Quantitative prediction of rate constants and its application to organic emitters

Abstract

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.

The quantitative prediction of rate constants and quantum yields is crucial to understand various phenomena and their mechanisms in nature. Here, the authors apply quantum chemical calculation to reproduce experimental data of multiple-resonance thermally activated delayed fluorescence emitters.

Introduction

Multiple-resonance thermally activated delayed fluorescence (MR–TADF) has attracted substantial attention in organic light-emitting diodes (OLEDs) research because it can realise high external quantum efficiency and narrow electroluminescence spectra with high colour purity^1–3. Hatakeyama et al. reported the first MR–TADF molecule (DABNA-1) in 2016¹ by incorporating B and N atoms into a carbon-based molecular structure. In DABNA-1, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) distributions are spatially separated on different atoms, resulting in a relatively small energy difference ΔE(T₁ → S₁) of 0.15–0.2 eV between the lowest excited singlet state (S₁) and lowest triplet state (T₁). The ΔE(T₁ → S₁) value is sufficiently small to cause TADF but rather large; the rate constant (k_RISC) for reverse intersystem crossing (RISC) is small (9.9 × 10³ s⁻¹)¹. Since the report of DABNA-1, a number of MR–TADF molecules have been developed; however, most of them exhibit k_RISC values on the order of 10⁴–10⁵ s^−1 1–7. The slow RISC is considered to cause triplet-related annihilation⁸, resulting in efficiency roll-off in OLEDs. The slow RISC also reduces the device lifetime of OLEDs. Thus, there is an urgent need to develop MR–TADF emitters with large k_RISC to solve these problems without sacrificing the rate constant of fluorescence from S₁ to the ground state (S₀) (k_F(S₁ → S₀) or simply k_F), the photoluminescence quantum yield (PLQY), and the colour purity. Recently, Yasuda’s group developed MR–TADF materials with k_RISC of 10⁸ s⁻¹ by utilising the heavy atom effect of selenium (Se) atoms⁹. However, the k_F (~10⁵ s⁻¹) was more than two orders of magnitude smaller than the oxygen (O) type analogue. Yang’s group developed Se-containing MR–TADF materials. The k_RISC of 10⁶ s⁻¹ and the k_F of ~10⁷ s⁻¹ are more well-balanced¹⁰. However, the efficiency roll-off problem has not been well solved; simultaneous realisation of k_RISC > 10⁷ s⁻¹ and k_F > 10⁷ s⁻¹ is desirable.

Among all the reported MR–TADF molecules, the RISC mechanisms have been analysed in detail for DABNA-1, DABNA-2, and ν-DABNA. Quantum chemical calculations^11–14 and time-resolved photoluminescence (PL) measurements¹⁵ indicate that the total RISC process of the three materials occurs via higher triplet states (T_n, n ≥ 2), typically T₂ (Kim et al.¹³ carried out an analysis including from T₁ to T₃), although only the direct T₁ → S₁ RISC has initially been considered. When RISC occurs via T₂ (T₁ → T₂ → S₁), k_RISC increases with decreasing T₂ → S₁ (especially when S₁ is higher in energy than T₂) and T₁ → T₂ energy gaps (denoted as ΔE(T₂ → S₁) and ΔE(T₁ → T₂), respectively), and increasing S₁–T₂ spin–orbit coupling (SOC(S₁–T₂)). A small |ΔE(T₂ → S₁)| and large SOC(S₁–T₂) accelerate the T₂ → S₁ transition, and a small ΔE(T₁ → T₂) accelerates the T₁ → T₂ internal up-conversion.

We have reported three methods to enhance RISC: (1) intervening in a locally excited state between charge transfer type singlet (¹CT) and triplet (³CT) states¹⁶, (2) using the fluctuational effect for ³CT → ¹CT RISC¹⁷, although it seemingly violates the El-Sayed rule¹⁸, and (3) using the heavy atom effect^19–21. Regarding the third method, a practical strategy for increasing k_RISC is assumed to incorporate third-, fourth-, or lower-row elements to enhance SOC by their heavy atom effect. Various O-, sulfur- (S-), and Se-containing molecules have been developed for conventional TADF emitters composed of donor and acceptor segments^10,22–30; k_RISC is enhanced by 2–20 times via O → S substitution^10,22,23,25, and up to 50 times via O → Se substitution¹⁰. The carbonyl (C = O) group can also enhance SOC because of the n–π* orbital; as evidenced by benzophenone, a representative carbonyl compound, having a large rate constant (k_ISC) for intersystem crossing (ISC) of ~10¹¹ s^−1 31. The C = O group is a promising alternative to S and Se for enhancing SOC with only C, H, and O atoms^5,6,32–35. A C = O-containing MR–TADF emitter developed by the Zysman–Colman group (DDiKTa⁶) exhibited a larger k_RISC of 6.3 × 10⁵ s⁻¹ than that of ν-DABNA (k_RISC = 2.0 × 10⁵ s⁻¹), although DDiKTa had a larger ΔE(T₁ → S₁) of 0.16 eV than ν-DABNA (0.07 eV), suggesting that the C = O groups in DDiKTa accelerated RISC.

Thus, understanding the excited-state decay mechanisms of MR-TADF emitters is important to design novel materials with enhanced TADF properties. To understand the decay mechanisms of a TADF emitter, it is common to determine the rate constants of electronic transitions from the experimental PLQY and transient photoluminescence decay curve fitted by a linear combination of exponential decay functions. A comprehensive understanding of the emission mechanism is achieved only when the rate constants for all elementary electronic transitions have been determined. However, it is difficult to determine all rate constants when the number of the rate constants is larger than that of experimentally determined fitting parameters (specifically, when singlet and triplet states energetically higher than S₁ and T₁ are involved). Our theoretical method allows us to quantitatively predict rate constants and quantum yields, including those inaccessible from experiments. Our method also clarifies the quantitative dynamics (time evolutions) of excitons, providing a comprehensive understanding of the TADF mechanism. Therefore, our method proposed in this study offers a guideline for designing TADF emitters with enhanced properties.

Here, we report a quantitative theoretical investigation of the emission mechanism of MR–TADF molecules, focusing on the acceleration of RISC. As with various phenomena, the emission here consists of multiple elementary processes. The quantitative description of the rate constants not only for RISC but also for all relevant elementary processes is of primary importance because such a description enables a comprehensive understanding and prediction of the emission mechanism. We investigate the RISC mechanism of previously synthesised MR–TADF emitters (BNOO, BNSS, and BNSeSe) (Fig. 1), reported by Yang’s group¹⁰. Here, we define a total ISC/RISC rate constant (k_toISC/k_toRISC) as the entire ISC/RISC process involving both the S₁ → T₁/T₁ → S₁ and S₁ → T₂ ↔ T₁/T₁ ↔ T₂ → S₁ transitions, which correspond to the experimentally obtained values (see the “Methods” section). Electronic states higher than S₁ and T₂ are not required to be considered because they are energetically well separated. All the calculated k_toISC (8.6 × 10⁷, 2.0 × 10⁸, and 1.0 × 10⁹ s⁻¹ for BNOO, BNSS, and BNSeSe, respectively) and k_toRISC (9.2 × 10³, 2.5 × 10⁵, and 1.5 × 10⁶ s⁻¹, respectively) in this study well-reproduce the experimental k_toISC (7.5 × 10⁷, 1.5 × 10⁸, and 4.9 × 10⁸ s⁻¹, respectively) and k_toRISC (4.3 × 10⁴, 1.9 × 10⁵, and 2.0 × 10⁶ s⁻¹, respectively). The calculations also clarify how the S and Se atoms accelerate RISC. All the calculated values of ΔE(T₁ → S₁), k_F(S₁ → S₀), and PLQY (Φ) as well as the prompt and TADF contributions, also reasonably reproduce the experimental results (Table 1); indicating the validity of our calculation method. The calculated k_toRISC of BNOO slightly deviates from the experimental value because of the overestimation of ΔE(T₁ → S₁) (when we use the experimental ΔE(T₁ → S₁) of 0.15 eV instead of the calculated value of 0.21 eV, k_toRISC is calculated to be 8.2 × 10⁴ s⁻¹, close to the experimental value of 4.3 × 10⁴ s⁻¹).

Fig. 1. Molecular structures and HOMO−1, HOMO, and LUMO distributions.

a BNOO, b BNSS, c BNSeSe, d BNTeTe, e BNPoPo, and f BNCOCO. HOMO and LUMO are the highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively.

Table 1.

Calculated energies, energy gaps, rate constants, quantum yields, spin-orbit couplings, and emission linewidths of BNOO, BNSS, BNSeSe, BNTeTe, BNPoPo, and BNCOCO

	BNOO	BNSS	BNSeSe	BNTeTe	BNPoPo	BNCOCO
E(S1) (eV)	2.51 (2.54)	2.44 (2.55)	2.51 (2.58)	2.52	2.41	2.55
E(T1) (eV)	2.30 (2.39)	2.30 (2.42)	2.37 (2.44)	2.38	2.26	2.41
ΔE(T1 → S1) (eV)	0.21 (0.15)	0.14 (0.13)	0.14 (0.14)	0.14	0.14	0.14
ΔE(T2 → S1) (eV)	0.08	−0.04	−0.05	−0.06	−0.12	0.03
ΔE(T1 → T2) (eV)	0.13	0.18	0.19	0.20	0.26	0.11
ktoISC (s−1)	8.6 × 107 (7.5 × 107)	2.0 × 108 (1.5 × 108)	1.0 × 109 (0.5 × 109)	6.6 × 109	4.3 × 1010	6.5 × 108
ktoRISC(T1) (s−1)	8.4	2.0 × 103	1.2 × 105	3.0 × 106	4.9 × 107	39
ktoRISC(T2) (s−1)	9.2 × 103	2.5 × 105	1.4 × 106	7.2 × 106	6.9 × 106	9.9 × 105
ktoRISC (s−1)	9.2 × 103 (4.3 × 104)	2.5 × 105 (1.9 × 105)	1.5 × 106 (2.0 × 106)	1.0 × 107	5.6 × 107	9.9 × 105
ktoRISC′ (s−1)	9.3 × 103	2.5 × 105	1.5 × 106	1.0 × 107	n. d.	9.9 × 105
kF(S1 → S0) (s−1)	2.7 × 108 (8.2 × 107)	2.2 × 108 (4.5 × 107)	2.5 × 108 (2.6 × 107)	2.4 × 108	1.5 × 108	4.7 × 108
kNR(S1 → S0) (s−1)	2.0 × 107	2.1 × 107	1.8 × 107	1.8 × 107	2.0 × 107	1.2 × 107
kISC(S1 → T1) (s−1)	7.9 × 104	1.6 × 106	8.7 × 107	1.9 × 109	3.7 × 1010	2.6 × 104
kISC(S1 → T2) (s−1)	8.6 × 107	2.0 × 108	9.4 × 108	4.7 × 109	5.3 × 109	6.5 × 108
kIC(T2 → T1) (s−1)	9.9 × 1012	2.4 × 1012	2.2 × 1012	1.4 × 1012	4.2 × 1011	4.7 × 1012
kRISC(T2 → S1) (s−1)	1.3 × 106	2.6 × 108	1.8 × 109	1.5 × 1010	1.8 × 1011	7.8 × 107
kNR(T2 → S0) (s−1)	0.68	2.7	1.9 × 102	2.1 × 103	2.9 × 104	0.66
kPhos(T2 → S0) (s−1)	92	1.5 × 103	9.6 × 103	5.7 × 104	3.4 × 105	3.4 × 103
kIC(T1 → T2) (s−1)	7.1 × 1010	2.4 × 109	1.7 × 109	6.5 × 108	1.6 × 107	6.1 × 1010
kRISC(T1 → S1) (s−1)	8.5	2.0 × 103	1.2 × 105	3.0 × 106	4.9 × 107	40
kNR(T1 → S0) (s−1)	3.8	14	1.4 × 102	1.1 × 103	3.5 × 104	0.44
kPhos(T1 → S0) (s−1)	1.6	10	4.9 × 102	6.6 × 103	9.7 × 104	0.25
kPrompt (s−1)	3.7 × 108 (1.9 × 108)	4.4 × 108 (2.0 × 108)	1.3 × 109 (0.5 × 109)	6.8 × 109	4.1 × 1010	1.1 × 109
kDelayed (s−1)	7.1 × 103 (26 × 103)	1.4 × 105 (0.5 × 105)	3.1 × 105 (1.0 × 105)	3.9 × 105	3.5 × 105	4.2 × 105
kTADF = ktoR(S1) (s−1)	6.6 × 103	1.3 × 105	2.9 × 105	3.6 × 105	1.9 × 105	4.1 × 105
ktoR(T1) (s−1)	1.6	10	4.9 × 102	6.6 × 103	9.7 × 104	0.25
ktoR(T2) (s−1)	0.65	1.4	7.1	28	13	43
ktoR (s−1)	6.6 × 103	1.3 × 105	2.9 × 105	3.7 × 105	2.9 × 105	4.1 × 105
Φ	0.93 (0.71)	0.91 (0.91)	0.93 (1.0)	0.93	0.83	0.97
ΦNR(S1)	0.07	0.09	0.07	0.07	0.07	0.03
ΦNR(T1)	0.00	0.00	0.00	0.00	0.10	0.00
ΦNR(T2)	0.00	0.00	0.00	0.00	0.00	0.00
ΦPrompt	0.71 (0.43)	0.50 (0.23)	0.19 (0.05)	0.04	0.003	0.41
ΦTADF	0.22 (0.28)	0.41 (0.68)	0.74 (0.95)	0.87	0.55	0.56
ΦPhos(T1)	0.00	0.00	0.00	0.02	0.28	0.00
ΦPhos(T2)	0.00	0.00	0.00	0.00	0.00	0.00
S0-T1 SOC (cm−1)	2.12	4.07	13.5	37.7	202	0.76
S0-T2 SOC (cm−1)	0.96	1.92	16.9	56.5	204	0.97
S1-T1 SOC (cm−1)	0.04	0.12	0.84	3.92	17.7	0.01
S1-T2 SOC (cm−1)	0.55	1.17	3.32	10.6	59.1	1.01
FWHM (nm)	50–53 (51)	50–53 (53)	47–50 (47)	47–50	52–56	43–46

Some values are time-dependent; the values here are those at the equilibrium states, which correspond to the experimentally observable ones. The values in the parentheses are experimental data reported by Hu et al.¹⁰. k_TADF for BNOO/BNSS/BNSeSe/BNTeTe/BNPoPo/BNCOCO is defined in the time domain longer than 100/100/10/ 10/1/10 ns. k_toRISC for BNOO/BNSS/BNSeSe/BNTeTe/BNPoPo/BNCOCO is defined in the time domain longer than 1/10/10/10/1/10 ns (Supplementary Figs. 2 and 3). k_toRISC′ is the total RISC rate constant calculated by our previously proposed method²⁰. n.d. means not determined. k_toR(S₁), k_toR(T₁), and k_toR(T₂) are the contributions from S₁ → S₀ fluorescence, T₁ → S₀ phosphorescence, and T₂ → S₀ phosphorescence to k_toR, respectively: k_toR = k_toR(S₁) + k_toR(T₁) + k_toR(T₂); k_toR(S₁) = k_F(S₁ → S₀) × [S₁]/([S₁] + [T₁] + [T₂]); k_toR(T₁) = k_Phos(T₁ → S₀) × [T₁]/([S₁] + [T₁] +[T₂]); k_toR(T₂) = k_Phos(T₂ → S₀) × [T₂]/([S₁] + [T₁] + [T₂]). k_toRISC(T₁) and k_toRISC(T₂) are the contributions from T₁ → S₁ and T₂ → S₁ ISCs to k_toRISC, respectively. k_toRISC = k_toRISC(T₁) + k_toRISC(T₂); k_toRISC(T₁) = k_RISC(T₁ → S₁) × [T₁]/([T₁] + [T₂]); k_toRISC(T₂) = k_RISC(T₂ → S₁) × [T₂]/([T₁] + [T₂]).

Next, we discuss the impacts of further heavy atom effects on RISC and the possibility of further increasing k_toRISC by using tellurium (Te), polonium (Po), and C = O substitutions (Fig. 1). Our calculations predict that a Te-containing emitter (BNTeTe) exhibits one order of magnitude larger k_toRISC of 10⁷ s⁻¹ than BNSeSe (k_toRISC ~ 10⁶ s⁻¹). Although Po is highly radioactive, we theoretically investigate a Po-containing emitter (BNPoPo) to understand the heavy atom effect. Our calculations indicate that BNPoPo exhibits both TADF and phosphorescence. We calculate the k_toRISC of a C = O-containing molecule (BNCOCO) to be 10⁶ s⁻¹. Compared with Se, the C = O group has a smaller SOC-enhancement ability but provides a sufficient T₁ → T₂ internal up-conversion, resulting in k_toRISC that is comparable to that of BNSeSe. The calculated k_F(S₁ → S₀)s are the same order of magnitude (10⁸ s⁻¹) for all the compounds investigated in this study. Their calculated PLQYs remain high, indicating that we can enhance RISC without sacrificing radiative decays nor PLQYs, which is otherwise a trade-off in TADF emitters. Finally, we theoretically predict the linewidths (full width at half maximum (FWHM)) of the PL spectra for these molecules. The calculated FWHM values are almost identical for BNOO, BNSS, and BNSeSe; agreeing well with the experimental values. Those for designed BNTeTe, BNPoPo, and BNCOCO are comparable to those for BNOO, BNSS, and BNSeSe; indicating that the acceleration of k_RISC in this study also does not sacrifice the FWHM values.

Results

Computational results for BNOO, BNSS, and BNSeSe

Our theoretical analysis is based on rate constant calculations. Details of the method of calculating the rate constants for fluorescence from S₁ to S₀ (k_F(S₁ → S₀)), T_n → S₀ phosphorescence (k_Phos(T_n → S₀)), S₁ → S₀ nonradiative decay (k_NR(S₁ → S₀)), T_n → S₀ nonradiative decay (k_NR(T_n → S₀)), T_m → T_n internal conversion (k_IC(T_m → T_n)), S₁ → T_n ISC (k_ISC(S₁ → T_n)), and T_n → S₁ RISC (k_RISC(T_n → S₁)) (m, n = 1, 2, m ≠ n) are described in the “Methods” section. As is well known, conventional time-dependent density functional theory (TD-DFT) methods substantially overestimate ΔE(T₁ → S₁) of MR–TADF emitters, which can be solved by calculations including double-excitation configurations³⁶. Several wave-function-based methods (including the second-order algebraic-diagrammatic construction ADC(2) and the spin-component-scaling second-order approximate coupled-cluster (SCS–CC2)) give more reliable ΔE(T₁ → S₁) than TD–DFT methods³⁶. Recently, it has become common to use different theoretical methods for different molecular properties^13,37,38. For example, Lin et al. calculated excitation energies with the TD-B3LYP/6-31G(d) method, and then, the T₁ energy was corrected using ΔE(T₁ → S₁) obtained from the SCS–CC2/def2-TZVP calculation³⁸. Tamm–Dancoff approximation (TDA)–DFT methods with double hybrid density functionals such as TDA–B2-PLYP are emerging alternative approaches for considering double-excitation configurations³⁹. TDA–DFT methods have the advantage of low computational cost compared with ADC(2) and SCS–CC2. Here, we compared the ΔE(T₁ → S₁) values calculated with the three methods as well as that by the conventional TD–DFT (B3LYP) method (blue texts in Supplementary Tables 13 and 14). Supplementary Table 13 also shows the experimental values. The TDA–B2-PLYP method provided the ΔE(T₁ → S₁) values closest to the experimental results for BNSS and BNSeSe. Therefore, we used the TDA–B2-PLYP method (TDA–DFT with the B2-PLYP double-hybrid functional and def2-TZVP basis set) for calculating ΔE(T₁ → S₁), which we combined with the energy levels of S₁ and T₂ calculated by the TD–DFT with the B3LYP functional and 6-31 G(d)+SDD basis set (TD–B3LYP method). All the other calculations (SOCs, vibronic coupling constants, transition dipole moments, and permanent dipole moments) were carried out by the TD–B3LYP method (see Methods section and Supplementary Method 2 for the details). We performed the TDA–B2-PLYP calculations with the ORCA 5.0.3 program package (FACCTs, Cologne, Germany)^40–42 and the ADC(2) and SCS–CC2 calculations with the TURBOMOLE program package⁴³. Table 1 shows the calculated and experimental data. Figure 2a–f shows the calculated excited-state energy diagrams, energy gaps, SOCs, and rate constants for BNOO, BNSS, and BNSeSe. Figure 2g–l shows the time evolutions of respective rate constants. The calculated ΔE(T₁ → S₁) for BNSS and BNSeSe agree with the experimental values (calculated and experimental ΔE(T₁ → S₁) are 0.14 and 0.13 eV, respectively, for BNSS and they are both 0.14 eV for BNSeSe), although we found a slight deviation by 0.06 eV for BNOO (the ΔE(T₂ → S₁) and ΔE(T₂ → T₁) values have not been determined experimentally). We can reasonably neglect the contributions of S_n (n ≥ 2) and T_m (m ≥ 3) as described above.

Fig. 2. Excited-state decay mechanism.

a–c Calculated excited-state energy diagram, energy differences (eV), and spin–orbit couplings (cm⁻¹) and d–f rate constants (s⁻¹) for BNOO, BNSS, and BNSeSe. g–i Calculated k_toR, k_toR(S₁), k_toR(T₁), and k_toR(T₂) for BNOO, BNSS, and BNSeSe. j–l Calculated k_toRISC, k_toRISC(T₁), and k_toRISC(T₂) for BNOO, BNSS, and BNSeSe. In d–f, the solid orange arrows depict the dominant S₁ → T₁ pathway, the solid green arrows depict the dominant T₁ → S₁ pathway, the solid black arrows depict the minor S₁ → T₁ and T₁ → S₁ pathways, the solid blue arrows depict the S₁ → S₀ fluorescence, the solid red arrows depict the T₁ → S₀ phosphorescence, and the dashed arrows depict S₁ → S₀ and T₁ → S₀ nonradiative decays. In g–i, the solid grey curves depict k_toR, the blue curves depict k_toR(S₁), the solid red curves depict k_toR(T₁), and the dashed red curves depict k_toR(T₂). In j–l, the solid grey curves depict k_toRISC, the solid red curves depict k_toRISC(T₁), and the dashed red curves depict k_toRISC(T₂). Source data for figure g-l are provided as a Source Data file.

For BNOO, BNSS, and BNSeSe, S₁ and T₂ are energetically close (|ΔE(T₂ → S₁)| <0.1 eV), and the S₁–T₂ SOCs are stronger than the S₁–T₁ SOCs (Table 1 and Fig. 2a–c). As a result, k_ISC(S₁ → T₂) is larger than k_ISC(S₁ → T₁) by 1000 times for BNOO, 100 times for BNSS, and 10 times for BNSeSe. S₁ and T₁ of BNOO, BNSS, and BNSeSe are predominantly described as the HOMO–LUMO transitions, whilst their T₂s are predominantly described as the HOMO−1-LUMO transitions (Fig. 1). The S₁–T₂ SOCs are more substantial than the S₁–T₁ SOCs because S₁–T₂ transitions are between different molecular orbital (MO) characters, whilst S₁–T₁ transitions are between similar MOs.

The larger S₁–T₂ SOCs compared with S₁–T₁ SOCs, as well as closer energy levels of S₁–T₂ compared with those of S₁–T₁, suggest faster transitions for T₂-mediated ISC and RISC compared with those for direct S₁–T₁ ISC and RISC. The calculated k_toISC agrees well with the experimental values for BNOO, BNSS, and BNSeSe (Table 1). Regarding BNOO and BNSS, the stepwise S₁ → T₂ → T₁ transition contributed to the entire ISC process and the direct S₁ → T₁ ISC was negligible (k_toISC ≈ k_ISC(S₁ → T₂) ≫ k_ISC(S₁ → T₁); k_IC(T₂ → T₁) is very large). Regarding BNSeSe, although the contribution from the direct S₁ → T₁ ISC was not negligible, the stepwise S₁ → T₂ → T₁ transition was still dominant (k_toISC ≈ k_ISC(S₁ → T₂) and k_ISC(S₁ → T₁) was one order smaller than k_ISC(S₁ → T₂)). k_toISC increased with increasing atomic number for the chalcogen atoms (8.6 × 10⁷ s⁻¹ < 2.0 × 10⁸ s⁻¹ < 1.0 × 10⁹ s⁻¹ for BNOO, BNSS, and BNSeSe, respectively), indicating that the O → S → Se substitution enhanced the S₁–T₂ SOC and accelerated the S₁ → T₂ ISC.

Next, we investigated RISC for BNOO, BNSS, and BNSeSe. Table 1 shows the calculated and experimental k_toRISC as well as Φ, Φ_Prompt, Φ_TADF, Φ_Phos(T₁), Φ_Phos(T₂), and k_TADF for BNOO, BNSS, and BNSeSe. Here, Φ_Prompt, Φ_TADF, Φ_Phos(T₁), and Φ_Phos(T₂) are the PLQYs of the prompt fluorescence, TADF, phosphorescence from T₁, and phosphorescence from T₂, respectively. k_TADF is the rate constant of TADF (delayed fluorescence). Because Φ_Phos(T₁) ≈ 0 for the three compounds, we attributed the delayed luminescence to TADF (Φ_Phos(T₂) ≈ 0 for all six compounds). Most importantly, the calculated k_toRISC increased with increasing atomic number for the chalcogen atoms, and the k_toRISC quantitatively agrees with the experimental values, which confirms the validity of our method of predicting k_toRISC. The calculations here enabled us to separate the contributions of direct RISC (T₁ → S₁) and RISC via T₂ (T₁ → T₂ → S₁). We denote them as k_toRISC(T₁) (=k_RISC(T₁ → S₁)) and k_toRISC(T₂), respectively. Regarding BNOO, BNSS, and BNSeSe, the rate constants for k_toRISC(T₂) are almost identical to k_toRISC (=k_toRISC(T₁) + k_toRISC(T₂)) (Table 1 and Fig. 2j–l), indicating that the T₁ → T₂ → S₁ transition is the dominant RISC pathway. It should also be noted that k_toRISC is much smaller than k_RISC(T₂ → S₁). This is because the downhill T₂ → T₁ IC is rapid compared with the uphill T₁ → T₂ IC and T₂ → S₁ RISC. Thus, in addition to increasing k_RISC(T₂ → S₁), decreasing ΔE(T₁ → T₂) is also crucial for accelerating T₂-mediated RISC.

Computational results for BNTeTe and BNPoPo

From the above discussion, we expected further enhanced RISC by further increasing the atomic number of the included atoms. Hence, we next replaced Se with Te or Po (Fig. 1d and e) to further increase the SOCs and accelerate RISC. HOMO − 1, HOMO, and LUMO of BNTeTe and BNPoPo are similar to those of BNOO, BNSS, and BNSeSe (Fig. 1). Regarding BNTeTe (Fig. 3a), the energy level alignment of S₁, T₁, and T₂ is almost identical to those of BNSS and BNSeSe (Fig. 2b, c). Hence, the S/Se→Te replacement affected k_toRISC by enhancing the S₁–T₁ and S₁–T₂ SOCs. We calculated the S₀–T₁, S₁–T₁, and S₁–T₂ SOCs of BNTeTe to be 37.7, 3.92, and 10.6 cm⁻¹, respectively, which are 3 to 4 times those of BNSeSe (13.5, 0.84, and 3.32 cm⁻¹, respectively). As a result, all SOC-related rate constants (k_ISC(S₁ → T₁), k_ISC(S₁ → T₂), k_RISC(T₂ → S₁), k_RISC(T₁ → S₁), k_NR(T₁ → S₀), and k_Phos(T₁ → S₀)) of BNTeTe were larger than those of BNSeSe. In contrast, k_F(S₁ → S₀) and k_NR(S₁ → S₀) of BNTeTe were comparable to those of BNSS and BNSeSe. Thus, acceleration of RISC is possible without sacrificing radiative decay and PLQY (Fig. 3d, g, j). The contributions of direct ISC and direct RISC to total ISC and total RISC processes, respectively, were more substantial for BNTeTe than for BNSeSe. However, the stepwise T₂-mediated process was still dominant (70% of the total process; Fig. 3j). The calculated k_toRISC of BNTeTe was on the order of 10⁷ s⁻¹. k_Phos also increased but on the order of 10³ s⁻¹; therefore, phosphorescence was still negligible (Φ_Phos(T₁) = 0.02), and T₁ excitons were preferentially converted into light as TADF (Fig. 3g). Thus, BNTeTe is a promising MR–TADF emitter with fast RISC.

Fig. 3. Excited-state decay mechanism.

a–c Calculated excited-state energy diagram, energy differences (eV), and spin–orbit couplings (cm⁻¹) and d–f rate constants (s⁻¹) for BNTeTe, BNPoPo, and BNCOCO. g–i Calculated k_toR, k_toR(S₁), k_toR(T₁), and k_toR(T₂) for BNTeTe, BNPoPo, and BNCOCO. j–l Calculated k_toRISC, k_toRISC(T₁), and k_toRISC(T₂) for BNTeTe, BNPoPo, and BNCOCO. In d–f, the solid orange arrows depict the dominant S₁ → T₁ pathway, the solid green arrows depict the dominant T₁ → S₁ pathway, the solid black arrows depict the minor S₁ → T₁ and T₁ → S₁ pathways, the solid blue arrows depict the S₁ → S₀ fluorescence, the solid red arrows depict the T₁ → S₀ phosphorescence, and the dashed arrows show S₁ → S₀ and T₁ → S₀ nonradiative decays. In g–i, the solid grey curves depict k_toR, the blue curves depict k_toR(S₁), the solid red curves depict k_toR(T₁), and the dashed red curves depict k_toR(T₂). In j–l, the solid grey curves depict k_toRISC, the solid red curves depict k_toRISC(T₁), and the dashed red curves depict k_toRISC(T₂). Source data for g–l are provided as a Source Data file.

Po has a more substantial heavy atom effect than Te (Fig. 3b); hence, one can expect that BNPoPo would also be an excellent TADF emitter with much faster RISC (although Po compounds are radioactive, we performed calculations to understand the heavy atom effect). In contrast to our expectation, BNPoPo exhibited substantial phosphorescence as well as TADF (Φ_Phos = 0.28 and Φ_TADF = 0.55; Table 1 and Fig. 3e show rate constants). Therefore, BNPoPo would not be a pure TADF emitter (Fig. 3h), although it exhibited the fastest k_toRISC of 5.6 × 10⁷ s⁻¹ because of the substantial heavy atom effect of Po. This is different from all the other compounds in this study (they are pure TADF emitters). Interestingly, the dominant RISC processes are found to change from T₁ → T₂ → S₁ to direct T₁ → S₁ process at 10^-10 s for BNPoPo (Fig. 3k). Our proposed method also provides such detailed exciton dynamics.

Computational results for BNCOCO

We investigated another approach to accelerating RISC: C = O substitution. Figure 1f shows the designed molecule (BNCOCO). We calculated ΔE(T₁ → S₁) of BNCOCO to be 0.14 eV (Fig. 3c); which is comparable to those of BNSS, BNSeSe, BNTeTe, and BNPoPo. S₁–T₂ SOC of BNCOCO (1.01 cm⁻¹, Fig. 3c) is only slightly smaller than that of BNSS (1.17 cm⁻¹, Fig. 2b). Although T₂ → S₁ RISC is downhill for BNSS and uphill for BNCOCO, the energy gaps for T₂ → S₁ were small in both cases. Consequently, the k_RISC(T₂ → S₁) of BNCOCO is only slightly smaller than that of BNSS. In contrast, the reduced ΔE(T₁ → T₂) induced a faster T₁ → T₂ transition than in BNSS, resulting in a larger k_toRISC (9.9 × 10⁵ s⁻¹) than BNSS (2.5 × 10⁵ s⁻¹). k_toISC and k_toRISC of BNCOCO are close to those of BNSeSe. The large spatial overlap between HOMO and LUMO + 1 around the C = O groups (Fig. 1f) lowers the T₂ energy level, minimising ΔE(T₁ → T₂) and leading to the large k_toRISC even without chalcogen atoms.

A method for minimising ΔE(T₁ → T₂)

Here, we would like to discuss another approach to minimise ΔE(T₁ → T₂), which is effective in increasing k_toRISC. Supplementary Fig. 8a shows a case when T₁ and T₂ consist of HOMO → LUMO and HOMO−1 → LUMO transitions, respectively. ΔE(T₁ → T₂) can be written as ΔE(T₁ → T₂) = (h_HH−h_H−1H−1) + (J_HH−J_H−1H−1) + (J_HL−J_H−1L), where h denotes the core integral (kinetic and potential energies), J denotes the Coulomb integral, and K denotes the exchange integral⁴⁴ (see Supplementary Method 6 for the detail). A simple approach of minimising ΔE(T₁ → T₂) is to decrease the first term h_HH−h_H−1H−1, which is possible by expanding the HOMO and HOMO−1 distributions (the second and third terms of ΔE(T₁ → T₂), expressed in terms of the Coulomb integrals, are not easy to control). Supplementary Fig. 8b shows a case where T₁ and T₂ consist of the HOMO → LUMO and HOMO → LUMO + 1 transitions, respectively. In this case, ΔE(T₁ → T₂) can be written as ΔE(T₁ → T₂) = (h_L+1L+1−h_LL) + (J_HH+1−J_HL) + (K_HL+1 − K_HL). For the same reason, expanding the LUMO + 1 and LUMO distributions results in an effective approach to minimise ΔE(T₁ → T₂) by decreasing h_L+1L+1−h_LL. Regardless of whether T₂ is described as the HOMO−1 → LUMO or HOMO → LUMO + 1 transition, expanding molecular orbitals relevant for T₁ and T₂ is a simple way to decrease ΔE(T₁ → T₂) and accelerate the T₂-mediated RISC process. Comparison of ν-DABNA-core and V-DABNA-core is a good example⁴⁵. V-DABNA has a larger π-conjugation than ν-DABNA and hence, V-DABNA shows a smaller ΔE(T₁ → T₂) of 98 meV than ν-DABNA-core (147 meV). This trend can be also seen in several other examples^46–50 (Supplementary Table 15).

Exciton dynamics

The above discussion indicates that the rate constants, including those of ISC and RISC, and quantum yields can be predicted quantitatively. This method also provides quantitative details on the exciton dynamics. Figures 2g–l and 3g–l show the time evolutions of respective rate constants. Supplementary Figs. 2 and 3 also show the time evolutions of respective exciton concentrations. These analyses enable complete elucidation of the emission mechanism. As shown in Figs. 2g–l and 3g–l, the total radiative, ISC, and RISC rate constants, corresponding to experimentally observed ones, are not constant and time-dependent because they are composed of multiple processes. Initially, k_toR is very large (10⁸−10⁹ s⁻¹) but decreases to 10⁴−10⁵ s⁻¹. These time evolutions, corresponding to prompt and delayed emissions, respectively, are automatically calculated by our method, including the transition states. Figures 2j–l and 3j–l show that k_toRISC’s are also time-dependent; initially, they are very large and stay constant after ~10⁻¹⁰−10⁻⁹ s. Figure 3k shows that the RISC mechanism of BNPoPo changed from the T₂-mediated RISC to direct T₁ → S₁ RISC at ~0.1 ns as described above.

Theoretically prediction of linewidths of PL spectra

Finally, we theoretically calculated and predicted the FWHM values of the emission spectra by the vertical gradient method implemented in the ORCA 5.0.3 program package^40–42. The calculated FWHM values of BNOO, BNSS, and BNSeSe agree well with the experimental values (Table 1 and Supplementary Figs. 4–6), confirming the validity of the calculations. Regarding BNTeTe, BNPoPo, and BNCOCO, the predicted FWHM values were 47–50, 52–56, and 43–46 nm, respectively (Table 1 and Supplementary Fig. 7); which are comparable to those for BNOO, BNSS, and BNSeSe. Thus, acceleration of RISC is possible without sacrificing the FWHM of the emission spectra.

Discussion

We comprehensively investigated the RISC mechanism of MR–TADF emitters, BNOO, BNSS, and BNSeSe by calculating all the relevant rate constants and quantum yields. The calculated values are in quantitative agreement with the experimental results. The ISC and RISC in BNOO, BNSS, and BNSeSe were found to occur predominantly via T₂. We also found that incorporating S and Se into the molecules was effective in accelerating RISC. Therefore, BNTeTe and BNPoPo were designed to further increase k_RISC. The strong heavy atom effect of Te enabled BNTeTe to exhibit k_RISC of 10⁷ s⁻¹, which is larger than those of BNOO (10⁴ s⁻¹), BNSS (10⁵ s⁻¹), and BNSeSe (10⁶ s⁻¹). Meanwhile, although BNPoPo had the largest k_RISC of 5.6 × 10⁷ s⁻¹, it exhibited a substantial contribution of phosphorescence because of the excessive heavy atom effect of Po. Our findings suggest that a moderately strong SOC that does not cause phosphorescence and suitable energy alignments are the keys to accelerating RISC in pure TADF. We also investigated BNCOCO to attain large k_RISC without heavy atoms. BNCOCO had an S₁–T₂ SOC comparable to BNSS but a smaller T₁–T₂ energy difference, resulting in a larger k_RISC of 10⁶ s⁻¹.

Distinct from conventional TADF, RISC has been rate-limiting in MR-TADF, and addressing this problem is currently the most important issue in further improving its characteristics. The calculated k_F(S₁ → S₀), Φ, and FWHM values of the emission spectra are nearly identical for all the compounds in this study. This study reveals the solution that the rate constants of RISC can be substantially improved without sacrificing the PLQY, rate constant of radiative decay, and emission linewidth.

Finally, we would like to discuss further acceleration of RISC. The k_RISC(T₂ → S₁) of BNTeTe was large (1.5 × 10¹⁰ s⁻¹) because of the large SOC and downhill transition. However, the k_toRISC was three orders of magnitude smaller (1.0 × 10⁷ s⁻¹). This is because in the dominant process of RISC (T₁ → T₂ → S₁), k_IC(T₂ → T₁) (1.4 × 10¹² s⁻¹) is much larger than k_IC(T₁ → T₂) (6.5 × 10⁸ s⁻¹) and k_RISC(T₂ → S₁). The small k_IC(T₁ → T₂) results in a slow pump-up of excitons from T₁ to T₂. Even if excitons are up-converted from T₁ to T₂, they quickly return to T₁ ([T₂] ≪ [T₁]). This problem can be solved by minimising the energy difference between T₁ and T₂ (eventually to zero) yet preserving the T₂ → S₁ transition downhill. This situation corresponds to a system in which S₁ is energetically lower than T₁. Recently, it has been revealed that molecules with an S₁ that is lower in energy than T₁ can be realised despite violating Hund’s rule^51–56. Such inverted S₁–T₁ (iST) systems have an ideal energy level diagram that enables the aforementioned downhill direct RISC process without the T₁ → T_n IC process. iST molecules have frontier orbital distributions that are similar to MR-TADF; HOMO and LUMO are localised on different atoms, having short-range CT characters. Our calculation method for MR–TADF can be directly applied to designing iST molecules, enabling further enhanced RISC and a comprehensive understanding of their emission mechanisms.

In this study, we proposed a theoretical method of predicting the energy level alignments, including higher lying states as well as S₁ and T_1, by combining conventional B3LYP and B2-PLYP double-hybrid functionals. Second, we devised a method of calculating RISC (and ISC) rate constants considering RISC (and ISC) mechanisms consisting of multiple pathways. Although only S₁, T₁, and T₂ are involved in the emission mechanism of molecules in this study, the method proposed here is robust enough to be applied to the case where many higher-lying states are involved. The theoretical advances enable us to quantitatively predict all the relevant rate constants and quantum yields. It is demonstrated that it is important to consider the entire system for quantitative understanding of the actual emission mechanism. To quantitatively understand RISC for all the compounds here, it is not sufficient to consider only the T₁→S₁ process; IC plays an important role, albeit indirect. 

Our method of calculating rate constants and quantum yields is not limited to such MR-TADFs and iSTs. The range of applications is vast, including electronic transitions in, e.g. biochemical, biomedical, pharmaceutical systems, chemical reactions, and surface science. We have also confirmed the applicability of this method to a wide range of TADF emitters other than MR-type molecules and to catalytic photooxygenation to inhibit aggregation of amyloid-β peptide as a therapeutic strategy for Alzheimer’s disease⁵⁷.

One phenomenon often consists of combinations of several elementary processes. Of these processes, only a key process has so far been focused on. However, as in the present example, all processes are often closely related. Our physics-based method is therefore important for obtaining a comprehensive understanding of phenomena and for quantitative predictions, including the time evolutions, which enable discovery of superior systems.

Methods

Calculations of rate constants for TADF, phosphorescence, total ISC, total RISC, and radiative decay based on excited-state populations

We calculated the rate constants for fluorescence from S₁ to S₀ (k_F(S₁ → S₀)), T_n → S₀ phosphorescence (k_Phos(T_n → S₀)), S₁ → S₀ nonradiative decay (k_NR(S₁ → S₀)), T_n → S₀ nonradiative decay (k_NR(T_n → S₀)), T_m → T_n internal conversion (k_IC(T_m → T_n)), S₁ → T_n ISC (k_ISC(S₁→T_n)), and T_n → S₁ RISC (k_RISC(T_n → S₁)) (m, n = 1, 2, m ≠ n) with Supplementary Eqs. (A1)–(A18) (Supplementary Method 1). S_n (n ≥ 2) and T_m (m ≥ 3) were located 0.3 eV higher in energy than S₁, resulting in very small contributions for all compounds in this study; therefore, we neglected their contributions to the TADF mechanism. We performed geometric optimisation and frequency analysis of S₁ for BNOO, BNSS, BNSeSe, BNTeTe, BNPoPo, and BNCOCO by the TD-TPSSh method (Supplementary Tables 1−6 and Supplementary Fig. 1 shows the optimised geometries). Then, we performed the excited-state calculations with the TD–B3LYP method using the optimised S₁ geometries (Supplementary Tables 7−12). For H, B, C, N, O, and S atoms, we used the 6–31G(d) basis set. For Se, Te, and Po atoms, we used the Stuttgart/Dresden pseudopotentials and basis set (SDD)⁵⁸. The geometrical optimisations, frequency analyses, and excited-state calculations were performed with the Gaussian 16 program package (Wallingford, CT, USA)⁵⁹.

Calculations of SOCs, vibronic coupling constants, transition dipole moments, and permanent dipole moments were carried out by the TD–B3LYP method (TD–DFT with the B3LYP functional and 6–31G(d)+SDD basis set), in which the SOCs, vibronic coupling constants, and T₁–T₂ transition dipole moments were calculated with the method proposed by McMurchie and Davidson⁶⁰ (Supplementary Method 2 and Code availability below), whereas the permanent dipole moments and S₀–S₁ transition dipole moment were calculated with the Gaussian 16 program package. We performed the TDA–B2-PLYP calculations with the ORCA 5.0.3 program package (FACCTs, Cologne, Germany)^40–42 and the ADC(2) and SCS–CC2 calculations with the TURBOMOLE program package⁴³.

The derivations of rate constants of individual elementary processes are shown in Supplementary Method 3. Here, we show the derivations of rate constants composed of multiple processes. Previously, rate constants for prompt and delayed fluorescence and RISC have been determined from transient photoluminescence (trPL) decay curves (number of photons counted vs. time plot)²⁰, which are difficult to use for separately analysing delayed fluorescence and phosphorescence, especially when their time scales are close. In this study, we propose a method of calculating these rate constants from the excited-state populations, [S_n] and [T_n] (n = 1, 2, 3, ...). Our method distinguishes delayed fluorescence and phosphorescence, which does not require calculating a trPL decay curve.

The rate for the total ISC from the excited singlet states (S_n (n = 1, 2, 3, ...)) to the triplet states (T_m (m = 1, 2, 3, ...)) can be expressed as

$${\sum }_{n\ge 1}\left({\sum }_{m\ge 1}{k}_{\mathrm{ISC}}\left({\mathrm{S}}_n\to {\mathrm{T}}_m\right)\right)\left[{\mathrm{S}}_n\right],$$ 1

which can be written in terms of the total population of the excited singlet states ${\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]$ as

$$\left\{{\sum }_{n\ge 1}\left({\sum }_{m\ge 1}{k}_{\mathrm{ISC}}\left({\mathrm{S}}_n\to {\mathrm{T}}_m\right)\right)\frac{\left[{\mathrm{S}}_n\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]}\right\}{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right].$$ 2

Therefore, the rate constant for the total ISC k_toISC can be defined as

$$k_{\mathrm{toISC}}={\sum }_{n\ge 1}\left({\sum }_{m\ge 1}{k}_{\mathrm{ISC}}\left({\mathrm{S}}_n\to {\mathrm{T}}_m\right)\right)\frac{\left[{\mathrm{S}}_n\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]}.$$ 3

Here, the effect of k_IC(S_n’→S_n”) on k_toISC are included in the [S_n] populations because [S_n] are calculated by solving the kinetic equations that include all the elementary transitions (see Supplementary Method 3).

Furthermore, the rate for the total RISC from the triplet states to the excited singlet states can be expressed as

$${\sum }_{m\ge 1}\left({\sum }_{n\ge 1}{k}_{\mathrm{RISC}}\left({\mathrm{T}}_m\to {\mathrm{S}}_n\right)\right)\left[{\mathrm{T}}_m\right],$$ 4

which can be written in terms of the total population of the triplet states ${\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]$ as

$$\left\{{\sum }_{m\ge 1}\left({\sum }_{n\ge 1}{k}_{\mathrm{RISC}}\left({\mathrm{T}}_m\to {\mathrm{S}}_n\right)\right)\frac{\left[{\mathrm{T}}_m\right]}{{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}\right\}{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right].$$ 5

Therefore, the rate constant for the total RISC k_toRISC can be defined as

$$k_{\mathrm{toRISC}}={\sum }_{m\ge 1}\left({\sum }_{n\ge 1}{k}_{\mathrm{RISC}}\left({\mathrm{T}}_m\to {\mathrm{S}}_n\right)\right)\frac{\left[{\mathrm{T}}_m\right]}{{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}.$$ 6

Note that k_toISC and k_toRISC are functions of time (t) through the time dependence of [S_n] and [T_m], respectively. As in the case of k_IC(S_n’→S_n”), the effect of k_IC(T_m’→T_m”) on k_toRISC are included in the [T_m] populations (see Supplementary Method 3). The values of k_toISC and k_toRISC depend on the time domain in which they are calculated. The rate for the total radiative decay from the excited singlet and triplet states is

$${\sum }_{n\ge 1}{k}_{\mathrm{F}}\left({\mathrm{S}}_n\to {\mathrm{S}}_0\right)\left[{\mathrm{S}}_n\right]+{\sum }_{m\ge 1}{k}_{\mathrm{Phos}}\left({\mathrm{T}}_m\to {\mathrm{S}}_0\right)\left[{\mathrm{T}}_m\right],$$ 7

which can be written in terms of the total population of the excited states ${\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]$ as

$$\left\{\right){\sum }_{n\ge 1}{k}_{\mathrm{F}}\left({\mathrm{S}}_n\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{S}}_n\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}+{\sum }_{m\ge 1}{k}_{\mathrm{Phos}}\left({\mathrm{T}}_m\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_m\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}\left\}\right)\times \left({\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]\right).$$ 8

Hence, the rate constant for the total radiative decay (k_toR) can be defined as

$$k_{\mathrm{toR}}={\sum }_{n\ge 1}{k}_{\mathrm{F}}\left({\mathrm{S}}_n\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{S}}_n\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}+{\sum }_{m\ge 1}{k}_{\mathrm{Phos}}\left({\mathrm{T}}_m\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_m\right]}{{\sum }_{l\ge 1}\left[{\mathrm{S}}_l\right]+{\sum }_{l\ge 1}\left[{\mathrm{T}}_l\right]}.$$ 9

As in k_toISC and k_toRISC, k_toR is a function of t.

As stated previously, regarding BNOO, BNSS, BNSeSe, BNTeTe, BNPoPo, and BNCOCO, it is sufficient to consider only S₁, T₁, and T₂. Hence, k_toISC, k_toRISC, and k_toR can be written as

$$k_{\mathrm{toISC}}=k_{\mathrm{ISC}}\left({\mathrm{S}}_1\to {\mathrm{T}}_1\right)+k_{\mathrm{ISC}}\left({\mathrm{S}}_1\to {\mathrm{T}}_2\right)$$ 10

$$k_{\mathrm{toRISC}}=k_{\mathrm{RISC}}\left({\mathrm{T}}_1\to {\mathrm{S}}_1\right)\frac{\left[{\mathrm{T}}_1\right]}{\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}+k_{\mathrm{RISC}}\left({\mathrm{T}}_2\to {\mathrm{S}}_1\right)\frac{\left[{\mathrm{T}}_2\right]}{\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}$$ 11

$$\begin{array}{c}\hfill k_{\mathrm{toR}}=k_{\mathrm{F}}\left({\mathrm{S}}_1\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{S}}_1\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}+k_{\mathrm{Phos}}\left({\mathrm{T}}_1\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_1\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}\hfill \\ \hfill +k_{\mathrm{Phos}}\left({\mathrm{T}}_2\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_2\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}.\hfill \end{array}$$ 12

The contributions from T₁ → S₁ and T₂ → S₁ RISCs to k_toRISC are defined as

$$k_{\mathrm{toRISC}}\left({\mathrm{T}}_1\right)=k_{\mathrm{RISC}}\left({\mathrm{T}}_1\to {\mathrm{S}}_1\right)\frac{\left[{\mathrm{T}}_1\right]}{\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]},$$ 13

$$k_{\mathrm{toRISC}}\left({\mathrm{T}}_2\right)=k_{\mathrm{RISC}}\left({\mathrm{T}}_2\to {\mathrm{S}}_1\right)\frac{\left[{\mathrm{T}}_2\right]}{\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}.$$ 14

The contributions from S₁ → S₀ fluorescence k_toR(S₁), T₁ → S₀ phosphorescence k_toR(T₁), and T₂ → S₀ phosphorescence k_toR(T₂) to k_toR, are defined as

$$k_{\mathrm{toR}}\left({\mathrm{S}}_1\right)=k_{\mathrm{F}}\left({\mathrm{S}}_1\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{S}}_1\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]},$$ 15

$$k_{\mathrm{toR}}\left({\mathrm{T}}_1\right)=k_{\mathrm{Phos}}\left({\mathrm{T}}_1\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_1\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]},$$ 16

$$k_{\mathrm{toR}}\left({\mathrm{T}}_2\right)=k_{\mathrm{Phos}}\left({\mathrm{T}}_2\to {\mathrm{S}}_0\right)\frac{\left[{\mathrm{T}}_2\right]}{\left[{\mathrm{S}}_1\right]+\left[{\mathrm{T}}_1\right]+\left[{\mathrm{T}}_2\right]}.$$ 17

Regarding fluorescent molecules with negligibly small k_Phos(T₁ → S₀) and k_Phos(T₂ → S₀), such as BNOO/BNSS/BNSeSe/BNTeTe/BNCOCO, k_toR ~ k_F(S₁ → S₀) when [T₁] ≪ [S₁] and [T₂]  ≪ [S₁] (this condition holds in the time domain immediately after the S₀ → S₁ photoexcitation). After sufficient time has passed following the excitation, S₁, T₁, and T₂ are thermally equilibrated, and [S₁]/([S₁] + [T₁] +[T₂]) becomes constant (Figs. 2 and 3, and Supplementary Figs. 2 and 3). In such a time domain, k_toR ~ k_F(S₁ → S₀) × [S₁]/([S₁] + [T₁] +[T₂]), which can be viewed as the rate constant for TADF (k_toR ~ k_TADF). For molecules that emit both fluorescence and phosphorescence, such as BNPoPo, k_toR is intrinsically the population-weighted average of k_F(S₁ → S₀), k_Phos(T₁ → S₀), and k_Phos(T₂ → S₀) (Eq. (12)). The lifetimes for the total radiative decay (τ_toR) and TADF (τ_TADF) can be calculated as τ_toR = 1/k_toR and τ_TADF = 1/k_TADF, respectively. Yersin et al. derived τ_toR for organometal complexes⁶¹. Equations (9) and (12) are corrected from their equations (Supplementary Method 3 shows a detailed comparison).

Calculations of rate constants for prompt fluorescence, TADF, and RISC based on transient photoluminescence decay curves

Experimentally, TADF properties have often been discussed in terms of the RISC rate constant (k_toRISC′) determined from a trPL decay curve. Here, k_toRISC′ denotes the trPL-based rate constant, whilst k_toRISC denotes the excited-state population-based rate constant (Eqs. (6) or (11)). We previously proposed an expression for k_toRISC′ under the assumption that the triplet states do not decay radiatively nor nonradiatively²⁰:

$${k_{\mathrm{toRISC}}}^{\prime }=\frac{k_{\mathrm{Prompt}}+k_{\mathrm{Delayed}}}{2}-\sqrt{{\left(\frac{k_{\mathrm{Prompt}}+k_{\mathrm{Delayed}}}{2}\right)}^2-k_{\mathrm{Prompt}}{k}_{\mathrm{Delayed}}\left(1+\frac{{\Phi }_{\mathrm{Delayed}}}{{\Phi }_{\mathrm{Prompt}}}\right)},$$ 18

where k_Prompt and k_Delayed denote the rate constants for prompt and delayed luminescence, respectively, determined by exponential fitting of a trPL decay curve. K_toRISC′ cannot be applied to emitters having phosphorescent contributions such as BNPoPo. Although k_toRISC is almost identical to k_toRISC′ for BNOO, BNSS, BNSeSe, BNTeTe, and BNCOCO (in which k_Phos(T₁ → S₀) and k_NR(T₁ → S₀) are negligibly small (Table 1)), k_toRISC is applicable to molecules that exhibit prompt fluorescence, delayed fluorescence, and phosphorescence simultaneously; which is more universal than k_toRISC′.

Author contributions

K.S. performed the theoretical calculations. H.K. planned and supervised the project. All authors contributed to the writing of this manuscript and have approved the final version.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

The Gaussian 16 and ORCA input and output files are deposited in the figshare data repository [10.6084/m9.figshare]. The source data underlying Figs. 2g–l and 3g–l are provided as a Source Data file. Source data are provided with this paper.

Code availability

The code used to generate Table 1 and Figs. 2 and 3 is available on GitHub [https://github.com/KatsuyukiShizu/d77) and Zenodo [10.5281/zenodo.11124543].

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-49069-4.