Double Harmonic Oscillator

Figure 5-13 A Double Harmonic Oscillator. Displacements jci and X2, shown positive, may also be negative or zero.

Fig. 4. Schematic representation of a 3-atom Double Harmonic Oscillator (DHO) with r, r2 interatomic distances and k, k2 force constants, and of TR-A, TR-B, TR-C temperature ranges corresponding to the DHO model.

Numerical Solution for SchrOdlnger s Equation for Double Harmonic Oscillator... 

Figure 17 QuickBASIC user interface for numerical solutions to Schrodinger s equation for the double harmonic oscillator.

Double harmonic oscillator. This consists of two particles, 1 and 2, that move only in the direction of the x-axis. The masses of 1 and 2 are m, and njj, respectively, and their positions are x, and Xj. Spring a is coimected to an unmovable wall and to particle 1. Spring b connects the two particles. 

Table 3.9 Vibrational contributions to molecular polarizabilities calculated within the double-harmonic oscillator approximation at the HF/6-31G level of theory for 1,2,3 and 11...

In Table3.9, the vibrational contributions for 11, 1, 2 and 3, calculated within the double-harmonic oscillator approximation are presented. The calculations were performed seminumerically, i.e. second derivatives of energy were calculated by differentiation of analytic first derivatives. Thus, the values of harmonic terms presented in Table 3.9 may serve as a reference point for numerical accuracy assessment of NR contributions discussed above. The relative error for [/uq ] ) term for 11 does not exceed 10%. It follows from Table3.9 that diagonal vibrational contributions to a... 

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)... 

RO, Fig. 3d) (2) higher-frequency, smaller amplitude, quasi-harmonic oscillations (QHO, Fig. 3a) and (3) double-frequency oscillations containing variable numbers of each of the two previous types. By far the most familiar feature of the BZ reaction, the relaxation oscillations of type 1 were explained by Field, Koros, and Noyes in their pioneering study of the detailed BZ reaction mechanism.15 Much less well known experimentally are the quasiharmonic oscillations of type 2,4,6 although they are more easily analyzed mathematically. The double frequency mode, first reported by Vavilin et al., 4 has been studied also by the present author and co-workers,6 who explained the phenomenon qualitatively on the basis of the Field-Noyes models of the BZ reaction. 

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 ... 

Fig. 13. Kinetic oscillations during the CO/O reaction on Pt(110) at I = 540 K, />0, = 7.5 x 10-5 torr, and for varying pm. (From Ref. 71.) (a) pco = 3.90 x 10 lorr constant behavior (fixed point), (b) pt0 = 3.K4 x I0"5 torr onset of harmonic oscillations with small amplitudes (Hopf bifurcation), (c)pco = 3.66 x 10 5 torr harmonic oscillation with increased amplitude, (d) pc0 = 3.61 x I0-5 torr first period doubling, (e) pc0 = 3.52 x 10 torr second period doubling, (f) pco = 3.42 x 10 5 torr aperiodic (chaotic) behavior.

Surprisingly, the enthalpy of combustion of isoxazole was determined only very recently.270 For isoxazole, AH° (298.15 K) = —(394.70 + 0.12) kcalth mol-1, from which the enthalpy of formation in the gas phase was derived as AHf (g) = 18.78 0.13 kcalth mol-1. The enthalpies of combustion of 3-amino-5-methylisoxazole and 5-amino-3,4-dimethylisoxazole have also been determined.271 Thermodynamic parameters for isoxazole have been derived from vibrational spectra using the harmonic oscillator-rigid rotor approximation.272,273 Analysis of the rotational spectra of isotopic forms of isoxazole, studied by double resonance modulated microwave spectroscopy, has given the molecular dimensions shown in Fig. 1.274,275... 

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. 

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