Harmonic characteristic oscillation frequency

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

A molecule with N atoms has a total of 37V degrees of freedom for its nuclear motions, since each nucleus can be independently displaced in three perpendicular directions. Three of these degrees of freedom correspond to translational motion of the center of mass. For a nonlinear molecule, three more degrees of freedom determine the orientation of the molecule in space and thus its rotational motion. This leaves 37V - 6 vibrational modes. For a linear molecule, there are just two rotational degrees of freedom, which leaves 3N -5 vibrational modes. For example, the nonlinear molecule H2O has three vibrational modes, while the linear molecule CO2 has four vibrational modes. The vibrations consist of coordinated motions of several atoms in such a way as to keep the center of mass stationary and nonrotating. These are called the normal modes. Each normal mode has a characteristic resonance frequency Vj (expressed in cm ), which is usually determined experimentally. To a reasonable approximation, each normal mode behaves as an independent harmonic oscillator of frequency u . The normal modes of H2O and CO2 are shown in Figs. 14.2 and 14.3. A normal mode will be infrared active only if it involves an oscillation of the dipole moment. All three modes of H2O are... 

Fairly good agreement exists between the calculated value of 1682 cm-1 and the experimental value of 1650 cm1. Direct correlation does not exist because Hooke s law assumes that the vibrational system is an ideal harmonic oscillator and, as mentioned before, the vibrational frequency for a single chemical moiety in a polyatomic molecule corresponds to the vibrations from a group of atoms. Nonetheless, based on the Hooke s law approximation, numerous correlation tables have been generated that allow one to estimate the characteristic absorption frequency of a specific functionality (13). It becomes readily apparent how IR spectroscopy can be used to identify a molecular entity, and subsequently physically characterize a sample or perform quantitative analysis. 

For the sake of simplicity, let us now set up the case of a second-order bandpass filter and a comparator with saturation levels This closed-loop system verifies the required premises the system is autonomous, the nonlinearity is both separable and frequency-independent, and the linear transfer function contains enough low-pass filtering to neglect the higher harmonics at the comparator output. Choosing adequately the band-pass filter, it can be forced that the first-order characteristic equation for the closed-loop system of Fig. 4 has an oscillation solution being and the oscillation frequency and... 

Another unusual spectral feature that is characteristic of the SL is observed in the field spectrum where resonances appear spaced from the central Lorentzian laser line by harmonics of the relaxation oscillation frequency. The intensities of these spectral sidebands fall off rapidly with increasing harmonic number and their relative integrated intensities decrease linearly as the laser power is increased. In addition, an asymmetry in the intensities of corresponding upper and lower sidebands typically occurs. This is a manifestation of the intrinsic correlation of amplitude and phase fluctuations. 2 in Fig. 1 the intensities of the fundamental upper and lower sidebands are plotted as functions of inverse power for a (GaAl)As transverse junction stripe (TJS) laser at room temperature. 

In order to evaluate the thermodynamic functions of process (3) it is necessary to know the interaction energy, equilibrium geometry, and frequencies of the normal modes of bases and base pairs involved in the process. The interaction energies and geometries were evaluated using procedures described in Sections 2.1-2.3 determination of harmonic normal vibration frequencies is described in Section 2.4. Partition functions, computed from the HF/6-31G and MP2/6-31G (0.25) characteristics, were evaluated within the rigid rotor-harmonic oscillator-ideal gas approximations. ... 

Infrared (ir) transmission depends on the vibrational characteristics of the atoms rather than the electrons (see Infrared and Raman spectroscopy). For a diatomic harmonic oscillator, the vibrational frequency is described by... 

Treating vibrational excitations in lattice systems of adsorbed molecules in terms of bound harmonic oscillators (as presented in Chapter III and also in Appendix 1) provides only a general notion of basic spectroscopic characteristics of an adsorbate, viz. spectral line frequencies and integral intensities. This approach, however, fails to account for line shapes and manipulates spectral lines as shapeless infinitely narrow and infinitely high images described by the Dirac -functions. In simplest cases, the shape of symmetric spectral lines can be characterized by their maximum positions and full width at half maximum (FWHM). These parameters are very sensitive to various perturbations and changes in temperature and can therefore provide additional evidence on the state of an adsorbate and its binding to a surface. 

In addition to the above prescriptions, many other quantities such as solution phase ionization potentials (IPs) [15], nuclear magnetic resonance (NMR) chemical shifts and IR absorption frequencies [16-18], charge decompositions [19], lowest unoccupied molecular orbital (LUMO) energies [20-23], IPs [24], redox potentials [25], high-performance liquid chromatography (HPLC) [26], solid-state syntheses [27], Ke values [28], isoelectrophilic windows [29], and the harmonic oscillator models of the aromaticity (HOMA) index [30], have been proposed in the literature to understand the electrophilic and nucleophilic characteristics of chemical systems. 

The concept of resonance was introduced into quantum mechanics by Heisenberg16 in connection with the discussion of the quantum states of the helium atom. He pointed out that a quantum-mechanical treatment somewhat analogous to the classical treatment of a system of resonating coupled harmonic oscillators can be applied to many systems. The resonance phenomenon of classical mechanics is observed, for example, for a system of two tuning forks with the same characteristic frequency of oscillation and attached to a common base, hich... 

Furthermore, to obtain a simple equation for two interacting molecules of gas (or liquid, subscript L) and solid (S) having the respective characteristic electronic vibrational frequency (or quantized harmonic oscillator of... 

The quantum-mechanical description of a polyatomic system may be extrapolated from the treatment of the diatomic molecule. Starting again with the harmonic oscillator, the energy levels for the entire system (E) can be given in terms of the characteristic frequencies (v,) and quantum numbers ( ,) of a series of independent harmonic oscillators ... 

Relation (2) is introduced in the theory of de Boer and Hamaker energy constant in London s theory in which the two atoms are represented by three-dimensional harmonic oscillators with a characteristic frequency v and a polarizability a. h is Planck s constant. 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

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