Repulsion, point centers

Ruggiero Giuseppe Boscovich (1711-1787 CE), a Jesuit priest, replaced corpuscles with force-atoms (1758 CE) or point-centers of alternating attractive and repulsive forces. The views of Father Boscovich were similar to those of Newton, the Hindu atomists of the Nyaya-Vaisheshika, and the Arab followers of the Kalam (49). 

The most frequently used model potentials are Rigid sphere, point center of repulsion, Sutherland s model, Lennard-Jones potential, modified Buckingham potential, Kihara potential, Morse potential. Their advantages and disadvantages are thoroughly discussed elsewhere [39] [28]. 

Fig. 2-1 Models of intermolecular potentials, (a) Forceless mass points (b) elastic hard spheres (c) elastic hard spheres with superposed central attractive forces (d) molecules with central finite repulsive and attractive forces (e) square-well model (f) point centers of inverse-power repulsion or attraction.

Two models, which are of more interest for their mathematical trac-tability than for their physical plausibility, are point centers of inverse power repulsion and point centers of inverse power attraction. Models with both central finite repulsion and attraction are more general cases of these interactions. 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

Thus, the thermodynamic basis functions (P, U, and can be readily determined for dilute gases from Eqs. (4.74), (4.80), and (4.85), respectively, for various interaction potentials such as hard-sphere, point centers of attraction or repulsion, and Lennard-Jones (L-J) interaction potentials. We now determine second virial coefficients for simple in-termolecular interactions that admit analytical results. 

Prove the formula for the second virial coefficient for point centers of attraction or repulsion. 

Because the repulsion interaction energy of two point charges is inversely proportional to the distance separating the two charges, Dewar and co-workers, for example, represent the (ssiss) two-center two-electron integral by ... 

Whereas the JKR model approached the topic of particle adhesion from a contact mechanics viewpoint, the DMT theory simply assumes that the adhesion-induced contact has the same shape as a Hertzian indentor. The normal pressure distribution Ph(p) for the Hertzian indentor is related to the repulsive force and the distance from the center of the contact circle to the point represented by r according to the relationship [49]... 

If the substituents are nonpolar, such as an alkyl or aryl group, the control is exerted mainly by steric effects. In particular, for a-substituted aldehydes, the Felkin TS model can be taken as the starting point for analysis, in combination with the cyclic TS. (See Section 2.4.1.3, Part A to review the Felkin model.) The analysis and prediction of the direction of the preferred reaction depends on the same principles as for simple diastereoselectivity and are done by consideration of the attractive and repulsive interactions in the presumed TS. In the Felkin model for nucleophilic addition to carbonyl centers the larger a-substituent is aligned anti to the approaching enolate and yields the 3,4-syn product. If reaction occurs by an alternative approach, the stereochemistry is reversed, and this is called an anti-Felkin approach. 

Inert Gases. The calculation of 7 should be relatively straightforward for crystals of inert gases, in which only one kind of interaction may be expected. These crystals have a face-centered cubic structure. If each atom is treated as a point source of attractive and repulsive forces, only the forces between the nearest pairs of atoms are considered, the zero point energy is neglected, and no re-arrangement of atoms in the surface region is permitted, then the calculated 7 still depends on the equation selected to represent the interatomic potential U. 

F centers may act as adsorption centers not only in the alkali halides, but in any other crystals as well. Take, for example, a crystal of ZnO, in which the F center is an oxygen valency with two (not one ) electrons localized near it, as depicted in Fig. 30. From the chemical point of view such a center represents two adjacent localized free valencies of like sign which on an ideal surface could never meet because of Coulomb repulsion between them. (This should be especially stressed.) As a result of this property, such an F center may play a specific role in catalysis acting as an active center for a number of reactions. 

When the electrostatic properties are evaluated by AF summation, the effect of the spherical-atom molecule must be evaluated separately. According to electrostatic theory, on the surface of any spherical charge distribution, the distribution acts as if concentrated at its center. Thus, outside the spherical-atom molecule s density, the potential due to this density is zero. At a point inside the distribution the nuclei are incompletely screened, and the potential will be repulsive, that is, positive. Since the spherical atom potential converges rapidly, it can be evaluated in real space, while the deformation potential A(r) is evaluated in reciprocal space. When the promolecule density, rather than the superposition of rc-modified non-neutral spherical-atom densities advocated by Hansen (1993), is evaluated in direct space, the pertinent expressions are given by (Destro et al. 1989)... 

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