Extended theoretical modeling of reverse intersystem crossing for thermally activated delayed fluorescence materials

Abstract

Thermally activated delayed fluorescence (TADF) materials and multi-resonant (MR) variants are promising organic emitters that can achieve an internal electroluminescence quantum efficiency of ~100%. The reverse intersystem crossing (RISC) is key for harnessing triplet energies for fluorescence. Theoretical modeling is thus crucial to estimate its rate constant (k_RISC) for material development. Here, we present a comprehensive assessment of the theory for simulating the RISC of MR-TADF molecules within a perturbative excited-state dynamics framework. Our extended rate formula reveals the importance of the concerted effects of nonadiabatic spin-vibronic coupling and vibrationally induced spin-orbital couplings in reliably determining k_RISC of MR-TADF molecules. The excited singlet-triplet energy gap is another factor influencing k_RISC. We present a scheme for gap estimation using experimental Arrhenius plots of k_RISC. Erroneous behavior caused by approximations in Marcus theory is elucidated by testing 121 MR-TADF molecules. Our extended modeling offers in-depth descriptions of k_RISC.

Converted spin-vibronic coupling effects enhance internal quantum transfer processes for molecular OLED materials.

INTRODUCTION

Organic light-emitting diodes (OLEDs) (1) have gained widespread interest owing to their remarkable features such as lightness and wide viewing range. Materials that emit light through fluorescence (2, 3) and phosphorescence (4–6) have been developed for their application in OLEDs. Electrical excitations that occur in OLED devices produce singlet and triplet excitons in a ratio of 1:3 through hole-electron recombination (4).

To increase the internal quantum efficiency (IQE) to ∼100%, all generated excitons with different spin multiplicities have to be used for electroluminescence. To meet this demand, Adachi and co-workers developed OLEDs based on thermally activated delayed fluorescence (TADF) (7). TADF is a process in which a nonradiative triplet exciton is converted into a radiative singlet exciton via reverse intersystem crossing (RISC), followed by delayed fluorescence (Fig. 1A). TADF materials show marked potential for fully using singlet and triplet excitons for fluorescence; materials have been reported to achieve nearly 100% IQE (8–10). Recently, TADF coupled with the multiple resonance (MR) effect, designated MR-TADF (11), has received considerable attention as synthesized MR-TADF luminophores demonstrate desirable narrowband emissions with high IQE and color purity (12–14).

Fig. 1. The reverse intersystem crossing process in TADF materials.

(A) Conceptual diagram of reverse intersystem crossing (RISC) process in TADF materials. (B) Treatments of RISC process using 1st+Condon, 2nd+Condon, 1st+HT, and 2nd+HT TVCF methods. Compared to the first method, the second approach additionally accounts for nonadiabatic spin-vibronic coupling (NA-SVC). The Herzberg-Teller (HT) treatment considers HT-SVC, which is discarded by the Condon treatment. (C) MR-TADF molecules: ν-DABNA, BOBO-Z (X₁=X₂=O), BOBS-Z (X₁=O, X₂=S), and BSBS-Z (X₁=X₂=S). (D) Arrhenius plots and activation energies $E_a^{\text{Exp}}$ obtained using the relation $\text{ln}{k}_{\text{RISC}}=-E_a^{\text{Exp}}/(k_{\mathrm{B}}T)+\text{ln}A$.

A large spin-orbit coupling (SOC) and small adiabatic singlet-triplet (ST) energy difference ΔE_ST are important physical factors for accelerating the rate-limiting RISC process. Theoretical modeling will assist in characterizing details of the RISC process and realizing chemical fine-tuning. The semiclassical Marcus formula (15) has been conventionally used to estimate the rate constant k_RISC and can describe state-transition processes based on ΔE_ST and SOC (16–19).

Recent studies have shown that the effects of spin-vibronic coupling (SVC) (20–28), which are discarded in Marcus theory, are impactful to modulating k_RISC for TADF molecules (24, 25, 29, 30).

Two types of SVC affect the predicted value of k_RISC. The first is expressed through Fermi’s golden rule as the second-order vibronic perturbation term (20–25) and is hereafter referred to as nonadiabatic SVC (NA-SVC). It is caused by the interplay between nonadiabatic coupling (NAC) and SOC. The second is the vibrationally induced SOC, which is related to the first derivatives of the SOC (20, 26–28, 30). This effect is called Herzberg-Teller SVC (HT-SVC).

Several previous studies (21–25) have confirmed that the NA-SVC effect can greatly enhance k_RISC by several orders of magnitude through indirect spin-flipping mediated by intermediate excited states such as T₂ and T₃. Moreover, the HT-SVC effect coupled with spin conversion is considered crucial for further influencing k_RISC. Marian and co-workers investigated critical effects of HT-SVC on spin-flipping processes (20, 26–28, 30). A recent study revealed that the simulated k_RISC for MR-TADF emitters could increase by more than one order of magnitude when incorporating HT-SVC into the rate formula (29). The derived formula was based on the direct sum of the individual rate contributions associated with the NA-SVC and HT-SVC separately. To the best of our knowledge, no existing rate formula considers the fully concerted coupling of both effects. Developing such a formula could help capture additional RISC channels opened by the HT-SVC effects coupled with the indirect spin-flipping process through low-lying intermediate T_n states (Fig. 1B).

The accuracy of ΔE_ST is crucial because k_RISC steeply varies depending on ΔE_ST, as in Marcus theory. There are two types of experimental approaches for measuring ΔE_ST: (i) fluorescence and phosphorescence energy difference and (ii) activation energy of the RISC process (31). In the latter, the activation energy $E_a^{\text{Exp}}$ is determined from the slope of the Arrhenius plot of the measured k_RISC versus temperature T using the Arrhenius equation $\text{ln}k=-\frac{E_a}{k_{\mathrm{B}}T}+\text{ln}A$. $E_a^{\text{Exp}}$ is then interpreted as ΔE_ST (7, 31). Computational quantum chemical approaches to gauge ΔE_ST should be an effective alternative; however, there are uncontrollable difficulties in reliably predicting small ΔE_ST for TADF molecules using the time-dependent density functional theory (TDDFT). More computationally demanding treatments involving correlated wave function theories have been shown to mitigate these challenges (32–34).

In this study, we aimed to elucidate in-depth aspects of the theory for simulating k_RISC for MR-TADF materials. A simplified excited-state dynamics approach is desirable for MR-TADF molecules with large π-conjugated systems. Thus, a perturbative treatment based on Fermi’s golden rule was considered. In addition, for their construction, the electronic-state potential energy surfaces were assumed to be harmonic in all dimensions. To facilitate formulation and computation, the thermal vibration correlation function (TVCF) formalism (21, 35) was used. Our major interest was to investigate the influence of the treatment of NA-SVC and HT-SVC and their concerted effects on k_RISC. This fully concerted SVC effect is denoted 2nd+HT (Fig. 1B). We developed an enhanced TVCF-based method considering the 2nd+HT effect (section S7).

Furthermore, we investigated the choice of ΔE_ST, which can be determined using experimental data or quantum chemical computations. We also introduced an approach for estimating ΔE_ST based on fitting the experimental Arrhenius plots. In addition, the reliability of Marcus theory was assessed by contrasting TVCF predictions for 121 MR-TADF molecules. We discussed approximation errors in Marcus theory.

RESULTS

Calculation of RISC rate constant using experimental ${\mathit{E}}_{\mathit{a}}^{\mathbf{\text{Exp}}}$ as ΔE_ST

ν-DABNA (36) and its element-substituted analogs, BOBO-Z, BOBS-Z, and BSBS-Z (37), are state-of-the-art MR-TADF materials with narrowband blue emissions (Fig. 1C).

We performed k_RISC calculations on these compounds as test cases. We assessed the predictions of the four levels of the TVCF theory, termed as 1st+Condon, 2nd+Condon, 1st+HT, and 2nd+HT (Fig. 1B); see the “Formulation of 2nd+HT TVCF Theory” section, and Marcus theory.

The rate equations of all these theories commonly form as a function of ΔE_ST and T. The ΔE_ST can be considered a free parameter. We set ΔE_ST to the measured $E_a^{\text{Exp}}$ corresponding to the slope of the experimental Arrhenius plots (Fig. 1D) and T to 300 K. TVCF simulation requires harmonic vibrational wave functions, which were computed at the DFT level. The results are shown in Fig. 2 (A and B) (see also table S1).

Fig. 2. Simulated RISC rate constants for four TADF molecules.

(A and B) k_RISC calculated by various rate constant formulas using $E_a^{\text{Exp}}$ as ΔE_ST, and the ratio of the predicted k_RISC to the experimental value for BXBX-Z and ν-DABNA. (C) Overview of the Arrhenius plot slope fitting (ARPSfit) method to determine $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ with which the rate constant calculation can reproduce the experimental Arrhenius plot with its slope $E_a^{\text{Exp}}$. (D and E) k_RISC calculated by various rate constant formulas using $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ obtained by the ARPSfit method, and the ratio of the predicted k_RISC to the experimental value. (F) Contributions of the direct process (T₁) and indirect processes mediated by the intermediate states, T₂, T₃, and T₄, to k_RISC.

Figure 2 (A and B) shows that, overall, the 2nd+Condon and 1st+HT methods predicted larger k_RISC values than the 1st+Condon method, showing good agreement with the experimental values. This result indicates that considering either the NA-SVC or HT-SVC effects substantially improves the accuracy. The 2nd+HT model, considering both effects, provided the most accurate results within one order of magnitude of error relative to the experimental values. A comparison of the 1st+Condon and 1st+HT results revealed that k_RISC was increased by the HT-SVC effect itself by a factor of 10 to 1000. The notable differences between the 1st+HT and 2nd+HT predictions confirmed the importance of the NA-SVC effect, which was most pronounced in BOBO-Z and ν-DABNA. The impact was not so large for either BOBS-Z or BSBS-Z. The former feature could be explained by the fact that ΔE_T1T2 was small (see also section S2).

Although the 2nd+HT model showed promising improvements, it still performed poorly in some cases. For BOBO-Z and ν-DABNA, the ratio of the predicted k_RISC at the 2nd+HT level to the experimental value was 0.16 and 0.09, respectively. These ratios were much smaller in an unbalanced manner than those for BOBS-Z and BSBS-Z, which were 1.2 and 0.93, respectively (Fig. 2B). We speculated that these errors arose from the condition that ΔE_ST was set to $E_a^{\text{Exp}}$. This setup for ΔE_ST was used in (7, 31).

Calculation of the RISC rate constant using ΔE_ST from slope fitting method

The observation shown above prompted us to explore an alternative strategy to define the relationship between the experimental $E_a^{\text{Exp}}$ and parameter ΔE_ST in the rate equations. The core of our scheme designated “Arrhenius plot slope fitting (ARPSfit)” is computationally simulating the Arrhenius plots by evaluating the rate equation as a function of 1/T for a given ΔE_ST. Let the slope of the simulated Arrhenius plots be denoted $E_a^{\text{Calc}}$, which varies depending on the given value of ΔE_ST (Fig. 2C). In the ARPSfit method, the objective value of ΔE_ST is determined such that the associated $E_a^{\text{Calc}}$ matches the preknowledge $E_a^{\text{Exp}}$. The ΔE_ST provided by the ARPSfit method is hereafter denoted $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ (Fig. 2C). To obtain a set of Arrhenius plots, the TVCF (or Marcus) formula was evaluated at 11 points of T in the range of 200 to 300 K. Note that ARPSfit coupled with TVCF requires harmonic vibrational analysis.

The resulting $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ for the MR-TADF systems are summarized in table S3. The $E_{\text{ST}}^{(\text{fit})}$ for BOBO-Z and ν-DABNA were predicted to be 0.035 and 0.037 eV, respectively, at the 2nd+HT TVCF level, which were smaller than the corresponding $E_a^{\text{Exp}}$, 0.102 and 0.087 eV, respectively.

By substituting $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ into the rate equations, we reevaluated k_RISC (Fig. 2, D and E, and table S4). A comparison between Fig. 2 (A and D) (or Fig. 2, B and E) enabled us to examine the influence of the redefined ΔE_ST on k_RISC. The use of the ARPSfit-based $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ effectively refined the RISC rate constant predictions for all cases compared to those obtained with the previous setting $\mathrm{\Delta }{E}_{\text{ST}}=E_a^{\text{Exp}}$. For BOBO-Z, BOBS-Z, BSBS-Z, ν-DABNA, the ratio of the 2nd+HT-based k_RISC prediction using $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ to the experimental value was 0.93, 1.8, 1.0, and 0.26, respectively, indicating a similar error order (Fig. 2E). Substantial improvements were observed in the Marcus results. These results demonstrated that the ARPSfit method offered a marked improvement in determining ΔE_ST from $E_a^{\text{Exp}}$.

As observed previously, the NA-SVC contribution appears to be prominent for BOBO-Z and ν-DABNA. To obtain detailed insights, we dissected k_RISC into state-wise contributions (Fig. 2F). Details of this analysis are presented in section S2.2. The RISC transition for BOBS-Z and BSBS-Z was 97% or more, characterized as a direct T₁ → S₁ process and almost unaffected by NA-SVC treatment. Conversely, a strong NA-SVC effect arose in BOBO-Z, in which the intermediate T₂ and T₃ states made large contributions via NA-SVC. Similarly, for ν-DABNA, the T₂-mediated NA-SVC channel accelerated the RISC process by a factor of 6.3, relative to the 1st+Condon/HT prediction. Figure 2E indicates that treatment at the 2nd+HT level was necessary to quantitatively predict k_RISC. This indicates that the NA-SVC and HT-SVC effects manifested in the intermediate T_n states are adequately coupled.

Unexpectedly, the Marcus predictions were greatly improved using $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$, closely approaching the experimental values, even for molecules where the NA-SVC effect discarded in Marcus formula was relevant.

Section S3 discusses the first-principles calculations of ΔE_ST using various levels of theory, PBE0/def2-SV(P) (38, 39), double-hybrid variant B2PLYP (40)/cc-pVDZ, wave function–based SCS-CC2 (41)/cc-pVDZ and its linear transformation (LT) correction method (34). The k_RISC was evaluated using these ΔE_ST.

We showed that the first-principles wave function methods, specifically SCS-CC2 and its LT methods, offered reliable $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ on the predictions of k_RISC. The resulting k_RISC values reproduced the experimental values within only a 10-fold error and consistent trends across molecules (fig. S3); for example, the 2nd+HT TVCF prediction at the SCS-CC2 LT level for BSBS-Z (4.85 × 10⁵ s⁻¹) was in good agreement with the experimental k_RISC (1.60 × 10⁶ s⁻¹). It is suggested that the first-principles calculations can be used for prescreening for molecular development where the experimental ΔE_ST is inaccessible.

Comparison of 1st+Condon TVCF and semiclassical Marcus formula

The Marcus results provided relatively good accuracy and better agreement with the experimental values than the 1st+Condon predictions (Fig. 2, D and E). This is remarkable because it contradicts the theoretical construction. Marcus formula corresponds to a model downgraded from the 1st+Condon theory. As discussed in the “Comparison of the 1st+Condon TVCF method and Marcus formula” section, the relationship between the two models is clearly formulated through four approximation steps (42): (i) Duschinsky rotation cutoff (Dus-off), (ii) displaced harmonic oscillator (DHO), (iii) short-time (st), and (iv) high-temperature (ht) approximations. These procedures result in five hierarchical levels of rate-constant equations (Fig. 3A).

Fig. 3. Rate constant equations based on 1st+Condon TVCF theory and its approximations.

(A) Five hierarchical levels of the rate constant equations, 1st+Condon TVCF, Duschinsky rotation cutoff (Dus-off), displaced harmonic oscillator (DHO), and short-time (st) approximated models, and Marcus theory from high-temperature (ht) approximation. (B) Plots of the real part of the correlation functions Re[ρ(t)] as a function of the time t evaluated at five levels of the theory for BOBO-Z. (C) Plots of k_RISC as a function of ∆E_ST evaluated at five levels of the theory for BOBO-Z. The gray vertical dashed line indicates the ARPSfit-based $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ determined at 1st+HT TVCF level. The red dot is the experimental value of k_RISC.

To determine why the Marcus model yielded such remarkable results, we numerically examined how these approximations affected the k_RISC computation. Here, let us focus on BOBO-Z. The results for the other molecules are presented in section S4. Figure 3B shows the real part of the correlation functions ρ(t) (Eq. 10) at individual levels (e.g., Eq. 19B). Applying a Fourier transform to these equations (Eq. 8) yields k_RISC (e.g., Eq. 19A), the values of which are displayed as a function of ΔE_ST in Fig. 3C. In this plot, the k_RISC values at a specific $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ (0.045 eV) for ΔE_ST can be seen. Monitoring these values revealed that the DHO and st approximations substantially affected k_RISC prediction.

Figure 3B illustrates that the DHO approximation changes Re[ρ(t)] from the 1st+Condon’s oscillatory damped shape to a monotonically decaying curve. Equation 8 can be reduced to the identity $k_{\mathrm{RISC}{\mathrm{\Delta }E}_{\text{ST}}=0}={\int }_{-\infty }^{\infty } \mathit{dt} \mathrm{Re}\left[\mathrm{\rho }\left(t\right)\right]$, indicating that it corresponds to the area of the region bounded by the curve Re[ρ(t)] in Fig. 3B (which appears as the intercepts of the curves plotted in Fig. 3C). Moreover, the DHO-approximated $k_{\mathrm{RISC}{\mathrm{\Delta }E}_{\text{ST}}=0}$ attains the largest value and is overestimated compared with the 1st+Condon counterpart. This trend reflects the fact that at the DHO level, Re[ρ(t)] is a positive function, while the 1st+Condon curve involves a positive-negative oscillation, which causes large-scale cancellation in its integration. The above analysis confirms that the DHO treatment spuriously increases k_RISC.

The Dus-off and DHO approximations to the 1st+Condon equation result in the following DHO-level expression

$$k_{\text{RISC}}^{\text{DHO}}(\mathrm{\Delta }{E}_{\text{ST}})=\frac{{\mid H_{\mathit{fi}}^{\text{SOC}}\mid }^2}{{\mathrm{\hslash }}^2}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}‍\mathit{dt} e^{i\mathrm{\Delta }{E}_{\text{ST}}t/\hslash }{\mathrm{\rho }}^{\text{DHO}}(t)$$ (1A)

$${\mathrm{\rho }}^{\text{DHO}}\left(t\right)=\text{exp}\left(-\sum _j^N{S}_j\left\{(1+2{\overline{n}}_j)\left[1-\text{cos}\left({\mathrm{\omega }}_{j}t\right)\right]+i\text{sin}\left({\mathrm{\omega }}_{j}t\right)\right\}\right)$$ (1B)

where the definitions of S_j, ${\overline{n}}_j$, and ω_j are given in the “Comparison of the 1st+Condon TVCF method and Marcus formula” section. Applying the st approximation, Eq. 1 is reduced to

$$k_{\text{RISC}}^{\text{st}}\left(\mathrm{\Delta }{E}_{\text{ST}}\right)=\frac{\mid H_{\mathit{fi}}^{\text{SOC}}{\mid }^2}{{\mathrm{\hslash }}^2}\underset{-\infty }{\overset{\infty }{\int }}\mathit{dt} e^{i(\mathrm{\Delta }{E}_{\text{ST}}-\mathrm{\hslash }\sum _j^N{S}_j{\omega }_j)t/\mathrm{\hslash }} {\mathrm{\rho }}^{\text{st}}\left(t\right)$$ (2A)

$${\mathrm{\rho }}^{\text{st}}(t)=\text{exp}\{-{\displaystyle \sum _j^N}{S}_j[({\overline{n}}_j+1/2){\mathrm{\omega }}_j^2]t^2\}$$ (2B)

The above analyses show that the st approximation transforms ρ(t) into a Gaussian shape. Its Fourier transform (Eq. 2) retains a Gaussian structure, as confirmed by the resulting st- and Marcus-level k_RISC curves (Fig. 3C). These curves exhibit negative quadratic decay on a logarithmic scale [$O(-\mathrm{\Delta }{E}_{\text{ST}}^2)$]. With small ΔE_ST, this shape thus plays a role in artificially enlarging k_RISC with a slow decay, compared to a monotonic [O(−ΔE_ST)] decay. Meanwhile, with larger ΔE_ST, the decay becomes increasingly steep and causes a near-diminishing k_RISC. Therefore, the favorable features of the Marcus results for the four MR-TADF systems are fortuitous owing to error cancellation.

1st+Condon TVCF and Marcus rate constants for 121 MR-TADF molecules

Given that the Marcus predictions coincide closely with the experimental values for the above four molecules, checking the applicability of this error cancellation to other MR-TADF molecules would be interesting. Therefore, we extended the test set to 121 MR-TADF molecules (14), on which k_RISC calculations were performed at 1st+Condon and Marcus levels. Figure 4 (A to C) shows scatter plots comparing the calculated values and the experimental data. The rate constant equations were evaluated using $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$, determined by the ARPSfit method. $\mathrm{\Delta }{E}_{\text{ST}}^{\text{Exp}}$ was obtained from the RISC activation energy $E_a^{\text{Exp}}$ based on the experimental Arrhenius plots or the difference between the measured fluorescence and phosphorescence photon energies (14) (table S9). The use of the spectroscopic fluorescence-phosphorescence difference for $E_a^{\text{Exp}}$ is discussed in section S5.

Fig. 4. Rate constant calculations for 121 MR-TADF molecules.

(A to C) Scatter plots to compare the experimental RISC rate constants k_RISC and the predicted rate constants using 1st+Condon and Marcus formulas with their parameter ∆E_ST set to $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ for 121 MR-TADF molecules. The red dashed line indicates the exact agreement y = x. The two gray dashed lines indicate the relations y = 0.1x and y = 10x. (D to F) Plots of k_RISC as a function of ΔE_ST for AZA-BN (large λ), BOBO-Z (small λ and small ΔE_ST), and BN-CP2 (small λ and large ΔE_ST). (G) The types of the MR-TADF molecules characteristics depending on ΔE_ST and λ: types 1, 2, and 3. The magnitude relation between Marcus and 1st+Condon results depends on the type. (H) Scatter plots of ΔE_ST versus reorganization energy λ for 121 MR-TADF molecules. The data points are classified into types 1, 2, and 3, based on the ratio of k_RISC obtained by Marcus and 1st+Condon theories [k_RISC(Marcus)/k_RISC(1st + Condon)].

Comparing Fig. 4 (B and C) with their comparative plots in fig. S6, we confirmed that the use of ARPSfit-based $\mathrm{\Delta }{E}_{\text{ST}}^{(\text{fit})}$ showed better agreement with the experimental k_RISC than $\mathrm{\Delta }{E}_{\text{ST}}^{\text{Exp}}$ for both models. The 1st+Condon model tended to underestimate k_RISC relative to the experimental data, with a Pearson correlation coefficient of 0.51. This value favorably exceeded that of Marcus theory versus the experiments (0.33). However, there were several cases in which Marcus theory predicted a larger k_RISC than the 1st+Condon, resulting in a better agreement with the experimental measurements. This trend appeared to be related to the error cancellation observed in the previous analysis. However, in several instances, the Marcus results were largely underestimated. Such large statistical variations in the Marcus results could be due to the Gaussian nature of the formula, which is sensitive to ΔE_ST alongside vibrational frequencies and reorganization energy. The underestimation of the 1st+Condon predictions may be rectified by the 2nd+HT treatment, which requires additional electronic structure computations.

Figure 4 (D to F) shows the plots of the Marcus and 1st+Condon results as a function of ΔE_ST for AZA-BN, BOBO-Z, and BN-CP2, respectively, as three typical classes. The reorganization energy λ plays a major role in determining the shape of the Marcus curve. In the large-λ case (e.g., AZA-BN) the ΔE_ST-dependent decay was as slow as that of the 1st+Condon model (Fig. 4D).

The small-λ cases—BOBO-Z and BN-CP2—led the Marcus prediction to largely depend on ΔE_ST (Fig. 4, E and F). The suited threshold for λ to be considered as large or small can be ~5 × 10² cm⁻¹ (or 0.06 eV). A small ΔE_ST yielded a large k_RISC (BOBO-Z), and vice versa (BN-CP2). A notable feature could be observed in the Marcus results with small λ, ascribed to small structural changes between S₁ and T₁; for BN-CP2 (type 3), the RISC transition was characterized as forbidden (Fig. 4C).

The above insights led us to classify the tested MR-TADF molecules into three types, depending on the degrees of ΔE_ST and λ. Figure 4G summarizes these characteristics. Type 1 is associated with large λ, whereas types 2 and 3 are associated with small λ. The ΔE_ST of types 2 and 3 were characterized as small and large, respectively. Marcus theory tended to overestimate k_RISC for types 1 and 2 and considerably underestimated k_RISC for type 3 compared to the 1st+Condon estimations. However, the overestimation was prone to error cancellation in the Marcus predictions, resulting in a near coincidence with the experimental k_RISC.

Figure 4H shows scatter plots of λ versus ΔE_ST for 121 data points. On the basis of the ratio of k_RISC—Marcus versus 1st+Condon, these points appeared to fall into three categories, types 1 to 3. This classification partitioned the data points approximately evenly. This implied that 30% of the Marcus predictions could be largely underestimated (type 3). Meanwhile, the other 30% might suffer from erroneous overestimation but resulted in values closer to the experimental data (type 1). This observation could also be confirmed in the cumulative histogram used to quantitatively analyze the distribution of the ratios relative to k_Exp. (figs. S7 and S8).

DISCUSSION

We presented an in-depth analysis of the spin-flipping process of MR-TADF materials accelerated by two types of spin-vibronic interactions: NA-SVC and HT-SVC. The importance of their concerted effects, 2nd+HT, was identified as a factor that reliably characterized the RISC process. Full incorporation of the 2nd+HT terms into the TVCF framework was developed. This extended approach markedly outperformed the previous perturbative models.

The accuracy of the predicted k_RISC is also susceptible to the precision of ΔE_ST. We developed the APRSfit method for effectively determining ΔE_ST based on the fitting to experimental Arrhenius plots. This method resulted in lower errors in the calculated k_RISC, compared with the cases using $E_a^{\text{Exp}}$.

Furthermore, we examined the extent to which each step of the approximation to the 1st+Condon theory, reduced to Marcus theory, affected k_RISC for MR-TADF systems. Our analyses indicate that the DHO and st approximations have a notable impact. These degradations cause fortuitous error cancellation in several Marcus results. By calculating k_RISC for 121 MR-TADF molecules, we identified the conditions that could closely align the Marcus predictions with the experimental values. One-third of the tested systems did not meet these conditions, and Marcus theory largely underestimated k_RISC for them.

Our extended modeling should benefit the reliable characterization of k_RISC for MR-TADF molecules despite its limitation by the harmonic approximation. Future work involves applying it to molecules underestimated by Marcus theory for molecular exploration.

MATERIALS AND METHODS

Computational details

The structural parameters of all the considered molecules were determined by performing TDDFT-level geometry optimizations for the S₁ and T₁ states separately. The TDDFT-level calculations, including the determination of vibrational states, were performed at the PBE0-D3BJ/def2-SV(P) level using Gaussian 16 (43). This level was compatible with those used in the precedent studies [e.g., (16–18)]. The NAC matrix elements were calculated using Q-Chem version 5.4.2 (44). The SOC matrix elements of the singlet and triplet states were evaluated at the ZORA-PBE0/ZORA-def2-SV(P) level (45) using ORCA version 5.0.2 (46, 47). For these SOC calculations, we used the SARC/J basis (48) for density fitting to the Coulomb term and the chain-of-sphere exchange (RIJCOSX) (49, 50). The derivatives of the SOC matrix elements were calculated via numerical differentiation at the T₁ structure. The numerical differentiation is computationally inefficient but cannot be avoided due to the absence of analytic derivative implementation. These costs limited the application of the 2nd+HT method to a very large number of molecules.

All rate constants were calculated using an in-house code. The vibrational modes calculated at the initial and final states were used for the TVCF rate constant calculations, considering the Duschinsky rotation [for details, see (51)]. The correlation function was evaluated over a time range of 0.05 to 10000.15 fs with a time step of 0.1 fs. The correlation function for the negative time was obtained from the complex conjugate of the positive time region. In the second TVCF treatment, the T₂, T₃, and T₄ states were incorporated as intermediate states. The SCS-CC2 calculations were performed using Turbomole version 7.6 (52, 53), and their LT-corrected energies were evaluated using the formula ΔE_ST(SCS − CC2 LT) = 0.94 ΔE_ST(SCS − CC2) − 0.01 (34).

Formulation of 2nd+HT TVCF theory

Let us here outline the formulation of the 2nd+HT TVCF rate constant equation. Section S7 shows full details of the derivation. The 2nd+HT theory is built upon the second-order Fermi’s golden rule given by

$$k_{f\leftarrow \mathrm{i}}=\frac{2\mathrm{\pi }}{\mathrm{\hslash }}\sum _{v,u}{P}_{\mathit{iv}}\mid H_{\mathit{fu},\mathit{iv}}^{\prime }+\sum _{n,w}\frac{H_{\mathit{fu},\mathit{nw}}^{\prime }{H}_{\mathit{nw},\mathit{iv}}^{\prime }}{E_{\mathit{iv}}-E_{\mathit{nw}}}{\mid }^2\mathrm{\delta }(E_{\mathit{iv}}-E_{\mathit{fu}})$$ (3)

which provides the transition rate between the initial (i) and final (f) states (42). It is written using the vibronic state energies E_iv and E_fu, the Boltzmann distribution $P_{\mathit{iv}}=\frac{e^{-\mathrm{\beta }{E}_{\mathit{iv}}}}{Z_i}$ with Z_i = ∑_v e^−βE_iv. We modeled the spin-flipping process using the perturbation $\widehat{H}\prime ={\widehat{H}}^{\text{SOC}}+{\widehat{H}}^{\text{nBO}}$ consisting of the SOC term ${\widehat{H}}^{\text{SOC}}$ and the NAC term ${\widehat{H}}^{\text{nBO}}={\sum }_k‍{\widehat{P}}_{\mathit{fk}}^2$ with ${\widehat{P}}_{\mathit{fk}}=-\mathrm{i}\mathrm{\hslash }{\nabla }_k$. The rate constants for the ISC and RISC transitions between S₁ and T₁ were evaluated as k_ISC = ∑_m=0,±1 k_T1,m←S1 and $k_{\text{RISC}}=\frac{1}{3}{\sum }_{m=0,\pm 1}‍k_{{\mathrm{S}}_1\leftarrow {\mathrm{T}}_{1,m}}$, respectively. As discussed in section S7, Eq. 3 can be partitioned into the first- and second-order components as $k_{f\leftarrow i}=k_{f\leftarrow i}^{1\text{st}}+k_{f\leftarrow i}^{2\text{nd}}$, where $k_{f\leftarrow i}^{1\text{st}}=\frac{2\mathrm{\pi }}{\mathrm{\hslash }}{\sum }_{v,u}{P}_{\mathit{iv}}\mid H_{\mathit{fu},\mathit{iv}}^{\prime }{\mid }^2\mathrm{\delta }(E_{\mathit{iv}}-E_{\mathit{fu}})$, which accounts for a direct transition process, and the rest, $k_{f\leftarrow i}^{2\text{nd}}$, is associated with an indirect process mediated by the intermediate states.

The SOC term was considered up to the HT level (20), which describes $H_{\mathit{fu},\mathit{iv}}^{\text{SOC}}$ via the first-order Taylor expansion with respect to the vibrational modes {Q_fk} as $H_{\mathit{fu},\mathit{iv}}^{\text{SOC}}\simeq H_{\mathit{fi}}^{\text{SOC}}\langle {\mathrm{\Theta }}_{\mathit{fu}}\mid {\mathrm{\Theta }}_{\mathit{iv}}\rangle +{\sum }_k\frac{\mathrm{\partial }{H}_{\mathit{fi}}^{\text{SOC}}}{\mathrm{\partial }{Q}_{\mathit{fk}}}\langle {\mathrm{\Theta }}_{\mathit{fu}}\mid Q_{\mathit{fk}}\mid {\mathrm{\Theta }}_{\mathit{iv}}\rangle$, where the zeroth- and first-order terms are denoted Condon and HT, respectively. This form leads us to decompose $k_{f\leftarrow i}^{1\text{st}}$ and $k_{f\leftarrow i}^{2\text{nd}}$ further into the Condon and HT elements as $k_{f\leftarrow i}^{1\text{st}}=k_{f\leftarrow i}^{1\text{st},\text{Condon}}+k_{f\leftarrow i}^{1\text{st},\text{HT}}$ and $k_{f\leftarrow i}^{2\text{nd}}=k_{f\leftarrow i}^{2\text{nd},\text{Condon}}+k_{f\leftarrow i}^{2\text{nd},\text{HT}}$, respectively. By combining these elements, four levels of the rate-constant equation—the 1st+Condon, 1st+HT, 2nd+Condon, and 2nd+HT—are defined as follows

$$k^{1\text{st}+\text{Condon}}=k_{f\leftarrow i}^{1\text{st},\text{Condon}}$$ (4)

$$k^{1\text{st}+\text{HT}}=k_{f\leftarrow i}^{1\text{st},\text{Condon}}+k_{f\leftarrow i}^{1\text{st},\text{HT}}$$ (5)

$$k^{2\text{nd}+\text{Condon}}=k_{f\leftarrow i}^{1\text{st},\text{Condon}}+k_{f\leftarrow i}^{2\text{nd},\text{Condon}}$$ (6)

$$k^{2\text{nd}+\text{HT}}=k_{f\leftarrow i}^{1\text{st},\text{Condon}}+k_{f\leftarrow i}^{1\text{st},\text{HT}}+k_{f\leftarrow i}^{2\text{nd},\text{Condon}}+k_{f\leftarrow i}^{2\text{nd},\text{HT}}$$ (7)

These equations were evaluated within the framework of the TVCF method (21, 24). In TVCF, the rate equation is expressed as a time-domain propagation of the normal-mode correlation function ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$, as follows

$$k_{f\leftarrow i}^{(\mathrm{X})}=\frac{1}{{\mathrm{\hslash }}^2}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}‍\mathit{dt} e^{i\mathrm{\Delta }{E}_{\text{ST}}t/\mathrm{\hslash }}{Z}_i^{-1}{\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$$ (8)

where X refers to the level of the formula: 1st+Condon, 1st+HT, 2nd+Condon, and 2nd+HT. Equation 8 is formed as a function of ΔE_ST via the following identity

$$\begin{array}{c}\mathrm{\delta }(E_{\mathit{iv}}-E_{\mathit{fu}})=\mathrm{\delta }(\mathrm{\Delta }{E}_{\text{ST}}+E_v^i-E_u^f)\\ =\frac{1}{2\mathrm{\pi }}\underset{-\infty }{\overset{\infty }{\int }}d\mathrm{\tau } e^{i\mathrm{\Delta }{E}_{\text{ST}}\mathrm{\tau }}{e}^{i\left(E_v^i-E_u^f\right)\mathrm{\tau }}\end{array}$$ (9)

which is based on the relations $E_{\mathit{iv}}=E_i+E_v^i$, $E_{\mathit{fu}}=E_f+E_u^f$, and ΔE_ST = E_i − E_f, where E_i and E_f are the adiabatic electronic state energies, and $E_v^i$ and $E_u^f$ correspond to the vibrational state energies. Note that ΔE_ST does not affect ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$.

A major approximation in TVCF considers the molecular vibrations of each electronic state to be harmonic oscillations. This enables us to obtain an analytical formula for ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$ that can be evaluated in a computationally feasible manner. The resulting ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$ can be divided into X-independent and X-dependent factors as follows

$${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)={\mathrm{\rho }}_{\mathit{fi}}^{(\text{core})}(t){\overline{\mathrm{\rho }}}_{\mathit{fi}}^{(\mathrm{X})}(t)$$ (10)

The ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{core})}(t)$ is given as

$${\mathrm{\rho }}_{\mathit{fi}}^{(\text{core})}(t)=\sqrt{\frac{\text{det}(\mathbf{a}\overline{\mathbf{a}})}{\text{det}(\mathbf{D})}}\text{exp}\left(\frac{i}{\mathrm{\hslash }}[{\mathbf{K}}^{\mathbf{T}}\mathbf{CK}-\frac{1}{2}{\mathbf{E}}^{\mathbf{T}}{\mathbf{D}}^{-\mathbf{1}}\mathbf{E}]\right)$$ (11)

which is built upon various types of matrices defined as

$$\mathbf{A}\equiv \overline{\mathbf{a}}+{\mathbf{J}}^{\mathbf{T}}\mathbf{aJ}$$ (12)

$$\mathbf{B}\equiv \overline{\mathbf{b}}+{\mathbf{J}}^{\mathbf{T}}\mathbf{bJ}$$ (13)

$$\mathbf{C}\equiv \mathbf{b}-\mathbf{a}$$ (14)

$$\mathbf{D}\equiv {\left[\begin{array}{cc}\mathbf{B}& -\mathbf{A}\\ -\mathbf{A}& \mathbf{B}\end{array}\right]}_{2N_{\text{vib}}\times 2N_{\text{vib}}}$$ (15)

$$\mathbf{E}\equiv {\left[\begin{array}{c}{\mathbf{K}}^{\mathrm{T}}\mathbf{CJ}\\ {\mathbf{K}}^{\mathrm{T}}\mathbf{CJ}\end{array}\right]}_{2N_{\text{vib}}\times 1}$$ (16)

The a and b are diagonal matrices of the dimension N_vib with their elements $a_{\mathit{jj}}=\frac{{\mathrm{\omega }}_j}{\mathrm{\hslash }\text{sin}({\mathrm{\omega }}_{j}t)}$ and $b_{\mathit{jj}}=\frac{{\mathrm{\omega }}_j}{\mathrm{\hslash }\text{tan}({\mathrm{\omega }}_{j}t)}$ based on the initial state frequency ω_j. Their variants denoted $\overline{\mathbf{a}}$ and $\overline{\mathbf{b}}$, respectively, are based on the final state frequency ${\overline{\mathrm{\omega }}}_j$ in the place of ω_j. J is the Duschinsky rotation matrix (54, 55), and K is the displacement vector between the initial and final states.

The integral in Eq. 8 can be evaluated using the Fourier transform of ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$ from the time (t) domain to the energy (ΔE_ST) domain, where ${\mathrm{\rho }}_{\mathit{fi}}^{(\mathrm{X})}(t)$ (Eq. 10) is evaluated at grid points of t ∈ [−t_max, t_max]. The resulting k_f←i is obtained as a function of ΔE_ST, which is represented on a grid in energy coordinates. The objective k_f←i can be determined by substituting ΔE_ST with the user-input value of the adiabatic energy difference, ΔE_fi.

Comparison of the 1st+Condon TVCF method and Marcus formula

In previous studies, the semiclassical Marcus formula (15) was mostly used to calculate the RISC rate constants, as follows

$$k_{\text{RISC}}^{\text{Marcus}}(\mathrm{\Delta }{E}_{\text{ST}})=\frac{2\mathrm{\pi }}{\mathrm{\hslash }}{\mid H_{\mathit{fi}}^{\text{SOC}}\mid }^2\frac{1}{\sqrt{4\mathrm{\pi }{\mathrm{\lambda }}_{\text{RISC}}{k}_{\mathrm{B}}T}}\text{exp}\left(\frac{-E_a^{\text{RISC}}}{k_{\mathrm{B}}T}\right)$$ (17)

where λ_RISC is the reorganization energy and $E_a^{\text{RISC}}$ is defined as

$$E_a^{\text{RISC}}=\frac{{(\mathrm{\Delta }{E}_{\text{ST}}+{\mathrm{\lambda }}_{\text{RISC}})}^2}{4{\mathrm{\lambda }}_{\text{RISC}}}$$ (18)

The Marcus equation can be derived from the 1st+Condon equation k^1st+Condon (Eq. 4), via stepwise approximations (42, 56).

Equation 4 is rewritten as

$$k_{f\leftarrow i}^{1\text{st}+\text{Condon}}(\mathrm{\Delta }{E}_{\text{ST}})=\frac{{\mid H_{\mathit{fi}}^{\text{SOC}}\mid }^2}{{\mathrm{\hslash }}^2}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}‍\mathit{dt} e^{i\mathrm{\Delta }{E}_{\text{ST}}t/\mathrm{\hslash }}{Z}_i^{-1}{\mathrm{\rho }}^{1\text{st}+\text{Condon}}(t)$$ (19A)

$${\mathrm{\rho }}^{1\text{st}+\text{Condon}}(t)=\sqrt{\frac{\text{det}(\mathbf{a}\overline{\mathbf{a}})}{\text{det}(\mathbf{D})}}\text{exp}\left(\frac{i}{\mathrm{\hslash }}[{\mathbf{K}}^{\mathbf{T}}\mathbf{CK}-\frac{1}{2}{\mathbf{E}}^{\mathbf{T}}{\mathbf{D}}^{-\mathbf{1}}\mathbf{E}]\right)$$ (19B)

The initial step to approximate $k_{f\leftarrow i}^{1\text{st}+\text{Condon}}$ toward Marcus theory is to cut off the Duschinsky rotation by setting J to an identity matrix (Dus-off). This enables ρ(t) to be factorized into separate entities associated with each vibrational mode, as follows

$$\begin{array}{c}{Z}_i^{-1}{\mathrm{\rho }}^{\text{Dus}-\text{off}}(t)={\prod }_j^N\left[\right(\frac{{\mathrm{\omega }}_j{\overline{\mathrm{\omega }}}_j}{\text{sin}\left({\mathrm{\omega }}_j{t}_i\right)\text{sin}\left({\overline{\mathrm{\omega }}}_j{t}_f\right)})\\ {\left[{\mathrm{\omega }}_j\text{cot}\left(\frac{{\mathrm{\omega }}_j{t}_i}{2}\right)+{\overline{\mathrm{\omega }}}_j\text{cot}\left(\frac{{\overline{\mathrm{\omega }}}_j{t}_f}{2}\right)\right]}^{-2}\\ {\left[-{\mathrm{\omega }}_j\text{tan}\left(\frac{{\mathrm{\omega }}_j{t}_i}{2}\right)-{\overline{\mathrm{\omega }}}_j\text{tan}\left(\frac{{\overline{\mathrm{\omega }}}_j{t}_f}{2}\right)\right]-1]}^{\frac{1}{2}}\\ \times \text{exp}(-{\mathit{iK}}_j^2\frac{{\mathrm{\omega }}_j{\overline{\mathrm{\omega }}}_j}{{\mathrm{\omega }}_j\text{cot}\frac{{\overline{\mathrm{\omega }}}_j{t}_f}{2}+{\overline{\mathrm{\omega }}}_j\text{cot}\frac{{\mathrm{\omega }}_j{t}_i}{2}})\end{array}$$ (20)

Furthermore, we consider the DHO model, which uses the common frequency for the initial-state ω_j and the final-state ${\overline{\mathrm{\omega }}}_j$, resulting in

$$\begin{array}{l}{Z}_i^{-1}{\mathrm{\rho }}^{\text{DHO}}(t)={\displaystyle \prod _j^N}\text{exp}\{-S_j[(1+2{\overline{n}}_j)\\ -(1+{\overline{n}}_j)\text{exp}(-{\mathit{i}\mathrm{\omega }}_{j}t)-{\overline{n}}_j\text{exp}({\mathit{i}\mathrm{\omega }}_{j}t)]\}\end{array}$$ (21)

where $S_j\equiv \frac{K_j^2{\mathrm{\omega }}_j}{2\mathrm{\hslash }}$ and ${\overline{n}}_j\equiv \frac{1}{\text{exp}(\mathrm{\hslash }\mathrm{\beta }{\mathrm{\omega }}_j)-1}$.

The st approximation, $\text{exp}(\pm i{\mathrm{\omega }}_{j}t)\simeq 1\pm i{\mathrm{\omega }}_{j}t-\frac{1}{2}{\mathrm{\omega }}_j^2{t}^2$, subject to the assumption ∑_j S_j ≫ 1, is applied to Eq. 21, which results in

$$Z_i^{-1}{\mathrm{\rho }}^{\text{st}}(t)=\text{exp}\{-i{\displaystyle \sum _j^N}{S}_j{\mathrm{\omega }}_{j}t-\sum _j(S_j{\overline{n}}_j+\frac{1}{2}){\mathrm{\omega }}_j^2{t}^2\}$$ (22)

Finally, the Marcus equation (Eq. 17) can be derived using Eq. 22 by applying the ht approximation that the temperature T is sufficiently large and ℏω_j/kT ≪ 1 holds

$$Z_i^{-1}{\mathrm{\rho }}^{\text{Marcus}}(t)=\text{exp}\{-i{\displaystyle \sum _j^N}‍S_j{\mathrm{\omega }}_{j}t-\sum _j‍S_j{\overline{n}}_j{\mathrm{\omega }}_j^2{t}^2\}$$ (23)

Supplementary Materials

This PDF file includes:

Sections S1 to S7

Figs. S1 to S8

Tables S1 to S10

Legend for Supplemental auxiliary files

Other Supplementary Material for this manuscript includes the following:

Supplemental auxiliary files