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) and 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 TD-DFT 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 (B2-PLYP ωand RSX-QIDH), spin-scaled (SCS/SOS-B2PLYP21, SCS-PBE-QIDH, andSOS-PBE-QIDH), and range-separated spin-scaled (SCS/SOS-B2-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.The work in Alicante is supported by the “Ministerio de Ciencia e Innovación” of Spain (Grant No. PID2019-106114GB-I00). Y.O. acknowledges funding by the “Fonds de la Recherche Scientifique-FNRS” under Grant n. F.4534.21 (MIS-IMAGINE). E.B. thanks ANR (Agence Nationale de la Recherche) and CGI (Commissariat à l’Investissement d’Avenir) for their financial support to this work through Labex SEAM (Science and Engineering for Advanced Materials and devices), Grant Nos. ANR-10-LABX-096 and ANR-18-IDEX-0001. G.R. acknowledges a grant from the “Fonds pour la formation a la Recherche dans l’Industrie et dans l’Agriculture” (F.R.I.A.) of the F.R.S.-F.N.R.S