Searching the Best\nDouble-Hybrid Density Functional\nto Correctly Predict the Singlet–Triplet Excited-State Inversion\nin Organic Systems

The theoretically disclosed, and experimentally confirmed,\nenergy\ninversion of the lowest singlet (<i>S</i><sub>1</sub>) and\ntriplet (<i>T</i><sub>1</sub>) excited states of organic\nmolecules (i.e., Hund’s rule violation) is investigated herein\nwith the aid of modern and nonempirically derived double-hybrid (DH)\ndensity functionals, in the search of the best trade-off between accuracy\nand computational cost of viable electronic structure methods. For\nthat purpose, we have selected a family of parameter-free expressions\ndiffering in their specific formulation (DFT-0DH, DFT-QIDH, DFT0-2,\nSOS1-DFT-0DH, SOS1-DFT-QIDH, SOS1-DFT0-2, RSX-DFT-0DH, RSX-DFT-QIDH,\nSOS1-RSX-DFT-0DH, and SOS1-RSX-DFT-QIDH) as well as in the underlying\nexchange–correlation functional used (PBE vs r<sup>2</sup>SCAN).\nFor the sake of evaluating which DH can correctly describe the singlet–triplet\nexcited-state energy inversion, second-order approximate with singles\nand doubles method with a spin-component scaling (SCS-CC2) and equation-of-motion\ncoupled cluster singles and doubles (EOM-CCSD) calculations are also\ncarried out. The results highlight the importance of the delicate\nbalance between all the energy terms composing DH density functionals,\nwith the correlation part being particularly significant for achieving\nthe most accurate results. We have also derived a new DH density functional\n(PBE-DH-INVEST) exploiting that relationship, providing low error\nmetrics and expected to yield robust results in, e.g., high-throughput\nstudies.