Counterexamples to the Convexity of the Energy for Coulomb Particles

We present examples where successive ionization potentials of an isolated electronic system are nonincreasing; ergo, the energy is a nonconvex function of particle number. The basic strategy, which is based on minimizing the energy of the N ± 1-electron systems by leveraging solutions to the Thomson problem, seems quite general, giving large families of systems with nonconvex behavior; we show examples for N = 5 and N = 7 charged particles. As with previous counterexamples in the literature, the key ingredient is a qualitative change in configuration upon ionization. We propose two families of molecular structures, $F_4{{\mathrm{Br}}_6}^{5-}$ and $F_6{{\mathrm{Br}}_8}^{7-}$, that are expected to show nonconvex behavior with respect to the number of electrons.