  We derive a rigorous optical potential for electron-molecule scattering
including the effects of nuclear dynamics by extending the common many-body
Green's function approach to optical potentials beyond the fixed-nuclei limit
for molecular targets. Our formalism treats the projectile electron and the
nuclear motion of the target molecule on the same footing whereby the dynamical
optical potential rigorously accounts for the complex many-body nature of the
scattering target. One central result of the present work is that the common
fixed-nuclei optical potential is a valid adiabatic approximation to the
dynamical optical potential even when projectile and nuclear motion are
(nonadiabatically) coupled as long as the scattering energy is well below the
electronic excitation thresholds of the target. For extremely low projectile
velocities, however, when the cross sections are most sensitive to the
scattering potential, we expect the influences of the nuclear dynamics on the
optical potential to become relevant. For these cases, a systematic way to
improve the adiabatic approximation to the dynamical optical potential is
presented that yields non-local operators with respect to the nuclear
coordinates.
