Adiabatic connection forms in density functional theory: $H2$ and the He isoelectronic series

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing two challenging problems in density functional theory—the singlet ground state potential energy curve of $H2$ in a restricted formalism and the energies of the helium isoelectronic series, $H−$ to $Ne8+$⁠. For $H2$⁠, an exponential-based form yields a potential energy curve that is virtually indistinguishable from the FCI curve, eliminating the unphysical barrier to dissociation observed previously with a [1,1]-Padé-based form and with the random phase approximation. For the helium isoelectronic series, the Padé-based form gives the best overall description, followed by the exponential form, with errors that are orders of magnitude smaller than those from a standard hybrid functional. Particular attention is paid to the limiting behavior of the AC forms with increasing bond distance in $H2$ and increasing atomic number in the isoelectronic series; several forms describe both limits correctly. The study illustrates the very high quality results that can be obtained using exchange-correlation functionals based on simple AC forms, when near-exact data are used to determine the parameters in the forms.