Spherically symmetric potential

This Schrodinger equation relates to the rotation of diatomic and linear polyatomic molecules. It also arises when treating the angular motions of electrons in any spherically symmetric potential... 

A much improved description of PE melts is obtained with a more elaborate potential energy function, based on a spherically symmetric potential energy... 

The starting point for the calculation of g(r) is the Ornstein-Zernike (OZ) equation, which, for a one-component system of liquids interacting via spherically symmetric potentials (e.g. Argon), is [89]... 

Exact solutions of the Schrddinger equation are, of course, impossible for atoms containing 90 electrons and more. The most common approximation used for solving Schrddinger s equation for heavy atoms is a Hartree-Fock or central field approximation. In this approximation, the individual electrostatic repulsion between the electron i and the N-1 others is replaced by a mean central field giving rise to a spherically symmetric potential... 

The spherically symmetric potential around a charged sphere is described by the Poisson-Boltzmann equation ... 

The Dirac-Pauli representation is most commonly used in all applications of the Dirac theory to studies on electronic structure of atoms and molecules. Apart of historical reasons, there are several features of this representation which make its choice quite natural. Probably the most important is a well defined symmetry of and in the case of spherically-symmetric potentials V. The Dirac Hamiltonian... 

Transformation (39) applied to the case of a non-Coulomb spherically-symmetric potential results in an equation in which residual terms proportional to W = V + Z/r would appear. Its algebraic representation reads ... 

When is expressed in spherical coordinates as t i(r, 0, cp), then reflection through the origin is accomplished by replacing 0 and cp by (it — 0) and (it + cp), respectively. (r cannot change sign as it is just a distance.) In other words, the parity of the wave function is determined only by its angular part. For spherically symmetric potentials, the value of l uniquely determines the parity as... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

The system studied consists of one solute molecule that is tagged and N solvent molecules, each of mass m. Since we are interested in a spherically symmetrical potential, the pair potential of the solvent-solvent pair and the solute-solvent pair is assumed to be given by the simple Lennard-Jones 12-6 potential... 

The potential energy of a guest molecule at a distance r from the cavity center is given by the spherically symmetrical potential m (r) proposed by Lennard-Jones and Devonshire (1932, 1938). 

The trajectory of an ion moving in such a potential presents a sequence of rectilinear sections placed between the points of elastic reflections of an ion from the walls of the well. We consider two variants of such a model related to one-dimensional and spatial motion of ion, depicted, respectively, in Figs. 47a and 47b. In the first variant the ion s motion during its lifetime59 presents periodic oscillations on the rectilinear section 2 lc between two reflection points. In the second variant we consider a spherically symmetric potential well, to which a spherical hollow cavity corresponds with the radius lc. 

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... 

Consider a doublet of electronic orbitals (fa,fp) localized around some reference atom taken at origin. We assume that the environment potential felt by an electron in the reference atom is the sum of atomic-like spherically symmetric potentials Va(r — R) centred at the other atomic positions R, and we take as z-axis the interatomic axis of the linear molecule in the equilibrium configuration. We consider thus the matrix... 

Fig. 4.1.7 Two-body classical scattering in a spherically symmetric potential. The relative motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass p = m-Ams/(rnA+rn-s), relative to a fixed center of force. Two trajectories are shown for the first trajectory, the final and initial relative velocity vector and the associated deflection angle x are shown. This trajectory corresponds to the impact point (b, = 0), whereas the second trajectory corresponds to the impact point (b, 4> = ir).

Fig. 4.1.8 Two-body classical scattering in a spherically symmetric potential. The effective potential I4ff(r), defined in Eq. (4.32), is shown for two different impact parameters, 6 = 0 where l eff (r) = U(r) and 6 > 0. The upper panel (a) is for a fixed total energy Ei, and the lower panel (b) for another total energy / h The intersection with Fefr(r) gives the distance of closest approach rc.

Specializing further to scattering in a spherically symmetric potential, the cross-section will be independent of the angle

and... 

This result is valid for elastic scattering in a spherically symmetric potential under the assumption that the target is at rest prior to the collision. 

The collision is described here by classical dynamics, and we assume that the motion takes place in a spherically symmetric potential U(r). It is well known that the relative motion of the atoms is equivalent to the motion of a particle with the reduced mass p, in an effective one-dimensional potential given by... 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

In the absence of external fields, we may take axes that move laterally with the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass ti=MaMjs/(Ma+Mb), moving in a spherically symmetrical potential U(R), where R is the intranuclear separation. The Schrodinger equation is therefore... 

Tang KT, Toennies JP (1977) A simple theoretical model for the Van der Waals potential at intermediate distances. I. Spherically symmetric potentials. J Chem Phys 66 1496-1506... 

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... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

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... 

Using a central field approximation in which it is assumed that each electron moves independently in an average spherically symmetric potential, it is possible to solve for the energies of the different configurations. Calculations of this type show that the / -configuration is the lowest energy configuration for the trivalent lanthanides and actinides. 

The concept can be illustrated with a simple one-dimensional problem, in which a spherically symmetric potential well is surrounded by a barrier (see Fig. 4). One might consider this problem as a model for dissociation of a diatomic molecule. According to the general theory, the spectrum is discrete for < 0 and continuous for E > 0. Let us now look for unbound solutions, x( ) = of fhe time-independent Schrodinger equation which behave like exp(ifci ) at large R. The desired wave functions have the form... 

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