( )j ), polarization energy (( )p), and components attributed to electrostatic interactions. 

In an aqueous system with large particles weU-separated by a distance, s, (D and 5 > t ) the electrostatic repulsion energy between two identical charged spheres may be approximated (1) ... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

That is, the sum of the electronic energy and nuclear repulsion energy of the molecule at the specified nuclear configuration. This quantity is commonly referred to as the total energy. However, more complete and accurate energy predictions require a thermal or zero-point energy correction (see Chapter 4, p. 68). 

Estimate the cost of nonbonded HH repulsion as < function of distance by plotting energy (vertical axis) vs HH separation (horizontal axis) for methane+metham (two methanes approaching each other with CH bond head on ). Next, measure the distance between the nearest hychogens in eclipsed ethane. What is the HI repulsion energy in the methane chmer at this distance Multiplied by three, does this approximate the rotatioi barrier in ethane ... 

This corresponds to determining a set of LMOs which maximize the self-repulsion energy. 

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

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