The vibrational de‐excitation probability, P_(10), is calculated quantum mechanically over a large energy range for models of three collision systems: O_2-O_2,Cl_2-Cl_2, and Br_2-Br_2. The vibrational de‐excitation cross section, σ_(10), is similarly calculated for the Cl_2-Cl_2 model. P_(10) and σ_(10) are obtained for the Lennard‐Jones intermolecular potential and three other "well‐less" potentials designed to duplicate the scattering of the Lennard‐Jones potential. The results emphasize the adiabatic nature of potentials with wells and indicate that the acceleration approximation for the effect of the well is not valid. The curves of P_(10) and σ_(10) as a function of initial translational energy are used to obtain exact collision numbers. These numbers are compared to the results of SSH theory. SSH theory is found to predict collision numbers with reasonable accuracy except at low temperatures. SSH theory is also not suitable for analyzing experimental collision numbers for the well depth potential parameter.