WKB Calculation of Vibrational Transition Probabilities in Molecular Collisions

A comparison is made between vibrational transition probabilities calculated by use of the WKB approximation with those obtained from other methods. For Cl₂–Cl₂, the transition matrix element for a purely repulsive exponential potential is within 1.5% of the Jackson–Mott exact value over the energy range from $\text{1500k–3900k}$ . For a “shifted exponential” potential, the WKB matrix element agrees satisfactorily with the exact value and with the Hartmann–Slawsky approximation at higher energies. It is also shown that the WKB matrix element for a Morse potential is larger than the Lennard‐Jones (12–6) matrix element by a factor of 1.3–1.5 over the energy range from $\text{900k–3900k}$ for Cl₂–Cl₂. Both for the exponential and inverse‐law potentials, the inclusion of attractive forces in the calculation leads to significantly larger matrix elements. Application of the WKB method to the calculation of the collision numbers of Cl²–Cl² and O₂–O₂ gives results within a factor of about 2 of the experimental data up to temperatures 2000° and 4000°K, respectively, when the effect of molecular orientation on the transition is properly taken into account.