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== Correlation functional ==
Analytic expressions for the correlation energy of the HEG are
:<math>\epsilon_{c} = A\ln(r_{s}) + B + r_{s}(C\ln(r_{s}) + D)\ ,</math>
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:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}\ .</math>
The analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The error as compared to the near-exact quantum Monte Carlo is on the order of milli-Hartree.
* Chachiyo's Correlation Functional :<math>\epsilon_{c} = a \ln \left( 1 + \frac{b}{r_s} + \frac{b}{r_s^2} \right) ,</math> <ref>{{cite journal | title = Simple and accurate uniform electron gas correlation energy for the full range of densities | author = Teepanis Chachiyo | journal = J. Chem. Phys. | volume = 145 | pages = 021101 | year = 2016 | doi = 10.1063/1.4958669 | issue = 2}}</ref>
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 | doi = 10.1103/PhysRevLett.45.566 | bibcode=1980PhRvL..45..566C | issue = 7}}</ref> The most popular LDA's to the correlation energy density interpolate these accurate values obtained from simulation while reproducing the exactly known limiting behavior. Various approaches, using different analytic forms for ''ε''<sub>c</sub>, have generated several LDA's for the correlation functional, including
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