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Molecules containing short-lived, radioactive nuclei are uniquely positioned to enable a wide range of scientific discoveries in the areas of fundamental symmetries, astrophysics, nuclear structure, and chemistry. Recent advances in the ability to create, cool, and control complex molecules down to the quantum level, along with recent and upcoming advances in radioactive species production at several facilities around the world, create a compelling opportunity to coordinate and combine these efforts to bring precision measurement and control to molecules containing extreme nuclei. In this manuscript, we review the scientific case for studying radioactive molecules, discuss recent atomic, molecular, nuclear, astrophysical, and chemical advances which provide the foundation for their study, describe the facilities where these species are and will be produced, and provide an outlook for the future of this nascent field. 

The nuclear shell model has perhaps been the most important conceptual and computational paradigm for the understanding of the structure of atomic nuclei. While the shell model has been used predominantly in a phenomenological context, there have been efforts stretching back more than half a century to derive shell model parameters based on a realistic interaction between nucleons. More recently, several ab initio many-body methods—in particular, many-body perturbation theory, the no-core shell model, the in-medium similarity renormalization group, and coupled-cluster theory—have developed the capability to provide effective shell model Hamiltonians. We provide an update on the status of these methods and investigate the connections between them and their potential strengths and weaknesses, with a particular focus on the in-medium similarity renormalization group approach. Three-body forces are demonstrated to be important for understanding the modifications needed in phenomenological treatments. We then review some applications of these methods to comparisons with recent experimental measurements, and conclude with some remaining challenges in ab initio shell model theory.