Connections between High-Density Scaling Limits of DFT Correlation Energies and Second-Order Z⁻¹ Quantum Chemistry Correlation Energy

In the continuing search for ever-better approximations to the full density-functional correlation energy functional Ec[n], we established the link between the second-order component of the correlation energy, [n] [which occurs through uniform scaling, [n] = Ec[nλ], where nλ(x,y,z) = λ3n(λx,λy,λz)], and the known result for the second-order Z-1 quantum chemistry correlation energy, . Except when certain degeneracies occur, [n] ≤ , with an equality only for two electrons. On the other hand, the correlation energy functional HFEc[n], whose functional derivative is meant to be added to the Hartree-Fock non-local effective potential to produce, via self-consistency, the exact ground-state density and ground-state energy, satisfies the equality HF [n]= , where HF [n] = HFEc[nλ], for any number of electrons, except when some degeneracies occur. Because quantities [n] and 2HF [n] are the initial slopes in the adiabatic connection formulas for Ec[n] and HFEc[n], respectively, the presented equalities involving [n] are especially significant. Five numerical tests are presented for closed- and open-shell densities obtained from hydrogenic orbitals. These tests are applied to widely used approximations to correlation energies.