Motivated by the stability problem for doubly charged negative atomic ions, we study the minimal charge Z cr (N) required for an atomic nucleus to bind N electrons. Employing large and diffuse basis sets and multireference configuration interaction computations, Z cr (N ) is determined for systems up to N = 19 corresponding to elements from the lowest three periods of the periodic table. For those elements X considered here, our results rule out the stability of X 2− , and give strong evidence that no other stable atomic dianions exist.