Analytic Born completion in the calculation of differential cross sections for electron scattering from a linear molecule

We present a closed-form expression for the infinity of high-order partial waves and symmetry classes required to converge differential cross sections for vibrationally elastic or inelastic electron scattering from a linear molecule. This expression combines correction terms based on the first Born approximation with terms that contain T-matrix elements from accurate, non-Born scattering calculations such as those of body-frame fixed-nuclear-orientation theory. The equations for analytic Born completion reduce the convergence requirements for a differential cross section from several potentially infinite sums over angular-momentum quantum numbers to a single parameter that determines the partial-wave order at which the switch to the Born approximation occurs. We demonstrate the efficacy and accuracy of these equations for low-energy collisions of electrons with H₂ and N₂.