Density functionals for the strong-interaction limit

The strong-interaction limit of density-functional (DF) theory is simple and provides information required for an accurate resummation of DF perturbation theory. Here we derive the point-charge-plus-continuum (PC) model for that limit, and its gradient expansion. The exchange-correlation (xc) energy ${E}_{\mathrm{xc}}[\ensuremath{\rho}]\ensuremath{\equiv}{\ensuremath{\int}}_{0}^{1}d\ensuremath{\alpha}{W}_{\ensuremath{\alpha}}[\ensuremath{\rho}]$ follows from the xc potential energies ${W}_{\ensuremath{\alpha}}$ at different interaction strengths $\ensuremath{\alpha}>~0$ [but at fixed density $\ensuremath{\rho}(\mathbf{r})].$ For small $\ensuremath{\alpha}\ensuremath{\approx}0,$ the integrand ${W}_{\ensuremath{\alpha}}$ is obtained accurately from perturbation theory, but the perturbation expansion requires resummation for moderate and large $\ensuremath{\alpha}.$ For that purpose, we present density functionals for the coefficients in the asymptotic expansion ${W}_{\ensuremath{\alpha}}\ensuremath{\rightarrow}{W}_{\ensuremath{\infty}}{+W}_{\ensuremath{\infty}}^{\ensuremath{'}}{\ensuremath{\alpha}}^{\ensuremath{-}1/2}$ for $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ in the PC model. ${W}_{\ensuremath{\infty}}^{\mathrm{PC}}$ arises from strict correlation, and ${W}_{\ensuremath{\infty}}^{\ensuremath{'}\mathrm{PC}}$ from zero-point vibration of the electrons around their strictly correlated distributions. The PC values for ${W}_{\ensuremath{\infty}}$ and ${W}_{\ensuremath{\infty}}^{\ensuremath{'}}$ agree with those from a self-correlation-free meta-generalized gradient approximation, both for atoms and for atomization energies of molecules. We also (i) explain the difference between the PC cell and the exchange-correlation hole, (ii) present a density-functional measure of correlation strength, (iii) describe the electron localization and spin polarization energy in a highly stretched ${\mathrm{H}}_{2}$ molecule, and (iv) discuss the soft-plasmon instability of the low-density uniform electron gas.