Failure of Kohn–Sham ensemble representation

1 · The essentials

A common question in describing electrons is whether a simpler model can reproduce where they are found in a molecule. The manuscript constructs a molecule with three electrons and two nuclei carrying equal positive integer charges. It considers the electrons’ lowest-energy state and the resulting picture of how their charge is spread through space. The manuscript claims that this picture cannot be reproduced by electrons that do not interact with one another, even when several lowest-energy arrangements may be mixed. This failure concerns models governed by one shared local potential that treats both electron spin states alike and belongs to a specified mathematical class. The required nuclear charge is given by an exact finite formula, but not as a numerical value. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine trying to reproduce a group photograph using people who pose independently under one shared set of instructions, with permission to blend several photographs. The manuscript describes an electron distribution for which the corresponding independent-electron construction cannot match the original, even with such blending.

Why it matters

The result claimed in the manuscript rules out a universal representation statement for the specified class of potentials. Allowing mixtures of noninteracting lowest-energy states does not remove the obstruction in the constructed example. Its scope is an exact representation question, rather than a claim about the accuracy of approximate calculations.

2 · Going further

The precise question is whether an interacting Coulomb molecule’s spin-summed ground-state density can also be the density of a ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential. Here the allowed potentials belong to $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ . The repository summary and abstract describe a counterexample: a finite three-electron molecule with two equal positive integer nuclear charges whose absolute ground-state density has no such representation.

The construction is the central approach identified in the abstract: exhibit a particular Coulomb molecule rather than merely discuss possible obstacles to representation. Its nuclear charge is specified by an exact finite nonnumerical formula. The supplied text does not give the formula, the steps of the argument, or the mechanism that prevents representation, so those details cannot be explained from this source. Only the main manuscript is supplied; no companion manuscripts or relationships among them are described.

Useful notions

Electron density as a spatial distribution
Ground states as lowest-energy states
Interacting versus noninteracting electron models
Mixtures of quantum states
Local potentials and function classes

What was known before / what is new

Before

The source frames the issue as Kohn–Sham ensemble representability: whether a density from an interacting ground state can be reproduced by a noninteracting ground-state ensemble. It provides no account of earlier results, historical development, or previously established special cases.

New

The manuscript claims an explicit type of Coulomb counterexample with three electrons and two equal positive-integer-charge nuclei. The failure persists even when ground-state ensembles are allowed, provided the noninteracting model uses one real, spin-independent local potential in the stated class. The nuclear charge is specified exactly by a finite formula rather than by a numerical value.

Limits and assumptions

The claim concerns the spin-summed absolute ground-state density of the constructed finite three-electron Coulomb molecule. The excluded representations use a single real, spin-independent local potential in $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ ; the supplied text does not establish failure for potentials outside that class or for other kinds of representation. It does not supply a numerical nuclear charge, the construction details, or a list of remaining open problems. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Spin-summed density: The spatial electron density obtained by combining the contributions from electron spin states.
Ground state: A state of a system with the lowest possible energy.
Ground-state ensemble: A mixture of ground states used to describe a density.
Kohn–Sham ensemble representation: A reproduction of an interacting system’s density by a ground-state ensemble of noninteracting electrons in a shared local potential.
Spin-independent local potential: A potential depending on spatial position that acts the same way on both electron spin states.
$L^{3/2} (R^{3}) + L^{\infty} (R^{3})$: The class of potentials expressible as the sum of a function with integrable absolute value to the power $3/2$ and an essentially bounded function.

Keywords : Kohn–Sham representability · Coulomb molecule · Ground-state density

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ . This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

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

25/09/2026

To be verified

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 $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ . The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.

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