A semiclassical calculation of transition probabilities for vibrational–vibrational–translational energy transfer in a collision of two diatomic molecules is presented. The collision model is approximate, utilizing a collinear collision of harmonic oscillators with an exponential repulsion between center atoms. The method of Kerner and Treanor is used to solve the Schrödinger equation with linearized potential in the oscillator coordinates. Our procedure is a logical extension of the Treanor method to diatomic–diatomic collisions. Closed form analytical results are obtained for this model. The general magnitudes and trends of the probabilities are expected to be indicative of the behavior of real molecules. General formulas for the probabilities of processes of the type
AB(n1) + BA(n2) → AB(n1′) + BA(n2′)
as a function of collision velocity are presented. Special attention is given to “resonant” energy transfer where n1′ + n2′ = n1 + n2.
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If ψS and ψA are the solutions to Eqs. (10) and (11) for g(t) = 0, then the solutions for g(t)≠0 will be ψSexp(− iφ) and ψAexp(iφ), respectively, where
φ = iħ−∞tg(t′)dt′
. Because only products of symmetric and antisymmetric normal mode wavefunctions enter in this problem, the phase factors due to g(t) cancel identically.
13.
A more precise interpretation of the word “high” can be made by considering the N2 molecule as an example. Here, ω = (k/μ)1/2 = 4.45×1014sec−1, and L = 0.2 A. The value of k′(t) varies during the collision, the maximum being reached at t = 0. The maximum value of (k′/μ)1/2 is 1.7×1014sec−1 for an initial velocity of 1.0×106cm/sec. Therefore, at this velocity, the maximum value of k′(t)/k is 0.15. During most of the collision k′(t)/k is considerably less than this value. We may then conclude that the approximation k′(t)≪k will not begin to break down until υ0 becomes large compared to 1×106cm/sec for N2.
14.
Strictly speaking, the lower limits of the integrals in this section should be −t0, where t0 is very large. We use the symbol −∞ with the understanding that Z and Λ are in phase before the collision.
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