A transferable exchange–correlation functional free of delocalization and self-interaction error
Difficulty 72: the exact functional is QMA-hard and decades of attempts have not removed delocalization error without tuning, though learned functionals (DM21, the Skala preprint) show a path for restricted domains; this sits above the 50 to 60 band of engineering-limited problems such as the Matbench-style MLIP gaps and well below the defining problems of mathematics. Application value 78: DFT labels underpin catalysis, battery and semiconductor screening and most ML-potential training sets, comparable to the surface-code problem (70) but broader. Leverage 78: every DFT-derived dataset and downstream model inherits its errors.
The problem
Nobody has a single, fast energy recipe for electrons that is free of the known systematic mistakes of density functional theory.
- 1Exact energy recipe unknown
- 2Approximations smear charge, mis-handle ties
- 3Exact rules can guide the design
- 4Learned recipes show promise
- 5Need metals, solids and independent tests
What is asked. An exchange–correlation functional that (1) obeys exact constraints, (2) reaches near-chemical accuracy across broad benchmarks and (3) costs no more than a hybrid functional.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Neural-network wavefunctions at chemical accuracy for 100+ electron molecules with transferable, amortised trainingComputational Chemistry · D54
- Step 2
- Step 3
- A transferable exchange–correlation functional free of delocalization and self-interaction errorThis problem
- Then
- Crystal structure prediction for flexible organic molecules: complete landscapes with sub-kJ/mol free-energy rankingComputational Chemistry · D56
- Absolute homogeneous nucleation rates from simulation within an order of magnitude of experimentComputational Chemistry · D62
- Universal ML potentials accurate on bond breaking, transition states and high-energy configurations
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Then
- A transferable exchange–correlation functional free of delocalization and self-interaction errorThe problem
State of the art
- Hand-designed functionals. On the GMTKN55 benchmark the WTMAD-2 error falls steadily up the rungs: the best range-separated hybrids and meta-GGAs (for example B97M-V) sit a few kcal/mol above the best double hybrids, which are the most accurate general-purpose functionals tested (Goerigk et al. 2017; Mardirossian and Head-Gordon 2017; exact WTMAD-2 values should be read from the primary tables, which are periodically extended). None removes delocalization error without system-dependent tuning: optimally tuned range-separated hybrids and Koopmans-compliant or localized orbital scaling corrections fix the fractional-charge curve for a given system, not universally (Cohen, Mori-Sánchez and Yang 2008; Medvedev et al. 2017 argue that recent functionals fit energies while drifting from the exact density).
- Learned functionals. DM21 (DeepMind, Science 2021) trained a neural local-hybrid functional on molecular data plus fractional-charge and fractional-spin constraints and reduced delocalization error on small systems, but is not size-consistent in all tests and was not built for periodic systems or transition metals.
- Skala (preprint). Microsoft Research (arXiv:2506.14665, June 2025, preprint; later versions revised) reports a deep-learned functional with semi-local cost that reaches chemical accuracy (about 1 kcal/mol) on atomization energies of small main-group molecules, trained on a large new set of wavefunction reference data, and that is competitive with hybrids on general main-group chemistry after extra data. The reported domain is main-group, closed-shell thermochemistry; performance on transition-metal chemistry, strong correlation, fractional-charge curves, charged and periodic systems has not been established in independent benchmarks that this entry has seen.
- Exact-condition status. No published functional is known to satisfy piecewise linearity, many-electron self-interaction freedom and the fractional-spin condition simultaneously while remaining accurate on GMTKN55.
Realistic applications
DFT is the workhorse for catalysis screening, battery and semiconductor materials, drug-target modelling and the labels used to train machine-learned potentials. A functional without delocalization error would improve barrier heights, charge-transfer and polaron energetics, band alignments and redox potentials, and would reduce the systematic errors that machine-learned potentials inherit from their DFT labels.
What counts as solved
A published functional (with code in at least two independent electronic-structure packages) whose exact-condition tests (fractional-charge and fractional-spin curves for a stated test set of at least 20 atoms and molecules) are satisfied to within 1 kcal/mol, and which meets the accuracy targets above on GMTKN55, a transition-metal benchmark such as MOR41 or SSE17, and a periodic benchmark set, as reproduced by independent groups on data not used in training or tuning.
How it gets verified
ReproductionA training run, experiment or computation rerun by someone else.
A published functional with code in at least two independent electronic-structure packages must satisfy fractional-charge and fractional-spin exact-condition tests to 1 kcal/mol on a stated test set of at least 20 atoms and molecules, and meet the accuracy targets on GMTKN55 and the transition-metal benchmarks in the pack. Anyone can rerun the benchmarks with the released code and basis sets.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Mostly new functional forms and exact-condition theory, with large reference datasets and training compute for learned functionals.
Bottlenecks
ranked by votes- #1TheorySatisfying exact conditions with a usable functional
Piecewise linearity, self-interaction freedom and the fractional-spin condition conflict in all known semi-local and hybrid forms, and the universal functional is QMA-hard, so only approximations are possible.
Community grades
Sign in to add your own grade.
Connected problems
Progress
- Jun 17, 2025Skala preprintsource
Deep-learned semi-local-cost functional reporting chemical accuracy on main-group atomization energies (preprint).
- Dec 10, 2021DM21source
A neural functional trained with fractional-charge and fractional-spin constraints.
- Jan 6, 2017Functionals drifting from the exact densitysource
Medvedev et al. show that many recent functionals improve energies while worsening densities.
- Oct 11, 2009Exact functional is QMA-hardsource
Schuch and Verstraete prove computing the universal functional is QMA-hard (date approximate).
- Jul 25, 2008Delocalization and static-correlation errors named