Exploring the parameter space for ionization and dissociation of H$_{2}^{+}$ in an intense laser pulse

We explore the dissociation and ionization of H$_{2}^{+}$ ions aligned with a 790~nm laser field of peak intensity in the range 1.0$\times $10$^{13}$ to 7.0$\times $10$^{14}$~W/cm$ ^{2} $. Calculated dissociation and ionization probabilities are reported for different initial vibrational states and for the initial state averaged over the Franck-Condon distribution. The dependence on the carrier-envelope phase difference for different initial states and for pulse durations from 5 to 30~fs FWHM is presented. These results --- from direct solution of the time-dependent Schr\"odinger equation --- are compared with solutions in the Born-Oppenheimer representation with two-channels for low peak laser intensities.