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Validity of DFT+U band gaps in all its known functional forms

Andrew C. Burgess1,2 and David D. O'Regan2,*

  • *Contact author: david.o.regan@tcd.ie
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Phys. Rev. B 114, 235127 – Published 26 October, 2026

DOI: https://doi.org/10.1103/9snx-cp83

Abstract

The density functional theory plus Hubbard U (DFT+U) technique is one of the most widely used tools by condensed matter physicists and solid-state chemists for the simulation of transition-metal and lanthanide-bearing crystals, and increasingly of much more diverse chemistries. Although often synonymous with the corrective functionals of Dudarev et al. [Phys. Rev. B 57, 1505 (1998)] and Liechtenstein et al. [Phys. Rev. B 52, R5467 (1995)], there exists a wide variety of DFT+U-type functionals ready to be utilized, and no doubt yet to be developed. Since the earliest days, the gap in the DFT+U single-particle eigenspectrum has been associated with the fundamental band gap, and the method has typically found more success for spectra than for total-energy derived properties. There has been some doubt, however, as to the conceptual validity of this association. Here, extending findings from recent years regarding local, semilocal, and hybrid functionals within the generalized Kohn-Sham framework, we prove and numerically demonstrate that the DFT+U eigenspectrum gap is conceptually valid, in the specific sense that it matches its own fundamental gap calculated using total-energy differences. We emphasize that this does not imply its agreement with experimental values, and indeed our argument is independent of the Hubbard U parameter. The result holds for pristine periodic systems with converged k-point sampling but not, however, for defective ones, isolated systems, or systems in which added charges exhibit spontaneous localization. We show that band-gap validity for pristine solids holds in the presence of pseudopotentials and PAW potentials, when using hybrid functionals, and in DFT+U(+J) irrespective of the level of subspace projection onto the band-edge states. We survey every collinear-spin DFT+U-type functional known to have been published to date, within a unified notation. We verify analytically under which conditions the eigenvalue gap equals its fundamental gap, for each functional. Returning to the related but different question of band-gap correction efficiency, we offer fresh analysis of DFT+U band-gap projection dependence, and each functional's effect on energies and gaps for the hydrogen lattice in the Mott-Hubbard limit.

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References (295)

  1. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  2. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  3. A. M. Teale, T. Helgaker, A. Savin, C. Adamo, B. Aradi, A. V. Arbuznikov, P. W. Ayers, E. J. Baerends, V. Barone, P. Calaminici, E. Cancès, E. A. Carter, P. K. Chattaraj, H. Chermette, I. Ciofini, T. Daniel Crawford, F. De Proft, J. F. Dobson, C. Draxl, T. Frauenheim, et al., DFT exchange: Sharing perspectives on the workhorse of quantum chemistry and materials science, Phys. Chem. Chem. Phys. 24, 28700 (2022).
  4. J. P. Perdew and A. Ruzsinszky, Fourteen easy lessons in density functional theory, Int. J. Quantum Chem. 110, 2801 (2010).
  5. R. Van Noorden, B. Maher, and R. Nuzzo, The top 100 papers, Nat. News 514, 550 (2014).
  6. R. Haunschild, A. Barth, and B. French, A comprehensive analysis of the history of DFT based on the bibliometric method RPYS, J. Cheminform. 11, 72 (2019).
  7. K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. Di Marco, C. Draxl, M. Dułak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, et al., Reproducibility in density functional theory calculations of solids, Science 351, aad3000 (2016).
  8. E. Dagotto, Complexity in strongly correlated electronic systems, Science 309, 257 (2005).
  9. S. Curtarolo, G. L. W. Hart, M. B. Nardelli, N. Mingo, S. Sanvito, and O. Levy, The high-throughput highway to computational materials design, Nat. Mater. 12, 191 (2013).
  10. J. K. Nørskov, T. Bligaard, J. Rossmeisl, and C. H. Christensen, Towards the computational design of solid catalysts, Nat. Chem. 1, 37 (2009).
  11. G. Goldbeck, The economic impact of molecular modelling, Tech. Rep. (Goldbeck Consulting Limited, 2012).
  12. G. Goldbeck, Industry interactions of the electronic structure research community in europe, Tech. Rep. (Goldbeck Consulting Limited, 2015).
  13. R. O. Jones, Density functional theory: Its origins, rise to prominence, and future, Rev. Mod. Phys. 87, 897 (2015).
  14. M. J. G. Peach, P. Benfield, T. Helgaker, and D. J. Tozer, Excitation energies in density functional theory: An evaluation and a diagnostic test, J. Chem. Phys. 128, 044118 (2008).
  15. S. Sharma, J. K. Dewhurst, N. N. Lathiotakis, and E. K. U. Gross, Reduced density matrix functional for many-electron systems, Phys. Rev. B 78, 201103(R) (2008).
  16. P. Ziesche, Pair density functional theory—A generalized density functional theory, Phys. Lett. A 195, 213 (1994).
  17. P. Geerlings, S. Fias, Z. Boisdenghien, and F. De Proft, Conceptual DFT: Chemistry from the linear response function, Chem. Soc. Rev. 43, 4989 (2014).
  18. L. Hedin, New method for calculating the one-particle Green's function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965).
  19. G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body Green's-function approaches, Rev. Mod. Phys. 74, 601 (2002).
  20. C. A. Taft and J. G. S. Canchaya, Review on simulation models for materials and biomolecular study and design, in Recent Advances in Complex Functional Materials: From Design to Application (Springer International Publishing, Cham, 2017), pp. 373–408.
  21. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006).
  22. The term ‘‘implicit’’ here is used to mean that while the potential operator is orbital index independent (there is only a single potential), its action upon each orbital may be, and typically will be, different. Functionals in this restricted class are functionals of the GKS density-matrix sufficiently, i.e., they don't use the full variational freedom made available by the orbitals.
  23. A. Seidl, A. Görling, P. Vogl, J. A. Majewski, and M. Levy, Generalized Kohn-Sham schemes and the band-gap problem, Phys. Rev. B 53, 3764 (1996).
  24. R. Garrick, A. Natan, T. Gould, and L. Kronik, Exact generalized Kohn-Sham theory for hybrid functionals, Phys. Rev. X 10, 021040 (2020).
  25. K. Nakamura, Y. Nohara, Y. Yosimoto, and Y. Nomura, Ab initio GW plus cumulant calculation for isolated band systems: Application to organic conductor (TMTSF)2PF6 and transition-metal oxide SrVO3, Phys. Rev. B 93, 085124 (2016).
  26. F. Aryasetiawan and O. Gunnarsson, The GW method, Rep. Prog. Phys. 61, 237 (1998).
  27. T. Kotani, M. van Schilfgaarde, and S. V. Faleev, Quasiparticle self-consistent GW method: A basis for the independent-particle approximation, Phys. Rev. B 76, 165106 (2007).
  28. S. Biermann and A. van Roekeghem, Electronic polarons, cumulants and doubly dynamical mean field theory: Theoretical spectroscopy for correlated and less correlated materials, J. Electron. Spectrosc. Relat. Phenom. 208, 17 (2016).
  29. M. van Schilfgaarde, T. Kotani, and S. Faleev, Quasiparticle self-consistent GW theory, Phys. Rev. Lett. 96, 226402 (2006).
  30. D. Vollhardt and A. I. Lichtenstein, Dynamical mean-field approach with predictive power for strongly correlated materials, Eur. Phys. J. Spec. Top. 226, 2439 (2017).
  31. K. Held, I. A. Nekrasov, G. Keller, V. Eyert, N. Blümer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, Realistic investigations of correlated electron systems with LDA+DMFT, Physica Status Solidi (b) 243, 2599 (2006).
  32. X. Deng, J. Mravlje, R. Žitko, M. Ferrero, G. Kotliar, and A. Georges, How bad metals turn good: Spectroscopic signatures of resilient quasiparticles, Phys. Rev. Lett. 110, 086401 (2013).
  33. A. Georges, L. de' Medici, and J. Mravlje, Strong correlations from Hund's coupling, Annu. Rev. Condens. Matter Phys. 4, 137 (2013).
  34. S. Biermann, F. Aryasetiawan, and A. Georges, First-principles approach to the electronic structure of strongly correlated systems: Combining the GW approximation and dynamical mean-field theory, Phys. Rev. Lett. 90, 086402 (2003).
  35. J. M. Tomczak, M. Casula, T. Miyake, F. Aryasetiawan, and S. Biermann, Combined GW and dynamical mean-field theory: Dynamical screening effects in transition metal oxides, Europhys. Lett. 100, 67001 (2012).
  36. C. Taranto, M. Kaltak, N. Parragh, G. Sangiovanni, G. Kresse, A. Toschi, and K. Held, Comparing quasiparticle GW+DMFT and LDA+DMFT for the test bed material SrVO3, Phys. Rev. B 88, 165119 (2013).
  37. G. Knizia and G. K.-L. Chan, Density matrix embedding: A simple alternative to dynamical mean-field theory, Phys. Rev. Lett. 109, 186404 (2012).
  38. S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis, Can. J. Phys. 58, 1200 (1980).
  39. U. von Barth and L. Hedin, A local exchange-correlation potential for the spin polarized case. I, J. Phys. C: Solid State Phys. 5, 1629 (1972).
  40. J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B 45, 13244 (1992).
  41. Perdew J. P., K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  42. C. Lee, W. Yang, and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Phys. Rev. B 37, 785 (1988).
  43. J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E.Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett. 100, 136406 (2008).
  44. A. D. Becke, Density-functional exchange-energy approximation with correct asymptotic behavior, Phys. Rev. A 38, 3098 (1988).
  45. I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari, and M. Cococcioni, Koopmans' condition for density-functional theory, Phys. Rev. B 82, 115121 (2010).
  46. G. Borghi, A. Ferretti, N. L. Nguyen, I. Dabo, and N. Marzari, Koopmans-compliant functionals and their performance against reference molecular data, Phys. Rev. B 90, 075135 (2014).
  47. N. Colonna, R. De Gennaro, E. Linscott, and N. Marzari, Koopmans spectral functionals in periodic boundary conditions, J. Chem. Theory Comput. 18, 5435 (2022).
  48. E. B. Linscott, N. Colonna, R. De Gennaro, N. L. Nguyen, G. Borghi, A. Ferretti, I. Dabo, and N. Marzari, Koopmans: An open-source package for accurately and efficiently predicting spectral properties with koopmans functionals, J. Chem. Theory Comput. 19, 7097 (2023).
  49. J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23, 5048 (1981).
  50. M. R. Pederson, A. Ruzsinszky, and J. P. Perdew, Communication: Self-interaction correction with unitary invariance in density functional theory, J. Chem. Phys. 140, 121103 (2014).
  51. S. Grimme, Semiempirical GGA-type density functional constructed with a long-range dispersion correction, J. Comput. Chem. 27, 1787 (2006).
  52. S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132, 154104 (2010).
  53. A. Tkatchenko and M. Scheffler, Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data, Phys. Rev. Lett. 102, 073005 (2009).
  54. M. Kim, W. J. Kim, T. Gould, E. K. Lee, S. Lebègue, and H. Kim, uMBD: A materials-ready dispersion correction that uniformly treats metallic, ionic, and van der Waals bonding, J. Am. Chem. Soc. 142, 2346 (2020).
  55. B. Himmetoglu, A. Floris, S. de Gironcoli, and M. Cococcioni, Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems, Int. J. Quantum Chem. 114, 14 (2014).
  56. V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, First-principles calculations of the electronic structure and spectra of strongly correlated systems: The LDA+U method, J. Phys.: Condens. Matter 9, 767 (1997).
  57. N. E. Kirchner-Hall, W. Zhao, Y. Xiong, I. Timrov, and I. Dabo, Extensive benchmarking of DFT+U calculations for predicting band gaps, Appl. Sci. 11, 2395 (2021).
  58. G. Sai Gautam and E. A. Carter, Evaluating transition metal oxides within DFT-SCAN and SCAN+U frameworks for solar thermochemical applications, Phys. Rev. Mater. 2, 095401 (2018).
  59. N. Artrith, J. A. Garrido Torres, A. Urban, and M. S. Hybertsen, Data-driven approach to parameterize SCAN+U for an accurate description of 3d transition metal oxide thermochemistry, Phys. Rev. Mater. 6, 035003 (2022).
  60. Y. Crespo and N. Seriani, Electronic and magnetic properties of α−MnO2 from ab initio calculations, Phys. Rev. B 88, 144428 (2013).
  61. A. Floris, S. de Gironcoli, E. K. U. Gross, and M. Cococcioni, Vibrational properties of MnO and NiO from DFT+U-based density functional perturbation theory, Phys. Rev. B 84, 161102(R) (2011).
  62. C. Franchini, R. Podloucky, J. Paier, M. Marsman, and G. Kresse, Ground-state properties of multivalent manganese oxides: Density functional and hybrid density functional calculations, Phys. Rev. B 75, 195128 (2007).
  63. Z. Hu and H. Metiu, Choice of U for DFT+U calculations for titanium oxides, J. Phys. Chem. C 115, 5841 (2011).
  64. A. Kiejna, T. Ossowski, and T. Pabisiak, Surface properties of the clean and Au/Pd covered Fe3O4(111): DFT and DFT+U study, Phys. Rev. B 85, 125414 (2012).
  65. J. S. Lim, D. Saldana-Greco, and A. M. Rappe, Improved pseudopotential transferability for magnetic and electronic properties of binary manganese oxides from DFT+U+J calculations, Phys. Rev. B 94, 165151 (2016).
  66. O. Y. Long, G. Sai Gautam, and E. A. Carter, Evaluating optimal U for 3d transition-metal oxides within the SCAN+U framework, Phys. Rev. Mater. 4, 045401 (2020).
  67. A. C. M. Padilha, A. R. Rocha, and G. M. Dalpian, DFT+U Simulation of the Ti4O7−TiO2 Interface, Phys. Rev. Appl. 3, 024009 (2015).
  68. C. Rödl, F. Fuchs, J. Furthmüller, and F. Bechstedt, Quasiparticle band structures of the antiferromagnetic transition-metal oxides MnO, FeO, CoO, and NiO, Phys. Rev. B 79, 235114 (2009).
  69. G. Rollmann, A. Rohrbach, P. Entel, and J. Hafner, First-principles calculation of the structure and magnetic phases of hematite, Phys. Rev. B 69, 165107 (2004).
  70. N. Tancogne-Dejean, M. A. Sentef, and A. Rubio, Ultrafast modification of Hubbard U in a strongly correlated material: Ab initio high-harmonic generation in NiO, Phys. Rev. Lett. 121, 097402 (2018).
  71. D. A. Tompsett, D. S. Middlemiss, and M. S. Islam, Importance of anisotropic Coulomb interactions and exchange to the band gap and antiferromagnetism of β−MnO2 from DFT+U, Phys. Rev. B 86, 205126 (2012).
  72. F. Zhou, M. Cococcioni, C. A. Marianetti, D. Morgan, and G. Ceder, First-principles prediction of redox potentials in transition-metal compounds with LDA+U, Phys. Rev. B 70, 235121 (2004).
  73. J.-J. Zhou, J. Park, I. Timrov, A. Floris, M. Cococcioni, N. Marzari, and M. Bernardi, Ab Initio electron-phonon interactions in correlated electron systems, Phys. Rev. Lett. 127, 126404 (2021).
  74. G. Carchini, M. García-Melchor, Z. Łodziana, and N. López, Understanding and tuning the intrinsic hydrophobicity of rare-earth oxides: A DFT+U study, ACS Appl. Mater. Interfaces 8, 152 (2016).
  75. C. W. M. Castleton, J. Kullgren, and K. Hermansson, Tuning LDA+U for electron localization and structure at oxygen vacancies in ceria, J. Chem. Phys. 127, 244704 (2007).
  76. J. A. Colón Santana, J. M. An, N. Wu, K. D. Belashchenko, X. Wang, P. Liu, J. Tang, Y. Losovyj, I. N. Yakovkin, and P. A. Dowben, Effect of gadolinium doping on the electronic band structure of europium oxide, Phys. Rev. B 85, 014406 (2012).
  77. T. Hamada and T. Ohno, A new constraint DFT technique for self-consistent determination of U values, J. Phys.: Condens. Matter 31, 065501 (2019).
  78. M. Huang and S. Fabris, CO Adsorption and Oxidation on Ceria Surfaces from DFT+U calculations, J. Phys. Chem. C 112, 8643 (2008).
  79. S. Li, Y. Li, M. Bäumer, and L. V. Moskaleva, Assessment of PBE+U and HSE06 methods and determination of optimal parameter U for the structural and energetic properties of rare earth oxides, J. Chem. Phys. 153, 164710 (2020).
  80. A. H. Reshak, M. Piasecki, S. Auluck, I. V. Kityk, R. Khenata, B. Andriyevsky, C. Cobet, N. Esser, A. Majchrowski, M. Świrkowicz, R. Diduszko, and W. Szyrski, Effect of U on the electronic properties of neodymium gallate (NdGaO3): Theoretical and experimental studies, J. Phys. Chem. B 113, 15237 (2009).
  81. Z. Yang, Q. Wang, S. Wei, D. Ma, and Q. Sun, The effect of environment on the reaction of water on the Ceria(111) surface: A DFT+U study, J. Phys. Chem. C 114, 14891 (2010).
  82. D. A. Scherlis, M. Cococcioni, P. Sit, and N. Marzari, Simulation of heme using DFT+U: A step toward accurate spin-state energetics, J. Phys. Chem. B 111, 7384 (2007).
  83. P. M. Panchmatia, M. E. Ali, B. Sanyal, and P. M. Oppeneer, Halide ligated iron porphines: A DFT+U and UB3LYP study, J. Phys. Chem. A 114, 13381 (2010).
  84. P. M. Panchmatia, B. Sanyal, and P. M. Oppeneer, GGA+U modeling of structural, electronic, and magnetic properties of iron porphyrin-type molecules, Chem. Phys. 343, 47 (2008).
  85. H. J. Kulik, L. C. Blasiak, N. Marzari, and C. L. Drennan, First-principles study of non-heme Fe(II) halogenase SyrB2 reactivity, J. Am. Chem. Soc. 131, 14426 (2009).
  86. H. J. Kulik, M. Cococcioni, D. A. Scherlis, and N. Marzari, Density functional theory in transition-metal chemistry: A self-consistent Hubbard U Approach, Phys. Rev. Lett. 97, 103001 (2006).
  87. I. E. Brumboiu, S. Haldar, J. Lüder, O. Eriksson, H. C. Herper, B. Brena, and B. Sanyal, Influence of electron correlation on the electronic structure and magnetism of transition-metal phthalocyanines, J. Chem. Theory Comput. 12, 1772 (2016).
  88. C. Cao, S. Hill, and H.-P. Cheng, Strongly correlated electrons in the [Ni(hmp)(ROH)X]4 single molecule magnet: A DFT+U study, Phys. Rev. Lett. 100, 167206 (2008).
  89. V. I. Anisimov, J. Zaanen, and O. K. Andersen, Band theory and Mott insulators: Hubbard U instead of Stoner I, Phys. Rev. B 44, 943 (1991).
  90. K. A. Hunnestad, H. Das, C. Hatzoglou, M. Holtz, C. M. Brooks, A. T. J. van Helvoort, D. A. Muller, D. G. Schlom, J. A. Mundy, and D. Meier, 3D oxygen vacancy distribution and defect-property relations in an oxide heterostructure, Nat. Commun. 15, 5400 (2024).
  91. O. Köksal and R. Pentcheva, Chern and Z2 topological insulating phases in perovskite-derived 4d and 5d oxide buckled honeycomb lattices, Sci. Rep. 9, 17306 (2019).
  92. Y. Wang, Z. Li, X. Luo, J. Gao, Y. Han, J. Jiang, J. Tang, H. Ju, T. Li, R. Lv, S. Cui, Y. Yang, Y. Sun, J. Zhu, X. Gao, W. Lu, Z. Sun, H. Xu, Y. Xiong, and L. Cao, Dualistic insulator states in 1T-TaS2 crystals, Nat. Commun. 15, 3425 (2024).
  93. E. M. Flores, M. L. Moreira, and M. J. Piotrowski, Structural and electronic properties of bulk ZnX (X = O, S, Se, Te), ZnF2, and ZnO/ZnF2: A DFT investigation within PBE, PBE + U, and hybrid HSE functionals, J. Phys. Chem. A 124, 3778 (2020).
  94. C. Loschen, J. Carrasco, K. M. Neyman, and F. Illas, First-principles LDA+U and GGA+U study of cerium oxides: Dependence on the effective U parameter, Phys. Rev. B 75, 035115 (2007).
  95. X. Ma, Y. Wu, Y. Lv, and Y. Zhu, Correlation effects on lattice relaxation and electronic structure of ZnO within the GGA+U formalism, J. Phys. Chem. C 117, 26029 (2013).
  96. B. J. Morgan and P. A. Madden, Lithium intercalation into TiO2(B): A comparison of LDA, GGA, and GGA+U density functional calculations, Phys. Rev. B 86, 035147 (2012).
  97. M.-T. Nguyen, N. Seriani, and R. Gebauer, Water adsorption and dissociation on α-Fe2O3(0001): PBE+U calculations, J. Chem. Phys. 138, 194709 (2013).
  98. D. Stoeffler and C. Etz, Ab initio electronic structure and magnetism in Sr2XMoO6 (X = Fe or Co) double perovskite systems: A GGA and GGA+U comparative study, J. Phys.: Condens. Matter 18, 11291 (2006).
  99. W. Yang and Y.-W. Son, Effects of self-consistent extended Hubbard interactions and spin-orbit couplings on energy bands of semiconductors and topological insulators, Phys. Rev. B 110, 155133 (2024).
  100. Y. Zeng, Q. Hu, M. Pan, K. Zhang, S. Grasso, C. Hu, and Q. Feng, Investigation of double perovskites Sr2SmNbO6 and X2CoNbO6 (X = Sr, Ba) with SCAN functional and plus U correction, Adv. Powder Mater. 1, 100019 (2022).
  101. S. Kim, Lattice and electronic properties of VO2 with the SCAN(+U) approach, J. Korean Phys. Soc. 78, 613 (2021).
  102. Google Scholar, https://scholar.google.com.
  103. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  104. S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. J. Probert, K. Refson, and M. C. Payne, First principles methods using CASTEP, Z. Krist. 220, 567 (2005).
  105. P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, et al., Advanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  106. T. D. Kühne, M. Iannuzzi, M. Del Ben, V. V. Rybkin, P. Seewald, F. Stein, T. Laino, R. Z. Khaliullin, O. Schütt, F. Schiffmann, D. Golze, J. Wilhelm, S. Chulkov, M. H. Bani-Hashemian, V. Weber, U. Borštnik, M. Taillefumier, A. S. Jakobovits, A. Lazzaro, H. Pabst, et al., CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations, J. Chem. Phys. 152, 194103 (2020).
  107. G. te Velde, F. M. Bickelhaupt, E. J. Baerends, C. Fonseca Guerra, S. J. A. van Gisbergen, J. G. Snijders, and T. Ziegler, Chemistry with ADF, J. Comput. Chem. 22, 931 (2001).
  108. P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k: An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties (Vienna University of Technology, Austria, 2001).
  109. E. Artacho, E. Anglada, O. Diéguez, J. D. Gale, A. García, J. Junquera, R. M. Martin, P. Ordejón, J. M. Pruneda, D. Sánchez-Portal, and J. M. Soler, The SIESTA method; developments and applicability, J. Phys.: Condens. Matter 20, 064208 (2008).
  110. X. Gonze, B. Amadon, P. M. Anglade, J. M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Côté, T. Deutsch, L. Genovese, Ph. Ghosez, M. Giantomassi, S. Goedecker, D. R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, et al., ABINIT: First-principles approach to material and nanosystem properties, Comput. Phys. Commun. 180, 2582 (2009).
  111. The Elk Code: See http://elk.sourceforge.net/ for The Elk Code (2026).
  112. J. J. R. Mortensen, A. H. Larsen, M. Kuisma, A. V. Ivanov, A. Taghizadeh, A. Peterson, A. Haldar, A. O. Dohn, C. Schäfer, E. Jónsson, E. D. Hermes, F. A. Nilsson, G. Kastlunger, G. Levi, H. Jónsson, H. Häkkinen, J. Fojt, J. Kangsabanik, J. Sødequist, J. Lehtomäki, et al., GPAW: An open Python package for electronic structure calculations, J. Chem. Phys. 160, 092503 (2024).
  113. A. Nakata, J. S. Baker, S. Y. Mujahed, J. T. L. Poulton, S. Arapan, J. Lin, Z. Raza, S. Yadav, L. Truflandier, T. Miyazaki, and D. R. Bowler, Large scale and linear scaling DFT with the CONQUEST code, J. Chem. Phys. 152, 164112 (2020).
  114. V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, and M. Scheffler, Ab Initio molecular simulations with numeric atom-centered orbitals, Comput. Phys. Commun. 180, 2175 (2009).
  115. N. Tancogne-Dejean, M. J. T. Oliveira, X. Andrade, H. Appel, C. H. Borca, G. Le Breton, F. Buchholz, A. Castro, S. Corni, A. A. Correa, U. De Giovannini, A. Delgado, F. G. Eich, J. Flick, G. Gil, A. Gomez, N. Helbig, H. Hübener, R. Jestädt, J. Jornet-Somoza, et al., Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems, J. Chem. Phys. 152, 124119 (2020).
  116. The FLEUR project, https://www.flapw.de/ (2026).
  117. K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme, Phys. Rev. B 59, 1743 (1999).
  118. J. C. A. Prentice, J. Aarons, J. C. Womack, A. E. A. Allen, L. Andrinopoulos, L. Anton, R. A. Bell, A. Bhandari, G. A. Bramley, R. J. Charlton, R. J. Clements, D. J. Cole, G. Constantinescu, F. Corsetti, S. M.-M. Dubois, K. K. B. Duff, J. M. Escartín, A. Greco, Q. Hill, L. P. Lee, et al., The ONETEP linear-scaling density functional theory program, J. Chem. Phys. 152, 174111 (2020).
  119. G. Geudtner, P. Calaminici, J. Carmona-Espíndola, J. M. del Campo, V. D. Domínguez-Soria, R. F. Moreno, G. U. Gamboa, A. Goursot, A. M. Köster, J. U. Reveles, T. Mineva, J. M. Vásquez-Pérez, A. Vela, B. Zúñinga-Gutierrez, and D. R. Salahub, deMon2k, WIREs Comput. Mol. Sci. 2, 548 (2012).
  120. J. M. Wills, M. Alouani, P. Andersson, A. Delin, O. Eriksson, and O. Grechnyev, Full-Potential Electronic Structure Method: Energy and Force Calculations with Density Functional and Dynamical Mean Field Theory (Springer Science and Business Media, New York, 2010).
  121. L. E. Ratcliff, W. Dawson, G. Fisicaro, D. Caliste, S. Mohr, A. Degomme, B. Videau, V. Cristiglio, M. Stella, M. D'Alessandro, S. Goedecker, T. Nakajima, T. Deutsch, and L. Genovese, Flexibilities of wavelets as a computational basis set for large-scale electronic structure calculations, J. Chem. Phys. 152, 194110 (2020).
  122. R. Sundararaman, K. Letchworth-Weaver, K. A. Schwarz, D. Gunceler, Y. Ozhabes, and T.A. Arias, JDFTx: Software for joint density-functional theory, SoftwareX 6, 278 (2017).
  123. P. R. Briddon and M. J. Rayson, Accurate Kohn–Sham DFT with the speed of tight binding: Current techniques and future directions in materials modelling, Physica Status Solidi (b) 248, 1309 (2011).
  124. P. Motamarri, S. Das, S. Rudraraju, K. Ghosh, D. Davydov, and V. Gavini, DFT-FE–A massively parallel adaptive finite-element code for large-scale density functional theory calculations, Comput. Phys. Commun. 246, 106853 (2020).
  125. M. F. Herbst, A. Levitt, and E. Cancès, Dftk: A julian approach for simulating electrons in solids, Proc. JuliaCon Conf. 3, 69 (2021).
  126. S. Ono, Y. Noguchi, R. Sahara, Y. Kawazoe, and K. Ohno, TOMBO: All-electron mixed-basis approach to condensed matter physics, Comput. Phys. Commun. 189, 20 (2015).
  127. P. Schulz and R. Muller, QUantum Electronic STructure (SeqQuest Electronic Structure Code), Sandia National Laboratories, Albuquerque (NM) (2004).
  128. O. Jepsen, The stuttgart tb-lmto-asa program version 47, http://www.mpi-stuttgart.mpg.de/andersen/ (2000).
  129. T. Ozaki, Variationally optimized atomic orbitals for large-scale electronic structures, Phys. Rev. B 67, 155108 (2003).
  130. N. Colonna, N. L. Nguyen, A. Ferretti, and N. Marzari, Koopmans-compliant functionals and potentials and their application to the GW100 test set, J. Chem. Theory Comput. 15, 1905 (2019).
  131. J. M. de Almeida, N. L. Nguyen, N. Colonna, W. Chen, M. C. Rodrigues, A. Pasquarello, and N. Marzari, Electronic structure of water from Koopmans-compliant functionals, J. Chem. Theory Comput. 17, 3923 (2021).
  132. N. L. Nguyen, N. Colonna, A. Ferretti, and N. Marzari, Koopmans-compliant spectral functionals for extended systems, Phys. Rev. X 8, 021051 (2018).
  133. D. S. Lambert and D. D. O'Regan, Use of DFT+U+J with linear response parameters to predict non-magnetic oxide band gaps with hybrid-functional accuracy, Phys. Rev. Res. 5, 013160 (2023).
  134. I. Timrov, F. Aquilante, L. Binci, M. Cococcioni, and N. Marzari, Pulay forces in density-functional theory with extended Hubbard functionals: From nonorthogonalized to orthogonalized manifolds, Phys. Rev. B 102, 235159 (2020).
  135. C. Ricca, I. Timrov, M. Cococcioni, N. Marzari, and U. Aschauer, Self-consistent DFT+U+V study of oxygen vacancies in SrTiO3, Phys. Rev. Res. 2, 023313 (2020).
  136. I. Timrov, P. Agrawal, X. Zhang, S. Erat, R. Liu, A. Braun, M. Cococcioni, M. Calandra, N. Marzari, and D. Passerone, Electronic structure of pristine and Ni-substituted LaFeO3 from near edge x-ray absorption fine structure experiments and first-principles simulations, Phys. Rev. Res. 2, 033265 (2020).
  137. B.-C. Shih, T. A. Abtew, X. Yuan, W. Zhang, and P. Zhang, Screened Coulomb interactions of localized electrons in transition metals and transition-metal oxides, Phys. Rev. B 86, 165124 (2012).
  138. S. Fabris, S. de Gironcoli, S. Baroni, G. Vicario, and G. Balducci, Taming multiple valency with density functionals: A case study of defective ceria, Phys. Rev. B 71, 041102(R) (2005).
  139. A. Bajaj and H. J. Kulik, Molecular DFT+U: A transferable, low-cost approach to eliminate delocalization error, J. Phys. Chem. Lett. 12, 3633 (2021).
  140. A. Bajaj, C. Duan, A. Nandy, M. G. Taylor, and H. J. Kulik, Molecular orbital projectors in non-empirical jmDFT recover exact conditions in transition-metal chemistry, J. Chem. Phys. 156, 184112 (2022).
  141. S. M. Griffin, Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite, arXiv:2307.16892.
  142. L. Wang, T. Maxisch, and G. Ceder, Oxidation energies of transition metal oxides within the GGA+U framework, Phys. Rev. B 73, 195107 (2006).
  143. M. Cococcioni and S. de Gironcoli, Linear response approach to the calculation of the effective interaction parameters in the LDA+U method, Phys. Rev. B 71, 035105 (2005).
  144. E. B. Linscott, D. J. Cole, M. C. Payne, and D. D. O'Regan, Role of spin in the calculation of Hubbard U and Hund's J parameters from first principles, Phys. Rev. B 98, 235157 (2018).
  145. O. K. Andersen and T. Saha-Dasgupta, Muffin-tin orbitals of arbitrary order, Phys. Rev. B 62, R16219 (2000).
  146. V. I. Anisimov and O. Gunnarsson, Density-functional calculation of effective coulomb interactions in metals, Phys. Rev. B 43, 7570 (1991).
  147. M. Springer and F. Aryasetiawan, Frequency-dependent screened interaction in Ni within the random-phase approximation, Phys. Rev. B 57, 4364 (1998).
  148. W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Reformulation of the LDA+U method for a local-orbital basis, Phys. Rev. B 58, 1201 (1998).
  149. B. Himmetoglu, R. M. Wentzcovitch, and M. Cococcioni, First-principles study of electronic and structural properties of CuO, Phys. Rev. B 84, 115108 (2011).
  150. F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Schönberger, Calculations of Hubbard U from first-principles, Phys. Rev. B 74, 125106 (2006).
  151. T. Miyake and F. Aryasetiawan, Screened Coulomb interaction in the maximally localized Wannier basis, Phys. Rev. B 77, 085122 (2008).
  152. T. Miyake, F. Aryasetiawan, and M. Imada, Ab initio procedure for constructing effective models of correlated materials with entangled band structure, Phys. Rev. B 80, 155134 (2009).
  153. E. Şaşıoğlu, C. Friedrich, and S. Blügel, Effective Coulomb interaction in transition metals from constrained random-phase approximation, Phys. Rev. B 83, 121101(R) (2011).
  154. T. Kotani, Ab initio random-phase-approximation calculation of the frequency-dependent effective interaction between 3d electrons: Ni, Fe, and MnO, J. Phys.: Condens. Matter 12, 2413 (2000).
  155. F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein, Frequency-dependent local interactions and low-energy effective models from electronic structure calculations, Phys. Rev. B 70, 195104 (2004).
  156. L. Vaugier, H. Jiang, and S. Biermann, Hubbard U and Hund exchange J in transition metal oxides: Screening versus localization trends from constrained random phase approximation, Phys. Rev. B 86, 165105 (2012).
  157. B. Amadon, T. Applencourt, and F. Bruneval, Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium, Phys. Rev. B 89, 125110 (2014).
  158. M. Casula, A. Rubtsov, and S. Biermann, Dynamical screening effects in correlated materials: Plasmon satellites and spectral weight transfers from a Green's function ansatz to extended dynamical mean field theory, Phys. Rev. B 85, 035115 (2012).
  159. L. Huang and Y. Wang, Dynamical screening in strongly correlated metal SrVO3, Europhys. Lett. 99, 67003 (2012).
  160. L. Boehnke, F. Nilsson, F. Aryasetiawan, and P. Werner, When strong correlations become weak: Consistent merging of GW and DMFT, Phys. Rev. B 94, 201106(R) (2016).
  161. G. Moynihan, G. Teobaldi, and D. D. O'Regan, Inapplicability of exact constraints and a minimal two-parameter generalization to the DFT+U based correction of self-interaction error, Phys. Rev. B 94, 220104(R) (2016).
  162. G. Moynihan, A self-contained ground-state approach for the correction of self-interaction error in approximate density-functional theory, Ph.D. thesis, School of Physics, Trinity College Dublin, 2018.
  163. P. Verma and D. G. Truhlar, Does DFT+U mimic hybrid density functionals? Theor. Chem. Acc. 135, 182 (2016).
  164. H. Eschrig, K. Koepernik, and I. Chaplygin, Density functional application to strongly correlated electron systems, J. Solid State Chem. 176, 482 (2003).
  165. D. D. O'Regan, N. D. M. Hine, M. C. Payne, and A. A. Mostofi, Projector self-consistent DFT+U using nonorthogonal generalized wannier functions, Phys. Rev. B 82, 081102(R) (2010).
  166. D. Novoselov, D. M. Korotin, and V. I. Anisimov, Hellmann–feynman forces within the DFT+U in wannier functions basis, J. Phys.: Condens. Matter 27, 325602 (2015).
  167. A. Carta, I. Timrov, S. Beck, and C. Ederer, Bridging constrained random-phase approximation and linear response theory for computing Hubbard parameters, Phys. Rev. Lett. 137, 046503 (2026).
  168. V. Ivády, R. Armiento, K. Szász, E. Janzén, A. Gali, and I. A. Abrikosov, Theoretical unification of hybrid-DFT and DFT+U methods for the treatment of localized orbitals, Phys. Rev. B 90, 035146 (2014).
  169. W. Yang, A. J. Cohen, and P. Mori-Sánchez, Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory, J. Chem. Phys. 136, 204111 (2012).
  170. J. P. Perdew, W. Yang, K. Burke, Z. Yang, E. K. U. Gross, M. Scheffler, G. E. Scuseria, T. M. Henderson, I. Y. Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Trushin, and A. Görling, Understanding band gaps of solids in generalized Kohn–Sham theory, Proc. Natl. Acad. Sci. USA 114, 2801 (2017).
  171. S. Kurth and S. Pittalis, The optimized effective potential method and LDA+U, in Computational Nanoscience: Do It Yourself!, edited by J. Grotendorst, S. Blüugel, and D. Marx, NIC Series Vol. 31 (John von Neumann Institute for Computing, Jülich, 2006), pp. 299–334.
  172. V. Ravindran, N. I. Gidopoulos, and S. J. Clark, Local exchange-correlation potentials by density inversion in solids, Phys. Rev. B 112, 085208 (2025).
  173. C.-O. Almbladh and U. von Barth, Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Phys. Rev. B 31, 3231 (1985).
  174. T. Gould and J. Toulouse, Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers, Phys. Rev. A 90, 050502(R) (2014).
  175. V. Vlček, H. R. Eisenberg, G. Steinle-Neumann, D. Neuhauser, E. Rabani, and R. Baer, Spontaneous charge carrier localization in extended one-dimensional systems, Phys. Rev. Lett. 116, 186401 (2016).
  176. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B 57, 1505 (1998).
  177. N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C. Foulkes, Finite-size errors in continuum quantum Monte Carlo calculations, Phys. Rev. B 78, 125106 (2008).
  178. D. Ceperley, Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions, Phys. Rev. B 18, 3126 (1978).
  179. Orbital-dependent functionals can be applied within strict Kohn-Sham DFT by means of the optimized effective potential (OEP) procedure. Studies involving OEP are very rare by comparison to those within the overwhelmingly popular, even ubiquitous GKS framework.
  180. N. Poilvert, Extending the Predictive Power and Scope of Electronic Structure Theory and Quantum Transport, Ph.D. Thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology, 2011, http://hdl.handle.net/1721.1/69668.
  181. M. R. Pederson, R. A. Heaton, and C. C. Lin, Local-density Hartree-Fock theory of electronic states of molecules with self-interaction correction, J. Chem. Phys. 80, 1972 (1984).
  182. D. Hait and M. Head-Gordon, Delocalization errors in density functional theory are essentially quadratic in fractional occupation number, J. Phys. Chem. Lett. 9, 6280 (2018).
  183. J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Density-functional theory for fractional particle number: Derivative discontinuities of the energy, Phys. Rev. Lett. 49, 1691 (1982).
  184. W. Yang, Y. Zhang, and P. W. Ayers, Degenerate ground states and a fractional number of electrons in density and reduced density matrix functional theory, Phys. Rev. Lett. 84, 5172 (2000).
  185. P. W. Ayers, The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons, J. Math. Chem. 43, 285 (2008).
  186. P. Mori-Sánchez, A. J. Cohen, and W. Yang, Discontinuous nature of the exchange-correlation functional in strongly correlated systems, Phys. Rev. Lett. 102, 066403 (2009).
  187. A. C. Burgess and D. D. O'Regan, Flat plane based double-counting free and parameter free many-body DFT+U, Phys. Rev. B 110, 205150 (2024).
  188. R. G. Parr and W. Yang, Density functional approach to the frontier-electron theory of chemical reactivity, J. Am. Chem. Soc. 106, 4049 (1984).
  189. P. W. Ayers and M. Levy, Perspective on “Density functional approach to the frontier-electron theory of chemical reactivity”, Theor. Chem. Acc. 103, 353 (2000).
  190. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1995).
  191. W. Yang, A. J. Cohen, F. De Proft, and P. Geerlings, Analytical evaluation of Fukui functions and real-space linear response function, J. Chem. Phys. 136, 144110 (2012).
  192. P. W. Ayers, F. De Proft, A. Borgoo, and P. Geerlings, Computing Fukui functions without differentiating with respect to electron number. I. Fundamentals, J. Chem. Phys. 126, 224107 (2007).
  193. B. Wang, P. Geerlings, S. Liu, and F. De Proft, Extending the scope of conceptual density functional theory with second order analytical methodologies, J. Chem. Theory Comput. 20, 1169 (2024).
  194. D. Zhang, X. Zheng, C. Li, and W. Yang, Orbital relaxation effects on Kohn–Sham frontier orbital energies in density functional theory, J. Chem. Phys. 142, 154113 (2015).
  195. P. Bultinck, D. Clarisse, P. W. Ayers, and R. Carbo-Dorca, The Fukui matrix: A simple approach to the analysis of the Fukui function and its positive character, Phys. Chem. Chem. Phys. 13, 6110 (2011).
  196. D. Peng and W. Yang, Fukui function and response function for nonlocal and fractional systems, J. Chem. Phys. 138, 184108 (2013).
  197. B. Wang, P. Geerlings, F. Heidar-Zadeh, P. W. Ayers, and F. De Proft, Slope of the delocalization function is proportional to analytical hardness, J. Phys. Chem. Lett. 15, 11259 (2024).
  198. P. Bultinck, D. Van Neck, G. Acke, and P. W. Ayers, Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods, Phys. Chem. Chem. Phys. 14, 2408 (2012).
  199. G. Acke, D. Van Hende, X. De Vriendt, and P. Bultinck, Extending conceptual density functional theory toward first-order reduced density matrices: An open subsystems viewpoint on the Fukui matrix, J. Chem. Theory Comput. 19, 5418 (2023).
  200. The ground-state electron density is equal to the diagonal elements of the ground-state first-order reference system reduced density matrix and, as such, their left- and right-hand derivatives with respect to electron count are equal so that F±(r,r)=F±(r).
  201. K. Terakura, T. Oguchi, A. R. Williams, and J. Kübler, Band theory of insulating transition-metal monoxides: Band-structure calculations, Phys. Rev. B 30, 4734 (1984).
  202. V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czyżyk, and G. A. Sawatzky, Density-functional theory and NiO photoemission spectra, Phys. Rev. B 48, 16929 (1993).
  203. M. T. Czyżyk and G. A. Sawatzky, Local-density functional and on-site correlations: The electronic structure of La2CuO4 and LaCuO3, Phys. Rev. B 49, 14211 (1994).
  204. D.-K. Seo, Self-interaction correction in the LDA+U method, Phys. Rev. B 76, 033102 (2007).
  205. I. V. Solovyev, P. H. Dederichs, and V. I. Anisimov, Corrected atomic limit in the local-density approximation and the electronic structure of d impurities in Rb, Phys. Rev. B 50, 16861 (1994).
  206. A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators, Phys. Rev. B 52, R5467 (1995).
  207. I. Dabo, Towards first-principles electrochemistry, Ph.D. thesis, Department of Materials Science and Engineering, Massachusetts Institute of Technology, 2008, http://hdl.handle.net/1721.1/44320.
  208. E. Macke, I. Timrov, N. Marzari, and L. C. Ciacchi, Orbital-resolved DFT+U for molecules and solids, J. Chem. Theory Comput. 20, 4824 (2024).
  209. A. G. Petukhov, I. I. Mazin, L. Chioncel, and A. I. Lichtenstein, Correlated metals and the LDA+U method, Phys. Rev. B 67, 153106 (2003).
  210. M. Shishkin and H. Sato, DFT+U in Dudarev's formulation with corrected interactions between the electrons with opposite spins: The form of Hamiltonian, calculation of forces, and bandgap adjustments, J. Chem. Phys. 151, 024102 (2019).
  211. M. Shishkin and H. Sato, Evaluation of redox potentials of cathode materials of alkali-ion batteries using extended DFT+U+U↑↓ method: The role of interactions between the electrons with opposite spins, J. Chem. Phys. 154, 114709 (2021).
  212. V. Leiria Campo Jr and M. Cococcioni, Extended DFT+U+V method with on-site and inter-site electronic interactions, J. Phys.: Condens. Matter 22, 055602 (2010).
  213. D. Soler-Polo, J. Ortega, and F. Flores, A local-orbital density functional formalism for a many-body atomic Hamiltonian: Hubbard–Hund's coupling and DFT+U functional, J. Phys.: Condens. Matter 33, 425604 (2021).
  214. G. K.-L. Chan, A fresh look at ensembles: Derivative discontinuities in density functional theory, J. Chem. Phys. 110, 4710 (1999).
  215. J. P. Perdew and E. Sagvolden, Exact exchange-correlation potentials in spin-density functional theory and their discontinuities at unit electron number, Can. J. Chem. 87, 1268 (2009).
  216. Y. Goshen and E. Kraisler, Ensemble ground state of a many-electron system with fractional electron number and spin: Piecewise-linearity and flat-plane condition generalized, J. Phys. Chem. Lett. 15, 2337 (2024).
  217. A. C. Burgess, E. Linscott, and D. D. O'Regan, Tilted-plane structure of the energy of finite quantum systems, Phys. Rev. Lett. 133, 026404 (2024).
  218. A. Bajaj, J. P. Janet, and H. J. Kulik, Communication: Recovering the flat-plane condition in electronic structure theory at semi-local DFT cost, J. Chem. Phys. 147, 191101 (2017).
  219. A. Bajaj, F. Liu, and H. J. Kulik, Non-empirical, low-cost recovery of exact conditions with model-Hamiltonian inspired expressions in jmDFT, J. Chem. Phys. 150, 154115 (2019).
  220. A. C. Burgess, E. Linscott, and D. D. O'Regan, DFT+U-type functional derived to explicitly address the flat plane condition, Phys. Rev. B 107, L121115 (2023).
  221. B. G. Janesko, Multiconfigurational correlation at DFT+U cost: On-site electron–electron interactions yield a block-localized configuration interaction Hamiltonian, J. Phys. Chem. A 128, 5077 (2024).
  222. S. L. Dudarev, P. Liu, D. A. Andersson, C. R. Stanek, T. Ozaki, and C. Franchini, Parametrization of LSDA+U for noncollinear magnetic configurations: Multipolar magnetism in UO2, Phys. Rev. Mater. 3, 083802 (2019).
  223. F. Bultmark, F. Cricchio, O. Grånäs, and L. Nordström, Multipole decomposition of LDA+U energy and its application to actinide compounds, Phys. Rev. B 80, 035121 (2009).
  224. E. Bousquet and N. Spaldin, J dependence in the LSDA+U treatment of noncollinear magnets, Phys. Rev. B 82, 220402(R) (2010).
  225. S. Ryee and M. J. Han, Comparative study of DFT+U functionals for non-collinear magnetism, J. Phys.: Condens. Matter 30, 275802 (2018).
  226. F. Gómez-Ortiz, N. Carral-Sainz, J. Sifuna, V. Monteseguro, R. Cuadrado, P. García-Fernández, and J. Junquera, Compatibility of DFT+U with non-collinear magnetism and spin-orbit coupling within a framework of numerical atomic orbitals, Comput. Phys. Commun. 286, 108684 (2023).
  227. L. Binci and N. Marzari, Noncollinear DFT+U and Hubbard parameters with fully relativistic ultrasoft pseudopotentials, Phys. Rev. B 108, 115157 (2023).
  228. J. P. Perdew, Density functional theory and the band gap problem, Int. J. Quantum Chem. 28, 497 (1985).
  229. V. Etacheri, R. Marom, R. Elazari, G. Salitra, and D. Aurbach, Challenges in the development of advanced Li-ion batteries: a review, Energy Environ. Sci. 4, 3243 (2011).
  230. M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, The evolution of multiferroics, Nat. Rev. Mater. 1, 16046 (2016).
  231. A. Kudo and Y. Miseki, Heterogeneous photocatalyst materials for water splitting, Chem. Soc. Rev. 38, 253 (2009).
  232. Y. Shiohara, M. Yoshizumi, Y. Takagi, and T. Izumi, Future prospects of high Tc superconductors-coated conductors and their applications, Physica C 484, 1 (2013).
  233. Q. Si, R. Yu, and E. Abrahams, High-temperature superconductivity in iron pnictides and chalcogenides, Nat. Rev. Mater. 1, 16017 (2016).
  234. W. Eerenstein, N. D. Mathur, and J. F. Scott, Multiferroic and magnetoelectric materials, Nature (London) 442, 759 (2006).
  235. E. Morosan, D. Natelson, A. H. Nevidomskyy, and Q. Si, Strongly correlated materials, Adv. Mater. 24, 4896 (2012).
  236. J. B. Sambur, T. Novet, and B. A. Parkinson, Multiple exciton collection in a sensitized photovoltaic system, Science 330, 63 (2010).
  237. M. Gatti and M. Guzzo, Dynamical screening in correlated metals: Spectral properties of SrVO3 in the GW approximation and beyond, Phys. Rev. B 87, 155147 (2013).
  238. M. Gatti, F. Bruneval, V. Olevano, and L. Reining, Understanding correlations in vanadium dioxide from first principles, Phys. Rev. Lett. 99, 266402 (2007).
  239. J. Chang, I. Eremin, P. Thalmeier, and P. Fulde, Theory of magnetic excitons in the heavy-fermion superconductor UPd2Al3, Phys. Rev. B 75, 024503 (2007).
  240. E. Fernández, P. Moses, A. Toftelund, H. Hansen, J. Martínez, F. Abild-Pedersen, J. Kleis, B. Hinnemann, J. Rossmeisl, T. Bligaard, and J. Nørskov, Scaling relationships for adsorption energies on transition metal oxide, sulfide, and nitride surfaces, Angew. Chem. Int. Ed. 47, 4683 (2008).
  241. S. L. Dudarev, A. I. Liechtenstein, M. R. Castell, G. A. D. Briggs, and A. P. Sutton, Surface states on NiO (100) and the origin of the contrast reversal in atomically resolved scanning tunneling microscope images, Phys. Rev. B 56, 4900 (1997).
  242. H. J. Kulik, Perspective: Treating electron over-delocalization with the DFT+U method, J. Chem. Phys. 142, 240901 (2015).
  243. M. Cococcioni, Accurate and efficient calculations on strongly correlated minerals with the LDA+U method: Review and perspectives, Rev. Mineral. Geochem. 71, 147 (2010).
  244. A. Iwaszuk and M. Nolan, Electronic structure and reactivity of Ce- and Zr-doped TiO2: Assessing the reliability of density functional theory approaches, J. Phys. Chem. C 115, 12995 (2011).
  245. M. D. Krcha and M. J. Janik, Challenges in the use of density functional theory to examine catalysis by M-doped ceria surfaces, Int. J. Quantum Chem. 114, 8 (2014).
  246. A. Janotti and Walle C. G. Van de, LDA+U and hybrid functional calculations for defects in ZnO, SnO2, and TiO2, Phys. Status Solidi B 248, 799 (2011).
  247. A. Jain, G. Hautier, C. J. Moore, S. Ping Ong, C. C. Fischer, T. Mueller, K. A. Persson, and G. Ceder, A high-throughput infrastructure for density functional theory calculations, Comput. Mater. Sci. 50, 2295 (2011).
  248. W. Setyawan and S. Curtarolo, High-throughput electronic band structure calculations: Challenges and tools, Comput. Mater. Sci. 49, 299 (2010).
  249. D. R. Bowler, T. Miyazaki, and M. J. Gillan, Recent progress in linear scaling ab initio electronic structure techniques, J. Phys.: Condens. Matter 14, 2781 (2002).
  250. H. J. Kulik and N. Marzari, Accurate potential energy surfaces with a DFT+U(R) approach, J. Chem. Phys. 135, 194105 (2011).
  251. A. Jain, G. Hautier, S. P. Ong, C. J. Moore, C. C. Fischer, K. A. Persson, and G. Ceder, Formation enthalpies by mixing GGA and GGA+U calculations, Phys. Rev. B 84, 045115 (2011).
  252. S. Lutfalla, V. Shapovalov, and A. T. Bell, Calibration of the DFT/GGA+U method for determination of reduction energies for transition and rare earth metal oxides of Ti, V, Mo, and Ce, J. Chem. Theory Comput. 7, 2218 (2011).
  253. M. Shishkin and H. Sato, Challenges in computational evaluation of redox and magnetic properties of Fe-based sulfate cathode materials of Li- and Na-ion batteries, J. Phys.: Condens. Matter 29, 215701 (2017).
  254. E. B. Isaacs and C. A. Marianetti, Compositional phase stability of strongly correlated electron materials within DFT+U, Phys. Rev. B 95, 045141 (2017).
  255. M. Shishkin and H. Sato, Self-consistent parametrization of DFT+U framework using linear response approach: Application to evaluation of redox potentials of battery cathodes, Phys. Rev. B 93, 085135 (2016).
  256. G. Hautier, S. P. Ong, A. Jain, C. J. Moore, and G. Ceder, Accuracy of density-functional theory in predicting formation energies of ternary oxides from binary oxides and its implication on phase stability, Phys. Rev. B 85, 155208 (2012).
  257. S. P. Ong, L. Wang, B. Kang, and G. Ceder, Li-Fe-P-O2 phase diagram from first principles calculations, Chem. Mater. 20, 1798 (2008).
  258. L. Wang, T. Maxisch, and G. Ceder, A first-principles approach to studying the thermal stability of oxide cathode materials, Chem. Mater. 19, 543 (2007).
  259. S. P. Ong, V. L. Chevrier, G. Hautier, A. Jain, C. Moore, S. Kim, X. Ma, and G. Ceder, Voltage, stability and diffusion barrier differences between sodium-ion and lithium-ion intercalation materials, Energy Environ. Sci. 4, 3680 (2011).
  260. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, Commentary: The materials project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  261. G. Hautier, A. Jain, and S. P. Ong, From the computer to the laboratory: materials discovery and design using first-principles calculations, J. Mater. Sci. 47, 7317 (2012).
  262. D. S. Lambert and D. D. O'Regan, Evaluation of first-principles Hubbard and Hund corrected DFT for defect formation energies in non-magnetic transition metal oxides, RSC Adv. 14, 38645 (2024).
  263. P. Mori-Sánchez and A. J. Cohen, The derivative discontinuity of the exchange-correlation functional, Phys. Chem. Chem. Phys. 16, 14378 (2014).
  264. X. De Vriendt, L. Lemmens, S. De Baerdemacker, P. Bultinck, and G. Acke, Quantifying delocalization and static correlation errors by imposing (spin) population redistributions through constraints on atomic domains, J. Chem. Theory Comput. 17, 6808 (2021).
  265. N. Q. Su, C. Li, and W. Yang, Describing strong correlation with fractional-spin correction in density functional theory, Proc. Natl. Acad. Sci. USA 115, 9678 (2018).
  266. A. J. Cohen, P. Mori-Sánchez, and W. Yang, Fractional spins and static correlation error in density functional theory, J. Chem. Phys. 129, 121104 (2008).
  267. J. C. Phillips, Generalized Koopmans' theorem, Phys. Rev. 123, 420 (1961).
  268. J. Hubbard, Electron correlations in narrow energy bands, Proc. Roy. Soc. Math. Phys. Eng. 276, 238 (1963).
  269. P. Mori-Sánchez, A. J. Cohen, and W. Yang, Many-electron self-interaction error in approximate density functionals, J. Chem. Phys. 125, 201102 (2006).
  270. A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals, J. Chem. Phys. 125, 194112 (2006).
  271. O. A. Vydrov, G. E. Scuseria, and J. P. Perdew, Tests of functionals for systems with fractional electron number, J. Chem. Phys. 126, 154109 (2007).
  272. A. J. Cohen, P. Mori-Sańchez, and W. Yang, Insights into current limitations of density functional theory, Science 321, 792 (2008).
  273. A. J. Cohen, P. Mori-Sánchez, and W. Yang, Fractional charge perspective on the band gap in density-functional theory, Phys. Rev. B 77, 115123 (2008).
  274. P. Borlido, T. Aull, A. W. Huran, F. Tran, M. A. Marques, and S. Botti, Large-scale benchmark of exchange–correlation functionals for the determination of electronic band gaps of solids, J. Chem. Theory Comput. 15, 5069 (2019).
  275. A. Svane and O. Gunnarsson, Localization in the self-interaction-corrected density-functional formalism, Phys. Rev. B 37, 9919 (1988).
  276. Z. Szotek, W. M. Temmerman, and H. Winter, Application of the self-interaction correction to transition-metal oxides, Phys. Rev. B 47, 4029 (1993).
  277. T. Sohier, M. Calandra, and F. Mauri, Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene, Phys. Rev. B 96, 075448 (2017).
  278. M. Schlipf and F. Gygi, Optimization algorithm for the generation of ONCV pseudopotentials, Comput. Phys. Commun. 196, 36 (2015).
  279. D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  280. N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Thermal contraction and disordering of the Al (110) surface, Phys. Rev. Lett. 82, 3296 (1999).
  281. O. V. Gritsenko and E. J. Baerends, The spin-unrestricted molecular Kohn–Sham solution and the analogue of Koopmans's theorem for open-shell molecules, J. Chem. Phys. 120, 8364 (2004).
  282. T. Gould, L. Kronik, and S. Pittalis, “Ensemblization” of density functional theory, J. Chem. Phys. 164, 040901 (2026).
  283. K. Capelle, G. Vignale, and C. A. Ullrich, Spin gaps and spin-flip energies in density-functional theory, Phys. Rev. B 81, 125114 (2010).
  284. T. Gál and P. Geerlings, Energy surface, chemical potentials, Kohn–Sham energies in spin-polarized density functional theory, J. Chem. Phys. 133, 144105 (2010).
  285. A. Hayman, N. Levy, Y. Goshen, M. Fraenkel, E. Kraisler, and T. Stein, Spin migration in density functional theory: Energy, potential, and density perspectives, J. Chem. Phys. 162, 114301 (2025).
  286. O. V. Gritsenko and E. J. Baerends, The analog of Koopmans' theorem in spin-density functional theory, J. Chem. Phys. 117, 9154 (2002).
  287. T. Gál, P. W. Ayers, F. De Proft, and P. Geerlings, Nonuniqueness of magnetic fields and energy derivatives in spin-polarized density functional theory, J. Chem. Phys. 131, 154114 (2009).
  288. M. Galvan and R. Vargas, Spin-potential in Kohn-Sham theory, J. Phys. Chem. 96, 1625 (1992).
  289. D. R. Hamann, M. Schlüter, and C. Chiang, Norm-conserving pseudopotentials, Phys. Rev. Lett. 43, 1494 (1979).
  290. L. Kleinman and D. M. Bylander, Efficacious form for model pseudopotentials, Phys. Rev. Lett. 48, 1425 (1982).
  291. D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys. Rev. B 41, 7892 (1990).
  292. K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Car-Parrinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials, Phys. Rev. B 47, 10142 (1993).
  293. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  294. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  295. I. Timrov, N. Marzari, and M. Cococcioni, Self-consistent Hubbard parameters from density-functional perturbation theory in the ultrasoft and projector-augmented wave formulations, Phys. Rev. B 103, 045141 (2021).

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Phys. Rev. B 114, 235127– Published 26 October, 2026
  • Received 29 April 2026
  • Revised 18 August 2026
  • Accepted 9 September 2026
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