Does Correlated Orbital Theory improve PBE-like functionals?

Correlated Orbital Theory (COT) provides an exact one-particle framework by imposing rigorous physical constraints on Kohn-Sham eigenvalues, and as a consequence, directly incorporates essential electron correlation into molecular orbitals. This approach paves the way toward a new class of approximations within Kohn–Sham Density Functional Theory (KS-DFT). However, since all existing Quantum Theory Project (QTP) functionals are derived from CAM-B3LYP, we pose the question: Can COT improve the hybrid versions of different exchange-correlation (XC) functionals as well? To that end, we explore two optimization strategies for adjusting the existing parameters within PBE0, TPSS0, and LC-PBE0: \textit{i}) the ionization potential (IP) condition and \textit{ii}) the HOMO–LUMO condition. As a result, in general, enforcing the IP condition consistently improves performance across the functionals from the PBE family. In contrast, the HOMO–LUMO condition yields meaningful improvements primarily for the range-separated XC functional considered (LC-PBE0).