Coupled double-well oscillator

This has the form of a double-well oscillator coupled to a transverse harmonic mode. The adiabatic approximation was discussed in great detail from a number of quantum-mechanical calculations, and it was shown how the two-dimensional problem could be reduced to a one-dimensional model with an effective potential where the barrier top is lowered and a third well is created at the center as more energy is pumped into the transverse mode. From this change in the reactive potential follows a marked increase in the reaction rate. Classical trajectory calculations were also performed to identify certain specifically quanta effects. For the higher energies, both classical and quantum calculations give parallel results. 

This Hamiltonian describes a reaction coordinate s in a symmetric double well as - bs that is coupled to a harmonic oscillator Q. The coupling is symmetric for the reaction coordinate and has the form cs Q, which would reduce the barrier height of a quartic double well. The origin of the Q oscillations is taken to be at Q = 0 when the reaction coordinate is at =fso (centers of the the reactant/product wells), which explains the presence of the term —cs Q. This potential has 2 minima at (s, Q) = ( So,0) and one saddle point at (s, Q) = (0, +cs2o/Mi22)... 

The two-dimensional PES shown in Figure 8.17 (as well as in Figures 8.3b and 8.7c) is typical of internal rotation coupled to inversion of the other part of the system. This situation is also realized in methylamine inversion, where the rotation barrier is modulated not by a harmonic oscillation but by motion in a double-well potential. The PES for these coupled motions can be modeled as follows ... 

The spin-related section mle can be proved by elastic neutron scattering measurements. In order to establish the specific fingerprint of the spin correlation, the scattering functions for the linear harmonic oscillator, for the double-well minimum function, and for pairs of coupled oscillators have been calculated in Ref. 119. 

To obtain some insight into the behavior of the solutions of the Hamiltonian equation (10), we performed a numerical simulation of a model system 23 we assumed that V(s) is a symmetric double well, we coupled, v to 1000 harmonic oscillators cok with frequencies ranging from 10 to 1000 cm, and symmetrically to one oscillator Qpv. Even though the simulation is completely classical, we obtained instructive results that illustrate several of the points we have mentioned in this section. 

Two different reactions have presently been studied in the Couette flow reactor, namely the variants of the Belousov-Zhabotinsky [27-30, 32] and chlorite-iodide [29-33] reactions. The BZ reaction has revealed a rich variety of steady, periodic, quasi-periodic, frequency-locked, period-doubled and chaotic spatio-temporal patterns [27, 28], well described in terms of the diffusive coupling of oscillating reactor cells, the frequency of which changes continuously along the Couette reactor as the result of the imposed spatial gradient of constraints. This experimental observation has been successfully simulated with a schematic model of the BZ kinetics [68] and the recorded bifurcation sequences of patterns resemble those obtained when coupling two nonlinear oscillators. 

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