Ground state energy should be convex in the number of electrons. Add electrons one at a time to a system of fixed nuclei: the energy should curve upward, each additional electron bound more loosely than the last, because electron-electron repulsion grows while the nuclear attraction stays constant. Convexity means that the energy of N electrons is at most the average of the energies for N-1 and N+1 electrons. This has been assumed, explicitly or implicitly, across decades of quantum chemistry.
It is false (arXiv:2409.08632). There exist nuclear configurations where the ground state energy is not convex in electron number. Specifically: nuclei that can bind 2 electrons and 4 electrons, but where the 3-electron system is more loosely bound than the 4-electron system. The third electron is harder to attach than the fourth.
The mechanism is electron correlation. In the 3-electron system, the electrons arrange themselves in a configuration where their mutual repulsion is not optimally screened by the nuclear potential. Adding a fourth electron changes the symmetry of the ground state, allowing a qualitatively different spatial arrangement where all four electrons are better screened than the three were. The fourth electron does not just add itself to the existing configuration — it restructures the entire electron cloud.
This violates a principle so deeply assumed that it rarely appears as a stated hypothesis. Ionization energies are “supposed to” decrease monotonically: each successive electron is easier to remove. The counterexample shows that ionization energy can increase — that removing one electron from a 4-electron system can require more energy than removing one from a 3-electron system, even though there are more electrons repelling each other.
The assumption was not baseless. For atoms — spherically symmetric nuclear potentials — convexity holds. The counterexample requires nuclei arranged in specific geometries that break the spherical symmetry. But molecules are not atoms, and molecular geometries routinely break spherical symmetry. The assumption that held for the simplest case does not generalize.