A model based on the strong-field and Born-Oppenheimer approximations qualitatively describes the distribution over vibrational states formed in a diatomic molecular ion following ionization of the neutral molecule by intense laser pulses. Good agreement is found with a recent experiment [X. Urbain et al., Phys. Rev. Lett. 92, 163004 (2004)]. In particular, the observed deviation from a Franck-Condon-like distribution is reproduced. Additionally, we demonstrate control of the vibrational distribution by a variation of the peak intensity or a change of frequency of the laser pulse.PACS numbers: 33.80. Rv,33.80.Eh,82.50.Hp Intense investigations during the last decades have led to quite a detailed understanding of the interaction between strong laser fields and atoms (see [1] for a review). This includes, e.g., the process of high-harmonic generation which is by now a valuable source for the production of coherent ultraviolet light. For molecules, however, the extra degrees of freedom introduced by the presence of more than one nucleus lead to a much more involved picture, and the description of the strong-field driving of such systems far from equilibrium is still a challenge in theoretical physics (for reviews see [2,3]).The main topic of this work is to investigate the partitioning of energy among the electronic and nuclear degrees of freedom when ionizing a diatomic molecule by a strong laser field. This is a topic of much current interest: until recently it was assumed that the distribution over vibrational states formed in the molecular ion following ionization of the neutral molecule follows a Franck-Condon (FC) distribution, and this assumption is still frequently used (see [4] and references therein). A recent experiment, however, reported a non-FC distribution and suggested that the use of the FC principle is inaccurate because the rate for tunneling ionization increases sharply with internuclear distance [4]. We will show that the rapid response of the electron to the laser field in fact ensures the applicability of the FC principle for the nuclear motion at each instant of time during the pulse. It is instead the variation of the ionization and excitation rate with intensity in the focal volume that leads to a departure from a conventional FC-like distribution. The theory is very versatile and readily adapted to a wide range of diatomic molecules.We note that as the strong field response of the molecular ion is very dependent on the distribution over vibrational levels, it is desirable to be able to control the latter. We show by explicit examples in H 2 , O 2 and N 2 that a simple change of the peak intensity of a 45 fs Gaussian pulse leads to a significant degree of control over the final vibrational distribution, and hence holds the promise for the production of a target of interest for state-specific studies.The energy absorbed by a molecule in the field is distributed among the electrons and nuclei. Ionization of molecules is accordingly accompanied by a vibrational excitation of the nuclear mo...