Ionization potentials in the limit of large atomic number

By extrapolating the energies of non-relativistic atoms and their ions with up to 3000 electrons within Kohn-Sham density functional theory, we find that the ionization potential remains finite and increases across a row, even as $Z\rightarrow\infty$. The local density approximation becomes chemically accurate (and possibly exact) in some cases. Extended Thomas-Fermi theory matches the shell-average of both the ionization potential and density change. Exact results are given in the limit of weak electron-electron repulsion.

💡 Research Summary

The paper investigates the behavior of the ionization potential (IP) in the limit of very large atomic number (Z → ∞) by performing extensive non‑relativistic Kohn‑Sham density‑functional theory (DFT) calculations on neutral atoms and their singly‑charged ions containing up to 3000 electrons. The authors first generate accurate total energies for each system using standard exchange‑correlation approximations, primarily the local density approximation (LDA) and, where needed, generalized gradient approximations (GGA). The ionization potential is obtained as the energy difference ΔE = E(N‑1) – E(N), where N = Z for the neutral atom and N = Z‑1 for the ion.

A central finding is that, contrary to naive expectations that the IP might vanish or diverge as the electron cloud becomes increasingly diffuse, the IP remains finite and, within a given period of the periodic table, actually increases monotonically with Z. This trend reflects the reduced screening of the nuclear charge as outer shells fill, leading to a stronger binding of the outermost electron. The authors demonstrate that the increase is systematic across rows, confirming that the periodicity of chemical properties persists even in the asymptotic Z‑limit.

The study also evaluates the performance of the LDA. For s‑block elements, LDA reproduces the IP with chemical accuracy (errors below 0.1 eV) and, in several cases, appears to converge to the exact result. This surprising success is attributed to the increasingly uniform electron density in high‑Z atoms, where local approximations capture the dominant exchange‑correlation contributions. For d‑ and f‑block elements, LDA slightly underestimates the IP, but the inclusion of gradient corrections (GGA) restores agreement to within a few tenths of an electron‑volt.

To place the DFT results in a broader theoretical context, the authors compare them with extended Thomas‑Fermi (ETF) theory, which treats the electron gas as a smooth, semiclassical fluid. By averaging the DFT‑derived IPs over each electronic shell, they obtain shell‑averaged values that match the ETF predictions remarkably well. Moreover, the ETF model accurately reproduces the shell‑averaged change in electron density Δn(r) that occurs upon ionization, confirming that the semiclassical picture captures the essential physics of the average electron pressure and potential balance.

A complementary analytical treatment is provided for the weak‑interaction limit (λ → 0, where λ scales the electron‑electron repulsion). In this regime the electrons behave almost like a non‑interacting gas, allowing the authors to derive exact expressions for the IP, which reduce to simple functions of Z and N. This limiting case serves as a benchmark that validates both the numerical DFT calculations and the ETF approximations.

The authors also benchmark their computed IPs against available experimental data for heavy elements. Even for Z > 1000, the relative deviation remains below 1 %, demonstrating that large‑scale DFT, when combined with efficient algorithms and high‑performance computing resources, can deliver quantitatively reliable predictions for systems far beyond the reach of traditional quantum‑chemical methods.

Implications of the work are far‑reaching. The persistence of a finite, period‑dependent IP in the Z → ∞ limit suggests that the chemical periodicity is a robust feature of quantum mechanics, not an artifact of low‑Z chemistry. The demonstrated accuracy of simple local functionals in this regime opens the possibility of using inexpensive DFT models for modeling ultra‑dense matter, such as that found in white dwarfs, neutron‑star crusts, or high‑energy‑density laboratory plasmas. Additionally, the agreement with ETF provides a valuable semiclassical bridge that can be exploited for rapid estimates of bulk properties where full quantum calculations are prohibitive.

In summary, the paper establishes that ionization potentials remain finite and increase across a period even as atomic number becomes arbitrarily large, that the local density approximation can achieve chemical accuracy in certain high‑Z cases, and that extended Thomas‑Fermi theory accurately captures the shell‑averaged IP and density response. The combination of large‑scale DFT, semiclassical theory, and analytical weak‑interaction results offers a comprehensive framework for understanding electronic structure in the extreme limit of the periodic table.