Herzberg-Teller Vibronic Coupling Effect on the Vibrationally-Resolved Electronic Spectra and Intersystem Crossing Rates of a TADF Emitter: 7-PhQAD

Assessing and improving the performance of organic light-emitting diode (OLED) materials require quantitative prediction of rate coefficients for the intersystem crossing (ISC) and reverse ISC (RISC) processes, which are determined not only by the singlet-triplet energy gap and the direct spin-orbit coupling (SOC) at a thermal equilibrium position of the initial electronic state but also by the non-Condon effects such as the Herzberg-Teller vibronic coupling (HTVC) and the spin-vibronic coupling (SVC). Here we applied the time-dependent correlation function approaches to calculate the vibronic absorption and fluorescence spectra and ∗To whom correspondence should be addressed


Introduction
The OLED, an organic optoelectronic device based on fluorescence or phosphorescence emitters, has been successfully applied in displays and lighting. [1][2][3][4][5] However, due to the limitation of spin statistics and the raising of exciton deactivation at high current density, 6,7 OLED devices still have low quantum yields. To break through the limitation of spin statistics and achieve nearly 100% internal quantum efficiency (IQE), scientists have proposed two strategies, one is based on the "singlet-trapping" and the other is based on the "triplet-trapping". The first strategy utilizes phosphorescence materials to convert the singlet excitons to triplet excitons through efficient intersystem crossing (ISC) induced by the strong spin-orbit coupling (SOC) effect, so as to emit light from the lowest triplet excited state. The second strategy utilizes the delayed fluorescence materials to convert the triplet excitons to singlet excitons through efficient reverse ISC (RISC) induced by thermal activation, so as to emit light from emissive singlet excited state. Upon careful exploitation of these strategies, near 100% IQE can be achieved. 8 In this regard, the performance of OLED device is inseparable from the ISC and RISC processes. Therefore, it is of particular importance for us to study the mechanism and quantitatively predict the rate coefficients. It is also essential to explore in depth what other factors besides the well-known singlet-triplet energy gap (∆E ST ) and SOC matrix elements (SOCMEs) have an impact on the rate coefficients. Powerful computing resources and effective algorithms provide a way to resolve this challenge. [9][10][11][12] One can gain a detailed insight into the mechanism of ISC and RISC processes through analysing the results of theoretical calculations.
Both the calculations of molecular vibronic spectra, [13][14][15][16][17][18][19][20][21][22] and the ISC and RISC rate coefficients require us to account for simultaneous changes in the vibrational and electronic states.One thus needs to combine both the electronic structure theories and quantum dynamics methods to obtain the structure parameters and to describe quantum dynamics, respectively. The earliest research examples to characterize the vibronic properties were to calculate the phosphorescence of small aromatic hydrocarbons such as benzene. [23][24][25] Later, the vibronic effects were involved in the calculations of ISC rates of some organic molecules. [26][27][28][29] Many recent studies on TADF materials (e.g.  demonstrated that the vibronic effect on the RISC rates is significant. The nonadiabatic couplings (NACs) between the low-lying excited singlet states and triplet states open the possibility for significant second-order coupling effects, and increase RISC rate by a few orders of magnitude in some TADF emitters. 30 So that, the quantitative predication of the ISC and RISC rates of a TADF emitter requires taking the vibronic states into account, rather than the pure electronic states. 26,31 The vibronic coupling effects on the S-T crossing rate coefficients should include both the contributions of HTVC and spin-vibronic coupling (SVC). Many previous works found that the second-order SVC (the last term of the right-hand side (RHS) of Eq. (6)) assisted triplet to singlet up-conversion, increasing RISC rates by a few orders of magnitude in some TADF emitters. They thus applied a full second-order SVC model to calculate S-T crossing rates (e.g. Refs 33,35,43).
Usually, the second-order vibronic coupling effect in most cases is much smaller than the firstorder one. However, due to a vibronic resonance that orchestrates three electronic states (S 1 , T 1 and T 2 ) together, S 1 −T 2 crossing plays a major role in enhancing ISC and RISC of some MR-type TDAF emitters. 33,42 Li et al. 36 synthesized a narrow-band, ultrapure blue TADF material,7-PhQAD, which is based on a rigid framework of quinolino-[3,2,1-de]acridine-5,9-dione. 7-PhQAD molecule's S 1 and T 1 states mainly possess (π, π * ) character leading to the transition of S 1 → T 1 nearly spin forbidden according to El-Sayed's rule. 37 Nevertheless, the experimentally-measured ISC rate actually reaches 10 8 s −1 , indicating the significant impact of the vibronic coupling effects. Here we thus calculate the vibrationally-resolved absorption and fluorescence spectra as well as its ISC and RISC rates to verify the microscopic mechanism of TADF process. We aim to describe its photophysical properties, quantitatively predict the ISC and RISC rates, and unveil the mechanism of the singlet-triplet (S-T) intersystem crossings. The time-dependent correlation function approaches, which have been implemented by our group to calculate the electronic absorption and emission spectra and resonant Raman scattering spectra 19,21,22 with including the Franck-Condon (FC), Herzberg-Teller vibronic coupling (HTVC) 38 and Duschinsky rotation (DR) 39,40 effects, have been extended to calculate the ISC and RISC rates of 7-PhQAD emitter.
In our calculations of S-T crossing rates, we include FC and HTVC contributions and ignore the second-order SVC effect, a similar way with Refs. 12,41. It is that only the first two terms of SOCME in Eq.(6) are involved. It is well known that can be approximately written as defined as the NAC vector between two excited states of the given spin multiplicity σ . Therefore, the HT-type vibronically-induced term provides mixing of the triplet and singlet excited states being coupled by the first order vibronic perturbation due to the nuclear displacement along with the 3N − 6 normal modes Q. Anyway, in this work, we account for the HTVC effect via directly evaluating the nuclear derivatives of SOC not via calculating NAC vectors between the excited states.
We aim to provide a demonstration of what proportion of ISC/RISC rate of a TADF emitter can be covered by the HT-type vibronic coupling effect.

Theoretical Methods and Computational Details
With respect to the Fermi-Golden rule, 44 the transition rate (k) from the initial state i to a dense manifold of final states f can be described as where |Φ i ⟩ and |Φ f ⟩ are the wavefunctions of the initial and final vibronic states, respectively, ω is the frequency of the external radiation.Ĥ ′ denotes the perturbation and can be written aŝ the rate equation becomes For a S-T intersystem crossing, if we restrict ourselves up to and second-order terms, the coupling matrix element can be expressed as 25,45 H where the summations extend over the complete sets of pure-spin Born-Oppenheimer (psBO) states of the given multiplicity. The first term is called as the direct SOC, and the last two terms originate from the mixed vibronic and SOC, usually called as the SVC. Although Eq. (3) specifically refers to a S → T crossing, the corresponding expression for the reverse crossing is readily written down.
One may evaluate Eq. (3) by making use of the HT expansion, i.e., expand the integrals about the nuclear equilibrium configuration Q = 0. Writing the psBO functions as products of an electronic wavefunction Ψ and a vibrational wavefunction The second term in Eq. (5) If the first term, the direct SOC, vanishes, this intersystem crossing is spin forbidden. In the above derivative, we apply the relation of In our calculations of S-T crossing rates, we only keep the first two terms of Eq. (6) and ignore the last second-order SVC term. If the harmonic oscillator approximation to the potential energy surfaces is adopted, then the calculations of S-T crossing rates are similar to the calculation of vibrationally-resolved absorption and fluorescence spectra. In the later cases, SOC operator is replaced by the dipole operatorμ and S 0 and S 1 states are concerned. We have implemented these time-dependent correlation function approaches to calculate the vibronic spectra including one-and two-photon absorption and emission 20,21 and resonance Raman scattering spectra. 19 A summary of our previous works on the vibronic spectra has been given in Ref. 22. Here we extend this time-dependent approach to calculate the S-T crossing rates.
Within the adiabatic approximation the molecular Hamiltonian can be written asĤ whereĤ i andĤ f are the vibrational Hamiltonians of the initial and final vibronic states, respectively. If we ignore the spin-vibronic interaction terms in H ′ i f , and account for the Boltzmann distribution of the initial-state vibronic manifold at the finite temperature and the molecular energy conservation for the nonradiative transition, i.e. ω = 0, Eq. (2) becomes the following thermal rate constant form for the transition from the initial to the finial electronic states with . Here β = 1/k B T , Tr(· · · ) represents the trace over nuclear degrees of freedom, and H i f SO denotes the SOCME between the initial and final electronic states, including the direct SOC and it's derivative term as expressed in Eq. (5). ∆E is the energy difference between two concerned electronic states of | Ψ i ⟩ and | Ψ f ⟩ (0-0 transition). γ represents the solvent reorganization energy, and the corresponding exponential form of γ comes from the high temperature approximation of solvent mode. 46 A detailed derivative about the exponential expression of the γ has been given in supporting information (SI).
We calculate the SOCMEs between a pair of singlet and triplet electronic states at the theoretical level of the time-dependent density functional theory (TDDFT) within Q-Chem 5.2 software package. 47 The root-mean-square SOCME between a pair states is defined as The derivatives of the SOCMEs with respect to the nuclear coordinates have been calculated numerically using a three-point finite-difference approximation with a step length of 0.001 Å. Considering the three degenerate electronic substates of the triplet (m s = 0, ±1), the calculated RISC rate is scaled by a factor of 1/3. TADF emerges from a balance of charge-transfer (CT) and local excited states. [48][49][50][51][52] The conventional approximate exchange-correlation functionals and kernels can lead to a large error in TDDFT calculations of a TADF emitter. The tuned range-separated XC functionals have thus been suggested to describe the excited-states properties with inclusion of CT character. [53][54][55][56][57] Here we tuned the range-separation parameter ω by minimizing the function 58,59

Results and Discussion
The geometric and electronic structures, and vibrationally-resolved absorption and fluorescence spectra The geometric structure of 7-PhQAD molecule is shown in Figure 1.
In   36 For example, the first three functionals produce the vertical and adiabatic energy gaps of ≥ 0.5 eV, and the LC-BLYP* yields a slightly smaller value of ∼ 0.4 eV.
The appreciable overestimation to ∆E S 1 T 1 by the single-reference TDDFT approach indicates that 7-PhQAD molecule is a strongly correlated system. To check this, we calculated VEEs of low-lying excited states using the algebraic-diagrammatic construction scheme of second order ADC(2). 71 The correlation consistent basis set, cc-pVDZ, was adopted. ADC(2)/cc-pVDZ overestimates the absorption and emission energies (see Table S1 in SI), but produces an energy gap of 0.27 eV, which is much closer to the experimental gap of 0.19 eV. The double substitutions contribute 13% and 11% to S 1 and T 1 states, respectively, as Table S1 in SI shows, meaning that the failure of TDDFT approach is attributed to its single-reference character, and the multireference quantum chemistry methods are needed to exactly describe 7-PhQAD molecule.
The electronic excitations of S 0 →S 1 and S 0 →T 1 are dominated by HOMO → LUMO transitions with a percentage of 95.1 and 96.1, respectively, evidenced by the nearly identical density differences shown in Figure S2. Both S 0 → S 1 and S 0 → T 1 transitions mainly have (π, π * ) and intramolecular CT characters as Figure 1 and Figure S2 in SI show. It can then be deduced that the  Table 2.
Unlike S 0 →S 1 and S 0 →T 1 excitations, more MO transitions contribute to T 2 and T 3 states.
For S 0 →T 2 , HOMO-3 → LUMO and HOMO-4 → LUMO+1 contribute 74.0% and 11.3%, respectively. Consequently, T 2 state has the character of (n, π * ). For S 0 →T 3 , HOMO → LUMO+1 makes a major contribution, accounted for 55.1%, followed by HOMO-4 → LUMO of 14.8%. As shown in panels (c) and (d) of Figure S2, the density difference plots of T 2 -S 0 and T 3 -S 0 show local excitation character, and are different from that of S 1 -S 0 . As a result, the SOCMEs of S 1 -T 2 and S 1 -T 3 should be larger than that of S 1 -T 1 .
According to the El-Sayed's rule, in order to compensate for the momentum change caused by electron spin reversal, it is necessary for an electron to jump in a mutually perpendicular orbit to balance the momentum change. Therefore, the initial and final states must have different transition properties to make ISC take place. But unfortunately, both S 1 and T 1 states of 7-PhQAD mainly have (π, π * ) character, indicating that if only direct SOC between S 1 and T 1 is considered, the ISC is hard to happen. In terms of the calculated excitation energies, T 2 and even T 3 are in energy close to S 1 , and these states are almost degenerate, indicating that the population transition via S 1 −T n (n>1) may potentially play a crucial role in ISC and RISC processes.
Considering the intramolecular CT and (π, π * ) characters of the excited states, after comprehensive consideration, we use the optimally-tuned XC functional CAM-B3LYP* in the later calculations.
The to the inaccurate S 1 PES calculated by the single-reference TDDFT. As the key geometrical parameters listed in Table S2 in SI show, TDDFT with and without TDA predicts a larger difference between the ground and excited-state geometries than ADC (2). TDDFT produces a more twisted conformation for S 1 state than ADC (2). The HTVC effect on the absorption and emission spectra is negligibly small.    Table 2 shows that the calculated SOCMEs between S 1 and T 1 are 0.14 and 0.15 cm −1 with respect to S 1 and T 1 geometries, respectively, which are negligibly small. The contribution of vibronic coupling effects on the rates of S-T crossings should be significant. Here we thus calculate both the ISC and RISC rates with the inclusion of the FC, HTVC, and DR effects. A result comparison is made with the VG approximation. In our rate calculations, for S 1 −T 1 crossings, we set ∆E S 1 T 1 = 1532 cm −1 , the experimental energy gap; and for S 1 −T 2 crossings, we set ∆E S 1 T 2 = 74 cm −1 , the calculated adiabatic energy gap between S 1 and T 2 by TDA-CAM-B3LYP*.  The theoretical calculations produce much larger ISC and RISC rate coefficients for S 1 −T 2 crossing than those for S 1 −T 1 crossing because the latter possesses a much smaller energy gap and larger SOCMEs. Considering the NAC vector between T 1 −T 2 , we deduce that the population transfer in 7-PhQAD mainly takes place via S 1 −T 1 and S 1 −T 2 −T 1 .

Concluding remarks
We performed a theoretical study on the geometric and electronic structures, the photophysical properties, and the rate coefficients of the S-T crossings for 7-PhQAD molecule, a newly synthesized MR-type TADF emitter. We found that the HTVC effect on the absorption and fluorescence spectra is negligibly small while it plays a crucial role in the rate coefficients of S-T crossings. The HTVC effect increases the ISC rate by more than one order of magnitude, and when it gets involved, the calculated ISC and RISC rates agree well with the experimental values. Therefore, we conclude that the experimentally-measured ISC rate of 7-PhQAD originates predominantly from the vibronic coupling effects, and the NACs between the triplet excited states play much more important role than those between the singlet ones. It is insufficient to reproduce such a fast experimental ISC rate if only the direct SOC coupling is incorporated because of the larger energy gap of ∆E ST and the negligibly small SOCME between S 1 and T 1 states. The vibronic coupling enhances the population transfer.
However, there exists a small deviation between the calculated and experimental rates, attributed to the neglect of the second-order SVC effect and the inaccurate excited-state PESs produced by the single-reference TDDFT. Like many other MR-type TADF emitters, 7-PhQAD is a strongly correlated electron system. Though its low-lying excited states are singe excitation dominated states, more than 10% double excitation characters indicate that the multi-reference quantum chemistry methods are required to exactly describe its excited states. The single-reference TDDFT not only overestimates ∆E ST but also the difference between the PESs of S 1 and S 0 states. The latter results in the significant enhancement of the vibrational motion, leading to the broadband absorption and emission produced by our theoretical calculation without VG approximation.
This work demonstrates a way to quantitatively predict the ISC and RISC rates of TADF emitters with including the HTVC effect. Our calculations unveil the dynamical mechanism for highly efficient OLED emission and open design routes that go beyond the FC approximation for the future development of high-performance systems.

Supporting Information Available
The excitation energies of 7-PhQAD calculated by ADC(2)/CC-pVDZ at the S 0 and S 1 geometries, the main geometrical parameters for the ground and excited-state geometries of 7-PhQAD, the charge density difference between the corresponding excited states and ground state by (TDA-)CAM-B3LYP*, and the differences between the geometric structures of S 0 and S 1 , of S 0 and T 1 , and between those of S 1 and T 1 optimized at TD-CAM-B3LYP*/6-311++G** level in toluene.
The treatment of HT effect in the time-dependent approach and the derivation of damping-related term are provided. This material is available free of charge via the Internet at http://pubs. acs.org/.