Anyons in an exactly solved model and beyond
Annals of Physics · 2006
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Self-Consistent Equations Including Exchange and Correlation Effects
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials
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Accurate and simple analytic representation of the electron-gas correlation energy
We propose a simple analytic representation of the correlation energy ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ for a uniform electron gas, as a function of density parameter ${\mathit{r}}_{\mathit{s}}$ and relative spin polarization \ensuremath{\zeta}. Within the random-phase approximation (RPA), this representation allows for the ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}3/4}$ behavior as ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}\ensuremath{\infty}. Close agreement with numerical RPA values for ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,0), ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,1), and the spin stiffness ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$)=${\mathrm{\ensuremath{\partial}}}^{2}$${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$, \ensuremath{\zeta}=0)/\ensuremath{\delta}${\mathrm{\ensuremath{\zeta}}}^{2}$, and recovery of the correct ${\mathit{r}}_{\mathit{s}}$ln${\mathit{r}}_{\mathit{s}}$ term for ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0, indicate the appropriateness of the chosen analytic form. Beyond RPA, different parameters for the same analytic form are found by fitting to the Green's-function Monte Carlo data of Ceperley and Alder [Phys. Rev. Lett. 45, 566 (1980)], taking into account data uncertainties that have been ignored in earlier fits by Vosko, Wilk, and Nusair (VWN) [Can. J. Phys. 58, 1200 (1980)] or by Perdew and Zunger (PZ) [Phys. Rev. B 23, 5048 (1981)]. While we confirm the practical accuracy of the VWN and PZ representations, we eliminate some minor problems with these forms. We study the \ensuremath{\zeta}-dependent coefficients in the high- and low-density expansions, and the ${\mathit{r}}_{\mathit{s}}$-dependent spin susceptibility. We also present a conjecture for the exact low-density limit. The correlation potential ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}^{\mathrm{\ensuremath{\sigma}}}$(${\mathit{r}}_{\mathit{s}}$,\ensuremath{\zeta}) is evaluated for use in self-consistent density-functional calculations.