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 ϵ_c(r) (a spatial function that yields the correlation energy E_c upon integration) is central to defining such DFAs. Unlike E_c, ϵ_c(r) is not uniquely defined, however. Indeed, there are infinitely many functions that integrate to the correct E_c for a given electron density ρ. The challenge for constructing useful DFAs is thus to find a suitable connection between ϵ_c(r) and ρ. Herein, we present a new such approach by deriving ϵ_c(r) directly from the coupled-cluster (CC) energy expression. The corresponding energy densities are analyzed for prototypical two-electron systems. As a proof-of-principle, 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. While the resulting functional is not transferable to many-electron systems (due to a lack of same-spin correlation), these results underscore the potential of the presented approach.