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 N_{2}, these errors only become important above 5000°K. For gases with lower vibrational frequencies such as O_{2}, or strong attractive forces such as NO, these effects become important at much lower temperatures.

## REFERENCES

*Thermodynamics and Physics of Matter*(Princeton University Press, Princeton, New Jersey, 1955).

*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 $(\u2212x/L),$ one must have exp $(\u22124\pi 2L/\lambda )\u226a1$ with λ the de Broglie wavelength in

*x*at $x\u2009=\u2009\u221e.$

^{3}Also, it can be shown

^{3}that the single velocity used in the semiclassical approach should be the arithmetic mean of the actual velocities before and after energy exchange. Although the approximation of a single classical trajectory is often poor at the most probable velocity for collision in a thermal gas, it is often a very good approximation at the most probable velocity for energy exchange, which is considerably higher.

^{1}obtained $L\u2009=\u2009r0/17.5,$ where $r0$ is the Lennard-Jones potential parameter tabulated for many gases. The effect of attractive forces can be crudely estimated by using a modified distribution of collision velocities “speeded up” by the energy ε where ε is the depth of the Lennard-Jones potential well.

*Tables of Integral Transforms*(Bateman Manuscript Project), edited by A. Erdély (McGraw-Hill Book Company, Inc., New York, 1954), Vol. II, p. 495, and Eq. (30), p. 292.

*Special Functions*(The Macmillan Company, New York, 1960), p. 201.