Towards Density Functional Approximations from Coupled Cluster Correlation Energy Densities

(Semi-)local density functional approximations (DFAs) are the workhorse electronic structure methods in condensed matter theory and surface science. The correlation energy density εc(r) (a spatial function that yields the correlation energy Ec upon integration) is central to defining such DFAs. Unlike Ec, εc(r) is not uniquely defined, however. Indeed, there are infinitely many functions that integrate to the correct Ec for a given electron density ρ. The challenge for constructing useful DFAs is thus to find a suitable connection between εc(r) and ρ. Herein, we present a new such approach by deriving εc(r) directly from the coupled- cluster (CC) energy expression. The corresponding energy densities are analyzed for prototypical two-electron systems. To explore their usefulness for designing DFAs, we construct a semilocal functional to approximate the numerical CC correlation energy densities. Importantly, the energy densities are not simply used as reference data, but guide the choice of the functional form, leading to a remarkably simple and accurate correlation functional for the Helium isoelectronic series.