Harmonic oscillator description

In this potential the energy levels are no longer equally spaced, and overtones, i.e. vibrational transitions with An > 1, become allowed. The overtone of gaseous CO at 4260 cm-1 (slightly less than 2x2143 = 4286 cm ) is an example. For small deviations of r from equilibrium, however, the Morse potential is successfully approximated by a parabola, and for the interpretation of IR spectra the harmonic oscillator description is usually sufficient. 

The spectral density (see also Sections (7-5.2) and (8-2.5)) plays a prominent role in models of thermal relaxation that use harmonic oscillators description of the thermal environment and where the system-bath coupling is taken linear in the bath coordinates and/or momenta. We will see (an explicit example is given in Section 8.2.5) that /(co) characterizes the dynamics of the thermal environment as seen by the relaxing system, and consequently determines the relaxation behavior of the system itself. Two simple models for this function are often used ... 

An important conceptual advantage of the double linear expansion of Equation 7 is the fact that, within the framework of first-order perturbation theory, the different terms correlate with different types of diatom monomer transitions. In particular, assuming a harmonic oscillator description of the diatom vibration, k=l terms are the primary source of direct Av l coupling, k 2 terms the main source of Av= 2 coupling, or more generally, direct transitions associated with any given value of Av are only driven by... 

For (ii) The simplest description of a vibration is a harmonic oscillator, which has been found to work reasonably well for most systems. Within this model, the oscillation frequency is given by... 

Quantization (the idea of quantums, photons, phonons, gravitons) is postulated in Quantum Mechanics, while the Theory of Relativity does not derive quantization from geometric considerations. In the case of the established phenomenon the quantized nature of portioned energy transfer stems directly from the mechanisms of the process and has a precise mathematical description. The quasi-harmonic oscillator obeys the classical laws to a greater extent than any other system. A number of problems, related to quasi-harmonic oscillators, have the same solution in classical and quantum mechanics. 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

This is a consequence of describing the harmonic oscillators with quantum mechanics. In a purely classical description, the average energy for each harmonic oscillator is simply k T, so the energy difference between two configurations is independent of T. 

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. 

These operators allow for the description of the quantum harmonic oscillator that is very parsimonious. The quantum harmonic oscillator has evenly spaced eigenstates, and the state of the system may be changed according to... 

We then have a description of an infinite number of harmonic oscillators with every possible mode at every point in space. The electromagnetic field is quantized in a cavity with a volume V by defining annihilation and creation operators by redefining these raising and lower operators as... 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

The hat-curved-harmonic oscillator model, unlike other descriptions of the complex permittivity available now for us [17, 55, 56, 64], gives some insight into the mechanisms governing the experimental spectra. Namely, the estimated relaxation time of a nonrigid dipole (xovib 0.2 ps) is close to that determined in the course of very accurate experimental investigations and of their statistical treatment [17, 54-56]. The reduced parameters presented in Tables XIVA and XIVB and the form of the hat-curved potential well (determined by the parameters u, (3, f) do not show marked dependence on the temperature, while the spectra themselves vary with T in greater extent. We shall continue discussion of these results in Section X.A. 

Both hat-curved-harmonic oscillator and hat-curved-cosine-squared potential composite models considered in this section give excellent description of wideband spectra of water H20 and D20 in the range from 0 to 1000 cm-1. However, it appears that the physical picture of fast vibrations of the H-bonded molecules differ for these two approaches. In the first one, where... 

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