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Clarify that rs is dimensionless, and add the use of the correlation functional in physics courses |
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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. 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>{{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>.
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. Its simplicity is also suitable for introductory density functional theory course material. <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]]
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