First beyond-LSDA density functional satisfying a tight lower bound for exchange

Universal constraints of density functional theory (DFT) play major roles in approximating its exchange-correlation energy ($E_{\mbox{xc}} )$. One of the prominent constraints is the Lieb-Oxford bound: $E_{xc}^{exact} [N]\ge \lambda_{xc} [N]E_{x}^{LDA} [N]$, where LDA stands for local density approximation, N is the electron number of systems, and $\lambda_{xc} [N]$ increases with N with an upper bound of 2.275. For ground-state 1-e systems, the above inequality reduces to $E_{x}^{exact} [N\mbox{=1}]\ge \lambda_{xc} [N\mbox{=1}]E_{x}^{LDA} [N\mbox{=1}]$ with a tight bound $\lambda_{xc} [N\mbox{=1}]=$1.48, shedding light on the exchange energy. Our recent study (John P. Perdew's talk) shows that, to avoid violating the tight bound for any possible 1-e densities, a semilocal functional should respect it locally. We further conjecture for exchange energies that $E_{x}^{exact} [N]\ge \gamma_{x} [N]E_{x}^{L\mbox{S}DA} [N]$ with$\gamma_{x} [N]$ decreasing with N and $\gamma_{x} [N=1]=\gamma_{x} [N=2]=\lambda _{xc} [N\mbox{=1}]$/2$^{1/3} =$1.174. Here, local spin density approximation (LSDA) is used as the reference since the exchange has a well-defined spin-scaling relation. Based on the tight Lieb-Oxford bound and the conjecture, we present a simple meta-generalized gradient approximation (MGGA) for exchange that interpolates different LSDAs for N$=$1 and uniform electron gas (N $\to$ infinity), respectively, and delivers excellent exchange energies for atoms. When combined with a modified PBE correlation, the MGGA yields good binding energies for molecules and lattice constants for solids.