Hookes atom

It is not easy to find a quantum many-body system for which the Schrodinger equation may be solved analytically. However, a useful example is provided by the problem of two electrons in an external harmonic-oscillator potential, called Hooke s atom. The Hamiltonian for this system is 

From Eq. (6), this yields a second-order differential equation for the interelectronic function /( ), namely 

For any value of the force constant k, at most a single differential equation needs to be solved to find the ground state of this system. This has been done numerically[8] for many values of k. More recently, Kais[9] and co-workers have studied the special case of k = 1/4, for which an analytic solution exists, while Taut has shown that analytic solutions exist for an infinite, discrete set of oscillator 

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So here is something really interesting. Indeed, we can manipulate the masses arbitrarily and therefore create links among some apparently unrelated systems. Note that if V is chosen as the Coulombic repulsion, Eq. (4.50) describes either two-electron Hooke atom (i.e., harmonium ), and then we put mi = m2 [Pg.215]

Harmonium represents the two-electron Hooke atom. A Hodce diaromic molecule means two heavy particles (nuclei) interacting by Coulomb forces. The same is true with electrons, but the heavy particle-light particle interactions are harmonic. 

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

As a simple example of a normal mode calculation consider the linear triatomic system ir Figure 5.16. We shall just consider motion along the long axis of the molecule. The displace ments of the atoms from their equilibrium positions along this axis are denoted by It i assumed that the displacements are small compared with the equilibrium values Iq and th( system obeys Hooke s law with bond force constants k. The potential energy is given by ... 

Dislocations are known to be responsible for die short-term plastic (nonelastic) properties of substances, which represents departure from die elastic behaviour described by Hooke s law. Their concentration determines, in part, not only dris immediate transport of planes of atoms drrough die solid at moderate temperatures, but also plays a decisive role in die behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondaiy creep, die dislocation distribution cannot be considered geometrically fixed widrin a solid because of die applied suess. 

Based upon Hooke s law, the vibrational frequency between these two atoms may be approximated as... 

The vibration of a diatomic molecule, or any vibrational mode in a polyatomic molecule, may be approximated by two atoms of mass m and m2 joined by a Hooke s law bond that allows vibration relative to the centre of mass. The frequency of such a two-body oscillator is given by... 

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 ... 

This long narrative about the propensity of carbon to connect with four other atoms partially.explains why there are so many carbon compounds there are lots of ways atoms can hook up with carbon atoms. 

When I was a child, I had three toys that provided endless possibilities a set of wooden blocks, a set of Tinkertoys, and an Erector Set. These three toys had a lot in common. Each had a nnmber of distinct strnctnral elements that could be assembled in an amazing nnmber of ways to create something new and different. This has a direct analogy to the concept of assembling molecnles from atoms the atoms are the strnctnral elements and they can be hooked together to make molecular constructs in a great many ways. 

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