A comparison of exact classical and quantum mechanical calculations of vibrational energy transfer

Abstract

Exact classical and quantal calculations of vibrational energy transfer were compared to determine the conditions under which classical mechanics is reliable. A detailed study of collinear collisions between an atom and a harmonic oscillator with a repulsive exponential interaction showed that the average classical and quantal energy transfer agree within a few percent, if the translational energy exceeds the vibrational spacing, and if the average energy transfer does not change sign as the collision energy is varied slightly. Additional collinear calculations with (i) a Morse oscillator and an exponential interaction and (ii) a harmonic oscillator with a Lennard-Jones interaction confirmed this result, except for the case of a deep potential well. In this case sticky collisions can occur, and the agreement is poorer. For collinear collisions of two harmonic oscillators with a repulsive exponential interaction, the agreement for T→V energy transfer was poorer than in the atom–diatom case. Good agreement was obtained for V→V transfer, with both classical and quantum calculations favoring resonant collisions. In order to calculate individual transition probabilities, a number of classical quantization methods were tested. The two moment method of Truhlar and Duff was found to work well, and was used to calculate integral cross sections for the vibrational relaxation of He+O2(v=1). For this system a potential energy surface consisting of a harmonic breathing sphere and a repulsive exponential interaction was used to calculate the integral cross sections σvv′ for the inelastic transitions v→v′. The classical values of σ10, σ12, σ13, and σ14 calculated by the two moment method were in satisfactory agreement with the quantum results for reduced translational energy ?t≡Et/h/ω?1, 2, 4, and 9, respectively. The resulting rate constants kvv′ were a factor of 2 too small for T≳300 K, and a factor of 1.25 too small for T≳500 K. The usual bin or histrogram method for quantizing the classical energy transfer produced an anomalous threshold of ?t≳3 for σ10 and σ12, and still higher values for the more endoergic transitions. The resulting k10 was a factor of 106 too small at 300 K and a factor of 2 too small at 1400 K. In addition we found that equating the P10 transition probability with the average reduced energy transfer gave accurate values of k10 for 300 <T<1000 K. However, equating P12(P10) with the fraction of trajectories resulting in energy gained (lost) by the oscillator produced excessively high values of k12 and k10. Finally, we found that the Landau–Teller model is in good agreement with the quantum mechanical calculations of k10, k12, and k13, up to a single arbitrary normalization constant.