Exchange in low-energy electron-molecule scattering: Free-electron-gas model exchange potentials and applications to $e$-${\mathrm{H}}_2$ and $e$-${\mathrm{N}}_2$ collisions

Calculations of cross sections for elastic $e$-${\mathrm{N}}_2$ collisions have been performed for impact energies from 0.01 to 1.0 Ry using model exchange potentials based on a free-electron-gas approximation in order to evaluate the utility of such local potentials in low-energy electron-molecule scattering. A body-frame fixed-nuclei formulation of the collision problem is used which treats the molecule as rigid and does not allow for electronic or vibrational excitation. Coupled integro-differential scattering equations which incorporate exchange rigorously are derived and the simplification of the nonlocal exchange potential to an approximate energy-dependent local potential is described. An initial study of $e$-${\mathrm{H}}_2$ collisions is reported in which results obtained with various model excahnge potentials are compared to those of exact static-exchange calculations. Cross sections for the $e$-${\mathrm{N}}_2$ system obtained using the model exchange potentials (with and without polarization) are reported and compared with measured cross sections and the results of other theoretical studies. For both systems, good agreement with more rigorous treatments of the exchange can be obtained.