Local-density approximation: Difference between revisions

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TheAn 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]] simulation 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 = 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}}</ref>

* The Chachiyo correlation functional

The parameters “a” and “b” do not come from fitting to the Monte Carlo data, but from a constraint that the functional approaches the high-density theoretical limit. The Chachiyo's formula yields more accurate results than the standard VWN formula. <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}}</ref> Keeping the same functional form, the parameter "b" has also been fitted to the Monte Carlo simulation, providing a better agreement. <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}}</ref>

* 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 = 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}}</ref>

The parameters “a”<math>a</math> and “b”<math>b</math> doare not come from fitting to the Monte Carlo data, but from athe constraint that the functional approaches the high-density theoretical limit. The Chachiyo's formula yieldsis more accurate results than the standard VWN formula. <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}}</ref>. KeepingIn the sameatomic functionalunit, <math> a = \frac{ \ln(2) - 1 } {2 \pi^2} </math>. The closed-form, theexpression parameterfor <math> "b" has</math> alsodoes beenexist; fittedbut it is more convenient to use the Montenumerical Carlovalue: simulation<math> b = 20.4562557 = \exp(\text{C}/2a) </math>. Here, providing<math>\text{C}</math> can be evaluated exactly using a betterclosed-form agreementintegral and a zeta function <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}}</ref>. Keeping the same simple and elegant 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}}</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 the simplest (also accurate) truly first-principle correlation function 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 }}</ref> yield sufficient accuracy compared to the experimental data. It is not clear, however, why the functional remains accurate for the full range of densities even though the values <math>a</math> and <math>b</math> are exclusively from the high-density limit. An alternative, more mathematically rigorous derivation of the functional form <math>\ln(1 + \cdots )</math> might be more theoretically desirable.

[[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}}</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''_c, have generated several LDA's for the correlation functional, including