Density functionals for the strong-interaction limit

Abstract

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 $E_{\mathrm{xc}}\left[\rho \right]\equiv {\int }_0^{1}d\alpha W_{\alpha }\left[\rho \right]$ follows from the xc potential energies $W_{\alpha }$ at different interaction strengths $\alpha >~0$ [but at fixed density $\rho \left(\mathbf{r}\right)].$ For small $\alpha \approx 0,$ the integrand $W_{\alpha }$ is obtained accurately from perturbation theory, but the perturbation expansion requires resummation for moderate and large $\alpha .$ For that purpose, we present density functionals for the coefficients in the asymptotic expansion $W_{\alpha }\to W_{\infty }{+W}_{\infty }^{\prime }{\alpha }^{-1/2}$ for $\overrightarrow{\alpha }\infty$ in the PC model. $W_{\infty }^{\mathrm{PC}}$ arises from strict correlation, and $W_{\infty }^{\prime \mathrm{PC}}$ from zero-point vibration of the electrons around their strictly correlated distributions. The PC values for $W_{\infty }$ and $W_{\infty }^{\prime }$ 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.