math-ph — Mathematical Physics

A Bound Helium Dianion Does Not Exist

Contributed by Alec Shelley, Daniel Tartakovsky

For a non-relativistic atom with $Z$ protons and $N$ bound electrons, Lieb proved in 1984 that $N<2Z+1$. Several improved bounds have been developed, but they all leave open the question of whether a bound helium dianion can exist. We prove that the answer is no. The ground-state energies of the four- and three-electron helium Hamiltonians coincide, $E_4(Z{=}2)=E_3(Z{=}2)$, so a helium nucleus binds at most three electrons. A weighted form of Lieb's identity reduces the problem to the nonnegativity of an explicit two-electron Schrodinger operator $m$, and Barta's inequality further reduces it to the pointwise nonnegativity of an explicit function of three variables. We prove the nonnegativity of this function by a combination of analytic estimates and a computer-assisted certification in interval arithmetic, and we publish the certificate with the paper.