The strictly correlated electrons (SCE) functional — the strong-interaction limit of the Hohenberg–Kohn functional, equivalent to a multimarginal optimal transport problem with Coulomb cost — captures static correlation exactly and dissociates molecules without symmetry breaking, but it over-correlates at physical coupling strength. Malet, Mirtschink, Giesbertz, Wagner and Gori-Giorgi (Phys. Chem. Chem. Phys. 16, 14551, 2014) left open how to turn this limit into a practical Kohn–Sham DFT exchange–correlation strategy: (i) an exact or systematically improvable evaluation of the co-motion functions and the SCE functional for general three-dimensional systems; (ii) approximate non-local functionals inspired by the SCE structure that work for real molecules and materials; (iii) corrections that are size consistent for arbitrary fragmentation, not only equal-fragment dissociation; and (iv) an interpolation between the weak- and strong-coupling limits robust from low-density (nanodevice) to chemically relevant regimes. A decade of follow-up work — optimal-transport numerics, local adiabatic-connection interpolation, the multiple radii functional, and ISI/ACII interpolation families — has built machinery for each strand, but as of August 2026 none is closed: exact general-3D SCE evaluation remains limited to roughly ten electrons, size-consistent local interpolations remain limited in accuracy, and the field's own 2024 statement is that a broadly applicable interpolation approach is still missing. The open core is the integrated objective: a computable, size-consistent, SCE-anchored exchange–correlation treatment accurate across both strong and weak correlation.

Background

Kohn–Sham density functional theory (KS DFT) computes ground-state energies and densities of interacting electrons from self-consistent single-particle equations; all many-body effects are packed into the exchange–correlation (XC) functional Exc[ρ]E_{xc}[\rho], which must be approximated as a functional of the electron density ρ\rho. A central failure mode of mainstream approximations (the local density and generalized-gradient approximations, and hybrid functionals) is strong, or static, correlation — stretched bonds, Wigner localization in low-density quantum dots, near-degenerate transition-metal states — where they either break spin or space symmetry or give qualitatively wrong energies.

The adiabatic connection (AC) rewrites ExcE_{xc} as an integral over a coupling constant λ\lambda that scales the electron–electron repulsion, Exc[ρ]=∫01Wλ[ρ] dλE_{xc}[\rho]=\int_0^1 W_\lambda[\rho]\,d\lambda, where Wλ[ρ]W_\lambda[\rho] is the interelectronic repulsion expectation value in the ground state of the λ\lambda-scaled Hamiltonian at fixed density. Conventional functionals model WλW_\lambda near weak coupling (λ≈0\lambda\approx 0). The opposite, strong-interaction limit (λ→∞\lambda\to\infty, the SCE limit) is also exactly defined: electrons become strictly correlated, and the leading term W∞[ρ]W_\infty[\rho] is the minimum of the classical repulsion over co-motion functions fi(r)f_i(\mathbf r) — maps that carry an electron at r\mathbf r to the positions of the other electrons while preserving the density. Seidl (1999) proposed this limit as a principle for constructing functionals, and Seidl, Gori-Giorgi and Savin (2007) gave the general formulation. Buttazzo, De Pascale and Gori-Giorgi (2012) and Cotar, Friesecke and Klüppelberg (2013) reformulated W∞W_\infty as a multimarginal optimal transport (OT) problem with Coulomb cost — an optimization over coupled electron positions that made SCE a rigorous and active interface between DFT, the calculus of variations, and numerical mathematics. The next-leading-order term in the large-λ\lambda expansion describes zero-point oscillations (ZPE) of the electrons around the rigid SCE configuration (Gori-Giorgi, Vignale and Seidl, 2009).

Malet, Mirtschink, Giesbertz, Wagner and Gori-Giorgi (2014) — the source of this problem — applied KS SCE to one-dimensional model chemical systems. Their findings define the question. The SCE functional dissociates the stretched H2_2 molecule essentially exactly, without symmetry breaking, and the SCE potential has the correct long-range asymptotics, so the KS HOMO eigenvalue gives good ionization energies; but chemical systems are far from the strong-interaction limit, so bare SCE total energies are far too low. A correction based on the next leading order, renormalized with exact exchange, largely repairs the total energies, and a simplified interaction-strength interpolation (ISI) between the weak- and strong-coupling limits also helps — but, as the authors show, that global ISI correction is size consistent (it reproduces E(A+B)=E(A)+E(B)E(A+B)=E(A)+E(B) for separated, non-overlapping fragments) only when the system dissociates into equal fragments, while full size consistency requires a local interpolation along the AC built from position-resolved energy densities (Mirtschink, Seidl and Gori-Giorgi, 2012). Size consistency is non-negotiable for dissociation, clusters, and thermochemistry, because a size-inconsistent functional accumulates errors that grow with system size.

What remains unresolved is the integrated objective these findings point to: an SCE-anchored XC strategy that is simultaneously computable for general three-dimensional systems (an exact or systematically improvable evaluation of the co-motion functions and the SCE functional, or a controlled surrogate), size consistent for arbitrary fragmentation, and accurate both in the strongly correlated, low-density regime where SCE is asymptotically right and in the weakly to moderately correlated regime where standard functionals excel.

Problem Statement

Construct an exchange–correlation treatment for Kohn–Sham DFT anchored in the strictly correlated electrons (strong-interaction, λ→∞\lambda\to\infty) limit that jointly satisfies the following three properties, each for the general case it names.

  1. Computability: provide an exact or systematically improvable evaluation of the SCE functional — equivalently, the co-motion functions, or the multimarginal optimal transport problem with Coulomb cost — for general three-dimensional electron densities of many-electron systems, not only spherically symmetric or effectively one-dimensional ones; or provide a surrogate whose approximation to W∞[ρ]W_\infty[\rho] and to the next-leading-order (zero-point-energy) term is controlled and systematically refinable, in the sense that its approximation error is under explicit control — a stated mechanism or parameter by which it can be reduced systematically — rather than accuracy demonstrated only on particular examples.

  2. Size consistency: the resulting functional, including the interpolation or correction between the weak- and strong-coupling limits, must be size consistent for arbitrary fragmentation of the system, not only for dissociation into equal fragments.

  3. Robustness across regimes: the same functional must remain accurate in the strongly correlated, low-density regime where SCE is asymptotically correct (for example, stretched and dissociating heteronuclear bonds, Wigner-localized low-density systems) while matching the accuracy of the best available approximations for standard weakly to moderately correlated chemistry (for example, main-group energetics) — that is, a weak-to-strong coupling interpolation that does not sacrifice one regime for the other.

References

  1. Francesc Malet, André Mirtschink, Klaas J. H. Giesbertz, Lucas O. Wagner, Paola Gori-Giorgi, 'Exchange–correlation functionals from the strong interaction limit of DFT: applications to model chemical systems', Phys. Chem. Chem. Phys. 16, 14551–14558 (2014). DOI: 10.1039/c4cp00407h, https://doi.org/10.1039/c4cp00407h (preprint: arXiv:1401.7822, https://arxiv.org/abs/1401.7822)
  2. Michael Seidl, 'Strong-interaction limit of density-functional theory', Phys. Rev. A 60, 4387–4395 (1999). DOI: 10.1103/PhysRevA.60.4387, https://doi.org/10.1103/PhysRevA.60.4387
  3. Michael Seidl, Paola Gori-Giorgi, Andreas Savin, 'Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities', Phys. Rev. A 75, 042511 (2007). DOI: 10.1103/PhysRevA.75.042511, https://doi.org/10.1103/PhysRevA.75.042511
  4. Paola Gori-Giorgi, Giovanni Vignale, Michael Seidl, 'Electronic Zero-Point Oscillations in the Strong-Interaction Limit of Density Functional Theory', J. Chem. Theory Comput. 5, 743–753 (2009). DOI: 10.1021/ct8005248, https://doi.org/10.1021/ct8005248
  5. André Mirtschink, Michael Seidl, Paola Gori-Giorgi, 'Energy Densities in the Strong-Interaction Limit of Density Functional Theory', J. Chem. Theory Comput. 8, 3097–3107 (2012). DOI: 10.1021/ct3003892, https://doi.org/10.1021/ct3003892
  6. Giuseppe Buttazzo, Luigi De Pascale, Paola Gori-Giorgi, 'Optimal-transport formulation of electronic density-functional theory', Phys. Rev. A 85, 062502 (2012). DOI: 10.1103/PhysRevA.85.062502, https://doi.org/10.1103/PhysRevA.85.062502
  7. Codina Cotar, Gero Friesecke, Claudia Klüppelberg, 'Density Functional Theory and Optimal Transportation with Coulomb Cost', Commun. Pure Appl. Math. 66, 548–599 (2013). DOI: 10.1002/cpa.21437, https://doi.org/10.1002/cpa.21437
  8. Huajie Chen, Gero Friesecke, Christian B. Mendl, 'Numerical Methods for a Kohn–Sham Density Functional Model Based on Optimal Transport', J. Chem. Theory Comput. 10, 4360–4368 (2014). DOI: 10.1021/ct500586q, https://doi.org/10.1021/ct500586q
  9. Stefan Vuckovic, Lucas O. Wagner, André Mirtschink, Paola Gori-Giorgi, 'Hydrogen Molecule Dissociation Curve with Functionals Based on the Strictly Correlated Regime', J. Chem. Theory Comput. 11, 3153–3162 (2015). DOI: 10.1021/acs.jctc.5b00387, https://doi.org/10.1021/acs.jctc.5b00387
  10. Yongxi Zhou, Hilke Bahmann, Matthias Ernzerhof, 'Construction of exchange-correlation functionals through interpolation between the non-interacting and the strong-correlation limit', J. Chem. Phys. 143, 124103 (2015). DOI: 10.1063/1.4931160, https://doi.org/10.1063/1.4931160
  11. Stefan Vuckovic, Tom J. P. Irons, Andreas Savin, Andrew M. Teale, Paola Gori-Giorgi, 'Exchange–Correlation Functionals via Local Interpolation along the Adiabatic Connection', J. Chem. Theory Comput. 12, 2598–2610 (2016). DOI: 10.1021/acs.jctc.6b00177, https://doi.org/10.1021/acs.jctc.6b00177
  12. Eduardo Fabiano, Paola Gori-Giorgi, Michael Seidl, Fabio Della Sala, 'Interaction-Strength Interpolation Method for Main-Group Chemistry: Benchmarking, Limitations, and Perspectives', J. Chem. Theory Comput. 12, 4885–4896 (2016). DOI: 10.1021/acs.jctc.6b00713, https://doi.org/10.1021/acs.jctc.6b00713
  13. Stefan Vuckovic, Paola Gori-Giorgi, 'Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy', J. Phys. Chem. Lett. 8, 2799–2805 (2017). DOI: 10.1021/acs.jpclett.7b01113, https://doi.org/10.1021/acs.jpclett.7b01113
  14. Derk P. Kooi, Paola Gori-Giorgi, 'Local and global interpolations along the adiabatic connection of DFT: a study at different correlation regimes', Theor. Chem. Acc. 137, 166 (2018). DOI: 10.1007/s00214-018-2354-5, https://doi.org/10.1007/s00214-018-2354-5
  15. Tim Gould, Stefan Vuckovic, 'Range-separation and the multiple radii functional approximation inspired by the strongly interacting limit of density functional theory', J. Chem. Phys. 151, 184101 (2019). DOI: 10.1063/1.5125692, https://doi.org/10.1063/1.5125692
  16. Yuehaw Khoo, Lin Lin, Michael Lindsey, Lexing Ying, 'Semidefinite Relaxation of Multimarginal Optimal Transport for Strictly Correlated Electrons in Second Quantization', SIAM J. Sci. Comput. 42, B1462–B1489 (2020). DOI: 10.1137/20M1310977, https://doi.org/10.1137/20M1310977
  17. Gero Friesecke, Andreas S. Schulz, Daniela Vögler, 'Genetic Column Generation: Fast Computation of High-Dimensional Multimarginal Optimal Transport Problems', SIAM J. Sci. Comput. 44, A1632–A1654 (2022). DOI: 10.1137/21M140732X, https://doi.org/10.1137/21M140732X
  18. Gero Friesecke, Augusto Gerolin, Paola Gori-Giorgi, 'The Strong-Interaction Limit of Density Functional Theory', in 'Density Functional Theory' (E. Cancès, G. Friesecke, eds.), Springer, pp. 183–266 (2022). DOI: 10.1007/978-3-031-22340-2_4, https://doi.org/10.1007/978-3-031-22340-2_4 (preprint: arXiv:2202.09760, https://arxiv.org/abs/2202.09760)
  19. Stefan Vuckovic, Augusto Gerolin, Kimberly J. Daas, Hilke Bahmann, Gero Friesecke, Paola Gori-Giorgi, 'Density functionals based on the mathematical structure of the strong-interaction limit of DFT', WIREs Comput. Mol. Sci. 13, e1634 (2022). DOI: 10.1002/wcms.1634, https://doi.org/10.1002/wcms.1634 (preprint: arXiv:2204.10769, https://arxiv.org/abs/2204.10769; the preprint byline uses the author's earlier arXiv name 'Timothy J. Daas')
  20. Lucian A. Constantin, Subrata Jana, Szymon Śmiga, Fabio Della Sala, 'Adiabatic connection interaction strength interpolation method made accurate for the uniform electron gas', J. Chem. Phys. 159, 244111 (2023). DOI: 10.1063/5.0178800, https://doi.org/10.1063/5.0178800
  21. Stefan Vuckovic, Hilke Bahmann, 'Nonlocal Functionals Inspired by the Strongly Interacting Limit of DFT: Exact Constraints and Implementation', J. Chem. Theory Comput. 19, 6172–6184 (2023). DOI: 10.1021/acs.jctc.3c00437, https://doi.org/10.1021/acs.jctc.3c00437
  22. Lucian A. Constantin, Szymon Śmiga, Fabio Della Sala, 'Towards adiabatic-connection interpolation model with broader applicability', Phys. Rev. B 109, 235129 (2024). DOI: 10.1103/PhysRevB.109.235129, https://doi.org/10.1103/PhysRevB.109.235129 (preprint: arXiv:2403.07391, https://arxiv.org/abs/2403.07391)
  23. Giulia Luise, Chin-Wei Huang, Thijs Vogels, Derk P. Kooi, Sebastian Ehlert, Stephanie Lanius, Klaas J. H. Giesbertz, Amir Karton, Deniz Gunceler, Stefano Battaglia, Gregor N. C. Simm, P. Bernát Szabó, Megan Stanley, Wessel P. Bruinsma, Lin Huang, Xinran Wei, José Garrido Torres, Abylay Katbashev, Rodrigo Chavez Zavaleta, Bálint Máté, Sékou-Oumar Kaba, Roberto Sordillo, Yingrong Chen, David B. Williams-Young, Christopher M. Bishop, Jan Hermann, Rianne van den Berg, Paola Gori-Giorgi, 'Accurate and scalable exchange-correlation with deep learning' (Skala functional), arXiv:2506.14665 (2025), https://arxiv.org/abs/2506.14665
  24. Lucian A. Constantin, Faisal Naeem, Eduardo Fabiano, Fulvio Sarcinella, Fabio Della Sala, 'Restoring the Point-and-Charge Gradient Expansion for the Strong Interaction Density Functionals', Phys. Rev. B 113, 085121 (2026). DOI: 10.1103/76sy-8bz4, https://doi.org/10.1103/76sy-8bz4 (preprint: arXiv:2603.06018, https://arxiv.org/abs/2603.06018)
  25. Thiago Carvalho Corso, 'Strictly correlated electrons in a quantum ring: from Kohn-Sham to Kantorovich potentials', arXiv:2604.09908 (2026), https://arxiv.org/abs/2604.09908