Hooke’s law model

SUSY offers us a new way of finding new analytically solvable potentials, either by finding a partner potential to one that already has an analytical solution, or finding under which condition a certain Hamiltonian can be factorized. As an example of the latter case, we will show that when the parameters satisfy certain relations, the Hooke s law model for a two-electron atom can be solved analytically. 

One of the main problems of quantum chemistry and solid-state theory is to develop computational techniques which accurately simulate the interaction of electrons in many-electron systems. In spite of numerous efforts by many theoreticians, the Schrodinger equation can be solved analytically only for one-electron systems. To solve the Schrddinger equation for electronic systems, one has to rely on approximate methods or models which can be studied analytically. The Hooke s law model provides an example of a soluble two-electron problem. In this model, the electrons are attracted to the nucleus by a harmonic potential but repel each other by the... 

Here, we will illustrate how to use SUSY to derive an analytic solution for the Hooke s law model. The Hamiltonian expressed in atomic units is given by ... 

There are many exactly solvable potentials in quantum mechanics and there was no intention to cover all of them in this article. We therefore decided to give a brief description of supersymmetry in quantum mechanics, the analytic solution of the Hooke s law model for two-electron atoms, and the large-dimension models for electronic structure problems. 

Hooke s law functional form is a reasonable approximation to the shape of the potential gy curve at the bottom of the potential well, at distances that correspond to bonding in md-state molecules. It is less accurate away from equilibrium (Figure 4.5). To model the se curve more accurately, cubic and higher terms can be included and the bond- ching potential can be written as follows ... 

Figure 1.11(b) illustrates the ball-and-spring model which is adequate for an approximate treatment of the vibration of a diatomic molecule. For small displacements the stretching and compression of the bond, represented by the spring, obeys Hooke s law ... 

In order to model viscoelasticity mathematically, a material can be considered as though it were made up of springs, which obey Hooke s law, and dashpots filled with a perfectly Newtonian liquid. Newtonian liquids are those which deform at a rate proportional to the applied stress and inversely proportional to the viscosity, rj, of the liquid. There are then a number of ways of arranging these springs and dashpots and hence of altering the... 

Theoretical models include those based on classical (Newtonian) mechanical methods—force field methods known as molecular mechanical methods. These include MM2, MM3, Amber, Sybyl, UFF, and others described in the following paragraphs. These methods are based on Hook s law describing the parabolic potential for the stretching of a chemical bond, van der Waal s interactions, electrostatics, and other forces described more fully below. The combination assembled into the force field is parameterized based on fitting to experimental data. One can treat 1500-2500 atom systems by molecular mechanical methods. Only this method is treated in detail in this text. Other theoretical models are based on quantum mechanical methods. These include ... 

In short, near-infrared spectra arise from the same source as mid-range (or normal ) infrared spectroscopy vibrations, stretches, and rotations of atoms about a chemical bond. In a classical model of the vibrations between two atoms, Hooke s Law was used to provide a basis for the math. This equation gave the lowest or base energies that arise from a harmonic (diatomic) oscillator, namely ... 

Fig. 6.2. Energy curve for Hooke s law versus Quantum Model of harmonic oscillator.

For such molecules, all of the vibrations are active in both the infrared and Raman spectra. Usually, certain of the vibrations give very weak bands or lines, others overlap, and some are difficult to measure, as they occur at very low wavenumber values.40 Because the vibrations cannot always be observed, a model of the molecule is needed, in order to describe the normal modes. In this model, the nuclei are considered to be point masses, and the forces between them, springs that obey Hooke s law. Furthermore, the harmonic approximation is applied, in which any motion of the molecule is resolved in a sum of displacements parallel to the Cartesian coordinates, and these are called fundamental, normal modes of vibration. If the bond between two atoms having masses M, and M2 obeys Hooke s law, with a stiffness / of the spring, the frequency of vibration v is given by... 

The force field approach assumes that it is possible to describe the geometry of a molecule by a set of mechanical equivalents. Changes in bond length are, for example, represented by- a modified Hook s law V(r) = 0.5 x 1 x (r — r0)2 for a spring or bending of a bond angle is modelled by a bending potential VB = kB x (0 — 0O)2, where kB, ks, r0, and 0O are parameters that are dependent on the incorporated atom... 

A simple way to appreciate the shape of fullerene is to construct a physical model in which rigid planar trivalent nodal connectors represent the atoms and flexible plastic bars (tubes) of circular cross-section represent the bonds. From a mechanical point of view the model may be considered as a polyhedron-like space frame whose equilibrium shape is due to self-stress caused by deformation of bars. We suppose that the bars are equal and straight in the rest position and that they are inclined relative to each other at every node with angle of 120°. The material of the bars is assumed to be perfectly elastic and that Hooke s law is valid. All the external loads and influences are neglected and only self-stress is taken into account. Then we pose the question What is the shape of the model subject to these conditions To answer this question we apply the idea used for coated vesicles by Tarnai Gaspar (1989). 

The axisymmetric nature of conical hoppers results in es = 0 and, according to Eq. (2.20), cre = (compatibility requirement, i.e., the relationship of strains. This relation, with the aid of constitutive relations between stress and strain (e.g., Hooke s law), provides an additional equation for stress so that the problem can be closed. However, the compatibility relation for a continuum solid may not be extendable to the cases of powders. Thus, additional assumptions or models are needed to yield another equation for stresses in powders. 

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