Asymptotics of Atomic Ionization Energy

Consider a non-relativistic atom in the Born–Oppenheimer approximation: a point nucleus of positive integer charge $Z$ is fixed at the origin, and $N$ electrons interact through Coulomb forces. In Hartree units, with two spin states, the electronic Hamiltonian on the antisymmetric space $\bigwedge^N L^2(\mathbb{R}^3;\mathbb{C}^2)$ is

$$H_{N,Z}=\sum_{j=1}^N\left(-\frac12\Delta_{x_j}-\frac{Z}{|x_j|}\right)+\sum_{1\leq i<j\leq N}\frac{1}{|x_i-x_j|}.$$

Define its ground-state energy as the bottom of its spectrum,

$$E(N,Z)=\inf\operatorname{spec}(H_{N,Z}).$$

For each positive integer $Z$, the atomic ionization energy in question is the energy required to remove one electron from the neutral atom:

$$\delta E(Z)=E(Z-1,Z)-E(Z,Z).$$

Determine the asymptotic behavior of $\delta E(Z)$ as the nuclear charge $Z$ tends to infinity through positive integers. In particular, this asks for the behavior of the exact quantum energy difference; it does not presume that the sequence has a limit. The Scott conjecture for large Coulomb systems: a review

Background

The leading large-$Z$ asymptotics of atomic total energies are much less informative about ionization: the neutral and singly charged energies have large common contributions, so their difference depends on terms that cancel in the total-energy expansions. This is why an accurate bulk energy approximation need not determine a chemically relevant energy difference. Thomas–Fermi theory gives a smooth description of the energy as the electron number varies, whereas the exact energies at integer electron numbers form discrete values; the ionization energy is a difference between those integer values, not simply the neutral Thomas–Fermi chemical potential. Ionization potentials in the limit of large atomic number

The Thomas–Fermi prediction provides an important model comparison: Lieb proved within that theory that the ionization energy does not grow with $Z$, and Englert obtained a finite limiting value for its singly ionized energy difference. These conclusions concern Thomas–Fermi theory, not a derivation of the asymptotics of the exact many-electron energies in the question. Ionization potentials in the limit of large atomic number

Known results

For the full quantum problem, the reported large-$Z$ picture in the 2010 study is based on calculations and extrapolation: it finds that ionization energies vary with periodic-table column, with a finite limit along each column and no single limit across all atomic numbers. The paper explicitly characterizes these conclusions as numerical and extrapolative, rather than rigorous proofs for the Coulomb Hamiltonian. It gives exact results in a weak-electron-repulsion limit, but that qualification does not establish the asserted behavior for the physical interaction strength. Ionization potentials in the limit of large atomic number

A rigorous uniform bound is known for the Hartree–Fock approximation: Solovej proved that the ionization energy of a neutral atom in Hartree–Fock theory is bounded by a universal constant, independent of $Z$. This establishes boundedness in that variational model, but it is not an asymptotic formula and does not by itself imply the same bound for the exact spectral energies $E(N,Z)$ defined above. Solovej (2003)