Ground state energy is not always convex in the number of electrons

We provide the first counterexample showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This convexity has been conjectured for decades and plays an important role in quantum chemistry. Our counterexample involves six nuclei with small fractional charges placed far apart. The ground state energy of 3 electrons is shown to be higher than the average of the energies for 2 and 4 electrons. We also show that the nuclei can bind 2 or 4 electrons, but not 3. This article raises the question of whether the energy convexity really holds for all possible molecules (with nuclei of integer charge).