Repulsion and attraction

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

Weeks J D, Katsov K and Vollmayr K 1998 Roles of repulsive and attractive forces in determining the structure of non uniform liquids generalized mean field theory Phys. Rev. Lett. 81 4400... 

DLVO Theory. The overall stabiUty of a particle dispersion depends on the sum of the attractive and repulsive forces as a function of the distance separating the particles. DLVO theory, named for Derjaguin and Landau (11) and Verwey and Overbeek (12), encompasses van der Waals attraction and electrostatic repulsion between particles, but does not consider steric stabilization. The net energy, AGp between two particles at a given distance is the sum of the repulsive and attractive forces ... 

The excess free energy is divided into terms representing the contributions due to repulsive and attractive nonassociative forces acting between molecules, as well as into a contribution arising from association [38,39]... 

The first step towards the development of appropriate expressions is the decomposition of the nonassociative pair potential into repulsive and attractive terms. In this work we apply the Weeks-Chandler-Andersen separation of interactions [117], according to which the attractive part of the Lennard-Jones potential is defined by... 

When two atoms approach each other so closely that their electron clouds interpenetrate, strong repulsion occurs. Such repulsive van der Waals forces follow an inverse 12th-power dependence on r (1/r ), as shown in Figure 1.13. Between the repulsive and attractive domains lies a low point in the potential curve. This low point defines the distance known as the van der Waals contact distance, which is the interatomic distance that results if only van der Waals forces hold two atoms together. The limit of approach of two atoms is determined by the sum of their van der Waals radii (Table 1.4). 

How large is an atom We cannot answer this question for an isolated atom. We can, however, devise experiments in which we can find how closely the nucleus of one atom can approach the nucleus of another atom. As atoms approach, they are held apart by the repulsion of the positively charged nuclei. The electrons of the two atoms also repel one another but they are attracted by the nuclei. The closeness of approach of two nuclei will depend upon a balance between the repulsive and attractive forces. It also depends upon the energy of motion of the atoms as they approach one another. If we think of atoms as spheres, we find that their diameters vary from 0.000 000 01 to 0.000 000 05 cm (from 1 X 10-8 to 5 X 10 8 cm). Nuclei are much smaller. A typical nuclear diameter is 10, s cm, about 1/100,000 the atom diameter. 

It is clear that systems of hard ellipsoids exhibit an intriguingly simple phase behaviour with some resemblance to that of real nematogens. However, such systems cannot form smectic or columnar phases and in addition the phase transitions are not thermally driven as they are for real mesogens. As we shall see in the following sections the Gay-Berne potential with its anisotropic repulsive and attractive forces is able to overcome both of these limitations. 

Following the approach similar to that of the atomic-scale model, the evolution of the system state and the lateral force on the asperity can be determined in terms of AUldrj = 0. If we chose V rj)= VQ cos ir rj) as the potential function for the repulsive and attractive pinning center, respectively, the lateral force F=-dy/d7 can be plotted as a function of the traveling distance p, as shown in Fig. 17. 

The bands of matrix-isolated molecules are frequently observed at the wavelengths which differ from those in gas-phase spectra. These matrix shifts are induced by the repulsive and attractive forces between the isolated molecules and the atoms which form the matrix site. Repulsions lead to small increases (1-15 cm ) of vibrational frequencies, and attractions decrease them. Matrix shifts depend on the type of matrix gas they rise in the sequence from neon to xenon. In general, the shifts are positive (the... 

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Balance between repulsive and attractive electrostatic effects depends on charges q +, q- (electronegativity differences of the C-X bonds) or the nature of the dipole (type of bonds) and their orientation. In the case of acetals, one needs to consider an ensemble of not only two dipoles but of, at least, four (Fig. 4). 

In order to obtain robust conformational assignments from vibrational spectra without rotational resolution, it is important to predict reliable monomer frequency shifts between conformations. Harmonic B3LYP predictions were shown to correlate reasonably well with experiment [69], and simple mles based on repulsive and attractive intra-monomer interactions were developed. However, the predicting power of the B3LYP method for the energy sequence... 

To what extent can theory predict the collision efficiency factor Two groups of researchers, Derjagin and Landau, and Verwey and Overbeek, independently of each other, have developed such a theory (the DLVO theory) (1948) by quantitatively evaluating the balance of repulsive and attractive forces that interact most effective tool in the interpretation of many empirical facts in colloid chemistry. 

The stability of the atomic nucleus depends upon a critical balance between the repulsive and attractive forces involving the protons and neutrons. For the lighter elements, a neutron to proton ratio (N P) of about 1 1 is required for the nucleus to be stable but with increasing atomic mass, the N P ratio for a stable nucleus rises to a value of approximately 1.5 1. A nucleus whose N P ratio differs significantly from these values will undergo a nuclear reaction in order to restore the ratio and the element is said to be radioactive. There is, however, a maximum size above which any nucleus is unstable and most elements with atomic numbers greater than 82 are radioactive. 

A related potential form, which was primarily developed to reproduce, structural energetics of silicon, was introduced by Tersoff and was based on ideas discussed by Abell . The binding energy in the AbeH-Tersoff expression is written as a sum of repulsive and attractive two-body interactions, with the attractive contribution being modified by a many-body term. 

According to the above mechanism, reverse osmosis separation is governed by two distinct factors, namely (i) an equilibrium effect which is concerned with the details of preferential sorption in the vicinity of the membrane surface, and (ii) a kinetic effect which is concerned with the mobilities of solute and solvent through membrane pores. While the former (equilibrium effect) is governed by repulsive and attractive potential gradients in the vicinity of the membrane surface, the latter (mobility effect) is governed both by the potential gradients (equilibrium effect) and the steric effects associated with the structure and size of molecules relative to those of pores on the membrane surface. 

Eliel et al. equilibrated the diastereotopic 2-iPr-5-R-l,3-dioxanes, determined their conformational preferences (Table VII) (77JOC1533), and discussed the results in terms of solvent effects and various repulsive and attractive interactions between the 5-substituent and the ring oxygen atoms (as visualized in Scheme 9). 

In this formula, m and n are the number of ligand and receptor atoms, respectively r is the interatomic distance between atoms i and j the q s are the point charges on the atom, and A and B are adjustable van der Waals repulsion and attraction parameters, and D is the dielectric function. They assumed that this scoring function could account for hydrogen bond energies in the electrostatic term. 

FIGURE 7.3 Variation of repulsion and attraction forces versus distance between two particles (schematic). 

Langmuir, . (1938) The role of the repulsive and attractive forces in the formation of tactoids, thixotropic gels, protein crystals and coacer-vates. J. Chem. Phys. 6 873-896... 

Most commonly, van der Waals interactions are represented as a sum of a repulsive and attractive terms. 

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