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:<math>\epsilon_{c} = \frac{1}{2}\left(\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + \dots\right)\ ,</math>
where the Wigner-Seitz parameter <math>r_{s}</math> is dimensionless. <ref>{{cite journal | title = Correlation Energy of an Electron Gas at High Density | author = Murray Gell-Mann and Keith A. Brueckner | journal = Phys. Rev. | volume = 106 | pages = 364 | year = 1957 | doi = 10.1103/PhysRev.106.364 | issue = 2| bibcode = 1957PhRv..106..364G }}</ref> It is defined as the radius of a sphere which encompasses exactly one electron, divided by the Bohr radius. The Wigner-Seitz parameter <math>r_s</math> is related to the density as
:<math>\frac{4}{3}\pi r_{s}^{3} = \frac{1}{\rho}\ .</math>
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* The Chachiyo correlation functional
:<math>\epsilon_{c} = a \ln \left( 1 + \frac{b}{r_s} + \frac{b}{r_s^2} \right) .</math> <ref>{{cite journal | title = Communication: 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| bibcode = 2016JChPh.145b1101C }}</ref>
The parameters <math>a</math> and <math>b</math> ''are not'' from empirical fitting to the Monte Carlo data, but from the theoretical constraint that the functional approaches high-density limit. The Chachiyo's formula is more accurate than the standard VWN fit function <ref>{{cite journal | title = A simpler ingredient for a complex calculation | author = Richard J. Fitzgerald | journal = Physics Today | volume = 69 | pages = 20 | year = 2016 | doi = 10.1063/PT.3.3288 | issue = 9| bibcode = 2016PhT....69i..20F }}</ref>. In the atomic unit, <math> a = \frac{ \ln(2) - 1 } {2 \pi^2} </math>. The closed-form expression for <math> b </math> does exist; but it is more convenient to use the numerical value: <math> b = 20.4562557 = \exp(\text{C}/2a) </math>. Here, <math>\text{C}</math> has been evaluated exactly using a closed-form integral and a zeta function (Eq. 21, G.Hoffman 1992) <ref>{{cite journal | title = Correlation energy of a spin-polarized electron gas at high density | author = Gary G. Hoffman | journal = Phys. Rev. B | volume = 45 | pages = 8370 | year = 1992 | doi = 10.1103/PhysRevB.45.8730 | issue = 15| bibcode = 1992PhRvB..45.8730H }}</ref>. <math>\text{C} = \tfrac{\ln(2)}{3} - \tfrac{3}{2\pi^2} \left [ \zeta(3) + \tfrac{22}{9} -\tfrac{\pi^2}{3} + \tfrac{32\ln(2)}{9} - \tfrac{8\ln^2(2)}{3} \right ] + \tfrac{2(1-\ln 2)}{\pi^2} \left [ \ln(\tfrac{4}{\alpha \pi}) + \left \langle \ln R_0 \right \rangle _{\text{av}} - \tfrac{1}{2} \right ].</math> Keeping the same functional form <ref>{{cite journal | title = Comment on “Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities” [J. Chem. Phys. 145, 021101 (2016)] | author = Valentin V. Karasiev | journal = J. Chem. Phys. | volume = 145 | pages = 157101 | year = 2016 | doi = 10.1063/1.4964758 | issue = 2| arxiv = 1609.05408 | bibcode = 2016JChPh.145o7101K }}</ref>, the parameter <math>b</math> has also been fitted to the Monte Carlo simulation, providing a better agreement.
As such, the Chachiyo formula is a simple (also accurate) first-principle correlation functional for DFT (uniform electron density). Tests on phonon dispersion curves <ref>{{cite journal | title = Study of the first-principles correlation functional in the calculation of silicon phonon dispersion curves | author = Ukrit Jitropas and Chung-Hao Hsu| journal = Japanese Journal of Applied Physics | volume = 56 | pages = 070313 | year = 2017 | doi = 10.7567/JJAP.56.070313 | bibcode = 2017JaJAP..56g0313J }}</ref> yield sufficient accuracy compared to the experimental data.
[[File:Correlation funtionals comparison.gif|thumb|Comparison between several LDA correlation energy functionals and the quantum Monte Carlo simulation]]
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