Fluorescence via Reverse Intersystem Crossing from Higher Triplet States in a Bisanthracene Derivative

To elucidate the high external quantum efficiency observed for organic light-emitting diodes using a bisanthracene derivative (BD1), non-radiative transition processes as well as radiative ones are discussed employing time-dependent density functional theory. It has been previously reported that the observed high external quantum efficiency of BD1 cannot be explained by the conventional thermally activated delayed fluorescence involving T1 exciton nor triplet-triplet annihilation. The calculated off-diagonal vibronic coupling constants of BD1, which govern the non-radiative transition rates, suggest a fluorescence via higher triplets (FvHT) mechanism, which entails the conversion of a high triplet exciton generated during electrical excitation into a fluorescent singlet exciton. This mechanism is valid as long as the relaxation of high triplet states to lower states is suppressed. In the case of BD1, its pseudo-degenerate electronic structure helps the suppression. A general condition is also discussed for the suppression of transitions in molecules with pseudo-degenerate electronic structures.


S1 Radiative and Non-Radiative Transition Rates
We consider transition from the initial vibronic state where |Ψ m ⟩ and |χ i ⟩ denote the initial electronic and vibrational states, and |Ψ n ⟩ and χ j ⟩ stand for the final electronic and vibrational states, respectively. Radiative and non-radiative transition rate constants between these vibronic states can be obtained as per the Fermi's golden rule, which describes a transition rate constant between quantum states in a general manner, as described in our previous papers 1,2 . Summing radiative transition rate constants over all the vibrational states, the radiative transition rate constant k r from the initial electronic state |Ψ m ⟩ to the final one |Ψ n ⟩ is given by where ω denotes an angular frequency of an emitted photon, c is the speed of light, P mi (T ) stands for the statistical weight of |Φ mi ⟩ at the temperature T , µ µ µ nm is the transition dipole moment between |Ψ m ⟩ and |Ψ n ⟩, andh is the reduced Planck Constant. On the other hand, the non-radiative transition rate constant k nr,α via mode α is given by where V mn α is the off-diagonal vibronic coupling constant between |Ψ m ⟩ and |Ψ n ⟩ with respect to mode α, and Q α denotes a mass-weighted normal coordinate of mode α.

S2 Vibronic Coupling Density and Transition Dipole Moment Density
Suppose that an operatorÔ consists of one-electron operatorsô without any differential operators, where r r r i denotes the spatial coordinate of electron i. A matrix element ofÔ is given by 1 where x x x i = (r r r i , s i ) with spatial coordinate r r r i and spin coordinate s i for electron i. ρ mn (r r r) ×ô(r r r) is a density form of the matrix element O mn . It should be noted that any approximation is not employed in this derivation.
An electric dipole moment operatorμ µ µ is an example ofÔ: Therefore, a transition dipole moment µ µ µ mn between electronic states m and n is given by the integral of a transition dipole moment density τ τ τ mn (r r r) = ρ mn (r r r) × (−er r r) 1,2 .
An off-diagonal vibronic coupling constant V mn α is where U ne denotes the sum of nuclear-electronic potentials, U nn stands for the sum of nuclear-nuclear potentials, and Q α is a mass-weighted normal coordinate of mode α. Since (∂U ne /∂ Q α ) R R R 0 can be written as the sum of one-electron potential derivatives: V mn α is given by the integral of a vibronic coupling density η mn α (r r r) = ρ mn (r r r) × v α (r r r) 1,2 .   Figure S3: Level shifts of triplet excited states of BD1 during geometry optimizations.

S7
Since all the optimized structures show D 2 symmetry, the selection rules for transition dipole moment, spinorbit coupling, and vibronic coupling within D 2 symmetry are discussed here. Table S2 is the character table of the D 2 point group. The direct products of the irreducible representations (irreps) are tabulated in Table   S3. From Table S1, the components of the electric dipole operator,μ x ,μ y ,μ z transform according to the B 3 , B 2 , and B 1 irreps, respectively, and the components of the orbital angular momentum operator,L x ,L y , L z transform according to the B 3 , B 2 , and B 1 irreps, respectively. Based on Table S3, the selection rules for transition dipole moment, spin-orbit coupling, and vibronic coupling were obtained and are listed in Tables S4, S5, and S6. According to Tables S5 and S6, an electric dipole transition or intersystem crossing between electronic states with the same irrep is symmetry forbidden.