### AIMS Materials Science

2017, Issue 6: 1372-1405. doi: 10.3934/matersci.2017.6.1372
Review

# Basics of the density functional theory

• Received: 11 September 2017 Accepted: 30 November 2017 Published: 15 December 2017
• The density functional theory (DFT) established itself as a well reputed way to compute the electronic structure in most branches of chemistry and materials science. In the formulation given by Kohn, Hohenberg and Sham in the 1960's, the many-electron wave function is replaced by the electron density, so that the energy is just a functional of the latter. The DFT is applied, with low computational cost and reasonable accuracy, to predict diverse properties as binding or atomization energies, shapes and sizes of molecules, crystal structures of solids, energy barriers to various processes, etc. In the mid 1980s, it became an attractive alternative to the well developed wave function techniques such as Hartree-Fock, when crucial developments in exchange-correlation energy has been taken into account, since the Hartree-Fock method treats exchange exactly but neglects correlation
This article is an introduction to the conceptual basis of the DFT in a language accessible for readers entering the field of quantum chemistry and condensed-matter physics. It begins with a presentation of the Thomas-Fermi atomic model and follows by the essentials of the density functional theory based on the works of Hohenberg, Kohn and Sham. With a discussion of the exchange and correlation effects, possible improvements are then presented. Lastly, mention is made of the main hybrid functionals and of the software packages successfully applied to diverse materials of chemical, physical and biological interest.

Citation: Jean-Louis Bretonnet. Basics of the density functional theory[J]. AIMS Materials Science, 2017, 4(6): 1372-1405. doi: 10.3934/matersci.2017.6.1372

### Related Papers:

• The density functional theory (DFT) established itself as a well reputed way to compute the electronic structure in most branches of chemistry and materials science. In the formulation given by Kohn, Hohenberg and Sham in the 1960's, the many-electron wave function is replaced by the electron density, so that the energy is just a functional of the latter. The DFT is applied, with low computational cost and reasonable accuracy, to predict diverse properties as binding or atomization energies, shapes and sizes of molecules, crystal structures of solids, energy barriers to various processes, etc. In the mid 1980s, it became an attractive alternative to the well developed wave function techniques such as Hartree-Fock, when crucial developments in exchange-correlation energy has been taken into account, since the Hartree-Fock method treats exchange exactly but neglects correlation
This article is an introduction to the conceptual basis of the DFT in a language accessible for readers entering the field of quantum chemistry and condensed-matter physics. It begins with a presentation of the Thomas-Fermi atomic model and follows by the essentials of the density functional theory based on the works of Hohenberg, Kohn and Sham. With a discussion of the exchange and correlation effects, possible improvements are then presented. Lastly, mention is made of the main hybrid functionals and of the software packages successfully applied to diverse materials of chemical, physical and biological interest.

 [1] Hartree DR (1928) The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods. Mathematical Proceedings of the Cambridge Philosophical Society, 24: 89–110. doi: 10.1017/S0305004100011919 [2] Fock V (1930) Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z Physik 61: 126–148. doi: 10.1007/BF01340294 [3] Slater JC (1930) Note on Hartree's Method. Phys Rev 35: 210–211. [4] Raimes S (1967) The Wave Mechanics of Electrons in Metals, Amsterdam: North-Holland. [5] Thomas LH (1927) The calculation of atomic fields. Mathematical Proceedings of the Cambridge Philosophical Society, 23: 542–548. doi: 10.1017/S0305004100011683 [6] Fermi E (1928) Eine Statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente. Z Physik 48: 73–79. doi: 10.1007/BF01351576 [7] Dirac PAM (1930) Note on Exchange Phenomena in the Thomas Atom. Mathematical Proceedings of the Cambridge Philosophical Society, 26: 376–385. doi: 10.1017/S0305004100016108 [8] Hohenberg P, Kohn W (1964) Inhomogeneous Electron Gas. Phys Rev 136: B864–B871. doi: 10.1103/PhysRev.136.B864 [9] Kohn W, Sham LJ (1965) Self-Consistent Equations Including Exchange and Correlation Effects. Phys Rev 140: A1133–A1138. doi: 10.1103/PhysRev.140.A1133 [10] Becke AD (1996) Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing. J Chem Phys 104: 1040–1046. [11] Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic behavior. Phys Rev A 38: 3098–3100. doi: 10.1103/PhysRevA.38.3098 [12] Lee C, YangW, Parr RG (1988) Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 37: 785–789. doi: 10.1103/PhysRevB.37.785 [13] Feynman RP, Metropolis N, Teller E (1949) Equations of State of Elements Based on the Generalized Fermi-Thomas Theory. Phys Rev 75: 1561–1573. doi: 10.1103/PhysRev.75.1561 [14] Schwinger J (1981) Thomas-Fermi model: the second correction. Phys Rev A 24: 2353–2361. doi: 10.1103/PhysRevA.24.2353 [15] Shakeshalf R, Spruch L (1981) Remarks on the existence and accuracy of the Z-1/3 expansion of the nonrelativistic ground-state energy of a neutral atom. Phys Rev A 23: 2118–2126. doi: 10.1103/PhysRevA.23.2118 [16] Englert BG, Schwinger J (1985) Atomic-binding-energy oscollations. Phys Rev A 32: 47–63. doi: 10.1103/PhysRevA.32.47 [17] Spruch L (1991) Pedagogic notes on Thomas-Fermi theory (and on some improvements): atoms, stars, and the stability of bulk matter. Rev Mod Phys 63: 151–209. doi: 10.1103/RevModPhys.63.151 [18] Mermin ND (1965) Thermal Properties of the Inhomogeneous Electron Gas. Phys Rev 137: A1441–A1443. (Concerns the generalization to electrons at finite temperatures). doi: 10.1103/PhysRev.137.A1441 [19] Levy M (1979) Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc Natl Acad Sci U S A 76: 6062–6065. doi: 10.1073/pnas.76.12.6062 [20] Becke AD (2014) Perspective: Fifty years of density-functional theory in chemical physics. J Chem Phys 140: 18A301. [21] Lindhard J (1954) On the properties of a gas of charged particles. Kgl Danske Videnskab Selskab Mat Fys Medd 28: 8. [22] Jones RO, Young WH (1971) Density functional theory and the vonWeizsacker method. J Phys C 4: 1322–1330. doi: 10.1088/0022-3719/4/11/007 [23] von Weizsäcker CF (1935) Zur Theorie der Kernmassen. Z Physik 96: 431–458 . doi: 10.1007/BF01337700 [24] Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61: 689–746. doi: 10.1103/RevModPhys.61.689 [25] Slater JC (1951) A Simplification of the Hartree-Fock Method. Phys Rev 81: 385–390. doi: 10.1103/PhysRev.81.385 [26] Robinson JE, Bassani F, Knox BS, et al. (1962) Screening Correction to the Slater Exchange Potential. Phys Rev Lett 9: 215–217. doi: 10.1103/PhysRevLett.9.215 [27] Wigner EP (1934) On the Interaction of Electrons in Metals. Phys Rev 46: 1002–1011. doi: 10.1103/PhysRev.46.1002 [28] Gell-Mann M, Brueckner K (1957) Correlation Energy of an Electron Gas at High Density. Phys Rev 106: 364–368. doi: 10.1103/PhysRev.106.364 [29] Ceperley DM (1978) Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions. Phys Rev B 18: 3126–3138. doi: 10.1103/PhysRevB.18.3126 [30] Ceperley DM, Alder BJ (1980) Ground State of the Electron Gas by a Stochastic Method. Phys Rev Lett 45: 566–569. doi: 10.1103/PhysRevLett.45.566 [31] Perdew JP, Wang Y (1992) Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B 45: 13244–13249. doi: 10.1103/PhysRevB.45.13244 [32] Levy M (1982) Electron densities in search of Hamiltonians. Phys Rev A 26: 1200–1208. doi: 10.1103/PhysRevA.26.1200 [33] Lieb EH (1983) Density functionals for coulomb-systems. Int J Quantum Chem 24: 243–277. doi: 10.1002/qua.560240302 [34] Levy M, Perdew JP (1985) 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. [35] Wigner E, Seitz F (1933) On the Constitution of Metallic Sodium. Phys Rev 43: 804–810. doi: 10.1103/PhysRev.43.804 [36] Hellmann H (1933) Zur Rolle der kinetischen Elektronenenergie für die zwischenatomaren Kräfte. Z Physik 85: 180–190. doi: 10.1007/BF01342053 [37] Feynman RP (1939) Forces in Molecules. Phys Rev 56: 340–343. doi: 10.1103/PhysRev.56.340 [38] Pupyshev VI (2000) The Nontriviality of the Hellmann-Feyman Theorem. Russ J Phys Chem 74: S267–S278. [39] Perdew JP, Kurth S (2003) Density Functionals for Non-relativistic Coulomb Systems in the New Century, In: Fiolhais C, Nogueira F, Marques MAL, A Primer in Density Functional Theory, Berlin Heidelberg: Springer-Verlag, 1–55. [40] Gunnarsson O, Lundqvist BI (1976) Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys Rev B 13: 4274–4298; Gunnarsson O, Lundqvist BI (1977) Erratum: Exchange and correlation in atoms, molecules, and solids by the spin-densityfunctional formalism. Phys Rev B 15: 6006. doi: 10.1103/PhysRevB.13.4274 [41] Gunnarsson O, Jonson M, Lundqvist BI (1979) Descriptions of exchange and correlation effects in inhomogeneous electron systems. Phys Rev 20: 3136–3165. doi: 10.1103/PhysRevB.20.3136 [42] Langreth DC, Perdew JP (1975) The Exchange-Correlation Energy of a Metallic Surface. Solid State Commun 17: 1425–1429. doi: 10.1016/0038-1098(75)90618-3 [43] Langreth DC, Perdew JP (1977) Exchange-correlation energy of a metallic surface: Wave-vector analysis. Phys Rev B 15: 2884–2901. doi: 10.1103/PhysRevB.15.2884 [44] Jones RO, Gunnarsson O (1989) The density functional formalism, its applications and prospects. Rev Mod Phys 61: 689–746. doi: 10.1103/RevModPhys.61.689 [45] Charlesworth JPA (1996) Weighted-density approximation in metals and semiconductors. Phys Rev B 53: 12666–12673. doi: 10.1103/PhysRevB.53.12666 [46] Gunnarsson O, Lundqvist BI (1976) Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys Rev B 13: 4274–4298. doi: 10.1103/PhysRevB.13.4274 [47] Alvarellos JE, Chacon E, Tarazona P (1986) Nonlocal density functional for the exchange and correlation energy of electrons. Phys Rev 33: 6579–6587. doi: 10.1103/PhysRevB.33.6579 [48] Chacon E, Tarazona P (1988) Self-consistent weighted-density approximation for the electron gas. I. Bulk properties. Phys Rev 37: 4020–4025. [49] Gupta AK, Singwi KS (1977) Gradient corrections to the exchange-correlation energy of electrons at metal surfaces. Phys Rev B 15: 1801–1810. [50] Perdew JP, Burke K, Ernzerhof M (1996) Generalized Gradient Approximation Made Simple. Phys Rev Lett 77: 3865–3868. doi: 10.1103/PhysRevLett.77.3865 [51] Perdew JP, Burke K, Wang Y (1996) Generalized gradient approximation for the exchangecorrelation hole of a many-electron system. Phys Rev B 54: 16533–16539. doi: 10.1103/PhysRevB.54.16533 [52] Perdew JP, Ruzsinszky A, Csonka GI, et al. (2008) Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys Rev Lett 100: 136406. (Modified PBE subroutines are available from: http://dft.uci.edu./pubs/PBEsol.html.) doi: 10.1103/PhysRevLett.100.136406 [53] Colle R, Salvetti O (1975) Approximate calculation of the correlation energy for the closed shells. Theoret Chim Acta 37: 329–334. doi: 10.1007/BF01028401 [54] Leung TC, Chan CT, Harmon BN (1991) Ground-state properties of Fe, Co, Ni, and their monoxides: Results of the generalized gradient approximation. Phys Rev B 44: 2923–2927. doi: 10.1103/PhysRevB.44.2923 [55] Singh DJ, Ashkenazi J (1992) Magnetism with generalized-gradient-approximation density functionals. Phys Rev B 46: 11570–11577. doi: 10.1103/PhysRevB.46.11570 [56] Körling M, Häglund J (1992) Cohesive and electronic properties of transition metals: The generalized gradient approximation. Phys Rev B 45: 13293–13297. doi: 10.1103/PhysRevB.45.13293 [57] Engel E, Vosko SH (1994) Fourth-order gradient corrections to the exchange-only energy functional: Importance of r2n contribution. Phys Rev B 50: 10498–10505. doi: 10.1103/PhysRevB.50.10498 [58] Andersson Y, Langreth DC, Lundqvist BI (1996) van derWaals Interactions in Density-Functional Theory. Phys Rev Lett 76: 102–105. doi: 10.1103/PhysRevLett.76.102 [59] Dobson JF, Dinte BP (1996) Constraint Satisfaction in Local and Gradient Susceptibility Approximations: Application to a van der Waals Density Functional. Phys Rev Lett 76: 1780–1783. doi: 10.1103/PhysRevLett.76.1780 [60] Patton DC, PedersonMR (1997) Application of the generalized-gradient approximation to rare-gas dimers. Phys Rev A 56: R2495–R2498. doi: 10.1103/PhysRevA.56.R2495 [61] Van Voorhis T, Scuseria GE (1998) A novel form for the exchange-correlation energy functional. J Chem Phys 109: 400–410. doi: 10.1063/1.476577 [62] Tao J, Perdew JP, Staroverov VN, et al. (2003) Climbing the Density Functional Ladder: Nonempirical Meta–Generalized Gradient Approximation Designed for Molecules and Solids. Phys Rev Lett 91: 146401. doi: 10.1103/PhysRevLett.91.146401 [63] Van Doren VE, Van Alsenoy K, Greelings P (2001) Density Functional Theory and Its Applications to Materials, Melville, NY: American Institute of Physics. [64] Gonis A, Kioussis N (1999) Electron Correlations and Materials Properties, New York: Plenum. [65] Perdew JP, Kurth S, Zupan A, et al. (1999) Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Phys Rev Lett 82: 2544– 2547; Perdew JP, Kurth S, Zupan A, et al. (1999) Erratum: Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Phys Rev Lett 82: 5179. doi: 10.1103/PhysRevLett.82.2544 [66] Becke AD (2000) Simulation of delocalized exchange by local density functionals. J Chem Phys 112: 4020–4026. doi: 10.1063/1.480951 [67] Becke AD (1993) A new mixing of Hartree–Fock and local density-functional theories. J Chem Phys 98: 1372–1377. doi: 10.1063/1.464304 [68] Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98: 5648–5652. [69] Seidl M, Perdew JP, Kurth S (2000) Simulation of All-Order Density-Functional Perturbation Theory, Using the Second Order and the Strong-Correlation Limit. Phys Rev Lett 84: 5070–5073. doi: 10.1103/PhysRevLett.84.5070 [70] Fuchs M, Gonze X (2002) Accurate density functionals: Approaches using the adiabaticconnection fluctuation-dissipation theorem. Phys Rev B 65: 235109. doi: 10.1103/PhysRevB.65.235109 [71] Seidl A, Görling A, Vogl P, et al. (1996) Generalized Kohn-Sham schemes and the band-gap problem. Phys Rev B 53: 3764–3774. doi: 10.1103/PhysRevB.53.3764 [72] Savin A (1996) On degeneracy, near degeneracy and density functional theory, In: Seminario JM, Recent Developments of Modern Density Functional Theory, Amsterdam: Elsevier, 327–357. [73] Sousa SF, Fernandes PA, Ramos MJ (2007) General Performance of Density Functionals. J Phys Chem A 111: 10439–10452. doi: 10.1021/jp0734474 [74] Curtiss LA, Raghavachari K, Redfern PC, et al. (1997) Assessment of Gaussian-2 and density functional theories for the computation of enthalpies of formation. J Chem Phys 106: 1063–1079. (This paper contains a set of 148 molecules having well-established enthalpies of formation at 298 K.) doi: 10.1063/1.473182 [75] Chan GKL, Handy NC (1999) Optimized Lieb-Oxford bound for the exchange-correlation energy. Phys Rev A 59: 3075–3077. (As an example, the exchange-correlation energy must satisfy the inegality $\left| {{E_{xc}}} \right| \le 2.27\left| {E_x^{LDA}} \right|$ .) doi: 10.1103/PhysRevA.59.3075 [76] Perdew JP, Ernzerhof M, Burke K (1996) Rationale for mixing exact exchange with density functional approximations. J Chem Phys 105: 9982–9985. doi: 10.1063/1.472933 [77] Adamo C, Barone V (1998) Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models. J Chem Phys 108: 664–675. doi: 10.1063/1.475428 [78] Wu Q, Yang W (2002) Empirical correction to density functional theory for van der Waals interactions. J Chem Phys 116: 515–524. doi: 10.1063/1.1424928 [79] Becke AD, Johnson ER (2005) Exchange-hole dipole moment and the dispersion interaction. J Chem Phys 122: 154104. doi: 10.1063/1.1884601 [80] 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: 073005. doi: 10.1103/PhysRevLett.102.073005 [81] Grimme S, Antony J, Ehrlich S, et al. (2010) 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. doi: 10.1063/1.3382344 [82] Dion M, Rydberg H, Schröder E, et al. (2004) Van der Waals Density Functional for General Geometries. Phys Rev Lett 92: 246401. doi: 10.1103/PhysRevLett.92.246401 [83] Thonhauser T, Cooper VR, Li S, et al. (2007) Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond. Phys Rev B 76: 125112 . doi: 10.1103/PhysRevB.76.125112 [84] Klimes J, Michaelides A (2012) Perspective: Advances and challenges in treating van der Waals dispersion forces in density functional theory. J Chem Phys 137: 120901. doi: 10.1063/1.4754130 [85] Car R, Parrinello M (1985) Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys Rev Lett 55: 2471–2474. doi: 10.1103/PhysRevLett.55.2471 [86] Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a screened Coulomb potential. J Chem Phys 118: 8207–8215. doi: 10.1063/1.1564060 [87] Heyd J, Scuseria GE (2004) Effcient hybrid density functional calculations in solids: Assessment of the Heyd–Scuseria–Ernzerhof screened Coulomb hybrid functional. J Chem Phys 121: 1187–1192. doi: 10.1063/1.1760074 [88] Ernzerhof M, Scuseria GE (1999) Assessment of the Perdew–Burke–Ernzerhof exchangecorrelation functional. J Chem Phys 110: 5029–5036. doi: 10.1063/1.478401 [89] Paier J, Marsman M, Kresse G (2007) Why does the B3LYP hybrid functional fail for metals? J Chem Phys 127: 024103. doi: 10.1063/1.2747249 [90] Burke K (2012) Perspective on density functional theory. J Chem Phys 136: 150901. doi: 10.1063/1.4704546 [91] Jones RO (2015) Density functional theory: Its origins, rise to prominence, and future. Rev Mod Phys 87: 897–923. doi: 10.1103/RevModPhys.87.897
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142