Local-density Approximation - Correlation Functional

Correlation Functional

Analytic expressions for the correlation energy of the HEG are not known except in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ρ, the high-density limit of the correlation energy density is

and the low limit

where the Wigner-Seitz radius is related to the density as

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. 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

• Vosko-Wilk-Nusair (VWN)

• Perdew-Zunger (PZ81)

• Cole-Perdew (CP)

• Perdew-Wang (PW92)

Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained perturbatively from the HEG model.