  Ultracold molecules with both electron spin and an electric dipole moment
offer new possibilities in quantum science. We use density-functional theory to
calculate hyperfine coupling constants for a selection of molecules important
in this area, including RbSr, LiYb, RbYb, CaF and SrF. We find substantial
hyperfine coupling constants for the fermionic isotopes of the alkaline-earth
and Yb atoms. We discuss the hyperfine level patterns and Zeeman splittings
expected for these molecules. The results will be important both to experiments
aimed at forming ultracold open-shell molecules and to their applications.
