Simple harmonic motion, classical

Mathematically, the movement of vibrating atoms at either end of a bond can be approximated to simple-harmonic motion (SHM), like two balls separated by a spring. From classical mechanics, the force necessary to shift an atom or group away from its equilibrium position is given by... 

Up to this point the treatment of molecular vibrations has been purely classical. Quantum mechanics does not allow the specification of the exact path taken by a particle hence the picture of each nucleus executing the appropriate simple harmonic motion for a given normal mode should not be taken literally. On the other hand, nuclei are relatively heavy (compared to electrons), so that the classical picture of the motion is not totally lacking in validity. 

Classically, in a polyatomic molecule, a normal mode of vibration occurs when each nucleus undergoes simple harmonic motion with the same frequency as and in phase with the other nuclei. The possible normal modes of vibrational of a non-linear polyatomic molecule with A atoms are 3A— 6 and are categorized as arising from stretching, bending, torsional and non-bonded interactions. Whether a particular mode is IR active depends on whether the symmetry of the vibration corresponds to one of the components of the dipole moment. 

The principle of classical equation solving, which is at the heart of MM, can be appreciated by imagining two objects, connected by a spring, executing simple harmonic motion collinear with the spring. If the force constant k of the spring is known, it is possible to calculate the potential energy V of the system at any separation x of the objects as... 

Vibrations are considered in terms of the classical expressions governing motion of nuclei vibrating about their equilibrium positions with a simple harmonic motion (40). The potential and kinetic potential energies of molecules are defined in terms of the coordinates most appropriate to the molecular structures. All relative motions of atoms about the center of mass (vibrations) are linear combinations of a set of coordinates, known as normal coordinates. For every normal mode of vibration, all coordinates vary periodically with the same frequency and pass through equilibrium simultaneously. 

The simplest possible assumption about the form of the vibrations in a diatomic molecule is that each atom moves toward or away from the other in a simple harmonic motion (2-5). Such a motion of the two atoms can be reduced to the harmonic vibration of a single mass point about an equilibrium position. In classic mechanics a harmonic oscillator can be defined as a mass point of mass m that is acted upon by a force F proportional to the distance Q from the equilibrium position and directed toward the equilibrium position ... 

A classical model o a one-electron atom consists of a positive charge of amount +e uniformly distributed throughout the volume of a sphere, radius R, together with a point electron of charge -e which is free to move within the sphere. Show that the electron will oscillate about its equilibrium position with simple harmonic motion and find the frequency of oscillation,... 

If the mechanical vibration of the simple harmonic oscillator by which we first represented the nuclear motion of the diatomic molecule in the preceding section is accompanied by an oscillation of the dipole moment of the molecule, then, according to classical physics, radiation will be emitted with the frequency of the oscillator. For small amplitudes of vibration we can take the oscillating part of the dipole moment as being proportional to the elongation cc of the molecule introduced in the preceding section, let us say equal to qx. The amount of radiation emitted by the oscillator in unit time is then given by ... 

There is a tendency to use the term molecular mechanics (MM) as opposite to quantum mechanics, therefore including all classical dynamics methods, such as energy minimization, Monte Carlo, and molecular dynamics. Sometimes it is used to describe only the energy minimization method using empirical force field (potential) or it is even used to refer to the quantum mechanical method specifically, emphasizing its use for molecular motion. Nevertheless, we will focus on the second definition of molecular mechanics (43-45). In this approach, a molecule is viewed as a collection of particles (atoms) held together by simple harmonic or elastic forces. Such forces are defined in terms of potential energy... 

At first sight it might be surprising that the Hessian matrix, which after all in oh initio molecular orbital theory is inherently quantum mechanical, is amenable to a purely classical treatment. This is because the Born-Oppenheimer approximation (Born and Oppenheimer 1927 Wikipedia 2010) allows for a pretty good separation of the electronic and nuclear motion, allowing the latter to be treated classically. A quantum mechanical description of the simple harmonic oscillator leads to quantized energy levels given by... 

An important application of Hamilton s classical equations of motion is the vibrational mechanics of systems of particles. We shall begin with application to an idealized, simple problem, the harmonic oscillator. The harmonic oscillator is a special model problem consisting of a mass able to move in one direction and connected by a spring to an infinitely heavy wall, as shown in Figure 7.1. The spring is special because it is harmonic, which means that the restoring force is linearly proportional to the value of the coordinate that... 

The motion in the classical domain corresponds to a harmonic oscillator of frequency v, with the displacement from equilibrium varying sinusoidally with time. The transcription of this problem into quantum mechanics is simple and straightforward it is a standard problem in introductory quantum mechanics texts. The energy levels of the quantum system are given by... 

A simple realization of the harmonic oscillator in classical mechanics is a particle which is acted upon by a restoring force proportional to its displacement from its equihbrium position. Considering motion in one dimension, this means... 

To construct, on a single-molecule basis, an (x(t), y t)) motion of rotation in the classical sense, the only solution is to find a superposition of / (()) rotor eigenstates for (9(t)), which is almost a solution of Eq. (2) in the time interval of interest. Furthermore, (0(f)) cannot be an exact solution of Eq. (2) for all times. This is again a manifestation that a classical motion such as a unidirectional rotation is not a natural movement for an isolated quantum object. Only a simple quantum harmonic oscillator is able to perform classical motion when correctly prepared [17]. 

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

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