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== Exchange functional ==
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 |year=1994|isbn=978-0-19-509276-9}}</ref><ref>{{cite journal|last=Dirac|first=P. A. M.|year=1930|title=Note on exchange phenomena in the Thomas-Fermi atom|journal=Proc. Cambridge Phil. Roy. Soc.|volume=26|pages=376–385|doi=10.1017/S0305004100016108|issue=3|bibcode = 1930PCPS...26..376D }}</ref>
:<math>E_{x}^{\mathrm{LDA}}[\rho] = - \frac{3}{4}\left( \frac{3}{\pi} \right)^{1/3}\int\rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}\ .</math>
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