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Ro-vibrational analysis of the XUV photodissociation of HeH$^+$ ions

We investigate the dynamics of the photodissociation of the hydrohelium cation HeH$^+$ by XUV radiation with the aim to establish a detailed comparison with a recent experimental work carried out at the FLASH free electron laser using both vibrationally hot and cold ions. As shown in previous theoretical works, the comparison is hindered by the fact that the experimental ro-vibrational distribution of the ions is unknown. We determine this distribution using a dissociative charge transfer set-up and the same source conditions as in the FLASH experiment. Using a non-adiabatic time-dependent wave packet method, we calculate the partial photodissociation cross sections for the $n=1-3$ coupled electronic states of HeH$^+$. We find a good agreement with the experiment for the total cross section into the He + H$^+$ dissociative channel. By performing an adiabatic calculation involving the $n=4$ states, we then show that the experimental observation of the importance of the electronic states with $n>3$ cannot be well explained theoretically, especially for cold ($v=0$) ions. We also calculate the relative contributions to the cross section of the $Σ$ and $Π$ states. The agreement with the experiment is excellent for the He$^+$ + H channel, but only qualitative for the He + H$^+$ channel. We discuss the factors that could explain the remaining discrepancies between theory and experiment.