  The direct laser cooling of neutral diatomic molecules in molecular beams
suggests that trapped molecular ions can also be laser cooled. The long storage
time and spatial localization of trapped molecular ions provides the
opportunity for multi-step cooling strategies, but also requires a careful
consideration of rare molecular transitions. We briefly summarize the
requirements that a diatomic molecule must meet for laser cooling, and we
identify a few potential molecular ion candidates. We then perform a detailed
computational study of the candidates BH+ and AlH+, including improved ab
initio calculations of the electronic state potential energy surfaces and
transition rates for rare dissociation events. Based on an analysis of
population dynamics, we determine which transitions must be addressed for laser
cooling and compare experimental schemes using continuous-wave and pulsed
lasers
