Abstract
The problem of collinear scattering of an atom from a homonuclear diatomic molecule is formulated in terms of a firstβorder nonlinear matrix differential equation for the variable coefficient of reflection. For a homonuclear molecule when the target Hamiltonian is invariant under the parity transformation, only transitions between even states or odd states are possible. This selection rule reduces the number of open or closed channels that contribute to the reflection and transmission coefficients. But for numerical calculation, under the conditions of the problem, one can approximate the target Hamiltonian by the Hamiltonian of a displaced harmonic oscillator. In this approximation, the reflectional symmetry of the Hamiltonian is not preserved and transitions between any two levels of the target are possible. To simplify the problem further, the interaction between the projectile and the target is assumed to be a sum of two Gaussian terms. For this combination of the potentials the manyβchannel interaction can be expressed analytically. By fitting the LennardβJones potential with a sum of two Gaussian potentials and solving the matrix differential equation, transition probabilities are obtained for the Heο£ΏH~2~ collision. The numerical results are compared with the results found by Secrest and Johnson, and by Clark and Dickinson.