Top-Down versus Bottom-Up Approaches for σ‑Functionals\nBased on the Approximate Exchange Kernel

Recently, we investigated\na number of so-called σ- and τ-functionals\nbased on the adiabatic-connection fluctuation–dissipation theorem\n(ACFDT); particularly, extensions of the random phase approximation\n(RPA) with inclusion of an exchange kernel in the form of an antisymmetrized\nHartree kernel. One of these functionals, based upon the approximate\nexchange kernel (AXK) of Bates and Furche, leads to a nonlinear contribution\nof the spline function used within σ-functionals, which we previously\navoided through the introduction of a simplified “top-down”\napproach in which the σ-functional modification is inserted\na posteriori following the analytic coupling strength integration\nwithin the framework of the ACFDT and which was shown to provide excellent\nperformance for the GMTKN55 database when using hybrid PBE0 reference\norbitals. In this work, we examine the analytic “bottom-up”\napproach in which the spline function is inserted a priori, i.e.,\nbefore evaluation of the analytic coupling strength integral. The\nnew bottom-up functionals, denoted σ↑AXK, considerably\nimprove upon their top-down counterparts for problems dominated by\nself-interaction and delocalization errors. Despite a small loss of\naccuracy for noncovalent interactions, the σ↑AXK@PBE0\nfunctionals comprehensively outperform regular σ-functionals,\nscaled σ-functionals, and the previously derived σ+SOSEX-\nand τ-functionals in the WTMAD-1 and WTMAD-2 metrics of the\nGMTKN55 database.