Harmonic oscillator trajectory calculation

Figure 1. Mathcad worksheet calculating the harmonic oscillator trajectory with parameters appropriate for the iodine molecule.

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

Fig. 4.1.2 Harmonic oscillator with the energy E = p2/(2m) + (1/2)kq2 (which is the equation for an ellipse in the (q,p)-space). In the quasi-classical trajectory approach, E is chosen as one of the quantum energies, and all points on the ellipse may be chosen as initial conditions in a calculation, i.e., corresponding to all phases a [0, 27r].

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. 

RRKM theory has been used widely to interpret measurements of unimolecular rate constants. However, harmonic state counting procedures are usually used in the RRKM calculations. This is not because enharmonic effects are thought to be unimportant, but because they are difficult to account for. The only comprehensive attempt to include the effect of anharmonicity has involved treating the vibrational degrees of freedom as separable Morse oscillators. However, since this correction is an obvious oversimplification it has not been widely used. The importance of anharmonicity is illustrated by comparing the trajectory unimolecular rate constant for C2H5 H + C2Hi dissociation at 100 kcal/mol (Fig. 4b), which is about 4.7 X 10 with that predicted by harmonic classical RRKM... 

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