Skip to main content
SpringerLink
Log in

Seven useful questions in density functional theory

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We explore a variety of unsolved problems in density functional theory, where mathematicians might prove useful. We give the background and context of the different problems, and why progress toward resolving them would help those doing computations using density functional theory. Subjects covered include the magnitude of the kinetic energy in Hartree–Fock calculations, the shape of adiabatic connection curves, using the constrained search with input densities, densities of states, the semiclassical expansion of energies, the tightness of Lieb–Oxford bounds, and how we decide the accuracy of an approximate density.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability

The datasets generated during and/or analyzed during the current study are available upon request from the corresponding authors.

References

  1. Jain, A., Shin, Y., Persson, K.A.: Computational predictions of energy materials using density functional theory. Nat. Rev. Mater. 1(1), 15004 (2016). https://doi.org/10.1038/natrevmats.2015.4

    Article  ADS  Google Scholar 

  2. Pickard, C.J., Errea, I., Eremets, M.I.: Superconducting hydrides under pressure. Annu. Rev. Condens. Matter Phys. 11(1), 57–76 (2020). https://doi.org/10.1146/annurev-conmatphys-031218-013413

    Article  ADS  Google Scholar 

  3. Nørskov, J.K., Abild-Pedersen, F., Studt, F., Bligaard, T.: Density functional theory in surface chemistry and catalysis. Proc. Natl. Acad. Sci. 108(3), 937–943 (2011). https://doi.org/10.1073/pnas.1006652108

    Article  ADS  Google Scholar 

  4. Zeng, L., Jacobsen, S.B., Sasselov, D.D., Petaev, M.I., Vanderburg, A., Lopez-Morales, M., Perez-Mercader, J., Mattsson, T.R., Li, G., Heising, M.Z., Bonomo, A.S., Damasso, M., Berger, T.A., Cao, H., Levi, A., Wordsworth, R.D.: Growth model interpretation of planet size distribution. Proc. Natl. Acad. Sci. 116(20), 9723–9728 (2019). https://doi.org/10.1073/pnas.1812905116

    Article  ADS  Google Scholar 

  5. Hendon, C.H., Colonna-Dashwood, L., Colonna-Dashwood, M.: The role of dissolved cations in coffee extraction. J. Agric. Food Chem. 62(21), 4947–4950 (2014). https://doi.org/10.1021/jf501687c

    Article  Google Scholar 

  6. Pribram-Jones, A., Gross, D.A., Burke, K.: Dft: a theory full of holes? Annu. Rev. Phys. Chem. 66(1), 283–304 (2015). https://doi.org/10.1146/annurev-physchem-040214-121420. (PMID: 25830374)

    Article  ADS  Google Scholar 

  7. Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., van den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., Dieleman, S., Grewe, D., Nham, J., Kalchbrenner, N., Sutskever, I., Lillicrap, T., Leach, M., Kavukcuoglu, K., Graepel, T., Hassabis, D.: Mastering the game of Go with deep neural networks and tree search. Nature 529(7587), 484–489 (2016). https://doi.org/10.1038/nature16961

    Article  ADS  Google Scholar 

  8. Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Žídek, A., Potapenko, A., Bridgland, A., Meyer, C., Kohl, S.A.A., Ballard, A.J., Cowie, A., Romera-Paredes, B., Nikolov, S., Jain, R., Adler, J., Back, T., Petersen, S., Reiman, D., Clancy, E., Zielinski, M., Steinegger, M., Pacholska, M., Berghammer, T., Bodenstein, S., Silver, D., Vinyals, O., Senior, A.W., Kavukcuoglu, K., Kohli, P., Hassabis, D.: Highly accurate protein structure prediction with AlphaFold. Nature 596(7873), 583–589 (2021). https://doi.org/10.1038/s41586-021-03819-2

    Article  ADS  Google Scholar 

  9. Lovelock, S.L., Crawshaw, R., Basler, S., Levy, C., Baker, D., Hilvert, D., Green, A.P.: The road to fully programmable protein catalysis. Nature 606(7912), 49–58 (2022). https://doi.org/10.1038/s41586-022-04456-z

    Article  ADS  Google Scholar 

  10. Scheffler, M., Aeschlimann, M., Albrecht, M., Bereau, T., Bungartz, H.-J., Felser, C., Greiner, M., Groß, A., Koch, C.T., Kremer, K., Nagel, W.E., Scheidgen, M., Wöll, C., Draxl, C.: FAIR data enabling new horizons for materials research. Nature 604(7907), 635–642 (2022). https://doi.org/10.1038/s41586-022-04501-x

    Article  ADS  Google Scholar 

  11. Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E., Ghasemipour, S.K.S., Ayan, B.K., Mahdavi, S.S., Lopes, R.G., Salimans, T., Ho, J., Fleet, D.J., Norouzi, M.: Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding. arXiv (2022). https://doi.org/10.48550/ARXIV.2205.11487

  12. Chowdhery, A., Narang, S., Devlin, J., Bosma, M., Mishra, G., Roberts, A., Barham, P., Chung, H.W., Sutton, C., Gehrmann, S., Schuh, P., Shi, K., Tsvyashchenko, S., Maynez, J., Rao, A., Barnes, P., Tay, Y., Shazeer, N., Prabhakaran, V., Reif, E., Du, N., Hutchinson, B., Pope, R., Bradbury, J., Austin, J., Isard, M., Gur-Ari, G., Yin, P., Duke, T., Levskaya, A., Ghemawat, S., Dev, S., Michalewski, H., Garcia, X., Misra, V., Robinson, K., Fedus, L., Zhou, D., Ippolito, D., Luan, D., Lim, H., Zoph, B., Spiridonov, A., Sepassi, R., Dohan, D., Agrawal, S., Omernick, M., Dai, A.M., Pillai, T.S., Pellat, M., Lewkowycz, A., Moreira, E., Child, R., Polozov, O., Lee, K., Zhou, Z., Wang, X., Saeta, B., Diaz, M., Firat, O., Catasta, M., Wei, J., Meier-Hellstern, K., Eck, D., Dean, J., Petrov, S., Fiedel, N.: PaLM: Scaling Language Modeling with Pathways. arXiv (2022). https://doi.org/10.48550/ARXIV.2204.02311

  13. Preskill, J.: Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). https://doi.org/10.22331/q-2018-08-06-79

  14. Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., Biswas, R., Boixo, S., Brandao, F.G.S.L., Buell, D.A., Burkett, B., Chen, Y., Chen, Z., Chiaro, B., Collins, R., Courtney, W., Dunsworth, A., Farhi, E., Foxen, B., Fowler, A., Gidney, C., Giustina, M., Graff, R., Guerin, K., Habegger, S., Harrigan, M.P., Hartmann, M.J., Ho, A., Hoffmann, M., Huang, T., Humble, T.S., Isakov, S.V., Jeffrey, E., Jiang, Z., Kafri, D., Kechedzhi, K., Kelly, J., Klimov, P.V., Knysh, S., Korotkov, A., Kostritsa, F., Landhuis, D., Lindmark, M., Lucero, E., Lyakh, D., Mandrà, S., McClean, J.R., McEwen, M., Megrant, A., Mi, X., Michielsen, K., Mohseni, M., Mutus, J., Naaman, O., Neeley, M., Neill, C., Niu, M.Y., Ostby, E., Petukhov, A., Platt, J.C., Quintana, C., Rieffel, E.G., Roushan, P., Rubin, N.C., Sank, D., Satzinger, K.J., Smelyanskiy, V., Sung, K.J., Trevithick, M.D., Vainsencher, A., Villalonga, B., White, T., Yao, Z.J., Yeh, P., Zalcman, A., Neven, H., Martinis, J.M.: Quantum supremacy using a programmable superconducting processor. Nature 574(7779), 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5

    Article  ADS  Google Scholar 

  15. Zhong, H.-S., Wang, H., Deng, Y.-H., Chen, M.-C., Peng, L.-C., Luo, Y.-H., Qin, J., Wu, D., Ding, X., Hu, Y., Hu, P., Yang, X.-Y., Zhang, W.-J., Li, H., Li, Y., Jiang, X., Gan, L., Yang, G., You, L., Wang, Z., Li, L., Liu, N.-L., Lu, C.-Y., Pan, J.-W.: Quantum computational advantage using photons. Science 370(6523), 1460–1463 (2020). https://doi.org/10.1126/science.abe8770

    Article  ADS  Google Scholar 

  16. Cerezo, M., Arrasmith, A., Babbush, R., Benjamin, S.C., Endo, S., Fujii, K., McClean, J.R., Mitarai, K., Yuan, X., Cincio, L., Coles, P.J.: Variational quantum algorithms. Nat. Rev. Phys. 3(9), 625–644 (2021). https://doi.org/10.1038/s42254-021-00348-9

    Article  Google Scholar 

  17. Madsen, L.S., Laudenbach, F., Askarani, M.F., Rortais, F., Vincent, T., Bulmer, J.F.F., Miatto, F.M., Neuhaus, L., Helt, L.G., Collins, M.J., Lita, A.E., Gerrits, T., Nam, S.W., Vaidya, V.D., Menotti, M., Dhand, I., Vernon, Z., Quesada, N., Lavoie, J.: Quantum computational advantage with a programmable photonic processor. Nature 606(7912), 75–81 (2022). https://doi.org/10.1038/s41586-022-04725-x

    Article  ADS  Google Scholar 

  18. Sun, J., Ruzsinszky, A., Perdew, J.P.: Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115(3), 036402 (2015)

    Article  ADS  Google Scholar 

  19. Maitra, N.T., Burke, K., Appel, H., Gross, E.K.U., van Leeuwen, R.: Ten Topical Questions in Time-dependent Density Functional Theory, pp. 1186–1225. https://doi.org/10.1142/9789812775702_0040

  20. Ruzsinszky, A., Perdew, J.P.: Twelve outstanding problems in ground-state density functional theory: a bouquet of puzzles. Comput. Theor. Chem. 963(1), 2–6 (2011). https://doi.org/10.1016/j.comptc.2010.09.002

    Article  Google Scholar 

  21. Bach, V., Delle Site, L.: In: Bach, V., Delle Site, L. (eds.) On Some Open Problems in Many-Electron Theory, pp. 413–417. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06379-9_23

  22. Frank, R.L.: The Lieb–Thirring inequalities: recent results and open problems (2020). https://doi.org/10.48550/ARXIV.2007.09326

  23. Frank, R., Hundertmark, D., Jex, M., Nam, P.T.: The Lieb–Thirring inequality revisited. J. Eur. Math. Soc. 23(8), 2583–2600 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wrighton, J., Albavera-Mata, A., Rodriguez, H.F., Tan, T.S., Cancio, A.C., Dufty, J.W., Trickey, S.B.: Some Problems in Density Functional Theory (2022). https://doi.org/10.48550/ARXIV.2207.02213

  25. Englert, B.-G., Siedentop, H., Trappe, M.-I.: Mathematical Elements of Density Functional Theory, Chap. 1. World Scientific, Singapore (2023). https://doi.org/10.1142/13303

  26. Kato, T.: Fundamental properties of Hamiltonian operators of Schrodinger type. Trans. Am. Math. Soc. 70(2), 195–211 (1951). (Accessed 2023-01-30)

    MathSciNet  MATH  Google Scholar 

  27. Kummel, H.G.: A Biography of the Coupled Cluster Method, pp. 334–348. World Scientific, Singapore (2002). https://doi.org/10.1142/9789812777843_0040

  28. Umrigar, C.J., Nightingale, M.P.: Quantum Monte Carlo Methods in Physics and Chemistry, vol. 525. Springer, Berlin (1999)

    MATH  Google Scholar 

  29. Foulkes, W.M.C., Mitas, L., Needs, R.J., Rajagopal, G.: Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 73, 33–83 (2001). https://doi.org/10.1103/RevModPhys.73.33

    Article  ADS  Google Scholar 

  30. 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. 76(12), 6062–6065 (1979). https://doi.org/10.1073/pnas.76.12.6062

    Article  ADS  MathSciNet  Google Scholar 

  31. Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136(3B), 864–871 (1964). https://doi.org/10.1103/PhysRev.136.B864

    Article  ADS  MathSciNet  Google Scholar 

  32. Englisch, H., Englisch, R.: Exact density functionals for ground-state energies. I. General results. Phys. Status Solidi (b) 123(2), 711–721 (1984). https://doi.org/10.1002/pssb.2221230238

  33. Englisch, H., Englisch, R.: Exact density functionals for ground-state energies ii. Details and remarks. Phys. Status Solidi (b) 124(1), 373–379 (1984). https://doi.org/10.1002/pssb.2221240140

  34. van Leeuwen, R.: Density functional approach to the many-body problem: key concepts and exact functionals. Adv. Quantum Chem. 43, 25–94 (2003). https://doi.org/10.1016/S0065-3276(03)43002-5

  35. Eberhard Engel, R.M.D.: Density Functional Theory. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-14090-7

  36. Thomas, L.H.: The calculation of atomic fields. Math. Proc. Camb. Philos. Soc. 23(05), 542–548 (1927). https://doi.org/10.1017/S0305004100011683

    Article  ADS  MATH  Google Scholar 

  37. Fermi, E.: Un Metodo Statistico per la Determinazione di alcune Prioprietà dell’Atomo. Rend. Acc. Naz. Lincei 6, 66 (1927)

    Google Scholar 

  38. Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, 1133–1138 (1965). https://doi.org/10.1103/PhysRev.140.A1133

    Article  ADS  MathSciNet  Google Scholar 

  39. Wagner, L.O., Stoudenmire, E.M., Burke, K., White, S.R.: Guaranteed convergence of the Kohn–Sham equations. Phys. Rev. Lett. 111, 093003 (2013). https://doi.org/10.1103/PhysRevLett.111.093003

    Article  ADS  Google Scholar 

  40. Perdew, J.P., Schmidt, K.: Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conf. Proc. 577(1), 1–20 (2001). https://doi.org/10.1063/1.1390175

    Article  ADS  Google Scholar 

  41. Toulouse, J.: Review of approximations for the exchange-correlation energy in density-functional theory. In: Cancès, E., Friesecke, G. (Eds.) Density-Functional Theory. Springer, Berlin (2022)

  42. Perdew, J.P., Zunger, A.: Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23(10), 5048 (1981)

    Article  ADS  Google Scholar 

  43. 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. 58(8), 1200–1211 (1980)

    Article  ADS  Google Scholar 

  44. Perdew, J.P.: Unified theory of exchange and correlation beyond the local density approximation. In: Ziesche, P., Eschrig, H. (eds.) Electronic Structure of Solids ’91. Physical Research, vol. 17, pp. 11–20. Akademie Verlag, Berlin (1991)

    Google Scholar 

  45. Mardirossian, N., Head-Gordon, M.: Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals. Mol. Phys.. 115(19), 2315–2372 (2017). https://doi.org/10.1080/00268976.2017.1333644

    Article  ADS  Google Scholar 

  46. Mardirossian, N., Head-Gordon, M.: How accurate are the Minnesota density functionals for noncovalent interactions, isomerization energies, thermochemistry, and barrier heights involving molecules composed of main-group elements? J. Chem. Theory Comput. 12(9), 4303–4325 (2016). https://doi.org/10.1021/acs.jctc.6b00637. (PMID: 27537680)

    Article  Google Scholar 

  47. Wagner, L.O., Yang, Z., Burke, K., Marques, M.A.L., Maitra, N.T., Nogueira, F.M.S., Gross, E.K.U., Rubio, A. (Eds.): Exact Conditions and Their Relevance in TDDFT, pp. 101–123. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-23518-4_5

  48. Perdew, J.P., Sun, J.: The Lieb–Oxford Lower Bounds on the Coulomb Energy, Their Importance to Electron Density Functional Theory, and a Conjectured Tight Bound on Exchange. The Elliott Lieb Anniversary Volume (2022)

  49. 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. 74(1), 66 (2023). https://doi.org/10.1146/annurev-physchem-062422-013259

    Article  Google Scholar 

  50. Lieb, E.H., Simon, B.: Thomas–Fermi theory revisited. Phys. Rev. Lett. 31, 681–683 (1973). https://doi.org/10.1103/PhysRevLett.31.681

    Article  ADS  Google Scholar 

  51. Weizsäcker, C.F.V.: Zur theorie der kernmassen. Zeitschrift für Physik 96(7), 431–458 (1935). https://doi.org/10.1007/BF01337700

    Article  ADS  MATH  Google Scholar 

  52. Harris, J., Jones, R.O.: The surface energy of a bounded electron gas. J. Phys. F Met. Phys. 4(8), 1170–1186 (1974). https://doi.org/10.1088/0305-4608/4/8/013

    Article  ADS  Google Scholar 

  53. Langreth, D.C., Perdew, J.P.: The exchange-correlation energy of a metallic surface. Solid State Commun. 17(11), 1425–1429 (1975). https://doi.org/10.1016/0038-1098(75)90618-3

    Article  ADS  Google Scholar 

  54. Gunnarsson, O., Lundqvist, B.I.: Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B 13, 4274–4298 (1976). https://doi.org/10.1103/PhysRevB.13.4274

    Article  ADS  Google Scholar 

  55. Hubbard, J., Flowers, B.H.: Electron correlations in narrow energy bands. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 276(1365), 238–257 (1963). https://doi.org/10.1098/rspa.1963.0204

    Article  ADS  Google Scholar 

  56. Aimi, T., Imada, M.: Does simple two-dimensional Hubbard model account for high-tc superconductivity in copper oxides? J. Phys. Soc. Jpn. 76(11), 113708 (2007). https://doi.org/10.1143/JPSJ.76.113708

    Article  ADS  Google Scholar 

  57. Arovas, D.P., Berg, E., Kivelson, S.A., Raghu, S.: The Hubbard model. Annu. Rev. Condens. Matter Phys. 13(1), 239–274 (2022). https://doi.org/10.1146/annurev-conmatphys-031620-102024

    Article  ADS  Google Scholar 

  58. Mott, N.F.: The transition to the metallic state. Philos. Mag. J. Theor. Exp. Appl. Phys. 6(62), 287–309 (1961). https://doi.org/10.1080/14786436108243318

    Article  Google Scholar 

  59. Shastry, B.S.: Mott transition in the Hubbard model. Mod. Phys. Lett. B 6(23), 1427–1438 (1992). https://doi.org/10.1142/S0217984992001137

  60. Assaraf, R., Azaria, P., Caffarel, M., Lecheminant, P.: Metal-insulator transition in the one-dimensional \(\rm SU (n)\) Hubbard model. Phys. Rev. B 60, 2299–2318 (1999). https://doi.org/10.1103/PhysRevB.60.2299

    Article  ADS  Google Scholar 

  61. Kohno, M.: Mott transition in the two-dimensional Hubbard model. Phys. Rev. Lett. 108, 076401 (2012). https://doi.org/10.1103/PhysRevLett.108.076401

    Article  ADS  Google Scholar 

  62. Fuks, J.I., Farzanehpour, M., Tokatly, I.V., Appel, H., Kurth, S., Rubio, A.: Time-dependent exchange-correlation functional for a Hubbard dimer: quantifying nonadiabatic effects. Phys. Rev. A 88, 062512 (2013). https://doi.org/10.1103/PhysRevA.88.062512

    Article  ADS  Google Scholar 

  63. Fuks, J.I., Maitra, N.T.: Challenging adiabatic time-dependent density functional theory with a Hubbard dimer: the case of time-resolved long-range charge transfer. Phys. Chem. Chem. Phys. 16, 14504–14513 (2014). https://doi.org/10.1039/C4CP00118D

    Article  Google Scholar 

  64. Carrascal, D.J., Ferrer, J., Smith, J.C., Burke, K.: The Hubbard dimer: a density functional case study of a many-body problem. J. Phys. Condens. Matter 27(39), 393001 (2015). https://doi.org/10.1088/0953-8984/27/39/393001

    Article  Google Scholar 

  65. Deur, K., Mazouin, L., Fromager, E.: Exact ensemble density functional theory for excited states in a model system: investigating the weight dependence of the correlation energy. Phys. Rev. B 95, 035120 (2017). https://doi.org/10.1103/PhysRevB.95.035120

    Article  ADS  Google Scholar 

  66. Carrascal, D.J., Ferrer, J., Maitra, N., Burke, K.: Linear response time-dependent density functional theory of the Hubbard dimer. Eur. Phys. J. B 91(7), 142 (2018). https://doi.org/10.1140/epjb/e2018-90114-9

    Article  ADS  MathSciNet  Google Scholar 

  67. Burke, K., Kozlowski, J.: In: Pavarini, E., Koch, E. (eds.) Lies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer, pp. 65–96. Forschungszentrum Jülich GmbH Institute for Advanced Simulation, Jülich (2021)

  68. Pemmaraju, C.D., Deshmukh, A.: Levy–Lieb embedding of density-functional theory and its quantum kernel: Illustration for the Hubbard dimer using near-term quantum algorithms. Phys. Rev. A 106, 042807 (2022). https://doi.org/10.1103/PhysRevA.106.042807

    Article  ADS  MathSciNet  Google Scholar 

  69. Schuch, N., Verstraete, F.: Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nat. Phys. 5(10), 732–735 (2009). https://doi.org/10.1038/nphys1370

    Article  Google Scholar 

  70. Penz, M., van Leeuwen, R.: Density-functional theory on graphs. J. Chem. Phys. 155(24), 244111 (2021). https://doi.org/10.1063/5.0074249

    Article  ADS  Google Scholar 

  71. Perdew, J.P., Ernzerhof, M., Burke, K.: Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys. 105(22), 9982–9985 (1996). https://doi.org/10.1063/1.472933

    Article  ADS  Google Scholar 

  72. 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 32, 2010–2021 (1985). https://doi.org/10.1103/PhysRevA.32.2010

    Article  ADS  Google Scholar 

  73. Fuchs, M., Niquet, Y.-M., Gonze, X., Burke, K.: Describing static correlation in bond dissociation by Kohn–Sham density functional theory. J. Chem. Phys. 122(9), 094116 (2005). https://doi.org/10.1063/1.1858371

    Article  ADS  Google Scholar 

  74. Puzder, A., Chou, M.Y., Hood, R.Q.: Exchange and correlation in the Si atom: a quantum Monte Carlo study. Phys. Rev. A 64, 022501 (2001). https://doi.org/10.1103/PhysRevA.64.022501

    Article  ADS  Google Scholar 

  75. Ernzerhof, M., Burke, K., Perdew, J.P.: Density functional theory, the exchange hole, and the molecular bond. Theor. Comput. Chem. 4, 207–238 (1996). https://doi.org/10.1016/S1380-7323(96)80088-4

    Article  Google Scholar 

  76. Teale, A.M., Coriani, S., Helgaker, T.: Accurate calculation and modeling of the adiabatic connection in density functional theory. J. Chem. Phys. 132(16), 164115 (2010). https://doi.org/10.1063/1.3380834

    Article  ADS  Google Scholar 

  77. Peach, M.J.G., Teale, A.M., Tozer, D.J.: Modeling the adiabatic connection in h\(_2\). J. Chem. Phys. 126(24), 244104 (2007). https://doi.org/10.1063/1.2747248

    Article  ADS  Google Scholar 

  78. Frydel, D., Terilla, W.M., Burke, K.: Adiabatic connection from accurate wave-function calculations. J. Chem. Phys. 112(12), 5292–5297 (2000). https://doi.org/10.1063/1.481099

    Article  ADS  Google Scholar 

  79. Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996). https://doi.org/10.1103/PhysRevLett.77.3865

    Article  ADS  Google Scholar 

  80. Lieb, E.H., Oxford, S.: Improved lower bound on the indirect coulomb energy. Int. J. Quantum Chem. 19(3), 427–439 (1981). https://doi.org/10.1002/qua.560190306

    Article  Google Scholar 

  81. Perdew, J.P., Ruzsinszky, A., Sun, J., Burke, K.: Gedanken densities and exact constraints in density functional theory. J. Chem. Phys. 140(18), 18–533 (2014). https://doi.org/10.1063/1.4870763

    Article  Google Scholar 

  82. Bloch, F.: Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit. Zeitschrift für Physik 57(7), 545–555 (1929). https://doi.org/10.1007/BF01340281

    Article  ADS  MATH  Google Scholar 

  83. Dirac, P.A.M.: Note on exchange phenomena in the Thomas atom. Math. Proc. Camb. Philos. Soc. 26(3), 376–385 (1930). https://doi.org/10.1017/S0305004100016108

    Article  ADS  MATH  Google Scholar 

  84. Levy, M., Perdew, J.P.: Tight bound and convexity constraint on the exchange-correlation-energy functional in the low-density limit, and other formal tests of generalized-gradient approximations. Phys. Rev. B 48, 11638–11645 (1993). https://doi.org/10.1103/PhysRevB.48.11638

    Article  ADS  Google Scholar 

  85. Kin-Lic Chan, G., Handy, N.C.: Optimized Lieb–Oxford bound for the exchange-correlation energy. Phys. Rev. A 59, 3075–3077 (1999). https://doi.org/10.1103/PhysRevA.59.3075

    Article  ADS  Google Scholar 

  86. Lewin, M., Lieb, E.H., Seiringer, R.: Floating Wigner crystal with no boundary charge fluctuations. Phys. Rev. B 100, 035127 (2019). https://doi.org/10.1103/PhysRevB.100.035127

    Article  ADS  Google Scholar 

  87. Lewin, M., Lieb, E.H., Seiringer, R.: Improved Lieb–Oxford bound on the indirect and exchange energies (2022). https://doi.org/10.48550/ARXIV.2203.12473

  88. Lieb, E.H.: Density functionals for coulomb systems. Int. J. Quantum Chem. 24(3), 243–277 (1983). https://doi.org/10.1002/qua.560240302

    Article  Google Scholar 

  89. Seidl, M., Benyahia, T., Kooi, D.P., Gori-Giorgi, P.: The Lieb-Oxford bound and the optimal transport limit of DFT (2022). https://doi.org/10.48550/ARXIV.2202.10800

  90. Seidl, M., Vuckovic, S., Gori-Giorgi, P.: Challenging the Lieb–Oxford bound in a systematic way. Mol. Phys. 114(7–8), 1076–1085 (2016). https://doi.org/10.1080/00268976.2015.1136440

    Article  ADS  Google Scholar 

  91. Burke, K., Cancio, A., Gould, T., Pittalis, S.: Locality of correlation in density functional theory. J. Chem. Phys. 145(5), 054112 (2016). https://doi.org/10.1063/1.4959126

    Article  ADS  Google Scholar 

  92. Perdew, J.P., Constantin, L.A., Sagvolden, E., Burke, K.: Relevance of the slowly varying electron gas to atoms, molecules, and solids. Phys. Rev. Lett. 97, 223002 (2006). https://doi.org/10.1103/PhysRevLett.97.223002

    Article  ADS  Google Scholar 

  93. Frank, R.L., Merz, K., Siedentop, H.: The Scott conjecture for large coulomb systems: a review. Lett. Math. Phys. 113(1), 11 (2023). https://doi.org/10.1007/s11005-023-01631-9

    Article  ADS  MathSciNet  MATH  Google Scholar 

  94. Scott, J.M.C.: Lxxxii, the binding energy of the Thomas–Fermi atom. Lond. Edinb. Dublin Philos. Mag. J. Sci.43(343), 859–867 (1952). https://doi.org/10.1080/14786440808520234

  95. Schwinger, J.: Thomas–Fermi model: the leading correction. Phys. Rev. A 22, 1827–1832 (1980). https://doi.org/10.1103/PhysRevA.22.1827

    Article  ADS  MathSciNet  Google Scholar 

  96. Schwinger, J.: Thomas–Fermi model: the second correction. Phys. Rev. A 24, 2353–2361 (1981). https://doi.org/10.1103/PhysRevA.24.2353

    Article  ADS  MathSciNet  Google Scholar 

  97. Englert, B.-G., Schwinger, J.: Thomas–Fermi revisited: the outer regions of the atom. Phys. Rev. A 26, 2322–2329 (1982). https://doi.org/10.1103/PhysRevA.26.2322

    Article  ADS  Google Scholar 

  98. Englert, B.-G., Schwinger, J.: Statistical atom: some quantum improvements. Phys. Rev. A 29, 2339–2352 (1984). https://doi.org/10.1103/PhysRevA.29.2339

    Article  ADS  MathSciNet  Google Scholar 

  99. Englert, B.-G., Schwinger, J.: New statistical atom: a numerical study. Phys. Rev. A 29, 2353–2363 (1984). https://doi.org/10.1103/PhysRevA.29.2353

    Article  ADS  MathSciNet  Google Scholar 

  100. Englert, B.-G., Schwinger, J.: Statistical atom: handling the strongly bound electrons. Phys. Rev. A 29, 2331–2338 (1984). https://doi.org/10.1103/PhysRevA.29.2331

    Article  ADS  MathSciNet  Google Scholar 

  101. Kunz, H., Rueedi, R.: Atoms and quantum dots with a large number of electrons: the ground-state energy. Phys. Rev. A 81, 032122 (2010). https://doi.org/10.1103/PhysRevA.81.032122

    Article  ADS  Google Scholar 

  102. Englert, B.-G., Schwinger, J.: Semiclassical atom. Phys. Rev. A 32, 26–35 (1985). https://doi.org/10.1103/PhysRevA.32.26

    Article  ADS  Google Scholar 

  103. Lieb, E.H., Simon, B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23(1), 22–116 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  104. Conlon, J.: Semi-classical limit theorems for Hartree–Fock theory. Commun. Math. Phys. 88(1), 133–150 (1983). https://doi.org/10.1007/BF01206884

    Article  ADS  MATH  Google Scholar 

  105. Fefferman, C., Seco, L.A.: On the Dirac and Schwinger corrections to the ground-state energy of an atom. Adv. Math. 107(1), 1–185 (1994). https://doi.org/10.1006/aima.1994.1060

    Article  MathSciNet  MATH  Google Scholar 

  106. Gontier, D., Hainzl, C., Lewin, M.: Lower bound on the Hartree–Fock energy of the electron gas. Phys. Rev. A 99, 052501 (2019). https://doi.org/10.1103/PhysRevA.99.052501

    Article  ADS  Google Scholar 

  107. Heilmann, O.J., Lieb, E.H.: Electron density near the nucleus of a large atom. Phys. Rev. A 52, 3628–3643 (1995). https://doi.org/10.1103/PhysRevA.52.3628

    Article  ADS  Google Scholar 

  108. Argaman, N., Redd, J., Cancio, A.C., Burke, K.: Leading correction to the local density approximation for exchange in large-\(z\) atoms. Phys. Rev. Lett. 129, 153001 (2022). https://doi.org/10.1103/PhysRevLett.129.153001

    Article  ADS  Google Scholar 

  109. Merz, K., Siedentop, H.: The atomic density on the Thomas—fermi length scale for the Chandrasekhar Hamiltonian. Rep. Math. Phys. 83(3), 387–391 (2019). https://doi.org/10.1016/s0034-4877(19)30057-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  110. Daas, T.J., Kooi, D.P., Grooteman, A.J.A.F., Seidl, M., Gori-Giorgi, P.: Gradient expansions for the large-coupling strength limit of the Møller–Plesset adiabatic connection. J. Chem. Theory Comput. 18(3), 1584–1594 (2022). https://doi.org/10.1021/acs.jctc.1c01206. (PMID: 35179386)

    Article  Google Scholar 

  111. Lieb, E.H.: Thomas–Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981). https://doi.org/10.1103/RevModPhys.53.603

    Article  ADS  MathSciNet  MATH  Google Scholar 

  112. Englert, B.-G.: Semiclassical Theory of Atoms vol. 300. Springer, Berlin (1988). https://doi.org/10.1007/3-540-19204-2

  113. Friesecke, G., Goddard, B.D.: Explicit large nuclear charge limit of electronic ground states for li, be, b, c, n, o, f, ne and basic aspects of the periodic table. SIAM J. Math. Anal. 41(2), 631–664 (2009). https://doi.org/10.1137/080729050

    Article  MathSciNet  MATH  Google Scholar 

  114. Friesecke, G., Goddard, B.D.: Atomic structure via highly charged ions and their exact quantum states. Phys. Rev. A 81, 032516 (2010). https://doi.org/10.1103/PhysRevA.81.032516

    Article  ADS  Google Scholar 

  115. Constantin, L.A., Snyder, J.C., Perdew, J.P., Burke, K.: Communication: ionization potentials in the limit of large atomic number. J. Chem. Phys. 133(24), 241103 (2010). https://doi.org/10.1063/1.3522767

    Article  ADS  Google Scholar 

  116. Burke, K., Perdew, J.P., Wang, Y.: In: Dobson, J.F., Vignale, G., Das, M.P. (Eds.) Derivation of a Generalized Gradient Approximation: The PW91 Density Functional, pp. 81–111. Springer, Boston, MA (1998). https://doi.org/10.1007/978-1-4899-0316-7_7

  117. Cancio, A., Chen, G.P., Krull, B.T., Burke, K.: Fitting a round peg into a round hole: asymptotically correcting the generalized gradient approximation for correlation. J. Chem. Phys. 149(8), 084116 (2018). https://doi.org/10.1063/1.5021597

    Article  ADS  Google Scholar 

  118. Frydel, D., Terilla, W.M., Burke, K.: Adiabatic connection from accurate wave-function calculations. J. Chem. Phys. 112(12), 5292–5297 (2000). https://doi.org/10.1063/1.481099

    Article  ADS  Google Scholar 

  119. Bach, V., Lieb, E.H., Loss, M., Solovej, J.P.: There are no unfilled shells in unrestricted Hartree–Fock theory. Phys. Rev. Lett. 72, 2981–2983 (1994). https://doi.org/10.1103/PhysRevLett.72.2981

    Article  ADS  Google Scholar 

  120. Bach, V.: Hartree–Fock theory, Lieb’s variational principle, and their generalizations. In: The Physics and Mathematics of Elliott Lieb, pp. 19–65. EMS Press, Berlin (2022)

  121. Gross, E.K.U., Petersilka, M., Grabo, T.: 3. Conventional Quantum Chemical Correlation Energy Versus Density-Functional Correlation Energy, pp. 42–53. https://doi.org/10.1021/bk-1996-0629.ch003

  122. Crisostomo, S., Levy, M., Burke, K.: Can the Hartree–Fock kinetic energy exceed the exact kinetic energy? J. Chem. Phys. 157(15), 154106 (2022). https://doi.org/10.1063/5.0105684

    Article  ADS  Google Scholar 

  123. Gill, P.M.W., Johnson, B.G., Pople, J.A., Frisch, M.J.: An investigation of the performance of a hybrid of Hartree–Fock and density functional theory. Int. J. Quantum Chem. 44(S26), 319–331 (1992). https://doi.org/10.1002/qua.560440828

    Article  Google Scholar 

  124. Song, S., Vuckovic, S., Sim, E., Burke, K.: Density-corrected DFT explained: questions and answers. J. Chem. Theory Comput. 18(2), 817–827 (2022). https://doi.org/10.1021/acs.jctc.1c01045. (PMID: 35048707)

    Article  Google Scholar 

  125. Nam, S., McCarty, R.J., Park, H., Sim, E.: Ks-pies: Kohn–Sham inversion toolkit. J. Chem. Phys. 154(12), 124122 (2021). https://doi.org/10.1063/5.0040941

    Article  ADS  Google Scholar 

  126. Nam, S., Song, S., Sim, E., Burke, K.: Measuring density-driven errors using Kohn–Sham inversion. J. Chem. Theory Comput. 16(8), 5014–5023 (2020). https://doi.org/10.1021/acs.jctc.0c00391. (PMID: 32667787)

    Article  Google Scholar 

  127. Garrigue, L.: Some properties of the potential-to-ground state map in quantum mechanics. Commun. Math. Phys. 386(3), 1803–1844 (2021). https://doi.org/10.1007/s00220-021-04140-9

    Article  ADS  MathSciNet  MATH  Google Scholar 

  128. Garrigue, L.: Building Kohn–Sham potentials for ground and excited states. Arch. Rational Mech. Anal. 245(2), 949–1003 (2022). https://doi.org/10.1007/s00205-022-01804-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  129. Shi, Y., Chávez, V.H., Wasserman, A.: n2v: a density-to-potential inversion suite. a sandbox for creating, testing, and benchmarking density functional theory inversion methods. WIREs Comput. Mol. Sci. 12(6), 1617 (2022). https://doi.org/10.1002/wcms.1617

    Article  Google Scholar 

  130. Lehtola, S., Steigemann, C., Oliveira, M.J., Marques, M.A.: Recent developments in libxc—a comprehensive library of functionals for density functional theory. SoftwareX 7, 1–5 (2018). https://doi.org/10.1016/j.softx.2017.11.002

    Article  ADS  Google Scholar 

  131. Medvedev, M.G., Bushmarinov, I.S., Sun, J., Perdew, J.P., Lyssenko, K.A.: Density functional theory is straying from the path toward the exact functional. Science 355(6320), 49–52 (2017). https://doi.org/10.1126/science.aah5975

    Article  ADS  Google Scholar 

  132. Kim, M.-C., Sim, E., Burke, K.: Understanding and reducing errors in density functional calculations. Phys. Rev. Lett. 111, 073003 (2013). https://doi.org/10.1103/PhysRevLett.111.073003

    Article  ADS  Google Scholar 

  133. Wasserman, A., Nafziger, J., Jiang, K., Kim, M.-C., Sim, E., Burke, K.: The importance of being inconsistent. Annu. Rev. Phys. Chem. 68(1), 555–581 (2017). https://doi.org/10.1146/annurev-physchem-052516-044957. (PMID: 28463652)

    Article  ADS  Google Scholar 

  134. Sim, E., Song, S., Vuckovic, S., Burke, K.: Improving results by improving densities: density-corrected density functional theory. J. Am. Chem. Soc. 144(15), 6625–6639 (2022). https://doi.org/10.1021/jacs.1c11506

    Article  Google Scholar 

  135. Kaplan, A.D., Shahi, C., Bhetwal, P., Sah, R.K., Perdew, J.P.: Understanding density-driven errors for reaction barrier heights. J. Chem. Theory Comput. 19(2), 532–543 (2023). https://doi.org/10.1021/acs.jctc.2c00953. (PMID: 36599075)

    Article  Google Scholar 

  136. Vuckovic, S., Song, S., Kozlowski, J., Sim, E., Burke, K.: Density functional analysis: the theory of density-corrected DFT. J. Chem. Theory Comput. 15(12), 6636–6646 (2019). https://doi.org/10.1021/acs.jctc.9b00826. (PMID: 31682433)

    Article  Google Scholar 

  137. Harriman, J.E.: Orthonormal orbitals for the representation of an arbitrary density. Phys. Rev. A 24, 680–682 (1981). https://doi.org/10.1103/PhysRevA.24.680

    Article  ADS  Google Scholar 

  138. Zumbach, G., Maschke, K.: New approach to the calculation of density functionals. Phys. Rev. A 28, 544–554 (1983). https://doi.org/10.1103/PhysRevA.28.544

    Article  ADS  MathSciNet  Google Scholar 

  139. Bokanowski, O., Grebert, B.: A decomposition theorem for wave functions in molecular quantum chemistry. Math. Models Methods Appl. Sci. 06(04), 437–466 (1996). https://doi.org/10.1142/S021820259600016X

    Article  MathSciNet  MATH  Google Scholar 

  140. Jones, R.O., Gunnarsson, O.: The density functional formalism, its applications and prospects. Rev. Mod. Phys. 61, 689–746 (1989). https://doi.org/10.1103/RevModPhys.61.689

    Article  ADS  Google Scholar 

  141. Kotochigova, S., Levine, Z.H., Shirley, E.L., Stiles, M.D., Clark, C.W.: Local-density-functional calculations of the energy of atoms. Phys. Rev. A 55, 191–199 (1997). https://doi.org/10.1103/PhysRevA.55.191

    Article  ADS  Google Scholar 

  142. Painter, G.S., Averill, F.W.: Bonding in the first-row diatomic molecules within the local spin-density approximation. Phys. Rev. B 26, 1781–1790 (1982). https://doi.org/10.1103/PhysRevB.26.1781

    Article  ADS  Google Scholar 

  143. Curtiss, L.A., Raghavachari, K., Redfern, P.C., Pople, J.A.: Assessment of Gaussian-2 and density functional theories for the computation of enthalpies of formation. J. Chem. Phys. 106(3), 1063–1079 (1997). https://doi.org/10.1063/1.473182

    Article  ADS  Google Scholar 

  144. Fahy, S., Wang, X.W., Louie, S.G.: Pair-correlation function and single-particle occupation numbers in diamond and silicon. Phys. Rev. Lett. 65, 1478–1481 (1990). https://doi.org/10.1103/PhysRevLett.65.1478

    Article  ADS  Google Scholar 

  145. Hood, R.Q., Chou, M.Y., Williamson, A.J., Rajagopal, G., Needs, R.J.: Exchange and correlation in silicon. Phys. Rev. B 57, 8972–8982 (1998). https://doi.org/10.1103/PhysRevB.57.8972

    Article  ADS  Google Scholar 

  146. Colonna, F., Savin, A.: Correlation energies for some two- and four-electron systems along the adiabatic connection in density functional theory. J. Chem. Phys. 110(6), 2828–2835 (1999). https://doi.org/10.1063/1.478234

    Article  ADS  Google Scholar 

  147. Filippi, C., Gonze, X., Umrigar, C.J.: Generalized Gradient Approximations to Density Functional Theory: Comparison with Exact Results, pp. 295–326. Elsevier, Amsterdam (1996). https://doi.org/10.48550/ARXIV.COND-MAT/9607046

  148. Wagner, L.O., Baker, T.E., Stoudenmire, E.M., Burke, K., White, S.R.: Kohn–Sham calculations with the exact functional. Phys. Rev. B 90, 045109 (2014). https://doi.org/10.1103/PhysRevB.90.045109

    Article  ADS  Google Scholar 

  149. Savin, A., Colonna, F., Pollet, R.: Adiabatic connection approach to density functional theory of electronic systems. Int. J. Quantum Chem. 93(3), 166–190 (2003). https://doi.org/10.1002/qua.10551

    Article  Google Scholar 

  150. Delle Site, L.: Levy–Lieb principle: the bridge between the electron density of density functional theory and the wavefunction of quantum Monte Carlo. Chem. Phys. Lett. 619, 148–151 (2015). https://doi.org/10.1016/j.cplett.2014.11.060

    Article  ADS  Google Scholar 

  151. Delle Site, L., Ghiringhelli, L.M., Ceperley, D.M.: Electronic energy functionals: Levy–Lieb principle within the ground state path integral quantum Monte Carlo. Int. J. Quantum Chem. 113(2), 155–160 (2013). https://doi.org/10.1002/qua.24321

    Article  Google Scholar 

  152. Cohen, A.J., Mori-Sánchez, P.: Landscape of an exact energy functional. Phys. Rev. A 93, 042511 (2016). https://doi.org/10.1103/PhysRevA.93.042511

    Article  ADS  Google Scholar 

  153. D’Amico, I., Coe, J.P., Franca, V.V., Capelle, K.: Quantum mechanics in metric space: wave functions and their densities. Phys. Rev. Lett. 106, 050401 (2011). https://doi.org/10.1103/PhysRevLett.106.050401

    Article  ADS  MathSciNet  Google Scholar 

  154. Sharp, P.M., D’Amico, I.: Metric-space approach to potentials and its relevance to density-functional theory. Phys. Rev. A 94, 062509 (2016). https://doi.org/10.1103/PhysRevA.94.062509

    Article  ADS  Google Scholar 

  155. Levy, M., Perdew, J.P., Sahni, V.: Exact differential equation for the density and ionization energy of a many-particle system. Phys. Rev. A 30, 2745–2748 (1984). https://doi.org/10.1103/PhysRevA.30.2745

    Article  ADS  Google Scholar 

  156. Lacombe, L., Maitra, N.T.: Embedding via the exact factorization approach. Phys. Rev. Lett. 124, 206401 (2020). https://doi.org/10.1103/PhysRevLett.124.206401

    Article  ADS  MathSciNet  Google Scholar 

  157. Requist, R., Gross, E.K.U.: Fock-space embedding theory: application to strongly correlated topological phases. Phys. Rev. Lett. 127, 116401 (2021). https://doi.org/10.1103/PhysRevLett.127.116401

    Article  ADS  MathSciNet  Google Scholar 

  158. Wesolowski, T.A., Wang, Y.A.: Recent Progress in Orbital-Free Density Functional Theory. World Scientific, Singapore (2013). https://doi.org/10.1142/8633

  159. Karasiev, V.V., Trickey, S.B.: Chapter Nine—Frank Discussion of the Status of Ground-State Orbital-Free DFT, vol. 71, pp. 221–245 (2015). https://doi.org/10.1016/bs.aiq.2015.02.004

  160. Teller, E.: On the stability of molecules in the Thomas–Fermi theory. Rev. Mod. Phys. 34, 627–631 (1962). https://doi.org/10.1103/RevModPhys.34.627

    Article  ADS  MATH  Google Scholar 

  161. Brack, M., Bhaduri, R.K.: Semiclassical physics. Front. Phys. 96, 458 (2003)

    MathSciNet  MATH  Google Scholar 

  162. Landry, B.R., Wasserman, A., Heller, E.J.: Semiclassical ground-state energies of many-electron systems. Phys. Rev. Lett. 103, 066401 (2009). https://doi.org/10.1103/PhysRevLett.103.066401

    Article  ADS  Google Scholar 

  163. March, N.H., Plaskett, J.S., Coulson, C.A.: The relation between the Wentzel–Kramers–Brillouin and the Thomas–Fermi approximations. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 235(1202), 419–431 (1956). https://doi.org/10.1098/rspa.1956.0094

    Article  ADS  MathSciNet  MATH  Google Scholar 

  164. Berry, M., Burke, K.: Exact and approximate energy sums in potential wells. J. Phys. A Math. Theor. 53(9), 095203 (2020). https://doi.org/10.1088/1751-8121/ab69a6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  165. Arendt, W., Nittka, R., Peter, W., Steiner, F.: 1. Weyl’s Law: Spectral Properties of the Laplacian in Mathematics and Physics, pp. 1–71. Wiley, New York (2009). https://doi.org/10.1002/9783527628025.ch1

  166. Bennewitz, C., Brown, M., Weikard, R.: Spectral and Scattering Theory for Ordinary Differential Equations. Vol. I: Sturm–Liouville Equations. Universitext, p. 379. Springer, Berlin (2020). https://doi.org/10.1007/978-3-030-59088-8

  167. Bhaduri, R.K., Brack, M.: Semiclassical Physics. Frontiers in Physics. Westview Press, Philadelphia (2003)

    MATH  Google Scholar 

  168. Weyl, H.: Über die asymptotische verteilung der eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1911, 110–117 (1911)

    MATH  Google Scholar 

  169. Hörmander, L.: The Analysis of Linear Partial Differential Operators. IV, Fourier Integral Operators. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, p. 352. Springer, Berlin (1985)

  170. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. London Mathematical Society Lecture Note Series, vol. 268, p. 227. Cambridge University Press, Cambridge (1999). https://doi.org/10.1017/CBO9780511662195

  171. Ivrii, V.: Microlocal Analysis, Sharp Spectral Asymptotics and Applications. I, p. 883. Springer, Berlin (2019). Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics

  172. Frank, R.L.: Cwikel’s theorem and the clr inequality. J. Spect. Theory 4(1), 1–21 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  173. Rozenblum, G., Solomyak, M.: In: Maz’ya, V. (Ed.) Counting Schrödinger Boundstates: Semiclassics and Beyond, pp. 329–353. Springer, New York (2009). https://doi.org/10.1007/978-0-387-85650-6_14

  174. Okun, P., Burke, K.: Uncommonly accurate energies for the general quartic oscillator. Int. J. Quantum Chem. 121(7), 26554 (2021). https://doi.org/10.1002/qua.26554

    Article  Google Scholar 

  175. Okun, P., Burke, K.: Semiclassics: the hidden theory behind the success of dft. In: Englert, B.-G. (Ed.) DFMPS 2019 Proceedings (2021). https://doi.org/10.48550/ARXIV.2105.04384

Download references

Acknowledgements

S.C., B. K. and K. B. acknowledge support from NSF grant No. CHE-2154371. R.P. acknowledges support from DOE DE-SC0008696. J.K. acknowledges support from DOE Award No. DE-FG02-08ER46496. K.D. acknowledges support from NSF grant DMS-1708511. A.W. acknowledges support from NSF grant CHE-1900301.

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of California, Irvine, CA, 92697, USA

    Steven Crisostomo, Ryan Pederson & Kieron Burke

  2. Department of Chemistry, University of California, Irvine, CA, 92697, USA

    John Kozlowski, Bhupalee Kalita, Suhwan Song & Kieron Burke

  3. Department of Physics and Astronomy, Ball State University, Muncie, IN, 47304, USA

    Antonio C. Cancio

  4. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Kiril Datchev

  5. Department of Chemistry, Purdue University, West Lafayette, IN, 47907, USA

    Adam Wasserman

  6. Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47907, USA

    Adam Wasserman

  7. Department of Chemistry, Yonsei University, Seoul, 03722, Korea

    Suhwan Song

Authors
  1. Steven Crisostomo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryan Pederson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John Kozlowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bhupalee Kalita
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Antonio C. Cancio
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kiril Datchev
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Adam Wasserman
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Suhwan Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Kieron Burke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Steven Crisostomo, Antonio C. Cancio, Kiril Datchev, Adam Wasserman or Kieron Burke.

Ethics declarations

Conflicts of interest

The authors have no conflicts to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the themed collection: Mathematical Physics and Numerical Simulation of Many-Particle Systems; V. Bach and L. Delle Site (eds.)..

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Crisostomo, S., Pederson, R., Kozlowski, J. et al. Seven useful questions in density functional theory. Lett Math Phys 113, 42 (2023). https://doi.org/10.1007/s11005-023-01665-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-023-01665-z

Keywords

Mathematics Subject Classification

Search

Navigation