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The '''local-density approximation''' (LDA) is an approximation of the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] (XC) energy functional in [[density functional theory]] (DFT) by taking the XC energy of an electron in a homogeneous [[Free electron model|electron gas]] of a density equal to the density at the electron in the system being calculated (which in general is inhomogeneous). This approximation was applied to DFT by [[Walter Kohn|Kohn]] and [[Lu Jeu Sham|Sham]] in an early paper.<ref name=kohn-sham>{{cite journal | author = W. Kohn and L. J. Sham | title =
Self-Consistent Equations Including Exchange and Correlation Effects | journal = Phys. Rev. | volume = 140 | pages =
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 =
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 =
The LDA functional assumes that the per-electron exchange-correlation energy at every point in space is equal to the per-electron exchange-correlation energy of a homogeneous electron gas.<ref name=kohn-sham />
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:<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>
in [[International System of Units|SI units]] where <math>n(\vec{r})</math> is the electron density per unit volume 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-Censity 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>
== Correlation ==
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There are several forms of correlation:
* Wigner <ref name=wigner>{{cite journal | title = On the Interaction of Electrons in Metals | author = E. Wigner | journal = Phys. Rev. | volume = 46 | pages =
* 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 (PZ) <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>
* Lee-Yang-Parr (LYP) <ref name=lyp>{{cite journal | title = Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density | author = Chengteh Lee, Weitao Yang, and Robert G. Parr | journal = Phys. Rev. B | volume = 37 | pages =
* 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 =
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 =
LYP is based on data fitted to the helium atom.<ref name=lyp />
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