Failure of Kohn–Sham ensemble representation

The explanation

A helium atom has two electrons in one shared orbital. Its electron density, the expected number of electrons near each point, integrates to 2. Kohn–Sham theory pretends those electrons do not repel each other. It then looks for one local potential whose noninteracting ground state has exactly the same density. For two electrons sharing an orbital, this is easy: take the orbital to be the square root of half the density.

Kohn–Sham density functional theory makes this replacement for any moleculeSource 1. A local potential is a single energy landscape felt by every electron at each point. The method assumes such a potential exists. Allowing ensembles, mixtures of several tied ground states, makes more densities reachableSource 2. So failure of one simple orbital guess does not settle the questionSource 2. Explicit prescribed densities were already known to fail in this potential classSource 3. Numerical studies of atoms also hinted at failuresSource 4. Neither gave a rigorous counterexample from an actual Coulomb ground state.

The paper claims such a counterexampleSource 5. It is a molecule with three electrons and two nuclei of equal positive integer chargeSource 5. The ground-state density cannot be matched by any noninteracting ground-state ensemble in one real, spin-independent potentialSource 5. The potential class is L^(3/2) + L^∞, a standard space that contains Coulomb potentials. No limit is placed on how degenerate the representing ground space isSource 6.

The molecule is extreme. In scaled units the nuclei sit at distance D = 10^200 on either side of a centre plane. The two lowest one-electron orbitals are an even orbital b and an odd orbital e. Their energy splitting is below 100/D, here less than 10^−198. Two electrons occupy b with opposite spins and one occupies e. The odd orbital vanishes on the centre plane. Gori-Giorgi, Gál and Baerends studied what happens next. Coupling to other ionic channels can make the density decay at the same exponential rate on and off the planeSource 7. On the plane it gets an extra polynomial suppressionSource 7. The paper argues that this tail forces any representing potential to leave the allowed class.

The exact charge Z is given by a formula, not a number. The paper asserts no effective numerical bound for itSource 8. The density also blocks a local-potential first variation of the ensemble exchange-correlation functionalSource 9. The result says nothing about larger classes of potentials, such as spin-dependent or nonlocal onesSource 10.

How sure is this? No Lean formalization accompanies this paper. The claim rests on the paper's long analytic estimates. No specialist review has been published. The example is a single, highly stretched molecule. It shows a limit in principle, not a failure of everyday chemistry calculations.

What it doesn't show. This does not rule out distributional, spin-dependent or nonlocal potentials, and it does not say Kohn–Sham calculations fail for typical molecules.