  Photoelectrons, the fast electrons produced in the photoionization of
planetary atmospheres, drive transformations in the atmospheric gas that are
often inhibited by energy considerations for thermal electrons. The
transformations include excitation and ionization of atoms and molecules, which
affect the detectability of these gases and constrain the fraction of incident
stellar radiation that transforms into heat. To gain insight into these
important questions, we build a Monte Carlo model that solves the slowing down
of photoelectrons in a gas with arbitrary amounts of H and He atoms and thermal
electrons. Our novel multi-score scheme differs from similar tools in that it
efficiently handles rare collisional channels, as in the case of low-abundance
excited atoms that undergo superelastic and inelastic collisions. The model is
validated and its performance demonstrated. Further, we investigate whether
photoelectrons might affect the population of the excited hydrogen H(2)
detected at some exoplanet atmospheres by transmission spectroscopy in the
H-$\alpha$ line. For the ultra-hot Jupiter HAT-P-32b, we find that
photoelectron-driven excitation of H(2) is inefficient at the pressures probed
by the line core but becomes significant (yet sub-dominant) deeper in the
atmosphere where the line wings form. The contribution of photoelectrons to the
destruction of H(2) either by collisional deexcitation or ionization is
entirely negligible, a conclusion likely to hold for exoplanet atmospheres at
large. Importantly, photoelectrons dominate the gas ionization at the altitudes
probed by the H-$\alpha$ line, a fact that will likely affect, even if
indirectly, the population of H(2) and other tracers such as metastable helium.
Future modeling of these excited levels should incorporate photoelectron-driven
ionization.
