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
1.
K. F. Herzfeld, “Relaxation Phenomena in Gases,” in Thermodynamics and Physics of Matter (Princeton University Press, Princeton, New Jersey, 1955).
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3.
D. Rapp, “Vibrational Energy in Quantum and Classical Mechanics,” Lockheed Missiles & Space Company Tech. Rept. 6-90-61-14 (Sunnyvale, California, July 1961).
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(a)
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6.
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 = ∞.
7.
It is assumed that the effect of the centrifugal potential due to the rotation of BC is small.
8.
However, since one must use a single collision trajectory before and after exchange of energy, this implies that the collisional kinetic energy m̃υ02/2 is large compared to the energy exchanged. When this condition applies, the classical trajectory is not appreciably altered by the energy transfer. An idea of the validity of this approximation can be obtained by comparing the classical limit of the strictly quantum treatment with the present method in the perturbation limit.3 Also, it can be shown3 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.
9.
By fitting the function exp (−X/L) to the bottom of the Lennard-Jones potential well, Herzfeld1 obtained L = r0/17.5, 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.
10.
More exact calculations of the effect of a potential well have also been attempted; see reference 4 and also
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11.
The appropriate expression for the potential in Eq. (7) is
Vnj(t) = E0sech2(υit/2L)Unj
.
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also,
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16.
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.
17.
E.g., E. D. Rainville, Special Functions (The Macmillan Company, New York, 1960), p. 201.
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© 1963 American Institute of Physics.
1963
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