Top-Down versus Bottom-Up Approaches for σ-Functionals Based on the Approximate Exchange Kernel

Recently, we investigated a number of so-called σ- and τ-functionals based on the adiabatic-connection fluctuation-dissipation theorem (ACFDT); particularly, extensions of the random phase approximation (RPA) with inclusion of an exchange kernel in the form of an antisymmetrized Hartree kernel. One of these functionals, based upon the approximate exchange kernel (AXK) of Bates and Furche, leads to a nonlinear contribution of the spline function used within σ-functionals, which we previously avoided through the introduction of a simplified "top-down" approach in which the σ-functional modification is inserted a posteriori following the analytic coupling strength integration within the framework of the ACFDT and which was shown to provide excellent performance for the GMTKN55 database when using hybrid PBE0 reference orbitals. In this work, we examine the analytic "bottom-up" approach in which the spline function is inserted a priori, i.e., before evaluation of the analytic coupling strength integral. The new bottom-up functionals, denoted σ↑AXK, considerably improve upon their top-down counterparts for problems dominated by self-interaction and delocalization errors. Despite a small loss of accuracy for noncovalent interactions, the σ↑AXK@PBE0 functionals comprehensively outperform regular σ-functionals, scaled σ-functionals, and the previously derived σ+SOSEX- and τ-functionals in the WTMAD-1 and WTMAD-2 metrics of the GMTKN55 database.