The usual calculations of vibrational energy exchange in a diatomic gas make use of first-order perturbation theory and are therefore limited to low transition probabilities. In actual practice, the thermally averaged transition probabilities in most gases are indeed small. However, this is partly because it is the relatively few molecules in the high-velocity ``tail'' of the velocity distribution which account for most of the energy exchange. The actual microscopic transition probabilities in these collisions may be too great to justify the perturbation method for many gases at high temperatures. We have therefore solved the time-dependent Schrödinger equation based on a semiclassical collision, subject to an assumed form of potential, to get exact transition probabilities in a molecular collision. At low velocities, our result reduces to the perturbation result. A thermal average of our transition probabilities should eliminate errors due to use of the perturbation solution in previous calculations. For N2, these errors only become important above 5000°K. For gases with lower vibrational frequencies such as O2, or strong attractive forces such as NO, these effects become important at much lower temperatures.

Topics
Atomic and molecular collisions, Vibrational energy transfer, Perturbation theory, Schrodinger equations, Markov processes
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This “semiclassical” method was briefly outlined in the final paragraph of reference 2 and is discussed at length in Sec. IV of reference 3. It is not the method of reference 2 where the motion of y was also treated classically. A more exact statement of the requirements for use of the semiclassical method is that if V(x) varies as exp (−x/L), one must have exp (−4π2L/λ)≪1 with λ the de Broglie wavelength in x at x = ∞.
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© 1963 American Institute of Physics.
1963
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