Near threshold scattering is a very important problem in quantum collision theory. The LFCC method, although accurate, is not applicable to all but simplest molecules (e.g. H2). The AN theory, on the other hand, seems to breakdown near threshold. We have devised a First-Order Non-Degenerate Adiabatic (FONDA) theory which is not as demanding as LFCC method but is more accurate than AN theory near threshold. The FONDA theory has been applied to H2 molecule, with model and separable exchange, and the results are very encouraging. The first phase of this proposal would be to improve the FONDA cross section for vibrational excitations of H2 near threshold by including more vibrational states of the target molecule in the construction of the K matrix. Reformulation the FONDA theory in terms of the T matrix boundary conditions will also be considered. The second phase is to implement direct exchange in the FONDA theory. Finally in the last phase of this project, this theory will be applied to N2 molecule.
