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 <i>N</i> ± 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 <i>N</i> = 5 and <i>N</i> = 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>Br</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>Br</mml:mi></mml:mrow><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>7</mml:mn><mml:mo>-</mml:mo></mml:mrow></mml:msup></mml:math>, that are expected to show nonconvex behavior with respect to the number of electrons.