  It was recently shown [Y. Suzuki, L. Lacombe, K. Watanabe, and N. T. Maitra,
Phys. Rev. Lett. 119, 263401 (2017)] that peak and valley structures in the
exact exchange-correlation potential of time-dependent density functional
theory are crucial for accurately capturing time-resolved dynamics of electron
scattering in a model one-dimensional system. Approximate functionals used
today miss these structures and con- sequently underestimate the scattering
probability. The dynamics can vary significantly depending on the choice of the
initial Kohn-Sham state, and, with a judicious choice, a recently-proposed
non-adiabatic ap- proximation provides extremely accurate dynamics on approach
to the target but this ultimately also fails to capture reflection accurately.
Here we provide more details, using a model of electron-He + as illustration,
in both the inelastic and elastic regimes. In the elastic case, the
time-resolved picture is contrasted with the time-independent picture of
scattering, where the linear response theory of TDDFT can be used to extract
transmission and reflection coefficients. Although the exact functional yields
identical scattering probabil- ities when used in this way as it does in the
time-resolved picture, we show that the currently-available approximate
functionals do not, even when they have the correct asymptotic behavior.
