  The Auger decay is a relevant recombination channel during the first few
femtoseconds of molecular targets impinged by attosecond XUV or soft X-ray
pulses. Including this mechanism in time--dependent simulations of
charge--migration processes is a difficult task, and Auger scatterings are
often ignored altogether. In this work we present an advance of the current
state-of-the-art by putting forward a real--time approach based on
nonequilibrium Green's functions suitable for first-principles calculations of
molecules with tens of active electrons. To demonstrate the accuracy of the
method we report comparisons against accurate grid simulations of
one-dimensional systems. We also predict a highly asymmetric profile of the
Auger wavepacket, with a long tail exhibiting ripples temporally spaced by the
inverse of the Auger energy.
