  We investigate the characteristic effects of nuclear motion on attosecond
transient absorption spectra in molecules by calculating the spectrum for
different model systems. Two models of the hydrogen molecular ion are
considered: one where the internuclear separation is fixed, and one where the
nuclei are free to vibrate. The spectra for the fixed nuclei model are similar
to atomic spectra reported elsewhere, while the spectra obtained in the model
including nuclear motion are very different and dominated by extremely broad
absorption features. These broad absorption features are analyzed and their
relation to molecular dissociation investigated. The study of the hydrogen
molecular ion validates an approach based on the Born-Oppenheimer approximation
and a finite electronic basis. This latter approach is then used to study the
three-dimensional hydrogen molecule including nuclear vibration. The spectrum
obtained from H$_2$ is compared to the result of a fixed-nuclei calculation. In
the attosecond transient absorption spectra of H$_2$ including nuclear motion
we find a rich absorption structure corresponding to population of different
vibrational states in the molecule, while the fixed-nuclei spectra again are
very similar to atomic spectra. We find that light-induced structures at
well-defined energies reported in atomic systems are also present in our fixed
nuclei molecular spectra, but suppressed in the ${\text{H}_2}^+$ and H$_2$
spectra with moving nuclei. We show that the signatures of light-induced
structures are closely related to the nuclear dynamics of the system through
the shapes and relative arrangement of the Born-Oppenheimer potential energy
curves.
