Harmonic oscillator vibration treatment

The cooidinates Qi are determined in the process of a semiclassical treatment of molecular vibrations [3-6]. The principal aim of these calculations is to define the specific coordinates Qi, in die basis of which the Schrddinger wave functions for the vibrational motion of a molecule are reduced to 3N-6 simple linear hannonic oscillator wave functions. One of the mathematical expressions of this result is Eq. (1.25). 3N-6i the munber of vibrations in an N-atomic molecule (3N-5 in the case of linear molecules). Described in terms of Qi the complex vibrational motion of a molecule is expressed as a superposition of 3N-6 linear harmonic oscillator vibrations, each having specific form as described by Qj and own frequency of oscillation. More comments about normal coordinates will be given in the following section. For a complete description of the theory of normal vibrations the reader is referred to a number of monogr hs [3-6]. 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is... 

So far we have illustrated the classic and quantum mechanical treatment of the harmonic oscillator. The potential energy of a vibrator changes periodically as the distance between the masses fluctuates. In terms of qualitative considerations, however, this description of molecular vibration appears imperfect. For example, as two atoms approach one another, Coulombic repulsion between the two nuclei adds to the bond force thus, potential energy can be expected to increase more rapidly than predicted by harmonic approximation. At the other extreme of oscillation, a decrease in restoring force, and thus potential energy, occurs as interatomic distance approaches that at which the bonds dissociate. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The Levich—Dogonadze—Kuznetsov (LDK) treatment [65] considers that the only source of activation is the polarization electrostatic fluctuations (harmonic oscillations) of the solvent around the reacting ion and uses essentially the same model as the Marcus—Hush approach. However, unlike the latter, it provides a quantum mechanical calculation of both the pre-exponential factor and the activation energy but neglects intramolecular (inner sphere) vibrations (1013—1014 s 1). 

In vibrational spectroscopy, where the treatment of molecular vibrations is based on the differential equation for an harmonic oscillator ... 

In the treatment of two atoms connected together, a simple harmonic oscillator model can be adopted involving the two masses connected with a spring having a force constant fk. Thus, the vibrational frequency in wavenumbers 2 depends from the reduced mass p, from fk with c being the velocity of light. 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

The vibration of a diatomic molecule may be of only one kind, an alternate expansion and contraction of the interatomic distance. The simplest mathematical treatment (useful, but approximate) of such a vibration assumes the molecule to be a harmonic oscillator, roughly analogous to a... 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

In the quantum mechanical treatment of vibrational normal modes, the vibrational Schrddinger equation is separated into individual harmonic oscillator equations by exactly the same transformation of variables [28]. 

A quantum-mechanical treatment has been given for the coherent excitation and detection of excited-state molecular vibrations by optical absorption of ultrashort excitation and probe pulses [66]. Here we present a simplified classical-mechanical treatment that is sufficient to explain the central experimental observations. The excited-state vibrations are described as damped harmonic oscillations [i.e., by Eq. (11) with no driving term but with initial condition Q(0) < 0.] We consider the effects of coherent vibrational oscillations in Si on the optical density OD i at a single wavelength k within the Sq -> Si absorption spectrum. Due to absorption from Sq to Si and stimulated emission from Si and Sq,... 

The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, AC circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter. 

An improved treatment of molecular vibration must account for anharmonicity, deviation from a harmonic oscillator. Anharmonicity results in a finite number of vibrational energy levels and the possibility of dissociation of the molecule at sufficiently high energy. A very successful approximation for the energy of a diatomic molecule is the Morse potential ... 

As indicated briefly above, the calculated entropy of Cg gas is higher than the experimental value even when a summation is performed only over the observed first six levels of the bending frequency. Treatment of the vibration as a harmonic oscillator or as a anharmonic oscillator gives values considerably higher. Strauss and Thiele (1 ) have also calculated functions based on a quartlc potential function whlich still yields values several units higher than the experiments. In order to reduce the entropy to the approximate range of the measurements we have made the assumption that the potential function is... 

In the treatment of a linear harmonic oscillator, we assume that a mass m, attached to a spring with a spring constant k is freely vibrating without loss of energy in the vertical (z) direction (see Fig. 1.8). 

Until now, our treatment has been built in exactly the same terms that might have been used in work on normal modes of vibration in the latter part of the nineteenth century. However, it is incumbent upon us to revisit these same ideas within the quantum mechanical setting. The starting point of our analysis is the observation embodied in eqn (5.19), namely, that our harmonic Hamiltonian admits of a decomposition into a series of independent one-dimensional harmonic oscillators. We may build upon this observation by treating each and every such oscillator on the basis of the quantum mechanics discussed in chap. 3. In light of this observation, for example, we may write the total energy of the harmonic solid as... 

The quantum-mechanical treatment of molecular vibrations leads to modifications of the harmonic oscillator model. While the Hooke s law treatment presented above would indicate a continuum of vibrational states, the molecular vibrational energy levels are quantised ... 

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