Result 278, Mathematical physics

Failure of Kohn–Sham ensemble representation

Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L3/2(R3)+L∞(R3)L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

Disproof or counterexample

The bigger picture

Why it matters

Replacing interacting electrons with independent ones can simplify the description of a molecule. The manuscript reports a case where even a statistical mixture of independent-electron ground states cannot reproduce the true electron density under specified restrictions.

What changes?

The claimed counterexample is a three-electron Coulomb molecule in three-dimensional space with two equal positive-integer nuclear charges, given by an exact finite but nonnumerical formula. Its absolute ground-state density, with spin contributions added together, admits no such representation using a single real, spin-independent local potential. The excluded potentials are sums of a bounded function and a function whose absolute value to the three-halves power has finite integral. A local potential assigns an energy at each point in space.

What does that help mathematicians do?

An ensemble allows statistical mixing of ground states, rather than requiring one ground state to match the density. The reported obstruction therefore cannot be removed merely by allowing such mixtures. Researchers cannot assume that every Coulomb ground-state density has this noninteracting representation within the stated potential class. Any argument relying on that universal assumption would need additional restrictions or a different representation.

Are there practical applications?

The immediate value is foundational for Kohn-Sham approaches to electronic structure: it identifies a limit on exact density matching by independent-electron models. This does not establish that approximate calculations fail in general. The nonnumerical specification of the nuclear charge also leaves the source without an explicit numerical example for computational testing.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Coulomb ground-state density without Kohn-Sham ensemble representation

September 25, 2026 45 pages

We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in L3/2(R3)+L∞(R3)L^{3/2}(\mathbf R^3)+L^\infty(\mathbf R^3). The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026,
  author = {{OpenAI}},
  title = {{A Coulomb ground-state density without Kohn--Sham ensemble representation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf}{OAI:A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.