Quantitative prediction of rate constants and its application to organic emitters

Many phenomena in nature consist of multiple elementary processes. If we can predict all the rate constants of respective processes quantitatively, we can comprehensively predict and understand various phenomena. Here, we report that it is possible to quantitatively predict all related rate constants and quantum yields without conducting experiments, using multiple-resonance thermally activated delayed fluorescence (MR–TADF) as an example. MR–TADFs are excellent emitters because of its narrow emission, high luminescence efficiency, and chemical stability, but they have one drawback: slow reverse intersystem crossing (RISC), leading to efficiency roll-off and reduced device lifetime. Here, we show a quantum chemical calculation method for quantitatively obtaining all the rate constants and quantum yields. This study reveals a strategy to improve RISC without compromising other important factors: radiative decay rate constants, photoluminescence quantum yields, and emission linewidths. Our method can be applied in a wide range of research fields, providing comprehensive understanding of the mechanism including the time evolution of excitons.


Method of calculating spin-orbit coupling, vibronic coupling constant, transition dipole moment, and permanent dipole moment
The formula for Tm-Tn vibronic coupling constant is obtained by replacing S1 and S0 with Tm and Tn, respectively.
(B4) Φ T 1 : electronic wave function of T1 Φ T 2 : electronic wave function of T2 : point in the three-dimensional space Φ T 1 and Φ T 2 were calculated with Gaussian 16 Rev C01 program package. 3 The integral ⟨Φ T 1 ||Φ T 2 ⟩ was calculated with the method proposed by McMurchie and Davidson. 4 S1-S0 transition dipole moment and permanent dipole moment of S0, T1, and T2 were calculated with Gaussian 16 Rev C01 program package 3 .

Method of calculating excited-state populations and the number of emitted photons
Transient photoluminescence decay curve is numerically calculated by the following kinetic equations for S0, S1, T1, and T2 populations.The kinetic equations were solved numerically using our own code.In the main text, we define the lifetime for the total radiative decay as τtoR = 1/ktoR and mention that τtoR is a simple extension of the averaged radiative decay time proposed by Yersin et al. 3 Here, we discuss this point in detail.From Equation 9, τtoR is written as The lifetimes for Sn→S0 fluorescence (τF(Sn→S0)) and Tm→S0 phosphorescence (τPhos(Tm→S0)) are written as τF(Sn→S0) = 1/kF(Sn→S0) and τPhos(Tm→S0) = 1/kPhos(Tm→S0), respectively (n, m = 1, 2, 3, …).Hence, When the excited singlet and triplet states are thermally equilibrated and E(T1) ≤ E(S1), [Sn] and [Tm] can be written in terms of [T1] and Sn-T1 and Tm-T1 energy differences τtoR is the population-weighted average of τF(Sn→S0) and τPhos(Tm→S0).When only S1 and T1 are considered, C13 can be written as Yersin et al. proposed the averaged radiative decay time (τav) under the assumption that S1 and T1 are thermally equilibrated where τ(S1) and τ(T1) are the decay times for S1 and T1, respectively, and ΔE(S1 − T1) is the S1-T1 energy gap.Note that the factor of 3 originating from the three triplet substates does not appear explicitly in the Yersin et al.'s formula (Equation C15).Comparing Equations C14 and C15 shows that τtoR defined in this study (Equation C14) is a simple extension of the Yersin et al.'s formula (Equation C15).
When E(S1) < E(T1) (in the case of inverted singlet-triplet excited states), [Sn] and [Tm] can be written in terms of [S1] and Sn-S1 and Tm-S1 energy differences Therefore, When only S1 and T1 are considered, C18 can be written as (C19) and (C14) are essentially the same.
In the main text, we define the total ISC ktoISC as When the excited singlet states are thermally equilibrated, from (C16) and (C20) Thus, (C20) contains the effect of thermal activation and deactivation between the excited singlet states on ktoISC.Likewise, when the triplet states are thermally equilibrated,
3h nucleus   : cartesian coordinates of the i th electron,   : cartesian coordinates of the A th nucleus, si: spin operator of the electron i Φ S 1 and Φ T   S were calculated with Gaussian 16 Rev C01 program package.3Theintegral 4 : electronic wave function of T1 state FSC: fine-structure constant   eff : effective nuclear charge of the A S ⟩ was calculated with the method proposed by McMurchie and Davidson.42.Vibronic coupling constant between S1 and S0 for the α th vibrational mode, Vα(S1-S0)

Table 1 |
Nuclear coordinates of S1 geometry of BNOO in gas phase optimised at the TD-TPSSh/6-31G(d) level of theory.

Table 2 |
Nuclear coordinates of S1 geometry of BNSS in gas phase optimised at the TD-TPSSh/6-31G(d) level of theory.

Table 3 |
Nuclear coordinates of S1 geometry of BNSeSe in gas phase optimised using the TD-TPSSh method.For the H, B, C, and N atoms, the 6-31G(d) basis set was used.For the Se atoms, the Stuttgart/Dresden pseudopotentials and basis set (SDD) were used.

Table 4 |
Nuclear coordinates of S1 geometry of BNTeTe in gas phase optimised using the TD-TPSSh method.For the H, B, C, and N atoms, the 6-31G(d) basis set was used.For the Te atoms, the Stuttgart/Dresden pseudopotentials and basis set (SDD) were used.

Table 5 |
Nuclear coordinates of S1 geometry of BNPoPo in gas phase optimised using the TD-TPSSh method.For the H, B, C, and N atoms, the 6-31G(d) basis set was used.For the Po atoms, the Stuttgart/Dresden pseudopotentials and basis set (SDD) were used.

Table 6 |
Nuclear coordinates of S1 geometry of BNCOCO in gas phase optimised at the TD-TPSSh/6-31G(d) level of theory.

Table 7 |
S1, S2, T1, T2, and T3 of BNOO calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.

Table 8 |
S1, S2, T1, T2, and T3 of BNSS calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.

Table 9 |
S1, S2, T1, T2, and T3 of BNSeSe calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.

Table 10 |
S1, S2, T1, T2, and T3 of BNTeTe calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.

Table 11 |
S1, S2, T1, T2, and T3 of BNPoPo calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.

Table 12 |
S1, S2, T1, T2, and T3 of BNCOCO calculated using the TD-B3LYP//TD-TPSSh method.Φ   is a Slater determinant that denotes the single-electron excitation from the ath occupied orbital to the rth unoccupied orbital.