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'''Local-density approximations''' ('''LDA''') are a class of approximations to the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] (XC) energy [[Functional (mathematics)|functional]] in [[density functional theory]] (DFT) that depend solely upon the value of the [[electronic density]] at each point in space (and not, for example, derivatives of the density or the [[Kohn-Sham equations|Kohn-Sham orbitals]]). Many approaches can yield local approximations to the XC energy. Overwhelming, however, successful local approximations are those that have been derived from the [[homogeneous electron gas]] (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, and which then applied to realistic systems (molecules and solids).
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as
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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>.
Local-density approximations are important in the construction of more sophisticated approximations to the exchange-correlation energy, such as [[generalized gradient approximation]]s or [[hybrid functional]]s, as a desirable property of any approximate exchange-correlation functional is that it reproduce the exact results of the HEG for non-varying densities. As such, LDA's are often an explicit component of such functionals.
== Homogeneous electron gas ==
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:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}.</math>
Accurate [[quantum Monte Carlo]] simulations 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
* Vosko-Wilk-Nusair (VWN) <ref name="vwn">{{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 (PZ81) <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
* 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
* 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
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>
== Spin polarization ==
The extension of density functionals to [[Spin polarization|spin-polarized]] systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed. A spin polarized system in DFT employs two spin-densities, ''ρ''<sub>α</sub> and ''ρ''<sub>β</sub> with ''ρ'' = ''ρ''<sub>α</sub> + ''ρ''<sub>β</sub>, and the form of the local-spin-density approximation (LSDA) is
:<math>E_{xc}^{\mathrm{LSDA}}[\rho_{\alpha},\rho_{\beta}] = \int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})\epsilon_{xc}(\rho_{\alpha},\rho_{\beta}).</math>
For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional<ref>{{cite journal|last=Oliver|first=G. L.|coauthors=Perdew, J. P. |date=1979|title=Spin-density gradient expansion for the kinetic energy|journal=Phys. Rev. A|volume=20|pages=397-403|doi=10.1103/PhysRevA.20.397}}</ref>:
:<math>E_{x}[\rho_{\alpha},\rho_{\beta}] = \frac{1}{2}\bigg( E_{x}[2\rho_{\alpha}] + E_{x}[2\rho_{\beta}] \bigg).</math>
The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:
:<math>\zeta(\mathbf{r}) = \frac{\rho_{\alpha}(\mathbf{r})-\rho_{\beta}(\mathbf{r})}{\rho_{\alpha}(\mathbf{r})+\rho_{\beta}(\mathbf{r})}.</math>
<math>\zeta = 0\,</math> corresponds to the paramagnetic spin-unpolarized situation with equal
<math>\alpha\,</math> and <math>\beta\,</math> spin densities whereas <math>\zeta = \pm 1</math> corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, ''ε''<sub>c</sub>(''ρ'',''ς''), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.<ref name="vwn"/><ref>{{cite journal|last=von Barth|first=U.|coauthors=Hedin, L.|date=1972|title=A local exchange-correlation potential for the spin polarized case|journal=J. Phys. C: Solid State Phys.|volume=5|pages=1629-1642|doi=10.1088/0022-3719/5/13/012}}</ref>
== Exchange-correlation potential ==
The exchange-correlation potential corresponding to the exchange-correlation energy for a local density approximation is given by<ref name="parryang"/>
:<math>v_{xc}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho(\mathbf{r})} = \epsilon_{xc}(\rho(\mathbf{r})) + \rho(\mathbf{r})\frac{\partial \epsilon_{xc}(\rho(\mathbf{r}))}{\partial\rho(\mathbf{r})}.</math>
== References ==
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