  Molecular ions formed in cold hybrid ion-atom experiments may find
interesting applications ranging from precision measurements to controlled
chemical reactions. Here, we investigate electronic structure of the Sr$_2^+$
molecular ion, which may be produced by photoassociation of laser-cooled Sr$^+$
ions immersed into an ultracold gas of Sr atoms or by ionization of ultracold
Sr$_2$ molecules. Using \textit{ab initio} electronic structure methods, such
as the coupled cluster and configuration interaction ones with small-core
relativistic energy-consistent pseudopotentials and large Gaussian basis sets,
we calculate potential energy curves for the ground and 41 excited electronic
states, and electric dipole transition moments between them. We show that
alkaline-earth molecular ions despite of their apparently simple structure with
three valence electrons only are challenging for state-of-the-art quantum
chemistry methods due to their multireference nature and high density of
states. Finally, we calculate and analyze Franck-Condon factors governing the
photoionization of ground-state Sr$_2$ molecules into $^2\Sigma^+_u$ and
$^2\Sigma^+_g$ states of Sr$_2^+$ molecular ions. The present results may be
useful for studying and guiding formation and spectroscopy of cold Sr$_2^+$
molecular ions.
