Searching the Best Double-Hybrid Density Functional to Correctly Predict the Singlet–Triplet Excited-State Inversion in Organic Systems

The theoretically disclosed, and experimentally confirmed, energy inversion of the lowest singlet (S₁) and triplet (T₁) excited states of organic molecules (i.e., Hund’s rule violation) is investigated herein with the aid of modern and nonempirically derived double-hybrid (DH) density functionals, in the search of the best trade-off between accuracy and computational cost of viable electronic structure methods. For that purpose, we have selected a family of parameter-free expressions differing in their specific formulation (DFT-0DH, DFT-QIDH, DFT0-2, SOS1-DFT-0DH, SOS1-DFT-QIDH, SOS1-DFT0-2, RSX-DFT-0DH, RSX-DFT-QIDH, SOS1-RSX-DFT-0DH, and SOS1-RSX-DFT-QIDH) as well as in the underlying exchange–correlation functional used (PBE vs r²SCAN). For the sake of evaluating which DH can correctly describe the singlet–triplet excited-state energy inversion, second-order approximate with singles and doubles method with a spin-component scaling (SCS-CC2) and equation-of-motion coupled cluster singles and doubles (EOM-CCSD) calculations are also carried out. The results highlight the importance of the delicate balance between all the energy terms composing DH density functionals, with the correlation part being particularly significant for achieving the most accurate results. We have also derived a new DH density functional (PBE-DH-INVEST) exploiting that relationship, providing low error metrics and expected to yield robust results in, e.g., high-throughput studies.