Quantum-Mechanical Treatment of Vibrations

When one considers a diatomic molecule, it can be compared to the ideal harmonic 

FIGURE 14.25 Potential energy diagram for an ideal harmonic oscillator. Usually, this diagram is applicable only for low-energy (that is, low qrantum number) vibrations. 

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By making this ideal harmonic oscillator assumption, we make the wavefunctions and energies for the ideal harmonic oscillator directly applicable to the diatomic molecule s vibrations In particular, because spectroscopy deals with differences in the energy states, we are particularly interested in the fact that 

Diatomic molecules are particularly easy to treat quantum-mechanically because they are easily described in terms of the classical harmonic oscillator. For example, the expression 

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We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

Phonon Elementary excitation in the quantum mechanical treatment of vibrations in a crystal lattice. 

It is important to remember that the reorganization energy is a composite parameter rather than a fundamental physical quantity. Refinements to the semiclassical theory usually arise from quantum mechanical treatments of vibrational motions. The increased rigor associated with these models, however, is rarely accompanied by the extra data required to cope with the influx of new parameters. The approximations involved in its definition, and the errors associated with its measurement dictate that k should never be expressed with great precision. 

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]. 

Although the harmonic model does not provide a complete description for the motional properties of a protein because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it does serve as a first approximation for which the theory is highly developed. Further, the harmonic model is essential for quantum mechanical treatments of vibrational contributions to the heat capacity and free energy [26, 27]. 

TABLE 1 Properties of Several Polymer Crystals Computed at 300K Using a Self-Consistent QHA Lattice Dynamics Calculation, with Quantum Mechanical Treatment of Vibrational Free Energy... 

The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

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 classical model predicts thermal motion to vanish at very low temperatures, in contradiction to the zero-point vibrations which follow from the quantum-mechanical treatment of oscillators. For temperatures at which hv % kBT, the spacing of the discrete energy levels cannot be neglected, so the classical model is no longer valid. 

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. 

The quantum mechanical treatment of non-adiabatic electron transfers are normally considered in terms of the formalism developed for multiphonon radiationless transitions. This formalism starts from Fermi s golden rule for the probability of a transition from a vibronic state Ay of the reactant (electronic state A with vibrational level v) to a set of vibronic levels B of the product... 

Since electrochemical surface reactions involve electron transfer to or from the surface, a quantum mechanical approach becomes necessary to account for electron tunneling in such processes. Quantum mechanical treatments of electron transfer and adsorption have been reviewed recently (67-77). The Gurney treatment (68, 72, 73) assumes the transfer of an electron at the Fermi level of the metal to an H3O at its ground state at the outer Helmholtz plane. The electrode potential changes the minimum vibrational energy of the bond necessary to induce tunneling. Levich (67) has... 

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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