  Tracing ultrafast processes induced by interaction of light with matter is
often very challenging. In molecular systems, the initially created electronic
coherence becomes damped by the slow nuclear rearrangement on a femtosecond
timescale which makes real-time observations of electron dynamics in molecules
particularly difficult. In this work, we report an extension of the theory
underlying the attosecond transient absorption spectroscopy (ATAS) for the case
of molecules, including a full account for the coupled electron-nuclear
dynamics in the initially created wave packet, and apply it to probe the
oscillations of the positive charge created after outer-valence ionization of
the propiolic acid molecule. By taking advantage of element-specific
core-to-valence transitions induced by X-ray radiation, we show that the
resolution of ATAS makes it possible to trace the dynamics of electron density
with atomic spatial resolution.
