A molecular density that no ordinary Kohn–Sham ensemble reproduces

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Family 278 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Density-functional theory replaces the full many-electron wavefunction with a simpler object: the electron density in ordinary three-dimensional space. Kohn–Sham theory then seeks a noninteracting system whose ground-state density matches that of the interacting electrons.

This raises an existence question. Given a density produced by a genuine Coulomb ground state, must some common local potential reproduce it using noninteracting electrons? Allowing mixtures of degenerate ground states makes the question more permissive than asking for one chosen orbital configuration.

Family 278 reports a counterexample: a finite three-electron molecule with two equal positive integer nuclear charges. Its spin-summed ground-state density is claimed to have no noninteracting ground-state ensemble representation in the potential class $L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3)$. The mechanism involves a small but persistent density tail on a symmetry plane and an obstruction applying to every possible ground-state ensemble.

2. Background and essential definitions

Density is less information than a wavefunction

A three-electron wavefunction depends on three positions and three spin labels and is antisymmetric under exchange of two electrons. Its spin-summed density $n_\Psi(r)$ is obtained by integrating the squared wavefunction over the other positions and summing spins. It satisfies

$$n_\Psi(r)\geq0, \qquad \int_{\mathbb R^3}n_\Psi(r)\,dr=3.$$

The density records the expected number of electrons per unit volume. It does not retain all correlations between their positions.

A noninteracting comparison system uses one scalar potential $v$ for every electron and both spin components:

$$H_s(v)=\sum_{i=1}^{3}\left(-\frac12\Delta_i+v(r_i)\right).$$

“Local” means that $v$ acts by multiplication at a position, rather than coupling different positions through an integral operator. The admissible real potential space is

$$\mathcal V=L^{3/2}(\mathbb R^3;\mathbb R)+L^\infty(\mathbb R^3;\mathbb R).$$

Thus $v=v_1+v_2$, where $\int|v_1|^{3/2}<\infty$ and $v_2$ is essentially bounded. This accommodates ordinary Coulomb singularities: near a nucleus $1/|r|$ belongs locally to $L^{3/2}$, while its far-away portion is bounded.

Representation by a ground-state ensemble

A ground-state ensemble is a positive trace-one operator $\Gamma$ supported in the eigenspace at the bottom of $H_s(v)$. It can mix several degenerate ground states. Its density $n_\Gamma$ is the corresponding average density.

The target property is $n_\Gamma=n_\Psi$ almost everywhere. All complex spinful states and arbitrary ground-space degeneracy are allowed. Merely finding orbitals whose squared magnitudes add to the desired density is insufficient: those orbitals must arise from ground states of one common local potential.

A Slater determinant is the antisymmetrized product of occupied spin-orbitals. Noninteracting ground states can be described through such occupations, but allowing ensembles introduces more freedom than a single determinant ansatz. The claimed counterexample must rule out that extra freedom too.

3. Development history and previous partial results

Hohenberg and Kohn established the density-functional viewpoint. Their density–potential uniqueness principle and existence of a noninteracting representing potential are separate questions. Kohn and Sham introduced the effective noninteracting construction that became central to electronic-structure calculations.

Levy’s and Lieb’s constrained-search formulations distinguish densities produced by admissible states from densities selected as ground states by a potential. Ullrich and Kohn emphasized how ground-state ensembles enlarge the latter class. Consequently failure of one guessed determinant does not establish failure of ensemble representation.

Earlier counterexamples could be specified directly as densities. Wagner and collaborators discuss a positive two-electron density with finite kinetic energy whose cusp forces an inverse potential outside the usual potential class. That construction does not itself make the density the ground-state density of a finite Coulomb molecule.

Trushin, Erhard and Görling reported numerical indications of noninteracting representability failures through atomic density inversion. Other work studied open-shell symmetry constraints. These are relevant predecessors, but the present manuscript claims an analytic exclusion for every ground-state ensemble of a specified interacting Coulomb system.

Positive results also depend on setting. Chayes, Chayes and Ruskai studied lattice systems. Corso treated spinless fermions on a bounded one-dimensional interval, while Sutter and collaborators obtained ensemble representation on a one-dimensional torus. Their hypotheses, including distributional potentials in the latter continuum settings, differ from three-dimensional Coulomb densities in $\mathcal V$.

The asymptotic idea has a particularly direct predecessor. Gori-Giorgi, Gál and Baerends analyzed how a nodal plane in a leading electron-removal amplitude can force unusual density tails and difficult Kohn–Sham potential behavior. Gori-Giorgi and Baerends later studied related effective density equations. The manuscript builds on that mechanism and adds a finite Coulomb construction, an all-ensemble kinetic certificate and an exclusion of the entire sum space $\mathcal V$.

4. What the new manuscript claims

The molecule has three electrons and two nuclei of equal charge $Z$, located at $\pm(D/Z)e_z$, where $e_z=(0,0,1)$ and the manuscript fixes $D=10^{200}$. Its electronic Hamiltonian is

$$\begin{aligned} H_C={}&\sum_{i=1}^{3}\left(-\frac12\Delta_i -\frac Z{|r_i-(D/Z)e_z|} -\frac Z{|r_i+(D/Z)e_z|}\right)\\ &+\sum_{1\leq i<j\leq3}\frac1{|r_i-r_j|}. \end{aligned}$$

The manuscript specifies a finite positive integer $Z$ by an exact nonnumerical formula in its equation (7.10). It uses auxiliary spectral data, densities and an angle minimization; it does not define the charge by searching over possible representing potentials. No effective numerical upper bound on $Z$ is asserted.

For this fixed finite system, the theorem claims a normalizable absolute ground state $\Psi$ in the sector $S_z=1/2$, and states that no $v\in\mathcal V$ and no ground-state ensemble $\Gamma$ of $H_s(v)$ satisfy $n_\Gamma=n_\Psi$ almost everywhere. The original state is an absolute ground state, not merely a state minimizing energy within an artificially selected spin sector.

A functional consequence concerns the noninteracting kinetic constrained search $T_{\mathrm{ens}}[n]$ and the interacting internal-energy search $F_{\mathrm{ens}}[n]$. Define

$$E_{\mathrm{Hxc}}[n]=F_{\mathrm{ens}}[n]-T_{\mathrm{ens}}[n].$$

This is the ensemble Hartree–exchange–correlation functional. At the constructed density, the manuscript excludes a local potential $w\in\mathcal V$ representing its one-sided first variation along every segment to every density in the common finite-energy domain. It also states that $T_{\mathrm{ens}}$ has no $\mathcal V$-valued subgradient there. A subgradient is a linear supporting functional lying below a convex functional at every competing density.

5. Why the advance matters

Importance: 5/5, provisional. The claimed example addresses an existence assumption underlying a central exact formulation of Kohn–Sham theory, while using a genuine finite Coulomb ground-state density and allowing all ground-state ensembles.

The distinction from a prescribed artificial density is significant. It would show that the representability obstruction can arise within the interacting molecular problem itself. The ensemble quantifier also prevents degeneracy and fractional occupations from automatically repairing the example.

This is an existence boundary, not a judgment on the practical success of approximate density-functional calculations. The molecule uses an exceptionally chosen charge and geometry, and the theorem excludes a specified potential class. The score concerns the conceptual significance if the claims hold, without independently assessing the proof.

6. The main idea

Make interaction weak after rescaling

Rescale positions by $y=Zr$ and divide energy by $Z^2$. The two nuclear charges become one, their half-separation becomes $D$, and electron repulsion has coefficient $\lambda=1/Z$.

The two lowest one-particle orbitals are an even positive orbital $b$ and an odd orbital $e$, whose sign changes across the median plane. At zero interaction the three electrons occupy $b$ with both spins and $e$ with one spin. The manuscript follows this state to small positive interaction.

An electron-removal amplitude describes what remains when one electron is separated from the two-electron ion. The leading amplitude has an odd angular zero on the median plane. Coupling to other ionic channels produces a smaller nonzero planar tail. Its exponential decay rate matches the leading off-plane rate, but it carries an additional inverse power of distance. The nonzero coefficient is essential: a vanished tail would not create the obstruction.

Certify the best noninteracting kinetic representation

For the scaled density $n$, the manuscript minimizes a weighted angle energy $\int n|\nabla\theta|^2$, with $\int n\sin^2\theta=1$ and an odd reflection condition on $\theta$. Set

$$g=\sqrt n\cos\theta, \qquad h=\sqrt n\sin\theta.$$

Their squared norms are two and one. They encode a doubly occupied spatial orbital and a singly occupied one. A weighted spectral estimate makes the resulting determinant a kinetic-energy minimum among all ensembles having density $n$.

That all-ensemble certificate is the bridge from one constructed pair of orbitals to every hypothetical representing potential. The manuscript concludes that any such local potential must equal $\Delta g/(2g)$, up to an additive constant, and that $g,h$ obey common potential equations with distinct orbital energies.

Compare the forced decay with the actual tail

Assume the forced potential belongs to $\mathcal V$. The manuscript translates bounded observation windows far out along the median plane and studies weak limits of the orbital equations. It derives an exterior energy bound forcing $g$ to decay exponentially faster than the density’s known leading rate.

That improved decay also controls the trace of $g$ on the plane. But oddness gives $\theta=0$ there, hence $g=\sqrt n$. The nonzero interaction-induced planar tail decays too slowly to satisfy the improved exponential integrability. This contradiction excludes $\mathcal V$.

The argument uses weak potential limits and a Sobolev trace, rather than assuming pointwise convergence of a candidate potential at infinity. That is why the stated obstruction covers the full integrable-plus-bounded class.

7. Scope, limitations, and open questions

The theorem concerns one common real, spin-independent local potential in $L^{3/2}+L^\infty$. It makes no assertion about larger distributional classes, spin-dependent potentials or nonlocal one-particle operators. Enlarging the comparison class changes the representability question.

The charge is finitely specified but not numerically bounded effectively by the theorem. No claim is made that this is a typical molecule or that its geometry is physically stable when nuclear motion is allowed. The result is for the stated clamped Coulomb Hamiltonian.

The functional corollary excludes a local first variation valid along every admissible segment. It should not be paraphrased as failure of every possible directional derivative or every weaker differentiability notion. Identifying broad positive representability classes and physically informative versions of this obstruction remains a separate task.

8. Sources and further reading

The foundational Hohenberg–Kohn and Kohn–Sham papers and the nodal-plane predecessor are linked above. For the constrained-search distinction, see Lieb, Density functionals for Coulomb systems. The source below supplies the finite molecule, ensemble certificate and potential-class exclusion, reported as stated.

Source manuscripts

• M1. OpenAI, A Coulomb ground-state density without Kohn-Sham ensemble representation (September 25, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.