Harmonic vibrations classical theory

In the classical theory of scattering (Cohen-Tannoudji et al. 1977, James 1982), atoms are considered to scatter as dipole oscillators with definite natural frequencies. They undergo harmonic vibrations in the electromagnetic field, and emit radiation as a result of the oscillations. 

According to classical theory the vibrational motion of a polyatomic molecule can be represented as a superposition of 3N-6 harmonic modes in each of which the atoms move synchronously (i.e. in phase) with a definite frequency v. These normal modes are characterized by time-dependent normal coordinates which indicate, on a mass-weighted scale, the relative displacement of the atoms from their equilibrium positions (Wilson et al., 1955). Figure 2 shows the general shape of the normal coordinates for a non-linear symmetric molecule AB2. The... 

Another factor of which a nonclassical theory must take account is the quantisation of the internal modes of D and A, and the consequent relaxation of the Bom-Oppenheimer constraint that the electron must transfer within a fixed nuclear framework. In classical theory, the vibrational modes of D and A are treated as classical harmonic oscillators, but in reality their quantisation is usually significant (that is, one or more of the vibration frequencies v is sufficiently high that the classical limit hv IcT does not apply). Electron transfer then requires the overlap, not only of the electronic wavefunctions of R and P, but also of their vibrational wavefunctions. It is then possible that nuclear tunnelling may assist electron transfer. As shown in Fig. 4.12, the vibrational wave-functions of R and P extend beyond the classical parabolas and overlap to some extent. This permits nuclear tunnelling from the R to the P surface, particularly in the region just below the classical intersection point. Part of the reorganisation of D and A, required prior to ET in the classical picture, may then occur simultaneously withET, by the nuclei tunnelling short (typically < 0.1 A) distances from their R to their P positions. 

Appendix II. Effect of Coupling Between Longitudinal and Rotational Vibrations of HB Molecules Appendix III. Applicability of Classical Theory for Harmonic Longitudinal Vibration... 

APPENDIX III. APPLICABILITY OF CLASSICAL THEORY FOR HARMONIC LONGITUDINAL VIBRATION... 

Regarding the longitudinal harmonic vibration of H-bonded charged molecules, we determine A from classical theory as a quantity, which concerns breaking of hydrogen bonds ... 

For a decade, a rival theory due to Slater (1955, 1959) provided considerable motivation for more detailed experimental as well as theoretical investigations. This, very interesting and elegant theory, which is discussed in more detail by Robinson and Holbrook (1972) and Nikitin (1974), as well as in chapter 8, is more akin to a dynamical than a statistical theory. Because the Slater theory treats the vibrations classically, it also requires the use of fewer oscillators to fit the experiment (sec fig. 1.1). Its flawed fundamental hypothesis that the molecule s modes were strictly harmonic, thereby preventing energy flow among them, and its failure to account quantitatively for the experimentally measured rates led to its being quickly overshadowed by the successes of the RRKM/QET theory. 

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. 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

Classically, Raman spectroscopy arises from an induced dipole in a molecule resulting from the interaction of an electromagnetic field with a vibrating molecule. In electromagnetic theory, an induced dipole is a first-rank tensor formed from the dot product of the molecular polarizability and the oscillating electric field of the photon, (jl = a-E. Assuming a harmonic potential for the molecular vibration, and that the polarizability does not deviate significantly from its equilibrium value (a0) as a result of the vibration... 

The classical RRK theory was proposed very soon after the quantum theory of Schrodinger, and so it is scarcely surprising that it uses a classical model of the vibrations. However, very soon afterwards Kassel proposed an alternative version of the RRK theory in which the oscillators were quantum harmonic oscillators [13]. In the simplest version of the theory all the oscillators have the same frequency v, although Kassel did also present a version in which the oscillators are divided into two classes with different frequencies. 

The reorganization free energy is usually split in two parts. The local mode contribution is obtained in standard routines which require local potentials (say harmonic potentials) and vibrational frequencies in the reactants and products states. The collective modes associated with the proteins and the solvent, however, pose complications. One complication arises because classical electrostatics needs modification when the spatial extension of the electric field and charge distributions are comparable with the local structure extensions of the environment. Other complications are associated with the presence of interfaces such as metal/solution, protein/solution, and metal/film/solution interfaces. These issues are only partly resolved, say by nonlocal dielectric theory and dielectric theory of anisotropic media. 

Before we finish the discussion on the corrections to the HLA it will be useful to say a few words on classical oscillator dipole theory. Mahan (10) was the first to do a self-consistent strong coupling theory for molecular solids (see also his review paper (11)). In this approximation the molecule is considered as a harmonic oscillator and not as a two-level model. This approach is correct for high-frequency intramolecular optical vibrations. However, Mahan calculated the... 

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