Violation of Hund's rule in molecules: Predicting the excited-state energy inversion by TD-DFT with double-hybrid methods

The energy difference (ΔEST) between the lowest singlet (S1) state and the triplet (T1) excited state of a set of azaphenalene compounds, which is theoretically and experimentally known to violate Hund's rule, giving rise to the inversion of the order of those states, is calculated here with a family of double-hybrid density functionals. That excited-state inversion is known to be very challenging to reproduce for time-dependent density functional theory employing common functionals, e.g., hybrid or range-separated expressions, but not for wavefunction methods due to the inclusion of higher-than-single excitations. Therefore, we explore here if the last developed family of density functional expressions (i.e., double-hybrid models) is able to provide not only the right excited-state energy order but also accurate ΔEST values, thanks to the approximate inclusion of double excitations within these models. We herein employ standard double-hybrid (B2-PLYP, PBE-QIDH, and PBE0-2), range-separated (ωB2-PLYP and RSX-QIDH), spin-scaled (SCS/SOS-B2PLYP21, SCS-PBE-QIDH, and SOS-PBE-QIDH), and range-separated spin-scaled (SCS/SOS-ωB2-PLYP, SCS-RSX-QIDH, and SOS-RSX-QIDH) expressions to systematically assess the influence of the ingredients entering into the formulation while concomitantly providing insights for their accuracy.