Ionization conjecture

The ionization conjecture asks whether the maximum bound-electron number is at most nuclear charge plus a constant, including for molecules. The full many-body molecular statement remains open.

Known results

• Ruskai and Sigal (1982) proved finiteness of $N_c(Z)$.
• Lieb (1984) proved the general bound $N_c(Z)<2Z+1$ for atoms, and $N_{\max}<2Z+M$ for molecules.
• Solovej (2003) proved the analogous $Z+O(1)$ result in Hartree–Fock theory.
• Müller theory also has a $Z+C$ bound, but this is a reduced theory.

October 2026 two-center estimate

Lu Chen and Yong Wu report, for the two-center Thomas–Fermi–Dirac–von Weizsäcker theory, the bound $N\leq Z+C Z^{153/154}(1+\log Z)^{3/8}$. This is a claimed advance toward neutrality, but it does not establish the conjectured $Z+O(1)$ bound for the full many-body problem.

Current status (as of October 2026): The full many-body ionization conjecture remains open; a two-center bound in a reduced theory is claimed but unverified.