The ionization of simple molecular targets, such as molecular hydrogen, or even the molecular hydrogen ion (\mbox{H}^{+}_{2}) by strong laser fields has become the focus of experimental research in the past few decades. On the theoretical side the problem presents two challenges: on the one hand one has to solve the problem numerically even in the one-electron case (\mbox{H}^{+}_{2}), since no analytic closed-form solution is possible; on the other hand there is the many-electron problem (\mbox{H}_{2} and other diatomic molecules, such as \mbox{N}_{2}, \mbox{O}_{2}, etc.), which currently is at the limit of computational feasibility (\mbox{H}_{2}), or exceeds it for molecules with more than two electrons.
In this thesis the single-electron problem of the hydrogen molecular ion in intense continuous-wave laser fields is addressed. The focus is on ionization rates of the molecule as a function of internuclear separation within the framework that the motion of the nuclei can be neglected (Born-Oppenheimer approximation). First, the problem of the DC limit is considered, i.e., a strong static electric field is applied along the internuclear axis. The field ionization rate is calculated by solving a stationary non-hermitean Schr\"odinger equation in a suitable coordinate system (prolate spheroidal coordinates). Some previously obtained values from the literature are reproduced; for larger internuclear separations improved values are obtained.
For the more interesting case of an infrared (continuous-wave) laser field Floquet theory is applied to transform the time-dependent Schr\"odinger equation for the electronic motion into a non-hermitean coupled-channel stationary problem. Ionization rates are found as a function of laser frequency (ω), and the low-frequency limit is pursued to understand how one can establish a connection to the DC limit. Results are obtained for the two lowest electronic states, which are named the {\it gerade} and {\it ungerade} (or even and odd) ground states in the field-free limit.
From the calculated results it is observed that the ionization rates peak at certain internuclear separations, such that a dissociating \mbox{H}^{+}_{2} molecule will be preferentially field ionized. In addition, the thesis reports on calculations of so-called high harmonic generation - a process where photo-electrons acquire energy from the laser field, are deflected back by the linearly polarized laser and recombine under the emission of photons with energies that correspond to odd-integer multiples of the laser photon energy
