Simple systematization of vibrational excitation cross-section calculations for resonant electron-molecule scattering in the boomerang and impulse models

Vibrational excitation $(νf←νi)$ cross-sections $σνf←νi(E)$ in resonant $e-N2$ and $e-H2$ scattering are calculated from transition matrix elements $Tνf,νi(E)$ obtained using Fourier transform of the cross correlation function $⟨ϕνf(R)∣ψνi(R,t)⟩$⁠, where $ψνi(R,t)≃e−iHA2−(R)t∕ℏϕνi(R)$ with time evolution under the influence of the resonance anionic Hamiltonian $HA2−(A2−=N2−∕H2−)$ implemented using Lanczos and fast Fourier transforms. The target $(A2)$ vibrational eigenfunctions $ϕνi(R)$ and $ϕνf(R)$ are calculated using Fourier grid Hamiltonian method applied to potential energy (PE) curves of the neutral target. Application of this simple systematization to calculate vibrational structure in $e-N2$ and $e-H2$ scattering cross-sections provides mechanistic insights into features underlying presence/absence of structure in $e-N2$ and $e-H2$ scattering cross-sections. The results obtained with approximate PE curves are in reasonable agreement with experimental/calculated cross-section profiles, and cross correlation functions provide a simple demarcation between the boomerang and impulse models.