Central Force Fields

The relationship (equation (5.81)) between M and L depends only on fundamental constants, the electronic mass and charge, and does not depend on any of the variables used in the derivation. Although this equation was obtained by applying classical theory to a circular orbit, it is more generally valid. It applies to elliptical orbits as well as to classical motion with attractive forces other than dependence. For any orbit in any central force field, the angular... 

The potential energy expressions used for force field calculations are all descendants of three basic types originating from vibrational spectroscopy (5) the generalized valence force field (GVFF), the central force field, and the Urey-Bradley force field. General formulations for the relative potential energy V in these three force fields are the following ... 

The final general form of the VFF- and UBFF-expressions (2) and (4) as modified for our purposes is thus as follows (Central force fields are less efficient than valence- and UB-force fields, and are not considered here.) ... 

Tn the Rohr model of the hydrogen atom, the proton is a massive positive point charge about which the electron moves. By placing quantum mechanical conditions upon an otherwise classical planetary motion of the electron, Bohr explained the lines observed in optical spectra as transitions between discrete quantum mechanical energy states. Except for hvperfine splitting, which is a minute decomposition of spectrum lines into a group of closely spaced lines, the proton plays a passive role in the mechanics of the hydrogen atom, It simply provides the attractive central force field for the electron,... 

Let us consider the description of a simple three-atom molecule by the three most frequently used spectroscopic parameterization schemes, the general central force field (GCF), the general valence force field (GVF) and the Urey-Bradley force field (UBF Fig. 2.14). 

In fact, there is increasing evidence that this transition is not continuous, and independent arguments have been presented that suggest the transition is first-order (Goodenough, 1992 Dagotto et al., 1998 Rozenberg et al., 1994). The simplest argument is based on the virial theorem of mechanics for central-force fields ... 

Consider the two-dimensional motion of a particle in a central force field. The Lagrange function in Cartesian coordinates is... 

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... 

The main difficulty with the valence-bond approach is that the atomic orbitals, whether hybridized or not, result from the interaction of electrons with a single central force field— that of the atomic nucleus. An electron that spends most of its time between two nuclei will find itself in a very different, two-center force field, and this will give rise to new types of orbitals that are better characterized as molecular, rather than as atomic orbitals. 

Note that for two spherical molecules with central force fields, the relative motion will take place in a plane. This plane will be defined by the centers of the two molecules and the direction of their relative velocity vector, which is parallel to the plane. No forces act perpendicularly to the plane. 

As a crude approximation for attractive fields wc can say that the effective radius fAB is increased to tab, where rin is the distance at which the potential energy of the pair AB in the solution is of the order of kT, For repulsive fields the effective radius Tab is diminished by the ratio rAB/rin- The more exact treatment for central force fields whose potential is U(r) gives rin as... 

Commentary on The statistical mechanics of systems with non-central force fields,... 

THE STATISTICAL MECHANICS OF SYSTEMS WITH NON-CENTRAL FORCE FIELDS... 

The development of a general theory of systems with non-central force fields can be divided into two parts. First the many types of directional interaction that may occur have to be classified within a general mathematical framework and then approximate methods of evaluating the partition function have to be devised. This paper summarizes some of the results of a method developed by the author 2 with particular reference to its application to the properties of liquid mixtures. 

The expansion (2.1) is particularly useful if the intermolecular forces are well represented by the interaction of permanent electrostatic moments. Attempts have been made, for example, to discuss the interaction of simple polar molecules in terms of a central force field representing the dispersion and repulsion forces, together with the interaction of point dipoles situated at the molecular centres.3-5 If the moments of the dipoles are /x, and /x, this directional field can be written in the form (2.1) where... 

In an exposition which aims to encompass general systems and ensembles, it is appropriate to make use of the Hamiltonian version of dynamics. In this view forces do not appear explicitly and the dynamics of the system evolve so as to keep the Hamiltonian function constant. In Newtonian dynamics forces appear explicitly and molecules move as a response to the forces they experience. For our purposes, the Newtonian view is sufficient since we will illustrate the large scale computational aspects with simplest possible particles, atoms with spherical, central force fields. The same principles hold for molecules with internal degrees of freedom as well. 

An early potential model used for zeolite framework modeling corresponds to a central force field model and was developed by Demontis et al. In this model the framework potential energy U is expanded in a Taylor series with respect to interatomic distances... 

We have been using computers in supposedly more sophisticated applications, such as representation of the pair radial distribution function g(R) (29). construction of the Voronoi diagram (30), simulation of light scattering in polymer solutions (31) or Monte Carlo simulation of a fluid with central force fields (32). The present application is anything but sophisticated we... 

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