Two-oscillator model

It was pointed out that contributions due to acoustical and optical phonons must be considered for ZnO [169-172]. This is done by the two-oscillator model [169,173]... 

Fig. 3.24. Experimental data (symbols) of the FWHM W of the discrete free exciton together with model calculations (solid line) according to the two-oscillator model (WioiT), 3.30) vs. temperature of a ZnO thin film...

Figure 4 Reflectance of K-TCNQ single crystal, for two polarizations. The dashed line is a fit of the reflectance for the electric field parallel to the chain axis by a single Lorentzian oscillator the solid line is a fit to the two-oscillator model. (From Ref. 34.)...

Gander, P.H., R.E. Kronauer, C.A. Czeisler M.C. Moore-Ede. 1984a. Simulating the action of zeitgebers on a coupled two-oscillator model of the human circadian system. Am. J. Physiol 247 R418-R426. 

The distance between the two interacting atoms will be denoted by r, and fg will denote the average distance (or the equilibrium distance in a hypothetical vibrationless state) of the unperturbed system. (The distance will be independent of the isotopic mass.) A displacement from the average distance will be denoted by a = r—rg. The two-oscillator model subdivides the displacement into two components a corresponds to the motion of the isotopic hydrogen relative to the carbon atom to which it is attached, and corresponds to the motion of the remainder of the molecular framework joining the two non-bonded atoms. The motion of the isotopic hydrogen relative to the carbon will be sensitive to the isotopic mass, whereas the framework motion will not. The two kinds of displacement are assumed to be governed by the probabihty distribution functions... 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anliannonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies cOq and coq 2> respectively.

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

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 simplest hydrogen bond X-H Y model may be viewed as composed of two oscillators. The first one corresponds to the stretching X-H-Y of the valence bond X-H. We will refer to this mode as the fast mode of the... 

Eisner Your model of how the oscillation is maintained requires that the NCX in the surface membrane be very close to the SERCA in the SR. But presumably you can t have the InsP3 and RyRs in this place, because if they were they would release Ca2+ into the space and this would make it impossible for backwards NCX to bring Ca2+ in. Later on, towards the end of your paper, the SERCA has moved away from the Na+/Ca2+ exchanger and the RyR is close to the Na+/Ca2+ exchange in order that Ca2+ could be pumped out of the cell. Do you require two different models to explain the two different sorts of experiments, and do either of them fit with what is known about the localization of the various receptors ... 

Table 5.1 Prediction of VPIE s for two rare gases and nitrogen using a crude oscillator model (Equation 5.23). Comparison with experiment at the melting point, TM, and boiling point, TB, and with experimental VPIE s for two hydrocarbons (Van Hook, W. A. Condensed matter isotope effects, in Kohen, A. and Limbach, H. H., Eds. Isotope Effects in Chemistry and Biology, CRC, Boca Raton, FL (2006))...

Let us next turn to Model II, representing the C —> B —> X internal-conversion process in the benzene cation. Figure 2 demonstrates that this (compared to the electronic two-state model, Model I) more complicated process is difficult to describe with a MFT ansatz. Although the method is seen to catch the initial fast C —> B decay quite accurately and can also qualitatively reproduce the oscillations of the diabatic populations of the C- and B-state, it essentially fails to reproduce the subsequent internal conversion to the electronic X-state. Jn particular, the MFT method predicts a too-slow population transfer from the C- and B-state to the electronic ground state. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

A two-variable model taking into account the allosteric (i.e. cooperative) nature of the enzyme and the autocatalytic regulation exerted by the product shows the occurrence of sustained oscillations. Beyond a critical parameter value, the steady state admitted by the system becomes unstable and the system evolves toward a stable limit cycle corresponding to periodic behavior. The model accounts for most experimental data, particularly the existence of a domain of substrate injection rates producing sustained oscillations, bounded by two critical values of this control parameter, and the decrease in period observed when the substrate input rate increases [31, 45, 46]. 

A number of theoretical models of VOA can be considered as generalizations of the coupled oscillator concept to more than two oscillators. For VCD, these models have the form... 

Two of the most severe limitations of the harmonic oscillator model, the lack of anharmonicity (i.e., non-uniform energy level spacings) and lack of bond dissociation, result from the quadratic nature of its potential. By introducing model potentials that allow for proper bond dissociation (i.e., that do not increase without bound as x=>°°), the major shortcomings of the harmonic oscillator picture can be overcome. The so-called Morse potential (see the figure below)... 

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