Removed information on the Chachiyo functional. The original paper is cited 22 times, making it far too irrelevant for an encyclopedia. Judging by the user name, Teepanis Chachiyo himself added the information on his work to this article without any disclaimer.
An analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The calculated correlation energies are in agreement with the results from [[quantum Monte Carlo]] simulation to within 2 milli-Hartree.
* 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 | pmid = 27421388 | doi-access = free }}</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 units|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 = 8730–8733 | year = 1992 | doi = 10.1103/PhysRevB.45.8730 | pmid = 10000713 | 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 | pmid = 27782483 | issue = 2| arxiv = 1609.05408 | bibcode = 2016JChPh.145o7101K | s2cid = 12118142 }}</ref> the parameter <math>b</math> has also been fitted to the Monte Carlo simulation, providing a better agreement. Also in this case, the <math>r_{s}</math> must either be in the atomic unit or be divided by the Bohr radius, making it a dimensionless parameter.<ref name="Murray Gell-Mann and Keith A. Brueckner 1957 364"/>
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 | issue = 7| pages = 070313 | year = 2017 | doi = 10.7567/JJAP.56.070313 | bibcode = 2017JaJAP..56g0313J }}</ref> yield sufficient accuracy compared to the experimental data. Its simplicity is also suitable for introductory density functional theory courses.<ref>{{cite book|last=Boudreau|first=Joseph|author2=Swanson, Eric |title=Applied Computational Physics|publisher=Oxford University Press|year=2017|isbn=978-0-198-70863-6|page=829}}</ref><ref>{{cite web |url=https://compphys.go.ro/dft-for-a-quantum-dot/ |title=DFT for a Quantum Dot |last=Roman |first=Adrian |date=November 26, 2017 |website=Computational Physics Blog |access-date=December 7, 2017}}</ref>
[[File:Correlation funtionals comparison.gif|thumb|Comparison between several LDA correlation energy functionals and the quantum Monte Carlo simulation]]
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| url = https://digital.library.unt.edu/ark:/67531/metadc1059358/ }}</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
