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Do any types of double-hybrid models render the correct order of excited state energies in inverted singlet-triplet emitters?

Organic emissive materials with the inverted singlet-triplet energy gaps, where in violation of Hund's multiplicity rule the lowest triplet excited-state is higher in energy than the lowest singlet excited-state, have recently come into the limelight. This unique feature is of important relevance, where the emitters meeting the singlet-triplet inversion have potential to usher in the next generation of organic light emitting diodes (OLEDs). Since experimental data in this context are currently sparse, necessity of the cost-effective theoretical tools able to provide reliable results seems to be evident. Following our recent endeavors on the spin-component-scaled (SCS), spin-opposite-scaled (SOS), and SOS-range separated exchange (SOS-RSX) double-hybrids (DHs) as well as other efforts revealing the superior performances of such models for time-dependent computations, in the present work, we develop and validate several models based on the SOS-configuration interaction singles with perturbative doubles correction [SOS-CIS(D)] devoid of any fitting procedure for describing the singlet-triplet inversion. Taking a series of emitters with the available reference values for the inverted singlet-triplet energy gaps as working models, it is unveiled that the extremes of the same-spin and opposite-spin parameters included in the direct and indirect terms of the SOS-CIS(D) correlation energy as well as the nonlocal exchange and correlation contributions do not necessarily work well for the inverted gaps, but particular proportions among them are needed to achieve a reliable accuracy. Perusing the results of our developed methods, the best one based on the Perdew-Burke-Ernzerhof (PBE) exchange and correlation terms and the quadratic integrand model, denominated as SOS0-CIS(D)-PBE-QIDH, is shown to be highly efficient and robust for computations of the inverted singlet-triplet energy gaps. Furthermore, through detailed comparisons, we have also evaluated the performances of a variety of the recently presented DHs, including parameterized, parameter-free, RSX, as well as spin-component and spin-opposite scaling models for the purpose. Dissecting all the findings, it is disclosed that the results of any type of the DHs cannot be reliable, leading to positive energy gaps in most cases. Nonetheless, there are still some approximations, including SCS-PBE-QIDH, dispersion corrected spin-component scaled double-hybrids (DSD) of DSD-PBEP86 and DSD-BLYP, SOS-PBE-QIDH, SOS-ωPBEPP86, and SOS-RSX-QIDH, that can predict the negative singlet-triplet energy gaps for all the considered emitters and provide comparable performances with respect to our proposed model. To wrap up, among the large panel of different families of DHs on the market, the newly proposed model herein alongside these latter functionals can be recommended as the currently best affordable methods for subsequent applications on the inverted singlet-triplet emitters in OLED materials.