# Density functional theory

DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. Computational costs are relatively low when compared to traditional methods, such as exchange only Hartree–Fock theory and its descendants that include electron correlation. Since, DFT has become an important tool for methods of nuclear spectroscopy such as Mössbauer spectroscopy or perturbed angular correlation, in order to understand the origin of specific electric field gradients in crystals.

Despite recent improvements, there are still difficulties in using density functional theory to properly describe: intermolecular interactions (of critical importance to understanding chemical reactions), especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces, dopant interactions and some strongly correlated systems; and in calculations of the band gap and ferromagnetism in semiconductors.[1] The incomplete treatment of dispersion can adversely affect the accuracy of DFT (at least when used alone and uncorrected) in the treatment of systems which are dominated by dispersion (e.g. interacting noble gas atoms)[2] or where dispersion competes significantly with other effects (e.g. in biomolecules).[3] The development of new DFT methods designed to overcome this problem, by alterations to the functional[4] or by the inclusion of additive terms,[5][6][7][8] is a current research topic. Classical density functional theory uses a similar formalism to calculate properties of non-uniform classical fluids.

Despite the current popularity of these alterations or of the inclusion of additional terms, they are reported[9] to stray away from the search for the exact functional. Further, DFT potentials obtained with adjustable parameters are no longer true DFT potentials,[10] given that they are not functional derivatives of the exchange correlation energy with respect to the charge density. Consequently, it is not clear if the second theorem of DFT holds[10][11] in such conditions.

Contents 1 Overview of method 1.1 Origins 2 Derivation and formalism 3 Relativistic density functional theory (ab initio functional forms) 4 Approximations (exchange–correlation functionals) 5 Generalizations to include magnetic fields 6 Applications 7 Thomas–Fermi model 8 Hohenberg–Kohn theorems 9 Pseudo-potentials 9.1 Ab initio pseudo-potentials 10 Electron smearing 11 Software supporting DFT 12 Classical density functional theory 13 See also 13.1 Lists 14 References 15 Sources 16 External links Overview of method In the context of computational materials science, ab initio (from first principles) DFT calculations allow the prediction and calculation of material behavior on the basis of quantum mechanical considerations, without requiring higher-order parameters such as fundamental material properties. In contemporary DFT techniques the electronic structure is evaluated using a potential acting on the system's electrons. This DFT potential is constructed as the sum of external potentials Vext, which is determined solely by the structure and the elemental composition of the system, and an effective potential Veff, which represents interelectronic interactions. Thus, a problem for a representative supercell of a material with n electrons can be studied as a set of n one-electron Schrödinger-like equations, which are also known as Kohn–Sham equations.[12] Origins Although density functional theory has its roots in the Thomas–Fermi model for the electronic structure of materials, DFT was first put on a firm theoretical footing by Walter Kohn and Pierre Hohenberg in the framework of the two Hohenberg–Kohn theorems (HK).[13] The original HK theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.[14][15] The first HK theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to three spatial coordinates, through the use of functionals of the electron density. This theorem has since been extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states.

The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional.

In work that later won them the Nobel prize in chemistry, the HK theorem was further developed by Walter Kohn and Lu Jeu Sham to produce Kohn–Sham DFT (KS DFT). Within this framework, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas–Fermi model, and from fits to the correlation energy for a uniform electron gas. Non-interacting systems are relatively easy to solve, as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange–correlation part of the total energy functional remains unknown and must be approximated.

Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original HK theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.

Derivation and formalism As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born–Oppenheimer approximation), generating a static external potential V, in which the electrons are moving. A stationary electronic state is then described by a wavefunction Ψ(r1, …, rN) satisfying the many-electron time-independent Schrödinger equation {displaystyle {hat {H}}Psi =left[{hat {T}}+{hat {V}}+{hat {U}}right]Psi =left[sum _{i=1}^{N}left(-{frac {hbar ^{2}}{2m_{i}}}nabla _{i}^{2}right)+sum _{i=1}^{N}V(mathbf {r} _{i})+sum _{i3.0.CO;2-3. ^ "Finite temperature approaches – smearing methods". VASP the GUIDE. Retrieved 21 October 2016. ^ Tong, Lianheng. "Methfessel–Paxton Approximation to Step Function". Metal CONQUEST. Retrieved 21 October 2016. ^ Evans, Robert (1979). "The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids". Advances in Physics. 281 (2): 143–200. Bibcode:1979AdPhy..28..143E. doi:10.1080/00018737900101365. ^ Evans, Robert; Oettel, Martin; Roth, Roland; Kahl, Gerhard (2016). "New developments in classical density functional theory". Journal of Physics: Condensed Matter. 28 (24): 240401. Bibcode:2016JPCM...28x0401E. doi:10.1088/0953-8984/28/24/240401. ISSN 0953-8984. PMID 27115564. ^ Singh, Yaswant (1991). "Density Functional Theory of Freezing and Properties of the ordered Phase". Physics Reports. 207 (6): 351–444. Bibcode:1991PhR...207..351S. doi:10.1016/0370-1573(91)90097-6. ^ ten Bosch, Alexandra (2019). Analytical Molecular Dynamics:from Atoms to Oceans. ISBN 978-1091719392. ^ Wu, Jianzhong (2006). "Density Functional Theory for chemical engineering:from capillarity to soft materials". AIChE Journal. 52 (3): 1169–1193. doi:10.1002/aic.10713. ^ Gelb, Lev D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. (1999). "Phase separation in confined systemss". Reports on Progress in Physics. 62 (12): 1573–1659. Bibcode:1999RPPh...62.1573G. doi:10.1088/0034-4885/62/12/201. S2CID 9282112. ^ Frisch, Harry; Lebowitz, Joel (1964). The equilibrium theory of classical fluids. New York: W. A. Benjamin. ^ Ornstein, L. S.; Zernike, F. (1914). "Accidental deviations of density and opalescence at the critical point of a single substance" (PDF). Royal Netherlands Academy of Arts and Sciences (KNAW). Proceedings. 17: 793–806. Bibcode:1914KNAB...17..793. ^ Lebowitz, J. L.; Percus, J. K. (1963). "Statistical Thermodynamics of Non-uniform Fluids". Journal of Mathematical Physics. 4 (1): 116–123. Bibcode:1963JMP.....4..116L. doi:10.1063/1.1703877. ^ Löwen, Hartmut (1994). "Melting, freezing and colloidal suspensions". Physics Reports. 237 (5): 241–324. Bibcode:1994PhR...237..249L. doi:10.1016/0370-1573(94)90017-5. ^ Hydrophobicity of ceria, Applied Surface Science, 2019, 478, pp.68-74. in HAL archives ouvertes ^ te Vrugt, Michael; Löwen, Hartmut; Wittkowski, Raphael (2020). "Classical dynamical density functional theory: from fundamentals to applications". Advances in Physics. 69 (2): 121–247. arXiv:2009.07977. Bibcode:2020AdPhy..69..121T. doi:10.1080/00018732.2020.1854965. S2CID 221761300. Sources Parr, R. G.; Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. ISBN 978-0-19-504279-5. Thomas, L. H. (1927). "The calculation of atomic fields". Proc. Camb. Phil. Soc. 23 (5): 542–548. Bibcode:1927PCPS...23..542T. doi:10.1017/S0305004100011683. S2CID 122732216. Hohenberg, P.; Kohn, W. (1964). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864. Bibcode:1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864. Kohn, W.; Sham, L. J. (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133. Becke, Axel D. (1993). "Density-functional thermochemistry. III. The role of exact exchange". The Journal of Chemical Physics. 98 (7): 5648. Bibcode:1993JChPh..98.5648B. doi:10.1063/1.464913. S2CID 52389061. Lee, Chengteh; Yang, Weitao; Parr, Robert G. (1988). "Development of the Colle–Salvetti correlation-energy formula into a functional of the electron density". Physical Review B. 37 (2): 785–789. Bibcode:1988PhRvB..37..785L. doi:10.1103/PhysRevB.37.785. PMID 9944570. S2CID 45348446. Burke, Kieron; Werschnik, Jan; Gross, E. K. U. (2005). "Time-dependent density functional theory: Past, present, and future". The Journal of Chemical Physics. 123 (6): 062206. arXiv:cond-mat/0410362. Bibcode:2005JChPh.123f2206B. doi:10.1063/1.1904586. PMID 16122292. S2CID 2659101. Lejaeghere, K.; Bihlmayer, G.; Bjorkman, T.; Blaha, P.; Blugel, S.; Blum, V.; Caliste, D.; Castelli, I. E.; Clark, S. J.; Dal Corso, A.; de Gironcoli, S.; Deutsch, T.; Dewhurst, J. K.; Di Marco, I.; Draxl, C.; Du ak, M.; Eriksson, O.; Flores-Livas, J. A.; Garrity, K. F.; Genovese, L.; Giannozzi, P.; Giantomassi, M.; Goedecker, S.; Gonze, X.; Granas, O.; Gross, E. K. U.; Gulans, A.; Gygi, F.; Hamann, D. R.; Hasnip, P. J.; Holzwarth, N. A. W.; Iu an, D.; Jochym, D. B.; Jollet, F.; Jones, D.; Kresse, G.; Koepernik, K.; Kucukbenli, E.; Kvashnin, Y. O.; Locht, I. L. M.; Lubeck, S.; Marsman, M.; Marzari, N.; Nitzsche, U.; Nordstrom, L.; Ozaki, T.; Paulatto, L.; Pickard, C. J.; Poelmans, W.; Probert, M. I. J.; Refson, K.; Richter, M.; Rignanese, G.-M.; Saha, S.; Scheffler, M.; Schlipf, M.; Schwarz, K.; Sharma, S.; Tavazza, F.; Thunstrom, P.; Tkatchenko, A.; Torrent, M.; Vanderbilt, D.; van Setten, M. J.; Van Speybroeck, V.; Wills, J. M.; Yates, J. R.; Zhang, G.-X.; Cottenier, S. (2016). "Reproducibility in density functional theory calculations of solids". Science. 351 (6280): aad3000. Bibcode:2016Sci...351.....L. doi:10.1126/science.aad3000. ISSN 0036-8075. PMID 27013736. External links Walter Kohn, Nobel Laureate – Video interview with Walter on his work developing density functional theory by the Vega Science Trust Capelle, Klaus (2002). "A bird's-eye view of density-functional theory". arXiv:cond-mat/0211443. Walter Kohn, Nobel Lecture Argaman, Nathan; Makov, Guy (2000). "Density Functional Theory -- an introduction". American Journal of Physics. 68 (2000): 69–79. arXiv:physics/9806013. Bibcode:2000AmJPh..68...69A. doi:10.1119/1.19375. S2CID 119102923. Electron Density Functional Theory – Lecture Notes Density Functional Theory through Legendre Transformationpdf Burke, Kieron. "The ABC of DFT" (PDF). 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