Short-Range Energy

Which denotes respectively the short-range penetration corrected electrostatic multipolar (EMTP ) energy, short-range repulsion (Erep ), polarization (Epoi), charge-transfer (Ed), and dispersion (EdiSp) contributions. In presence of an open-shell cation, a ligand field correction is introduced (Elf)-... 

The summations are over all atoms i and / with separations and C,y, and Pij are the adjustable parameters of the model. The interaction energy of the system as a whole is then the sum of the Coulombic energies, short-range repulsive energies, and the weakly attractive energy components for all constituents. As we shall see, the individual components are typically the atomic centers and the points representing the polarization centers of the system. The successes of the simple ionic models introduced by Born and coworkers have been well documented and cover a wide range of applications. 

Generally speaking, intermolecular forces act over a short range. Were this not the case, the specific energy of a portion of matter would depend on its size quantities such as molar enthalpies of formation would be extensive variables On the other hand, the cumulative effects of these forces between macroscopic bodies extend over a rather long range and the discussion of such situations constitutes the chief subject of this chapter. 

Such attractive forces are relatively weak in comparison to chemisorption energies, and it appears that in chemisorption, repulsion effects may be more important. These can be of two kinds. First, there may be a short-range repulsion affecting nearest-neighbor molecules only, as if the spacing between sites is uncomfortably small for the adsorbate species. A repulsion between the electron clouds of adjacent adsorbed molecules would then give rise to a short-range repulsion, usually represented by an exponential term of the type employed... 

The details of the second-order energy depend on the fonn of exchange perturbation tiieory used. Most known results are numerical. However, there are some connnon features that can be described qualitatively. The short-range mduction and dispersion energies appear in a non-expanded fonn and the differences between these and their multipole expansion counterparts are called penetration tenns. 

A charge transfer contribution is often identified in perturbative descriptions of intennolecular forces. This, however, is not a new effect but a part of the short-range induction energy. It is possible to separate the charge transfer part from the rest of the induction energy [80]. It turns out to be relatively small and often negligible. Stone [28] has explained clearly how charge transfer has often been a source of confusion and error. 

A few ab initio calculations are the main source of our current, very meagre knowledge of non-additive contributions to the short-range energy [91], It is unclear whether the short-range non-additivity is more or less important than the long-range, dispersion non-additivity in the rare-gas solids [28, 92],... 

Flere u. j(r,T,P) is the short-range potential for ions, and e is the dielectric constant of the solvent. The solvent averaged potentials are thus actually free energies that are fimctions of temperature and pressure. The... 

This expression contains the contribution of the short-range potential included earlier in, so that the excess free energy, to this level of approximation, is... 

A short-range eontribution, w x0, whieh takes into aooount the nearest distanee of approaoh of the ion to the eleetrode surfaee. For ions that do not speoifioally adsorb this will be the OFIP, distanee h from the eleetrode. For ions that do speoifioally adsorb w ixj will be more oomplex, having oontributions both from short-range attraotive foroes and from the energy of de-solvation. 

In addition, the energy of interaotion between any two ions will oontain a eontribution from the mirror potential of the seoond ion u(r.p is now given by a short-range tenn and a tenn of the fonn... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

As with SCRF-PCM only macroscopic electrostatic contribntions to the Gibbs free energy of solvation are taken into account, short-range effects which are limited predominantly to the first solvation shell have to be considered by adding additional tenns. These correct for the neglect of effects caused by solnte-solvent electron correlation inclnding dispersion forces, hydrophobic interactions, dielectric saturation in the case of... 

The second excitation mechanism, impact scattering, involves a short range interaction between the electron and the molecule (put simply, a collision) which scatters the electrons over a wide range of angles. The usefiil feature of impact scattering is that all vibrations may be excited and not only the dipole active ones. As in Raman spectroscopy, the electron may also take an amount of energy hv away from excited molecules and leave the surface with an energy equal to Eq + hv. 

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

IS added to the short-range molecule-molecule interaction. Problems with the reaction ethod may arise from discontinuities in the energy and/or force when the number of les j rvithin the cavity of the molecule i changes. These problems can be avoided by dng a switching function for molecules that are near the reaction field boundary. 

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. 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

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