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Spherical distribution potential

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is... [Pg.199]

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)... [Pg.174]

The first term is the potential caused by the spherically distributed charge Q, the second term is the potential caused by redistribution of charge Q in response to the nonhomogeneous field of point charges —Q/n, and the third is the ordinary coulombic potential caused by the charges —Q/n. The potential Vl(r,0,[Pg.205]

The discovery of confinement resonances in the photoelectron angular distribution parameters from encaged atoms may shed light [36] on the origin of anomalously high values of the nondipole asymmetry parameters observed in diatomic molecules [62]. Following [36], consider photoionization of an inner subshell of the atom A in a diatomic molecule AB in the gas phase, i.e., with random orientation of the molecular axis relative to the polarization vector of the radiation. The atom B remains neutral in this process and is arbitrarily located on the sphere with its center at the nucleus of the atom A with radius equal to the interatomic distance in this molecule. To the lowest order, the effect of the atom B on the photoionization parameters can be approximated by the introduction of a spherically symmetric potential that represents the atom B smeared over... [Pg.37]

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. [Pg.180]

In writing this equation, we have made use of Van Vleck s pure precession hypothesis [12], in which the molecular orbital /.) is approximated by an atomic orbital with well-defined values for the quantum numbers n, l and /.. Such an orbital implies a spherically symmetric potential and its use is most appropriate when the electronic distribution is nearly spherical. Examples of this situation occur quite often in the description of Rydberg states. It is also appropriate for hydrides like OH where the molecule is essentially an oxygen atom with a small pimple, the hydrogen atom, on its side. Accepting the pure precession hypothesis allows the matrix elements of the orbital operators to be evaluated since... [Pg.359]

With such a model, the ion-ion interactions are obtained by calculating the most probable distribution of ions around any central ion and then evaluating the energy of the configuration. If (r) is the spherically symmetrical potential in the solution at a distance r from a central ion i of charge z 8, then (r) will be made up of two parts ZxZlDv the coulombic field due to the central ion, and an additional part, ai(r), due to the distribution of the other ions in the solution around t. The potentials at(r) and must satisfy Poisson s equation = — 47rp/D at every point... [Pg.522]

There are many other indications that the electrostatic effects of non-spherical features of the charge distribution, such as lone pairs and n electrons, can be important in determining molecular crystal structures. At the extreme of homonuclear diatomics (X2), the electrostatic potential outside the molecule arises from the non-spherical distribution of the valence electrons. Just as there are considerable variations in the bonding orbitals in the diatomics, there are also considerable variations in the lowest temperature ordered crystal structure. [Pg.276]

The procedure for calculating the final trajectory and the product angular and energy distributions is as follows. First, the incoming trajectory (deflection Xr ) is determined classical mechanically as a function of the initial impact parameter 60 of A with respect to the centre-of-mass of BC, the initial relative kinetic energy Tq, and the long-range spherically symmetric potential n c ( ") which acts between the centres-of-mass of A and BC. [Pg.341]

For a quantitative discussion of the model, the bubble is regarded as a spherically symmetric potential well in the interior of which the Ps atom moves in a potential-free space. The parameters of the bubble may be determined by quantum-mechanical methods on the basis of the experimentally measured lifetime and angular distribution data [Le 73b]. [Pg.172]

Exact solutions of Schrddinger s equation are impossible for atoms containing more than one electron. The most common approximation used for solving SchrSdinger s equation for complex atoms is the central field approximation . In this approximation, each electron is considered to move in the field of the nucleus and a mean central field due to the charge distribution of the other electrons. If electron i moves in the spherically symmetric potential -lHrt)le, the Hamiltonian Ho for the central field approximation will be... [Pg.5]

To get a single-molecule potential, first the average over all orientations of the intermolecular vector f 2 is obtained, then the average over all orientations of molecule 2, and finally the average over the intermolecular separation ri2- If a spherical distribution of the f 2 vector is assumed for the moment, the average over all orientations of ri2 involves only the Wigner rotation matrices... [Pg.60]

In this section we consider, as an example of non-spherical distribution, needle ions. We perform our construction as before, by writing down a mean potential from which we obtain a number and charge densities. A needle ion consists of a charge q uniformly distributed along a line of length d. The charge distribution of a needle ion is... [Pg.239]

Many related effects have been studied in detail. Dissociation energies have been derived from metastable decay rates of alkali cluster ions. The gaps between electronic shells have been probed by measuring the size dependence of the ionization energy. Reduced ionization energies are, indeed, found for systems with 9, 21, 41, etc. valence electrons. The anomalies are much less than predicted, but refined effective (spherically nonsymmetric) potentials can explain this trend as well as the observation of some additional fine structure in the size distribution and in the ionization energy. Additional information... [Pg.332]

For many particles, the diffuse-charge layer can be characterized adequately by the value of the zeta potential. For a spherical particle of radius / o which is large compared with the thickness of the diffuse-charge layer, an electric field uniform at a distance from the particle will produce a tangential electric field which varies with position on the particle. Laplace s equation [Eq. (22-22)] governs the distribution... [Pg.2006]

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. [Pg.154]


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See also in sourсe #XX -- [ Pg.233 ]




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Distribution potential

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