Double-hybrid density functionals are currently the most accurate density functionals for ground-state properties and electronic excitations. Nevertheless, the lack of a long-range correction scheme makes them unreliable when it comes to long-range excitations. For this reason, we propose the first two time-dependent double-hybrid functionals with correct asymptotic long-range behavior named ωB2PLYP and ωB2GPPLYP. Herein, we demonstrate their excellent performance and show that they are the most accurate and robust time-dependent density functional theory methods for electronic excitation energies. They provide a balanced description of local-valence, Rydberg, and charge-transfer states. They are also able to tackle the difficult first two transitions in polycyclic aromatic hydrocarbons and show very promising results in a preliminary study on transition-metal compounds, exemplified for titanium dioxide clusters. This work shows that double hybrids can be systematically improved also for excitation energies, and further work in this field is warranted.

The development of non-empirical double-hybrid density functionals (DHDFs) is a very active research area with the number of approaches in this field having increased rapidly. At the same time, there is a lack of published work that provides a fair assessment and comparison between non-empirical and semi-empirical DHDFs on an equal footing. Herein, we close this gap and present a thorough analysis of both classes of DHDFs on the large GMTKN55 benchmark database for general main-group thermochemistry, kinetics, and noncovalent interactions [Goerigk et al., Phys. Chem. Chem. Phys., 2017, 19, 32184-32215]. In total, 115 variations of dispersion-corrected and -uncorrected DHDFs are tested, which will be condensed to an in-depth assessment of 31 methods: 19 non-empirical and 12 semi-empirical DHDFs. As such, our study represents the largest DHDF study ever conducted and can serve as an important benchmark informing method developers and users alike. Our results show that semi-empirical DHDFs are the most robust density functional approximations and more reliable and accurate than non-empirical ones. In fact, some non-empirical approaches are even outperformed by hybrid approaches or even dispersion-corrected and -uncorrected MP2 and SCS-MP2. SOS0-PBE0-2-D3(BJ) is the only exception and the only non-empirical DHDF that we can safely recommend for general applicability. However, it is still outperformed by six semi-empirical DHDFs, of which we would like to particularly recommend the following five: ωB97X-2-D3(BJ), DSD-BLYP-D3(BJ), DSD-PBEP86-D3(BJ), B2NC-PLYP-D3(BJ), and B2GPPLYP-D3(BJ). Our findings seriously question current trends in the field and they highlight that novel strategies have to be found in order to outperform the currently best density functional theory methods on the market. We hope that our study can function as an important cornerstone inspiring such a change of direction in the field.

We continue our work on the long-range corrected double-hybrid density functionals (LC-DHDFs) ωB2PLYP and ωB2GP-PLYP that we developed in the context of time-dependent (TD) Density Functional Theory (DFT) to enable the robust description of singlet–singlet excitations [M. Casanova-Páez, M. B. Dardis, and L. Goerigk, J. Chem. Theory Comput. 15, 4735 (2019)]. In our initial study, we only assessed the impact of a LC on BLYP-based DHDFs, and herein, we extend our understanding by providing the first test of PBE-based LC-DHDFs within the established TD-DHDF scheme. Moreover, this study is one of few that provides a direct comparison between TD-DHDFs and their faster Tamm–Dancoff-approximation variants (TDA-DHDFs). Most importantly, this is the first TDA-DHDF study since Grimme and Neese’s TDA-B2PLYP [J. Chem. Phys. 127, 154116 (2007)] and the first work on TD-DHDFs that addresses singlet–triplet excitations. We show how the difference between TD-DHDFs and TDA-DHDFs is often negligible for singlet–singlet excitations, but how one has to apply TDA-DHDFs for triplet excitations. For both excitation types, the LC is beneficial to the BLYP-based DHDFs, but detrimental to the PBE-based ones. For local-valence and Rydberg excitations, ωB2PLYP and ωB2GP-PLYP as well as the global DHDF PBE-QIDH can be recommended. If a transition exhibits charge-transfer character, ωB2PLYP and ωB2GP-PLYP should be applied. An analysis of the gaps between the first singlet and triplet excited states of our systems revealed that there is room for further improvements to reach better robustness. Until that goal has been achieved, we recommend ωB2PLYP and ωB2GP-PLYP as some of the currently best TDA-DFT methods.

Following the work on spin-component and spin-opposite scaled (SCS/SOS) global double hybrids for singlet-singlet excitations by Schwabe and Goerigk [J. Chem. Theory Comput. 2017, 13, 4307-4323] and our own works on new long-range corrected (LC) double hybrids for singlet-singlet and singlet-triplet excitations [J. Chem. Theory Comput. 2019, 15, 4735-4744; J. Chem. Phys. 2020, 153, 064106], we present new LC double hybrids with SCS/SOS that demonstrate further improvement over previously published results and methods. We introduce new unscaled and scaled versions of different global and LC double hybrids based on Becke88 or PBE exchange combined with LYP, PBE or P86 correlation. For singlet-singlet excitations, we cross-validate them on six benchmark sets that cover small to medium-sized chromophores with different excitation types (local valence, Rydberg, and charge transfer). For singlet-triplet excitations, we perform the cross-validation on three different benchmark sets following the same analysis as in our previous work in 2020. In total, 203 unique excitations are analyzed. Our results confirm and extend those of Schwabe and Goerigk regarding the superior performance of SCS and SOS variants compared to their unscaled parents by decreasing mean absolute deviations, root-mean-square deviations or error spans by more than half and bringing absolute mean deviations closer to zero. Our SCS/SOS variants show to be highly efficient and robust for the computation of vertical excitation energies, which even outperform specialized double hybrids that also contain an LC in their perturbative part. In particular, our new SCS/SOS-ωPBEPP86 and SCS/SOS-ωB88PP86 functional are four of the most accurate and robust methods tested in this work and we fully recommend them for future applications. However, if the relevant SCS and SOS algorithms are not available to the user, we suggest ωB88PP86 as the best unscaled method in this work.

This account is written for general users of time-dependent density functional theory (TD-DFT) methods as well as chemists who are unfamiliar with the field. It includes a brief overview of conventional TD-DFT approaches and recommendations for applications to organic molecules based on our own experience. The main emphasis of this work, however, lies in providing the first in-depth review of time-dependent double-hybrid density functionals. They were first established in 2007 with very promising follow-up studies in the subsequent four years before developments or applications became scarce. The topic has regained more interest since 2017, and this account reviews those latest developments led by our group. These developments have shown unprecedented robustness for a variety of different types of electronic excitations when compared to more conventional TD-DFT methods. In particular, time-dependent double hybrids do not suffer from artificial ghost states and are able to reproduce exciton-coupled absorption spectra. Our latest methods include range separation and belong to the currently best TD-DFT methods for singlet-singlet excitations in organic molecules. While there is still room for improvement and further development in this space, we hope that this account encourages users to adjust their computational protocols to such new methods to provide more real-life testing and scenarios.