Local-density approximation: Difference between revisions

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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>

== Correlation functional ==