REFERENCES
2. Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133-8.
3. Langreth, D.; Perdew, J. The exchange-correlation energy of a metallic surface. Solid. State. Commun. 1975, 17, 1425-9.
4. Gunnarsson, O.; Lundqvist, B. I. Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B. 1976, 13, 4274-98.
5. Langreth, D.; Perdew, J. The gradient approximation to the exchange-correlation energy functional: A generalization that works. Solid. State. Commun. 1979, 31, 567-71.
6. Langreth, D. C.; Mehl, M. J. Beyond the local-density approximation in calculations of ground-state electronic properties. Phys. Rev. B. 1983, 28, 1809-34.
7. Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6062-5.
8. Levy, M.; Perdew, J. P. Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. Phys. Rev. A. 1985, 32, 2010-21.
9. Ceperley, D. M.; Alder, B. J. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 1980, 45, 566-9.
10. Vosko, S. H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 1980, 58, 1200-11.
11. Perdew, J. P.; Wang, Y. Erratum: Accurate and simple analytic representation of the electron-gas correlation energy [Phys. Rev. B 45, 13244 (1992)]. Phys. Rev. B. 2018, 98, 079904.
12. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865-8.
13. Sun, J.; Ruzsinszky, A.; Perdew, J. Strongly Constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 2015, 115, 036402.
14. Furness, J. W.; Kaplan, A. D.; Ning, J.; Perdew, J. P.; Sun, J. Accurate and numerically efficient r2SCAN meta-generalized gradient approximation. J. Phys. Chem. Lett. 2020, 11, 8208-15.
15. Kaplan, A. D.; Levy, M.; Perdew, J. P. The predictive power of exact constraints and appropriate norms in density functional theory. Annu. Rev. Phys. Chem. 2023, 74, 193-218.
16. Perdew, J. P.; Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B. 1981, 23, 5048-79.
17. Slater, JC. The self-consistent field for molecules and solids. In Quantum theory of molecules and solids, Vol. 4; McGraw-Hill, 1974; pp 35-55.
18. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Density-functional theory for fractional particle number: derivative discontinuities of the energy. Phys. Rev. Lett. 1982, 49, 1691-4.
19. Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648-52.
20. Ruzsinszky, A.; Nepal, N. K.; Pitarke, J. M.; Perdew, J. P. Constraint-based wave vector and frequency dependent exchange-correlation kernel of the uniform electron gas. Phys. Rev. B. 2020, 101, 245135.
21. Perdew, J. P.; Ruzsinszky, A.; Sun, J.; Nepal, N. K.; Kaplan, A. D. Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories. Proc. Natl. Acad. Sci. U.S.A. 2021, 118, e2017850118.
22. Anderson, P. W. More is different: broken symmetry and the nature of the hierarchical structure of science. Science. 1972, 177, 393-6.
23. Fu, Y.; Singh, D. J. Density functional methods for the magnetism of transition metals: SCAN in relation to other functionals. Phys. Rev. B. 2019, 100, 045126.
24. Lebeda, T.; Aschebrock, T.; Kümmel, S. Balancing the contributions to the gradient expansion: accurate binding and band gaps with a nonempirical meta-GGA. Phys. Rev. Lett. 2024, 133, 136402.
25. Pederson, M. R.; Ruzsinszky, A.; Perdew, J. P. Communication: Self-interaction correction with unitary invariance in density functional theory. J. Chem. Phys. 2014, 140, 121103.
26. Santra, B.; Perdew, J. P. Perdew-Zunger self-interaction correction: How wrong for uniform densities and large-Z atoms? J. Chem. Phys. 2019, 150, 174106.
27. Zope, R. R.; Yamamoto, Y.; Diaz, C. M.; et al. A step in the direction of resolving the paradox of Perdew-Zunger self-interaction correction. J. Chem. Phys. 2019, 151, 214108.
28. Shahi, C.; Maniar, R.; Ning, J.; et al. Local spin density approximation strongly improved by a better-informed local scaling of its self-interaction correction. J. Chem. Theory. Comput. 2026, 22, 5514-22.
29. Bhattarai, P.; Wagle, K.; Shahi, C.; et al. A step in the direction of resolving the paradox of Perdew–Zunger self-interaction correction. II. Gauge consistency of the energy density at three levels of approximation. J. Chem. Phys. 2020, 152, 214109.
30. Perdew, J. P.; Chowdhury, S. T. U. R.; Shahi, C.; Kaplan, A. D.; Song, D.; Bylaska, E. J. Symmetry breaking with the SCAN density functional describes strong correlation in the singlet carbon dimer. J. Phys. Chem. A. 2022, 127, 384-9.
31. Perdew, J. P. SCAN meta-GGA, strong correlation, symmetry breaking, self-interaction correction, and semi-classical limit in density functional theory: Hidden connections and beneficial synergies? APL. Computational. Physics. 2025, 1, 010903.
32. Zunger, A.; Xiong, J. X.; Perdew, J. P. Symmetry breaking transforms strong to normal correlation and false metals to true insulators. arXiv 2026. arXiv:2512.18236. Available online: https://doi.org/10.48550/arXiv.2512.18236 (accessed 17 August 17 2026).
33. Perdew, J. P. Symmetry breaking with a proper approximate density functional: exact ground-state energy as a close lower bound on the computed energy. ChemRxiv 2026. Available online: https://chemrxiv.org/doi/abs/10.26434/chemrxiv.15004601/v2 (accessed 17 August 17 2026).
34. Liu, Z.; Shang, S. Revealing symmetry-broken superconducting configurations by density functional theory. Supercond. Sci. Technol. 2025, 38, 075021.
35. Woods, C. H.; Li, Y.; Yao, W.; Li, C.; Perdew, J. P. Exactness of symmetry-broken self-interaction correction in the strongly-correlated or classical limit: harmonium as a demonstration. arXiv 2026. arXiv:2607.22488. Available online: https://doi.org/10.48550/arXiv.2607.22488 (accessed 17 August 17 2026).
36. Mermin, N. D. Thermal properties of the inhomogeneous electron gas. Phys. Rev. 1965, 137, A1441-3.
37. Runge, E.; Gross, E. K. U. Density-functional theory for time-dependent systems. Phys. Rev. Lett. 1984, 52, 997-1000.
38. Petersilka, M.; Gossmann, U. J.; Gross, E. K. U. Excitation energies from time-dependent density-functional theory. Phys. Rev. Lett. 1996, 76, 1212-5.
39. Yang, W.; Ayers, P. W. Foundation for the ∆SCF approach in density functional theory. arXiv 2024. arXiv:2403.04604. Available online: https://doi.org/10.48550/arXiv.2403.04604 (accessed 17 August 17 2026).
40. Linscheid, A.; Sanna, A.; Essenberger, F.; Gross, E. K. U. Ab initio theory of superconductivity in a magnetic field. I. Spin density functional theory for superconductors and Eliashberg equations. Phys. Rev. B. 2015, 92, 024505.
41. Liu, Z. K.; Hew, N. L. E.; Shang, S. L. Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters. Microstructures 2024, 4, 2024009.
42. Wang, S.; Shang, S.; Liu, Z.; Hao, W. ZENN: a thermodynamics-inspired computational framework for heterogeneous data-driven modeling. Proc. Natl. Acad. Sci. U.S.A. 2026, 123, e2511227122.
43. S. Liu, Z. Li, and Z.K. Liu, Thermodynamic GeoAI reveals regime dependent mechanisms in heterogeneous spatial systems. arXiv 2026; arXiv:2604.04339. Available online: https://doi.org/10.48550/arXiv.2604.04339 (accessed 17 August 17 2026).


