An analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The errorcalculated ascorrelation comparedenergies toare in agreement with the near-exactresults from [[quantum Monte Carlo]] simulation is on theto orderwithin of2 milli-Hartree.
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As such, the Chachiyo formula is the simplest (also accurate) truly first-principle correlation function forin the 50-year history of 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]]
