Editing Density functional theory (section)

==Approximations (exchange–correlation functionals)==
The major problem with DFT is that the exact functionals for exchange and correlation are not known, except for the [[Fermi gas|free-electron gas]]. However, approximations exist which permit the calculation of certain physical quantities quite accurately.<ref>{{cite journal |title=DFT in a nutshell |first1=Kieron |last1=Burke |first2=Lucas O. |last2=Wagner |journal=International Journal of Quantum Chemistry |volume=113 |page=96 |year=2013 |doi=10.1002/qua.24259 |issue=2|doi-access=free }}</ref> One of the simplest approximations is the [[local-density approximation]] (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

: <math>E_\text{XC}^\text{LDA}[n] = \int \varepsilon_\text{XC}(n) n(\mathbf r) \,\mathrm d^3 \mathbf r.</math>

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron [[Spin (physics)|spin]]:

: <math>E_\text{XC}^\text{LSDA}[n_\uparrow, n_\downarrow] = \int \varepsilon_\text{XC}(n_\uparrow, n_\downarrow) n(\mathbf r) \,\mathrm d^3 \mathbf r.</math>

In LDA, the exchange–correlation energy is typically separated into the exchange part and the correlation part: {{math|''ε''<sub>XC</sub> {{=}} ''ε''<sub>X</sub> + ''ε''<sub>C</sub>}}. The exchange part is called the Dirac (or sometimes Slater) [[Local-density approximation#Exchange functional|exchange]], which takes the form {{math|''ε''<sub>X</sub> ∝ ''n''<sup>1/3</sup>}}. There are, however, many mathematical forms for the correlation part. Highly accurate formulae for the correlation energy density {{math|''ε''<sub>C</sub>(''n''<sub>↑</sub>, ''n''<sub>↓</sub>)}} have been constructed from [[quantum Monte Carlo]] simulations of [[jellium]].<ref>{{cite journal |title=Prescriptions for the design and selection of density functional approximations: More constraint satisfaction with fewer fits |first1=John P. |last1=Perdew |first2=Adrienn |last2=Ruzsinszky|author2-link=Adrienn Ruzsinszky |first3=Jianmin |last3=Tao |first4=Viktor N. |last4=Staroverov |first5=Gustavo |last5=Scuseria |first6=Gábor I. |last6=Csonka |s2cid=13097889 |journal=Journal of Chemical Physics |volume=123 |page=062201 |year=2005 |doi=10.1063/1.1904565 |pmid=16122287 |issue=6 |bibcode=2005JChPh.123f2201P }}</ref> A simple first-principles [[Local-density approximation#Correlation functional|correlation functional]] has been recently proposed as well.<ref>{{cite journal | title = Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities | first = Teepanis | last = Chachiyo | journal = Journal of Chemical Physics | volume = 145 | page = 021101 | year = 2016 | doi = 10.1063/1.4958669 | pmid = 27421388 | issue = 2| bibcode = 2016JChPh.145b1101C | doi-access = free }}</ref><ref>{{cite journal | title = A simpler ingredient for a complex calculation | first = Richard J. | last = Fitzgerald  | journal = Physics Today | volume = 69 | page = 20 | year = 2016 | doi = 10.1063/PT.3.3288 | issue = 9 | bibcode = 2016PhT....69i..20F }}</ref> Although unrelated to the Monte Carlo simulation, the two variants provide comparable accuracy.<ref>{{cite journal | title = Study of the first-principles correlation functional in the calculation of silicon phonon dispersion curves | first1 = Ukrit | last1 = Jitropas | first2 = Chung-Hao | last2 = Hsu| journal = Japanese Journal of Applied Physics | volume = 56 | issue = 7 | page = 070313 | year = 2017 | doi = 10.7567/JJAP.56.070313 | bibcode = 2017JaJAP..56g0313J | s2cid = 125270241 }}</ref>

The LDA assumes that the density is the same everywhere. Because of this, the LDA has a tendency to underestimate the exchange energy and over-estimate the correlation energy.<ref>{{Cite journal |last=Becke |first=Axel D. |s2cid=33556753 |date=2014-05-14 |title=Perspective: Fifty years of density-functional theory in chemical physics |journal=The Journal of Chemical Physics |volume=140 |issue=18 |pages=A301 |doi=10.1063/1.4869598 |pmid=24832308 |issn=0021-9606 |bibcode = 2014JChPh.140rA301B |doi-access=free }}</ref> The errors due to the exchange and correlation parts tend to compensate each other to a certain degree. To correct for this tendency, it is common to expand in terms of the gradient of the density in order to account for the non-homogeneity of the true electron density. This allows corrections based on the changes in density away from the coordinate. These expansions are referred to as generalized gradient approximations (GGA)<ref>{{cite journal |last1=Perdew |first1=John P. |last2=Chevary |first2=J. A. |last3=Vosko |first3=S. H. |last4=Jackson |first4=Koblar A. |last5=Pederson |first5=Mark R. |last6=Singh |first6=D. J. |last7=Fiolhais |first7=Carlos |title=Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation |journal=Physical Review B |date=1992 |volume=46 |issue=11 |pages=6671–6687 |doi=10.1103/physrevb.46.6671 |pmid=10002368 |bibcode = 1992PhRvB..46.6671P |hdl=10316/2535 |s2cid=34913010 |hdl-access=free}}</ref><ref>{{cite journal |last1=Becke |first1=Axel D. |title=Density-functional exchange-energy approximation with correct asymptotic behavior |journal=Physical Review A |date=1988 |volume=38 |issue=6 |pages=3098–3100 |doi=10.1103/physreva.38.3098 |bibcode=1988PhRvA..38.3098B |pmid=9900728}}</ref><ref>{{cite journal |last1=Langreth |first1=David C. |last2=Mehl |first2=M. J. |title=Beyond the local-density approximation in calculations of ground-state electronic properties |journal=Physical Review B |date=1983 |volume=28 |issue=4 |page=1809 |doi=10.1103/physrevb.28.1809 |bibcode=1983PhRvB..28.1809L }}</ref> and have the following form:

: <math>E_\text{XC}^\text{GGA}[n_\uparrow, n_\downarrow] = \int \varepsilon_\text{XC}(n_\uparrow, n_\downarrow, \nabla n_\uparrow, \nabla n_\downarrow) n(\mathbf r) \,\mathrm d^3 \mathbf r.</math>

Using the latter (GGA), very good results for molecular geometries and ground-state energies have been achieved.

Potentially more accurate than the GGA functionals are the meta-GGA functionals, a natural development after the GGA (generalized gradient approximation). Meta-GGA DFT functional in its original form includes the second derivative of the electron density (the Laplacian), whereas GGA includes only the density and its first derivative in the exchange–correlation potential.

Functionals of this type are, for example, TPSS and the [[Minnesota Functionals]]. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the [[Laplacian]] ([[second derivative]]) of the density.

Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated from [[Hartree–Fock]] theory. Functionals of this type are known as [[hybrid functional]]s.