Impact parameter and semi-classical treatments of atomic collisions

The wavefunction describing the colliding systems being expanded in the usual manner, $$\Psi(\mathbf{r}, \mathbf{R}) = \mathbf{S}_n\psi_n (\mathbf{r})F_n(\mathbf{R}),$$ it is shown that great simplification may be effected by writing $$F_n(\mathbf{R}) = \mu_n (\mathbf{R})\mathbf{F}^a_n(\mathbf{R}),$$ where F$^a_n$(R) is the three-dimensional J.W.K.B. approximation to the solution of an equation of the form $$(\nabla^2 + \kappa_n(\mathbf{R})^2) F_n(\mathbf{R}) = 0,$$ and where $\mu_n$(R) is a modulating function the asymptote to which determines the excitation cross-section. This semi-classical treatment is more accurate than the standard impact parameter treatment. The two treatments are closely related. Indeed if certain additional approximations are made the modulating function $\mu_n$(R) of the former becomes identical with the corresponding expansion coefficient a$_n$(R) of the latter. The semi-classical treatment is also applicable when the wavefunction is expanded in terms of the eigenfunctions of the quasimolecule formed by the colliding systems. Detailed consideration is given to the case of symmetrical charge transfer. It is demonstrated that on the two-state approximation the semi-classical treatment, like the full quantal treatment, predicts damping of the oscillations in the probability of charge transfer in close collisions. The four-state approximation is discussed.