Morse Oscillator Model Research Articles

Semiclassical calculation for collision induced dissociation. II. Morse oscillator model

A recently developed semiclassical procedure for calculating collision induced dissociation probabilities Pdiss is applied to the collinear collision between a particle and a Morse oscillator diatomic. The particle–diatom interaction is described with a repulsive exponential potential function. Pdiss is reported for a system of three identical particles, as a function of collision energy Et and initial vibrational state of the diatomic n1. The results are compared with the previously reported values for the collision between a particle and a truncated harmonic oscillator. The two studies show similar features, namely: (a) there is an oscillatory structure in the Pdiss energy profiles, which is directly related to n1; (b) Pdiss becomes noticeable (≳10−3) for Et values appreciably higher than the energetic threshold; (c) vibrational enhancement (inhibition) of collision induced dissociation persists at low (high) energies; and (d) good agreement between the classical and semiclassical results is found above the classical dynamic threshold. Finally, the convergence of Pdiss for increasing box length is shown to be rapid and satisfactory.

Steady state dissociation of shock heated HF, HCl, and CO in excess Ar

The steady state dissociation rate constants of shock heated HF,HCl, and CO in the presence of large excess of Ar are calculated, assuming that dissociation can occur from all the vibrational levels, at several temperatures and compared with experimental results. Both the truncated harmonic oscillator (THO) model and Morse oscillator (MO) model are considered. Both the models fit the dissociation rates satisfactorily in the temperature ranges considered. The results are in accord with a ’’weak bias’’ mechanism and large depletion in the higher vibrational levels.

Studies of a model of hydrogen recombination

A Morse oscillator model for recombination of hydrogen and its isotopes was investigated computationally. The model is very similar to one recently studied by Pritchard and co-workers. The aims of the study were the test of certain approximations and the determination of the range of validity of the model. It was found that at low temperatures the steady state approximation is an increasingly accurate and increasingly convenient approximation. A way of estimating the error arising from this approximation was proposed. Other approximations which were tested gave unsatisfactory results. The rate constant vs. temperature curve for the model does not agree well with the experimental curve for hydrogen in argon except insofar as it reproduces the observed levelling-off at high temperatures. For the model this result may be ascribed to increasing relative rates for multiquantum transitions among the levels of the diatom. The model does give the experimentally observed ratios of the rate constants for H- and D-atom recombination. This result indicates that the important factor in determining the isotope ratio is the effect of mass change on the density of states rather than its effect on the collision dynamics.

Master equation theory for steady-state chemical reactions: Dissociation of diatomic molecules in gases

A steady-state form of the master equation is solved to determine rate constants for dissociation of diatomic molecules. A truncated Morse oscillator model describes the vibrational energies and the (SSH) transition probabilities. Single-quantum vibrational–translational energy exchanges are shown to dominate the rate of dissociation. The theory provides order of magnitude agreement with experimental data for H2, N2, O2, NO, F2, Cl2, Br2, and I2. Anharmonicity of the potential contributes to the lowering of the energy of activation below the bond dissociation energy. Deviations from the Boltzmann distribution cause the energy of activation to decrease at higher temperatures.

Comparisons of Morse and harmonic oscillator models for vibration-rotation excitation of H2 by Li+

Matrix elements of a potential for atom-diatomic molecule collision problems in three dimensions have been calculated using two models to describe the rotating-vibrating molecule. These models are the undistorted three dimensional harmonic oscillator model of Eastes and Secrest and a rotating-vibrating Morse oscillator. The most important difference between the two models is the presence of centrifugal stretching in the Morse oscillator model. A comparison of matrix elements is made which is applicable to any atom-diatom potential with a dependence on the vibrational coordinate in the form ξ, ξ2, eαξ, and ξeαξ. In addition, the specific example of Lester's ab initio Hartree-Fock potential for Li+–H2 is considered. Cross sections are calculated for Li+–H2 with limited basis sets for the harmonic and Morse oscillator at 0.1, 2.0, and 3.26 eV relative kinetic energy.

Effect of VV Transfer on the Rate of Diatomic Dissociation

The effect of VV transfer processes on the rate of thermal dissociation of diatomic molecules is considered within the steady-state approximation for a single-quantum step master equation. An accurate rate law is obtained for the truncated harmonic oscillator through replacement of the agents for VV transfer by the vibrational energy. Approximate numerical solutions for a Morse oscillator model of the representative species O2, H2, and HCl are in quite satisfactory agreement with the measured rates and provide at least qualitative explanations for nearly all the unusual features of the measurements. The rate enhancement effect of VV transfer is found to be slight; the primary, and quite general consequence of VV transfer is rather a depression of the rate, especially at high temperatures, which then appears as a considerable reduction in activation energy. This effect is shown to arise from a violation of detailed balance for the VV processes whenever the vibrational energy is depleted by rapid dissociation.

On the Theory of Radiationless Electronic Transitions. II. The Morse Oscillator Model for Arbitrary Values of the Parameter Z

Abstract In a previous paper1 it has been shown that the harmonic oscillator model is incapable of providing sufficiently large Franck-Condon factors F(E) if realistic parameter values are to be used. In consequence the calculated rate constants v=i/τ of radiationless electronic transitions at E ∼ 3 ev do not agree with those obtained experimentally. It has further been shown that this problem can be solved by using the Morse oscillator model. The calculations, however, were confined to a specific, though frequently encountered, case of the general expression for the Franck-Condon factor of a Morse oscillator, viz., the case of a small parameter =, where δω1-ω, ω1 and ω being vibrational frequencies in a higher and lower electronic states respectively, Δr = rei - re is the difference of coordinates of oscillator equilibrium positions in the respective electronic states mentioned, and α is a parameter of the Morse curve for the lower electronic state.

On the Temperature Dependence of Radiationless Electron Transitions. II. The Morse Oscillator Model in case of an “Arbitrary” Parameter Z

Abstract In our previous work1 a formula is presented for the rate constant of a radiationless electronic transition as a function of temperature τ for the Morse oscillator model. The formula was derived for the particular case of a sufficiently small parameter . In the above formula v o = v (T = o), A is a constant, Δω = ω1 - ω1 where ω and to are vibrational frequencies in the higher and lower electronic states respectively, and α is a parameter of the Morse curve where re1 is analogous to re but refers to an excited electronic state. The expression for probability v(T) can be used to account for a number of experimental results on the temperature quenching of luminescence. However, the probability expression is valid for low values of only, and, besides, in practice the usual Arrhenius dependence of v(τ) is much more frequent than the one indicated by the expression. Therefore, it seems desirable to analyze the Morse oscillator model for a more or less arbitrary value of .

On the Temperature Dependence of Radiationless Electron Transitions. I. The Morse Oscillator Model in case of a Small Value of the Parameter Z

Abstract A number of experimental results have been recently made available on the temperature dependence of lifetimes of excited states of polyatomic molecules in condenced1–6 and gaseous phases. The dependence was shown to be quite substantial in experiments with both fluorescence5,6 and phosphorescence1–4 temperature quenching. The decay-rate constant of electronic excitation is 1/τ′ = β + 1/τ, where b is determined by radiative processes and is but slightly temperature-dependent5,6. As for the rate constant 1/τ of radiationless electronic transitions, it might be the very parameter that governs the temperature dependence of 1/τ.

Scmielassical Calculation of the Vibrational Transition Probabilities of Diatomic Molecules in Collision. The Effect of Vibrational Anharmonicity

Abstract The multi-state semiclassical method proposed by Sharp and Rapp for calculating the vibrational transition probabilities of harmonic oscillators in collision was extended to Morse oscillators. Model calculations have been carried out for the excitation of a vibrational ground-state hydrogen molecule colliding with a helium atom at varying relative velocity. It has been shown that the Morse oscillator model gives greater transition probabilities than the corresponding harmonic oscillator model.

Transient Phenomena in Dissociative Reactions

The ``diffusion theory'' of dissociation is used to analyze vibration-dissociation coupling in the early phases of dissociation behind a shock wave. It is shown that deviations from vibrational equilibrium introduce ``incubation times'' into the long-time behavior of macroscopic molecular observables, such as the number density. For example, if Q(t) is some macroscopic observable whose value is Q0 in vibrational equilibrium behind the shock wave with no dissociation, then at long times Q(t) = Q0 exp [−(t − τQ)/τDiss], where τDiss is the dissociation time and τQ is the incubation time for the observable Q. The incubation times are experimentally observable quantities and may be positive or negative, depending on the observable, Q. Approximate methods are used to calculate incubation times according to a simplified Morse oscillator model. The results agree well with experimental measurements made by Wray in O2-argon mixtures. Incubation effects are found to increase rapidly in importance as the temperature increases, with |τQ/τDiss| ≳ 10% when kT ≳ 15% of the dissociation energy.