A density functional free of self-interaction that keeps thermochemical accuracy
In plain words
Common approximations let an electron feel a push from its own charge, which spreads electrons out too much and spoils predictions of reaction barriers and charge transfer. Corrections that remove this self-repulsion usually make ordinary bond energies worse, and no approximation yet fixes the first without hurting the second.
Precise statement
Construct an exchange-correlation functional $E_{\mathrm{xc}}[n]$ that is exact for every one-electron density ($E_{\mathrm{xc}}[n_1] = -J[n_1]$, with J the classical Coulomb self-energy), gives total energies $E(N)$ piecewise linear in fractional electron number N between integers, and retains main-group thermochemistry and barrier-height accuracy at least equal to the best hybrid functionals on a broad benchmark such as GMTKN55 (a set of 55 standard chemistry test databases). An answer is an explicit functional with these properties demonstrated numerically, or a proof that no functional of a given class (semilocal, hybrid, or orbital-dependent of fixed form) can satisfy them together.
What would settle it
An explicit functional passing the one-electron, fractional-charge linearity and benchmark-accuracy tests together, or a no-go theorem for a stated functional class.
Status in the literature
Perdew-Zunger-type self-interaction corrections fix the one-electron limit but degrade equilibrium thermochemistry; the machine-learned DM21 functional (Kirkpatrick et al., Science 2021, https://doi.org/10.1126/science.abj6511) imposes fractional-charge and fractional-spin constraints approximately while keeping hybrid-level accuracy, but it is not exact for one-electron densities and its transferability was disputed (Gerasimov et al., Science 2022 comment, https://doi.org/10.1126/science.abq3385).