  In this work we report ab initio calculations for the $H_2^+$ molecule
interacting with ultrashort intense laser pulses. We analyse several
observables that can, in principle, be available experimentally, in order to
get a deeper understanding of the strong field molecular dynamics. In
particular, we will focus our attention to the interplay between electronic and
nuclear dynamics and how the two motions are correlated.
  We have extended the Resolvent Technique to molecules and we have extracted
the correlated energy spectra in different regimes of ionization: from the
perturbative regime to the tunneling ionization regime. We have applied this
new method, called resolvent operator method (ROM), to a 1+1D model of the
$H_2^+$ molecule. We have applied this method to several photoionization
regimes and we have compared with previous results in the literature. We show
that the correlated spectra can provide us much more information than the
integrated spectra on the electron or nuclear energy. We report the correlated
energy spectrum from the multiphoton ionization to the tunneling ionization
regime and how the sharing of the excess energy is different for the two
regimes.
  We have also applied the ROM to a 3D model of the $H_2^+$ molecule in which
angular distributions can be obtained. With these angular correlated spectra we
show that the contributions of electrons coming from absorption of different
number of photons can be disentangled.
  We have calculated the high harmonic spectra for a 3D model of the $H_2^+$
molecule and its isotopes. In these calculations we have observed that for
sufficiently long pulses and for light molecules we observe even harmonic
generation, a phenomenon that is not expected in a centrosymmetric molecule.
