A practical functional that obeys the flat-plane condition
In plain words
The exact energy changes in straight lines when a fraction of an electron is added or when spin is fractionally shared, and approximate functionals bend those lines. Bending one way spreads electrons out too much; bending the other way mistreats stretched bonds.
Precise statement
Construct an exchange-correlation approximation, at cost no higher than a global hybrid, whose energy $E(N, S_z)$ is piecewise linear in fractional electron number N (no delocalization error) and constant in fractional spin at fixed N (no static correlation error), as tested on H2+ and $\mathrm{H_2}$ dissociation, fractional-charge atoms and the stretched-bond benchmark sets, while keeping main-group thermochemistry errors below 3 kcal/mol. An answer is such a functional, or a proof that no functional of a given class can satisfy both conditions.
What would settle it
A functional passing fractional-charge, fractional-spin and standard thermochemistry benchmarks simultaneously, or a no-go theorem for a defined functional class.
Status in the literature
In 2021 the DM21 learned functional, trained on fractional-charge and fractional-spin data, reduced both errors for small systems; no functional satisfies both conditions in general.
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