Editing Density functional theory (section)

{{Short description|Computational quantum mechanical modelling method to investigate electronic structure}}
{{distinguish|Discrete Fourier transform}}
{{Electronic structure methods}}

'''Density functional theory''' ('''DFT''') is a computational [[quantum mechanics|quantum mechanical]] modelling method used in [[physics]], [[chemistry]] and [[materials science]] to investigate the [[electronic structure]] (or [[nuclear structure]]) (principally the [[ground state]]) of [[Many-body problem|many-body systems]], in particular atoms, molecules, and the [[condensed phase]]s. Using  this theory, the properties of a many-electron system can be determined by using [[Functional (mathematics)|functionals]] - that is, functions that accept a function as input and output a single real number.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Functional |url=https://mathworld.wolfram.com/Functional.html |access-date=2024-10-05 |website=mathworld.wolfram.com |language=en}}</ref>  In the case of DFT, these are functionals of the spatially dependent [[electronic density|electron density]]. DFT is among the most popular and versatile methods available in [[condensed-matter physics]], [[computational physics]], and [[computational chemistry]].

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 interaction|exchange]] and [[Electronic correlation|correlation]] interactions. Computational costs are relatively low when compared to traditional methods, such as exchange only [[Hartree–Fock method|Hartree–Fock theory]] and [[post-Hartree–Fock|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 gradient]]s in crystals.

Despite recent improvements, there are still difficulties in using density functional theory to properly describe: [[intermolecular force|intermolecular interactions]] (of critical importance to understanding chemical reactions), especially [[van der Waals force]]s (dispersion); charge transfer excitations; [[transition state]]s, global [[potential energy surface]]s, dopant interactions and some [[strongly correlated material|strongly correlated]] systems; and in calculations of the [[band gap]] and [[ferromagnetism]] in [[semiconductor]]s.<ref name="dftmag">{{cite journal| last1=Assadi| first1=M. H. N.| title= Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs| journal= Journal of Applied Physics| year=2013| volume=113| issue=23| pages= 233913–233913–5 |doi=10.1063/1.4811539|arxiv = 1304.1854 |bibcode = 2013JAP...113w3913A | s2cid=94599250|display-authors=etal}}</ref> 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)<ref>{{cite journal |first1=Tanja |last1=Van Mourik |last2=Gdanitz |first2=Robert J. |year=2002 |title=A critical note on density functional theory studies on rare-gas dimers |journal=Journal of Chemical Physics |volume=116 |issue=22 |pages=9620–9623 |doi=10.1063/1.1476010|bibcode = 2002JChPh.116.9620V }}</ref> or where dispersion competes significantly with other effects (e.g. in [[biomolecule]]s).<ref>{{cite journal |first1=Jiří |last1=Vondrášek |last2=Bendová |first2=Lada |last3=Klusák |first3=Vojtěch |last4= Hobza |first4=Pavel  |year=2005 |title=Unexpectedly strong energy stabilization inside the hydrophobic core of small protein rubredoxin mediated by aromatic residues: correlated ab initio quantum chemical calculations |journal=Journal of the American Chemical Society |pmid=15725017 |volume=127 |issue=8 |pages=2615–2619 |doi=10.1021/ja044607h}}</ref> The development of new DFT methods designed to overcome this problem, by alterations to the functional<ref>{{cite journal |first=Stefan |last=Grimme |s2cid=28234414 |year=2006 |title=Semiempirical hybrid density functional with perturbative second-order correlation |journal=Journal of Chemical Physics |pmid=16438568 |volume=124 |issue=3 |page=034108 |doi=10.1063/1.2148954|bibcode = 2006JChPh.124c4108G }}</ref> or by the inclusion of additive terms,<ref>{{cite journal |first1=Urs |last1=Zimmerli |last2=Parrinello |first2=Michele |last3=Koumoutsakos |first3=Petros  |s2cid=20826940 |year=2004 |title=Dispersion corrections to density functionals for water aromatic interactions |journal=Journal of Chemical Physics |pmid=15268413 |volume=120 |issue=6 |pages=2693–2699 |doi=10.1063/1.1637034|bibcode = 2004JChPh.120.2693Z }}</ref><ref>{{cite journal |first=Stefan |last=Grimme |year=2004 |title=Accurate description of van der Waals complexes by density functional theory including empirical corrections |journal=Journal of Computational Chemistry |pmid=15224390 |volume=25 |issue=12 |pages=1463–1473 |doi=10.1002/jcc.20078|s2cid=6968902 }}</ref><ref>{{cite journal |last1=Jurečka |first1=P. |last2=Černý |first2=J. |last3=Hobza |first3=P. |last4=Salahub |first4=D. R. |date=2006 |title=Density functional theory augmented with an empirical dispersion term. Interaction energies and geometries of 80 noncovalent complexes compared with ab initio quantum mechanics calculations |url=https://doi.org/10.1002/jcc.20570 |journal=Journal of Computational Chemistry |volume=28 |issue=2 |pages=555–569 | pmid=17186489 | doi=10.1002/jcc.20570|s2cid=7837488 }}</ref><ref>{{cite journal |first1=O. Anatole |last1=Von Lilienfeld |last2=Tavernelli |first2=Ivano |last3=Rothlisberger |first3=Ursula |last4=Sebastiani |first4=Daniel |year=2004 |title=Optimization of effective atom centered potentials for London dispersion forces in density functional theory |journal=Physical Review Letters |pmid=15524874 |volume=93 |issue=15 |page=153004 |doi=10.1103/PhysRevLett.93.153004 |bibcode=2004PhRvL..93o3004V|url=https://edoc.unibas.ch/43374/1/PhysRevLett.93.153004%281%29.pdf }}</ref><ref>{{cite journal |first1=Alexandre |last1=Tkatchenko |last2=Scheffler |first2=Matthias|year=2009 |title=Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data |journal=Physical Review Letters |volume=102 |page=073005 |doi=10.1103/PhysRevLett.102.073005 |issue=7 |pmid=19257665|bibcode = 2009PhRvL.102g3005T |doi-access=free |hdl=11858/00-001M-0000-0010-F9F2-D |hdl-access=free }}</ref> is a current research topic. <!-- These references are quite dated; 10-15 years old. Are there new ones that better indicate the current state of research --> Classical density functional theory uses a similar formalism to calculate the properties of non-uniform classical fluids.

Despite the current popularity of these alterations or of the inclusion of additional terms, they are reported<ref>{{Cite journal|last1=Medvedev|first1=Michael G.|last2=Bushmarinov|first2=Ivan S.|last3=Sun|first3=Jianwei|last4=Perdew|first4=John P.|last5=Lyssenko|first5=Konstantin A.|date=2017-01-05|title=Density functional theory is straying from the path toward the exact functional|journal=Science|volume=355|issue=6320|pages=49–52|doi=10.1126/science.aah5975|pmid=28059761|issn=0036-8075|bibcode=2017Sci...355...49M|s2cid=206652408}}</ref> to stray away from the search for the exact functional. Further, DFT potentials obtained with adjustable parameters are no longer true DFT potentials,<ref name=":0">{{Cite journal|last=Jiang|first=Hong|date=2013-04-07|title=Band gaps from the Tran-Blaha modified Becke-Johnson approach: A systematic investigation|journal=The Journal of Chemical Physics|volume=138|issue=13|pages=134115|doi=10.1063/1.4798706|pmid=23574216|bibcode=2013JChPh.138m4115J|issn=0021-9606}}</ref> 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<ref name=":0" /><ref name="Bagayoko 127104">{{Cite journal|last=Bagayoko|first=Diola|date=December 2014|title=Understanding density functional theory (DFT) and completing it in practice|journal=AIP Advances|volume=4|issue=12|pages=127104|doi=10.1063/1.4903408|bibcode=2014AIPA....4l7104B|issn=2158-3226|doi-access=free}}</ref> in such conditions.