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The '''localLocal-density approximationapproximations''' ('''LDA''') isare ana approximationclass of approximations to the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] (XC) energy functional in [[density functional theory]] (DFT) bythat takingdepend solely upon the XC energyvalue of an electron in a homogeneousthe [[Freeelectronic electron model|electron gasdensity]] of a density equal to the density at theeach electronpoint in the system being calculatedspace (whichand innot, generalfor isexample, inhomogeneous).derivatives of Thisthe approximationdensity wasor applied to DFT bythe [[WalterKohn-Sham Kohnequations|Kohn]]-Sham and [[Lu Jeu Sham|Shamorbitals]]). inMany anapproaches earlycan paper.<refyield name=kohn-sham>{{citelocal journalapproximations |to authorthe =XC Wenergy. KohnOverwhelming, andhowever, L.successful J.local Shamapproximations |are titlethose =that Self-Consistenthave Equationsbeen Includingderived Exchangefrom andthe Correlation[[homogeneous Effectselectron |gas]] journal(HEG) = Physmodel. Rev.In |this volumeregard, =LDA 140is |generally pagessynonymous =with A1133–A1138functionals |based yearon =the 1965HEG |approximation, urland =which http://link.aps.org/abstract/PR/v140/pA1133then |applied doito =realistic 10.1103/PhysRev.140.A1133systems |(molecules formatand = abstract }}</ref>solids).
In general, a local-density approximation for the exchange-correlation energy is written as
The Hohenberg-Kohn theorem states that the energy of the [[Stationary state|ground state]] of a system of electrons is a [[Functional (mathematics)|functional]] of the [[electronic density]], in particular the exchange and correlation energy is also a functional of the density (this energy can be seen as the quantum part of the electron-electron interaction). This XC functional is not known exactly and must be approximated.<ref>{{cite journal | author = P. Hohenberg and W. Kohn | title = Inhomogeneous Electron Gas | journal = Phys. Rev. | volume = 136 | pages = B864–B871 | year = 1964 | url = http://link.aps.org/abstract/PR/v136/pB864 | doi = 10.1103/PhysRev.136.B864 | format = abstract }}</ref>
:<math>E_{xc}^{\mathrm{LDA}}[\rho] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{xc}(\rho),</math>
LDA is the simplest approximation for this functional, it is ''local'' in the sense that the electron exchange and correlation energy at any point in space is a function of the electron density at that point only.<ref>{{cite journal | author = John R. Smith | title = Beyond the Local-Density Approximation: Surface Properties of (110) W | journal = Phys. Rev. Lett. | volume = 25 | issue = 15 | pages = 1023–1026 | year = 1970 | url = http://link.aps.org/abstract/PRL/v25/p1023 | doi = 10.1103/PhysRevLett.25.1023 | format = abstract}}</ref>
where ''ρ'' is the [[electronic density]] and ''ε''<sub>xc</sub>, the exchange-correlation energy density, is a function of the density alone. The exchange-correlation energy is decomposed in to exchange and correlation terms linearly,
The XC energy functional is the sum of an exchange functional <math>E_x\,\;</math> and a correlation functional <math>E_c\,\;</math> <ref name=kohn-sham />
:<math>E_{xc} = E_x + E_c, \,\;</math>
so that separate expressions for ''ε''<sub>x</sub> and ''ε''<sub>c</sub> are sought. The exchange term takes on a simple analytic form for the HEG. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for ''ε''<sub>c</sub>.
== Homogeneous electron gas ==
LDA uses the exchange for the uniform electron gas of a density equal to the density at the point where the exchange is to be evaluated,
Approximation for ''ε''<sub>xc</sub> depending only upon the density can be developed in numerous ways. The most successful approach is based on the homogeneous electron gas. This is constructed by placing ''N'' interacting electrons in to a volume, ''V'', with a positive background charge keeping the system neutral. ''N'' and ''V'' are then taken to infinity in the manner that keeps the density (''ρ'' = ''N'' / ''V'') finite. This is a useful approximation as the total energy consists of contributions only from the kinetic energy and exchange-correlation energy, and that the wavefunction is expressible in terms of planewaves. In particular, for a constant density ''ρ'', the exchange energy density is proportional to ''ρ''<sup>⅓</sup>.
:<math>E_x = \int d^3r \, n(\vec{r}) \left( {{-3e^2}\over{4\pi}} \right) \left(3 \pi^2 n(\vec{r})\right)^{1 \over 3}</math>
▲== Exchange functional ==
in [[International System of Units|SI units]] where <math>n(\vec{r})</math> is the electron density at the point <math>\vec{r} \,\;</math>and <math>e\,\;</math> is the charge of an electron.<ref>{{cite journal | title = Nonempirical Construction of Current-Density Functionals from Conventional Density-Functional Approximations | author = Jianmin Tao and John P. Perdew | journal = Phys. Rev. Lett. | volume = 95 | pages = 196403 | year = 2005 | url = http://link.aps.org/abstract/PRL/v95/p196403 | doi = 10.1103/PhysRevLett.95.196403 | format = abstract }}</ref>
The exchange-energy density of a HEG is known analytically. The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density in not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression<ref name="parryang">{{cite book|last=Parr|first=Robert G|coauthors=Yang, Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|___location=Oxford |date=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal|last=Dirac|first=P. A. M.|date=1930|title=Note on exchange phenomena in the Thomas-Fermi atom|journal=Proc. Cambridge Phil. Roy. Soc.|volume=26|pages=376–385}}</ref>
:<math>E_x = \int\mathrm{d}\mathbf{r} \, \rho(\mathbf{r}) \left(\frac{3}{4}\left( \frac{3}{\pi} \right)^{1/3} \rho^{1/3}(\mathbf{r})\right) = C_{x}\int\mathrm{d}\mathbf{r}\ \rho^{4/3}(\mathbf{r}).</math>
There are several forms of correlation:
▲== Correlation functional ==
* Wigner <ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages = 1002–1011 | year = 1934 | url = http://link.aps.org/abstract/PR/v46/p1002 | doi = 10.1103/PhysRev.46.1002 | format = abstract }}</ref><ref>{{cite journal | title = Theory of Metal Surfaces: Charge Density and Surface Energy | author = N. D. Lang and W. Kohn | journal = Phys. Rev. B | volume = 1 | pages = 4555–4568 | year = 1970 | url = http://link.aps.org/abstract/PRB/v1/p4555 | doi = 10.1103/PhysRevB.1.4555 | format = abstract }}</ref>
Analytic expressions for the correlation energy of the HEG are not known except in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ''ρ'', the high-density limit of the correlation energy density is<ref name="parryang"/>
:<math>\epsilon_{c} = A\ln(r_{s}) + B + r_{s}(C\ln(r_{s}) + D),\,</math>
and the low limit
:<math>\epsilon_{c} = \frac{1}{2}\left(\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + \dots\right),</math>
where the Wigner-Seitz radius is related to the density as
:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}.</math>
VWN, PZ and PW92 are fitted to aAccurate [[quantum Monte Carlo]] calculationsimulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566–569 | year = 1980 | url = http://link.aps.org/abstract/PRL/v45/p566 | doi = 10.1103/PhysRevLett.45.566 | format = abstract}}</ref> ofThe most popular LDA's to the electroncorrelation gasenergy atdensity varyinginterpolate densitiesthese accurate values obtained from simulation while reproducing the exactly known limiting behavior. <ref name=pw92Various approaches, using different analytic forms for ''ε''<sub>c</ sub> , have generated several LDA's for the correlation functional, including▼
* Vosko-Wilk-Nusair (VWN) <ref>{{cite journal | title = Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis | author = S. H. Vosko, L. Wilk and M. Nusair | journal = Can. J. Phys. | volume = 58 | pages = 1200 | year = 1980 }}</ref>
* Perdew-Zunger (PZPZ81) <ref>{{cite journal | title = Self-interaction correction to density-functional approximations for many-electron systems | author = J. P. Perdew and A. Zunger | journal = Phys. Rev. B | volume = 23 | pages = 5048 | year = 1981 | url = http://link.aps.org/abstract/PRB/v23/p5048 | doi = 10.1103/PhysRevB.23.5048 | format = abstract }}</ref>
* Cole-Perdew (CP) <ref>{{cite journal | title = Calculated electron affinities of the elements | author = L. A. Cole and J. P. Perdew | journal = Phys. Rev. A | volume = 25 | pages = 1265 | year = 1982 | url = http://link.aps.org/abstract/PRA/v25/p1265 | doi = 10.1103/PhysRevA.25.1265 | format = abstract }}</ref>
* Perdew-Wang (PW92) <ref name=pw92>{{cite journal | title = Accurate and simple analytic representation of the electron-gas correlation energy | author = John P. Perdew and Yue Wang | journal = Phys. Rev. B | volume = 45 | pages = 13244–13249 | year = 1992 | url = http://link.aps.org/abstract/PRB/v45/p13244 | doi = 10.1103/PhysRevB.45.13244 | format = abstract }}</ref>
Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained [[Møller-Plesset_perturbation_theory#Rayleigh-Schr.C3.B6dinger_perturbation_theory|perturbatively]] from the HEG model.<ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages = 1002–1011 | year = 1934 | url = http://link.aps.org/abstract/PR/v46/p1002 | doi = 10.1103/PhysRev.46.1002 | format = abstract }}</ref>
Wigner correlation is obtained by using equally spaced electrons and applying perturbation theory.<ref name=wigner />
▲VWN, PZ and PW92 are fitted to a [[quantum Monte Carlo]] calculation<ref>{{cite journal | title = Ground State of the Electron Gas by a Stochastic Method | author = D. M. Ceperley and B. J. Alder | journal = Phys. Rev. Lett. | volume = 45 | pages = 566–569 | year = 1980 | url = http://link.aps.org/abstract/PRL/v45/p566 | doi = 10.1103/PhysRevLett.45.566 | format = abstract}}</ref> of the electron gas at varying densities.<ref name=pw92 />
== References ==
{{reflist}}
<references />
[[Category:Density functional theory]]
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