No, Density Functional Theories Do Not Fail for Inverted Singlet-Triplet Gap

Soumen Ghosh

In organic light-emitting diodes (OLEDs), charge recombination is an important step that produces singlet or triplet excitons from spatially separated holes and electrons. Charge recombination process produce 75% triplet and 25% singlet excitons. Generation of photons from singlet excitons happen through a spin-allowed de-excitation process. However, due to the spin forbidden nature of the de-excitation process of triplet excitons, these often contribute to energy loss. Several strategies have been developed for transforming triplet excitons produced during the recombination process to singlet excitons. In many OLEDs, triplet state (T1) to singlet state (S1) reverse intersystem crossing (RISC) is achieved by tuning the energy gap (ΔEST) between energetically close S1 and T1 excited states. The process of RISC followed by de-excitation is referred to as thermally activated delayed fluorescence (TADF). Requirement of thermal activation for the RISC process can be eliminated if triplet excited state is energetically lower in energy than the first singlet excited state. In recent years, a new class of molecules have been found that show inverted singlet-triplet gaps.

Mechanism for radiative decay in inverted singlet-triplet molecules and seven molecules from our test set

I was introduced to inverted singlet-triplet molecules when Kalishankar sent a draft of a manuscript to me which was later published in Chem. Phys. Lett. 2021, 779, 138827. He clearly showed in that draft that time-dependent density functional theory (TDDFT) with different types of exchange-correlation functionals fail to produce inverted singlet-triplet gaps while all types of coupled-cluster methods were very successful for these systems. To me, it was a very interesting observation as it was clear that this is not an issue of choosing right exchange-correlation functionals. Like TDDFT, CIS also fails for inverted singlet-triplet gap. However, doubles-corrected CIS [CIS(D)] produce inverted singlet-triplet gaps qualitatively correctly. These results indicated to us that adding second order perturbative correction to TDDFT can be helpful to solve this problem. Luckily, there was already such a theory that was developed for double-hybrid (DH) functionals, available in Orca. However, choice of functionals were limited at that time. So we tested most popular DH functional B2PLYP and it showed mixed success. It was encouraging to us because it meant that our initial hypothesis was right, now we just have to find the correct functional form. 

It was clear to us from the beginning that B2PLYP has too little MP2 correlation to tackle this problem. Luckily for us, very soon a large number of well-optimized DH functionals were added to the new version of Orca and we started testing them for this problem immediately. We soon found out that it is essential to choose a functional with high non-local correlation and high Hartree-Fock (HF) exchange to produce qualitatively correct inverted singlet-triplet gaps.

         Our observation about DH functionals were interesting but I was still not fully satisfied with this solution. In principle with a proper local exchange-correlation functional form, one should be able to capture the same physics that the non-local MP2 correlation provides. We also analyzed multireference wave functions of ground and excited- states. We found that even though there were non-zero contributions of doubly excited configurations in the first singlet excited-states of inverted singlet-triplet gap molecules that we studied, the contribution was small (~5%). Moreover, CIS(D) or TDDFT(D) methods do not treat doubly excited configurations explicitly. So their success has to be due to an interplay of exchange and correlation energy and not due to proper treatment of doubly excited configurations. Up to then, many of the previous studies were predicting that one need to go beyond adiabatic approximation to solve this problem but we were not convinced. As a next step, I decided to give variational excited state methods a try. We only had one system where ground state and first singlet excited states has different symmetry. When I calculated ground and excited states variationally for that molecule with PBE0 functional, I got inverted singlet-triplet gap. We immediately knew this was the correct solution to this problem. However, we have to generalize it, so that it can be applied to any molecule with any symmetry or no symmetry. We took advantage of a method called excited state DFT where one can fix the occupation numbers of orbital for a KS-DFT variational solution to prevent variational collapse. It was made simple by the fact that first singlet excited state was predominately a HOMO to LUMO transition. With this approach, we were able to get inverted singlet-triplet gap for all the molecules in our test set using functionals like PBE, B3LYP and PBE0. Higher HF exchange did increase the gap which was expected. With this, we have proved that we don’t need to go beyond adiabatic approximation in TDDFT to solve this problem, it can be solved if we add orbital optimization to the excited state solution.

Our findings will significantly reduce the cost of large-scale screening needed to find new inverted singlet-triplet molecules as now one don’t need to use correlated wave function methods for such screening.