Recent Developments in Density Functional Approximations

Reference work entry


We survey some of the standard approximations used in density functional calculations, most of which are at least 20 years old, and some new approaches that have been developed since.


  1. Aryasetiawan F, Gunnarsson O (1998) The GW method. Rep Prog Phys 61(3):237. ADSCrossRefGoogle Scholar
  2. Bartók AP, Payne MC, Kondor R, Csányi G (2010) Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. Phys Rev Lett 104:136403. ADSCrossRefGoogle Scholar
  3. Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38(6):3098–3100. ADSCrossRefGoogle Scholar
  4. Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98(7):5648–5652. Google Scholar
  5. Berland K, Cooper VR, Lee K, Schröder E, Thonhauser T, Hyldgaard P, Lundqvist BI (2015) Van der Waals forces in density functional theory: a review of the vdW-DF method. Rep Prog Phys 78(6):066501ADSCrossRefGoogle Scholar
  6. Bloch F (1929) Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für Physik A Hadrons and Nuclei 52(7):555–600zbMATHGoogle Scholar
  7. Brockherde F, Vogt L, Li L, Tuckerman ME, Burke K, Mller KR (2017) Bypassing the Kohn-Sham equations with machine learning. Nature Commun. CrossRefGoogle Scholar
  8. Burke K (2012) Perspective on density functional theory. J Chem Phys 136.
  9. Burke K (2016) Viewpoint: improving electronic structure calculations. Physics 9(108).
  10. Burke K, Perdew JP, Wang Y (1997) Derivation of a generalized gradient approximation: the PW91 density functional, Plenum, p 81.
  11. Burke K, Werschnik J, Gross EKU (2005) Time-dependent density functional theory: past, present, and future. J Chem Phys 123(6):062206. ADSCrossRefGoogle Scholar
  12. Burke K, Cancio A, Gould T, Pittalis S (2016) Locality of correlation in density functional theory. J Chem Phys 145(5):054112.,
  13. Burns LA, Faver JC, Zheng Z, Marshall MS, Smith DG, Vanommeslaeghe K, MacKerell AD Jr, Merz KM Jr, Sherrill CD (2017) The biofragment database (BFDb): an open-data platform for computational chemistry analysis of noncovalent interactions. J Chem Phys 147(16):161727ADSCrossRefGoogle Scholar
  14. Carrascal DJ, Ferrer J (2012) Exact Kohn-Sham eigenstates versus quasiparticles in simple models of strongly correlated electrons. Phys Rev B 85:045110. ADSCrossRefGoogle Scholar
  15. Carrascal DJ, Ferrer J, Smith JC, Burke K (2015) The Hubbard dimer: a density functional case study of a many-body problem. J Phys Condens Matter 27(39):393001. CrossRefGoogle Scholar
  16. Casida ME (1996) Time-dependent density functional response theory of molecular systems: theory, computational methods, and functionals. In: Seminario JM (ed) Recent developments and applications in density functional theory. Elsevier, AmsterdamGoogle Scholar
  17. Ceperley DM, Alder BJ (1980) Ground state of the electron gas by a stochastic method. Phys Rev Lett 45:566ADSCrossRefGoogle Scholar
  18. Chen GP, Voora VK, Agee MM, Balasubramani SG, Furche F (2017) Random-phase approximation methods. Ann Rev Phys Chem 68(1):421–445., pMID: 28301757ADSCrossRefGoogle Scholar
  19. Cohen AJ, Mori-Sánchez P, Yang W (2008) Insights into current limitations of density functional theory. Science 321(5890):792–794ADSCrossRefGoogle Scholar
  20. Curtiss LA, Redfern PC, Raghavachari K (2005) Assessment of gaussian-3 and density-functional theories on the g3/05 test set of experimental energies. J Chem Phys 123:124107ADSCrossRefGoogle Scholar
  21. Dion M, Rydberg H, Schröder E, Langreth DC, Lundqvist BI (2004) Van der Waals density functional for general geometries. Phys Rev Lett 92(24):246401. ADSCrossRefGoogle Scholar
  22. Elliott P, Burke K (2009) Non-empirical derivation of the parameter in the B88 exchange functional. Can J Chem Ecol 87(10):1485–1491. CrossRefGoogle Scholar
  23. Engel E, Dreizler RM (2011) Density functional theory: an advanced course. Springer, BerlinzbMATHCrossRefGoogle Scholar
  24. Erhard J, Bleiziffer P, Görling A (2016) Power series approximation for the correlation kernel leading to Kohn-Sham methods combining accuracy, computational efficiency, and general applicability. Phys Rev Lett 117:143002. ADSMathSciNetCrossRefGoogle Scholar
  25. Ernzerhof M, Scuseria GE (1999) Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional. J Chem Phys 110:5029ADSCrossRefGoogle Scholar
  26. Eshuis H, Yarkony J, Furche F (2010) Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. The Journal of Chemical Physics 132(23):234114ADSCrossRefGoogle Scholar
  27. Flick J, Ruggenthaler M, Appel H, Rubio A (2015) Kohn-Sham approach to quantum electrodynamical density-functional theory: exact time-dependent effective potentials in real space. Proc Nat Acad Sci 112(50):15285–15290ADSCrossRefGoogle Scholar
  28. Furche F (2001) Molecular tests of the random phase approximation to the exchange-correlation energy functional. Phys Rev B 64:195120ADSCrossRefGoogle Scholar
  29. Furche F (2008) Developing the random phase approximation into a practical post-Kohn–Sham correlation model. J Chem Phys 129(11):114105ADSCrossRefGoogle Scholar
  30. Gill PMW, Johnson BG, Pople JA, Frisch MJ (1992) An investigation of the performance of a hybrid of hartree-fock and density functional theory. Int J Quantum Chem 44(S26):319–331. CrossRefGoogle Scholar
  31. Grimme S (2006) Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J Comput Chem 27(15):1787–1799CrossRefGoogle Scholar
  32. Gross EKU, Kohn W (1985) Local density-functional theory of frequency-dependent linear response. Phys Rev Lett 55:2850ADSCrossRefGoogle Scholar
  33. Groth S, Dornheim T, Sjostrom T, Malone FD, Foulkes WMC, Bonitz M (2017) Ab initio exchange-correlation free energy of the uniform electron gas at warm dense matter conditions. Phys Rev Lett 119:135001. ADSCrossRefGoogle Scholar
  34. Grüning M, Marini M, Rubio A (2006) Density functionals from many-body perturbation theory: the band gap for semiconductors and insulators. J Chem Phys 124:154108ADSCrossRefGoogle Scholar
  35. Gunnarsson O, Lundqvist B (1976) Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys Rev B 13:4274ADSCrossRefGoogle Scholar
  36. Harris J, Jones R (1974) The surface energy of a bounded electron gas. J Phys F 4:1170ADSCrossRefGoogle Scholar
  37. Heitler W, London F (1927) Interaction between neutral atoms and homopolar binding according to quantum mechanics. Z Physik 44:455ADSCrossRefGoogle Scholar
  38. Hermann J, DiStasio RA, Tkatchenko A (2017) First-principles models for van der Waals interactions in molecules and materials: concepts, theory, and applications. Chem Rev 117(6):4714–4758., pMID: 28272886CrossRefGoogle Scholar
  39. Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a screened coulomb potential. J Chem Phys 118(18):8207–8215. ADSCrossRefGoogle Scholar
  40. Janesko BG, Henderson TM, Scuseria GE (2009) Screened hybrid density functionals for solid-state chemistry and physics. Phys Chem Chem Phys 11(3):443–454CrossRefGoogle Scholar
  41. Johnson ER, Becke AD (2006) Van der waals interactions from the exchange hole dipole moment: application to bio-organic benchmark systems. Chem Phys Lett 432(4–6):600–603ADSCrossRefGoogle Scholar
  42. Johnson ER, Becke AD (2017) Communication: DFT treatment of strong correlation in 3D trans- ition-metal diatomics. J Chem Phys 146(21):211105ADSCrossRefGoogle Scholar
  43. Jones R, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61:689ADSCrossRefGoogle Scholar
  44. Jurecka P, Sponer J, Cerny J, Hobza P (2006) Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs. Phys Chem Chem Phys 8:1985–1993CrossRefGoogle Scholar
  45. Klimeš J, Bowler DR, Michaelides A (2009) Chemical accuracy for the Van der Waals density functional. J Phys Condens Matter 22(2):022201ADSCrossRefGoogle Scholar
  46. Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140(4A):A1133–A1138. ADSMathSciNetCrossRefGoogle Scholar
  47. Kümmel S, Kronik L (2008) Orbital-dependent density functionals: theory and applications. Rev Mod Phys 80(1):3–60. ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. Langreth D, Mehl M (1981) Easily implementable nonlocal exchange-correlation enery functional. Phys Rev Lett 47:446. ADSCrossRefGoogle Scholar
  49. Langreth D, Mehl M (1983) Beyond the local-density approximation in calculations of ground-state electronic properties. Phys Rev B 28:1809. ADSCrossRefGoogle Scholar
  50. Langreth D, Perdew J (1975) The exchange-correlation energy of a metallic surface. Solid State Commun 17:1425ADSCrossRefGoogle Scholar
  51. Langreth D, Perdew J (1977) Exchange-correlation energy of a metallic surface: wave-vector analysis. Phys Rev B 15:2884ADSCrossRefGoogle Scholar
  52. Lee C, Yang W, Parr RG (1988) Development of the colle-salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 37(2):785–789. ADSCrossRefGoogle Scholar
  53. Leininger T, Stoll H, Werner HJ, Savin A (1997) Combining long-range configuration interaction with short-range density functionals. Chem Phys Lett 275(3-4):151–160ADSCrossRefGoogle Scholar
  54. Li L, Baker TE, White SR, Burke K (2016a) Pure density functional for strong correlation and the thermodynamic limit from machine learning. Phys Rev B 94:245129. ADSCrossRefGoogle Scholar
  55. Li L, Snyder JC, Pelaschier IM, Huang J, Niranjan UN, Duncan P, Rupp M, Mller KR, Burke K (2016b) Understanding machine-learned density functionals. Int J Quantum Chem 116(11):819–833. CrossRefGoogle Scholar
  56. Ma SK, Brueckner K (1968) Correlation energy of an electron gas with a slowly varying high density. Phys Rev 165:18ADSCrossRefGoogle Scholar
  57. Maitra NT (2016) Perspective: Fundamental aspects of time-dependent density functional theory. J Chem Phys 144(22):220901. ADSCrossRefGoogle Scholar
  58. Maitra NT, Zhang F, Cave RJ, Burke K (2004) Double excitations within time-dependent density functional theory linear response. J Chem Phys 120(13):5932–5937. ADSCrossRefGoogle Scholar
  59. Martin RM, Reining L, Ceperley DM (2016) Interacting electrons. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  60. Medvedev MG, Bushmarinov IS, Sun J, Perdew JP, Lyssenko KA (2017) Response to comment on density functional theory is straying from the path toward the exact functional. Science 356(6337):496–496ADSCrossRefGoogle Scholar
  61. Motta M, Ceperley DM, Chan GKL, Gomez JA, Gull E, Guo S, Jiménez-Hoyos CA, Lan TN, Li J, Ma F, Millis AJ, Prokof’ev NV, Ray U, Scuseria GE, Sorella S, Stoudenmire EM, Sun Q, Tupitsyn IS, White SR, Zgid D, Zhang S (2017) Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods. Phys Rev X 7:031059. Google Scholar
  62. Onida G, Reining L, Rubio A (2002) Electronic excitations: density-functional versus many-body green’s-function approaches. Rev Mod Phys 74(2):601–659. ADSCrossRefGoogle Scholar
  63. Perdew JP (1985) What do the Kohn-Sham orbitals mean? How do atoms dissociate? Plenum, Density functional methods in physics. Springer, Boston, MA pp 265–308.
  64. Perdew JP (1991) Electronic structure of solids ‘91, Ziesche P, Eschrig H (eds) (Berlin: Akademie-Verlag) p. 11 Perdew JP and Wang Y (1992) Phys Rev B 45(13):244Google Scholar
  65. Perdew JP, Schmidt K (2001) Jacobs ladder of density functional approximations for the exchange-correlation energy. In: AIP conference proceedings, AIP, vol 577, pp 1–20ADSCrossRefGoogle Scholar
  66. Perdew JP, Wang Y (1986) Accurate and simple density functional for the electronic exchange energy: generalized gradient approximation. Phys Rev B 33:8800ADSCrossRefGoogle Scholar
  67. Perdew JP, Wang Y (1992) Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B 45(23):13244–13249. ADSCrossRefGoogle Scholar
  68. Perdew JP, Parr RG, Levy M, Balduz JL (1982) Density-functional theory for fractional particle number: derivative discontinuities of the energy. Phys Rev Lett 49:1691–1694. ADSCrossRefGoogle Scholar
  69. Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C (1992) Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation. Phys Rev B 46:6671ADSCrossRefGoogle Scholar
  70. Perdew JP, Burke K, Ernzerhof M (1996a) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868., ibid 78:1396(E) (1997)
  71. Perdew JP, Ernzerhof M, Burke K (1996b) Rationale for mixing exact exchange with density functional approximations. J Chem Phys 105(22):9982–9985. 1.472933 ADSCrossRefGoogle Scholar
  72. Perdew JP, Yang W, Burke K, Yang Z, Gross EKU, Scheffler M, Scuseria GE, Henderson TM, Zhang IY, Ruzsinszky A, Peng H, Sun J (2017) Understanding band gaps of solids in generalized Kohn-Sham theory. Proc Nat Acad Sci. CrossRefGoogle Scholar
  73. Petersilka M, Gossmann UJ, Gross EKU (1996) Excitation energies from time-dependent density-functional theory. Phys Rev Lett 76:1212ADSCrossRefGoogle Scholar
  74. Pribram-Jones A, Gross DA, Burke K (2015) DFT: a theory full of holes? Ann Rev Phys Chem 66(1):283–304. ADSCrossRefGoogle Scholar
  75. Qiu Y, Henderson TM, Zhao J, Scuseria GE (2017) Projected coupled cluster theory. J Chem Phys 147(6):064111ADSCrossRefGoogle Scholar
  76. Requist R, Gross EKU (2016) Exact factorization-based density functional theory of electrons and nuclei. Phys Rev Lett 117(19):193001ADSCrossRefGoogle Scholar
  77. Runge E, Gross EKU (1984) Density-functional theory for time-dependent systems. Phys Rev Lett 52(12):997. ADSCrossRefGoogle Scholar
  78. Rupp M, Tkatchenko A, Müller KR, von Lilienfeld OA (2012) Fast and accurate modeling of molecular atomization energies with machine learning. Phys Rev Lett 108:058301. ADSCrossRefGoogle Scholar
  79. Savin A (1996) On degeneracy, near-degeneracy and density functional theory. Technical report, Louisiana State University, Baton RougeCrossRefGoogle Scholar
  80. Seidl A, Görling A, Vogl P, Majewski JA, Levy M (1996) Generalized Kohn-Sham schemes and the band-gap problem. Phys Rev B 53:3764–3774. ADSCrossRefGoogle Scholar
  81. Smith JC, Sagredo F, Burke K (2018) Warming up density functional theory. In: Frontiers of quantum chemistry. Wójcik MJ, Nakatsuji H, Kirtman B, Ozaki Y (eds) Springer Nature, Singapore, pp 249–271. CrossRefGoogle Scholar
  82. Snyder JC, Rupp M, Hansen K, Mueller KR, Burke K (2012) Finding density functionals with machine learning. Phys Rev Lett 108:253002ADSCrossRefGoogle Scholar
  83. Snyder JC, Rupp M, Hansen K, Blooston L, Müller KR, Burke K (2013) Orbital-free bond breaking via machine learning. J Chem Phys 139(22):224104ADSCrossRefGoogle Scholar
  84. Stadele M, Moukara M, Majewski JA, Vogl P, Görling A (1999) Exact exchange Kohn-Sham formalism applied to semiconductors. Phys Rev B 59:10031ADSCrossRefGoogle Scholar
  85. Stein T, Kronik L, Baer R (2009) Reliable prediction of charge transfer excitations in molecular complexes using time-dependent density functional theory. J Amer Chem Soc 131(8): 2818–2820. CrossRefGoogle Scholar
  86. Stephens PJ, Devlin FJ, Chabalowski CF, Frisch MJ (1994) Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J Phys Chem 98:11623CrossRefGoogle Scholar
  87. Sun J, Marsman M, Csonka GI, Ruzsinszky A, Hao P, Kim YS, Kresse G, Perdew JP (2011) Self-consistent meta-generalized gradient approximation within the projector-augmented-wave method. Phys Rev B 84(3):035117ADSCrossRefGoogle Scholar
  88. Sun J, Ruzsinszky A, Perdew JP (2015) Strongly constrained and appropriately normed semilocal density functional. Phys Rev Lett 115:036402. ADSCrossRefGoogle Scholar
  89. Tkatchenko A, Scheffler M (2009) Accurate molecular Van der Waals interactions from ground-state electron density and free-atom reference data. Phys Rev Lett 102:073005ADSCrossRefGoogle Scholar
  90. Toulouse J, Gerber IC, Jansen G, Savin A, Ángyán JG (2009) Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation. Phys Rev Lett 102:096404. ADSCrossRefGoogle Scholar
  91. Vosko SH, Wilk L, Nusair M (1980) Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can J Phys 58(8):1200–1211. ADSCrossRefGoogle Scholar
  92. Vu K, Snyder JC, Li L, Rupp M, Chen BF, Khelif T, Mller KR, Burke K (2015) Understanding kernel ridge regression: common behaviors from simple functions to density functionals. Int J Quantum Chem 115(16):1115–1128. CrossRefGoogle Scholar
  93. Wagner LO, Stoudenmire EM, Burke K, White SR (2013) Guaranteed convergence of the Kohn-Sham equations. Phys Rev Lett 111:093003. ADSCrossRefGoogle Scholar
  94. Wagner LO, Baker TE, Stoudenmire M E, Burke K, White SR (2014) Kohn-Sham calculations with the exact functional. Phys Rev B 90:045109. ADSCrossRefGoogle Scholar
  95. Zhao Y, Truhlar DG (2006) A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J Chem Phys 125(19):194101. ADSCrossRefGoogle Scholar
  96. Zhao Y, Truhlar D (2008) The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor Chem Accounts 120:215–241CrossRefGoogle Scholar
  97. Zhao Y, Schultz NE, Truhlar DG (2006) Design of density functionals by combining the method of constraint satisfaction with parametrization for thermochemistry, thermochemical kinetics, and noncovalent interactions. J Chem Theory Comput 2(2):364–382CrossRefGoogle Scholar
  98. Zheng X, Cohen AJ, Mori-Sánchez P, Hu X, Yang W (2011) Improving band gap prediction in density functional theory from molecules to solids. Phys Rev Lett 107:026403. ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of California IrvineIrvineUSA
  2. 2.Departments of Physics and of ChemistryUniversity of California IrvineIrvineUSA

Personalised recommendations