Vibrationally Assisted Direct Intersystem Crossing between the Same Charge-Transfer States for Thermally Activated Delayed Fluorescence: Analysis by Marcus–Hush Theory Including Reorganization Energy

Thermally activated delayed fluorescence (TADF) has recently become an extensively investigated phenomenon due to its high potential for application in organic optoelectronics. Currently, there is still lack of a model describing correctly basic photophysical parameters of organic TADF emitters. This article presents such a photophysical model describing the rates of intersystem crossing (ISC), reverse ISC (rISC), and radiative deactivation in various media and emphasizing key importance of molecular vibrations on the example of a popular TADF dye 9,10-dihydro-9,9-dimethyl-10-(4-(4,6-diphenyl-1,3,5-triazin-2-yl)phenyl)-acridine (DMAC-TRZ). The presented experimental and theoretical investigations prove that ISC and rISC can occur efficiently between the singlet and triplet states of the same charge-transfer nature (1CT and 3CT, respectively). In emitters with the orthogonal donor and acceptor fragments, such spin-forbidden 1CT ↔ 3CT transitions are activated by molecular vibrations. Namely, the change of dihedral angle between the donor and the acceptor affords reasonable spin–orbit coupling, which together with a small energy gap and reorganization energy enable 1CT ↔ 3CT transition rates reaching 1 × 107 s–1. Evidence of direct 1CT ↔ 3CT spin-flip and negligible role of a second triplet state, widely believed as a key parameter in the design of (r)ISC materials, change significantly the current understanding of TADF mechanism. In authors’ opinion, photophysics, and molecular design principles of TADF emitters should be revised considering the importance of vibrationally enhanced 1CT ↔ 3CT transitions.


Table of contents
Determination of experimental photophysical parameters Page S2 Table S1. Photoluminescent and photophysical parameters in various media Page S3 Analysis of molecular orbitals and nature of electronically excited states Page S4 Table S2. Computational data for various rotational isomers is the 3 CT-state geometry and rates of the 3 CT→ 1 CT transition Page S6 Table S3. Rates of ISC and rISC as statistical sums for rotamers for all investigated media Page S7 Prediction of the rate of radiative deactivation Page S8 Table S4. TD-DFT predicted energy (λ S1-S0 ) and oscillator strengths (f) of S 1 -S 0 transition, and calculated radiative deactivation constants (k r ) for various dihedral angles.
Page S8 Figure S1. Dependence of the TD-DFT predicted energy gaps between singlet and triplet states on the dihedral angle between donor and acceptor fragments Page S8 Theoretical model including rotational isomers and vibrationally-enhanced SOC Page S9

Determination of photophysical parameters
Intensity weighted mean lifetime values were used for polyexponential decays of prompt (PF) or delayed (DF) fluorescence. The ratio of DF and PF quantum yields (φ DF /φ PF ) was determined as following: , (S1) where A DF and A PF are pre-exponential factors of prompt and delayed fluorescence lifetimes, respectively; τ PF and τ DF are lifetimes of prompt and delayed fluorescence, respectively. Rate constants of radiative (k r ) and nonradiative (k nr ) decay, and intersystem crossing (k ISC ) were given by equations: and PLQY -photoluminescence maximum and quantum yield, respectively; E 1CT -energy of the 1 CT state obtained as onset of photoluminescence spectra; E 3CT -energy of the 3 CT state calculated using E 1CT and ΔE 1CT-3CT ; ΔE 1CT-3LE -energy gap between 1 CT and 3 LE states obtained from the onsets of respective emissions; negative values indicate that E 1CT < E 3LE . ΔE 1CT-3CT -energy gap between 1 CT and 3 CT states obtained from Marcus-Hush equation and experimental k rISC values corresponding to the dihedral angle of 90° (see text). φ DF /φ PF -ratio of quantum yields of delayed (DF) and prompt (PF) fluorescence; τ PF , τ DF -lifetimes of PF and DF, respectively; k ISC , k rISC , k r , k nr -rates of intersystem crossing, reverse intersystem crossing, radiative and nonradiative deactivation, respectively. a Values obtained by extrapolation of the ΔE 1CT-3CT -E 1CT dependence. b Values obtained using Marcus-Hush equation and experimental k rISC values. c Value predicted by TD-DFT calculations.

Analysis of molecular orbitals and nature of electronically excited states
The nature of states was assigned based on the orbitals involved in respective transitions available from single-point calculations using optimal geometry in corresponding state. Thus, based on the optimized geometry of T 1 -state, calculated highest occupied (HOMO) and lowest unoccupied molecular orbitals (LUMO) involved in the T 1 -S 0 transition are localized on separate fragments: 9,9-dimethyl-9,10-dihydroacridine (DMAC) and phenyls-triazine, respectively (Scheme S1). The T 1 state was thus assigned as a 3 CT state. The same conclusion was made for the S 1 state ( 1 CT).
Molecular orbitals, both HOMO and LUMO, involved in the T 2 -S 0 transition calculated using the optimized geometry of T 2 -state, are localized on the phenyl-s-triazine fragment (Scheme S1). Therefore, the T 2 state was assigned as a 3 LE state. Molecular orbitals involved in the T 3 -S 0 transition calculated using the optimized geometry of T 3 -state, are localized on the N-phenyl-DMAC fragment. The T 3 state was thus assigned as a second 3 LE state. Opt.

Assigned
Orbitals involved S 1 and T 1 * CT T 2 3 LE T 3 3 LE *geometries of the 1 CT and 3 CT states are almost identical so only the 1 CT one is shown.
Scheme S1. Molecular orbitals involved in the electronic trasitions of DMAC-TRZ Experimentally, the CT nature of S 1 state was confirmed by the strong positive solvatofluorochromism ( Figure 1, the main text). As was reported previously, 1 at 78 K, the phosphorescence spectra of DMAC-TRZ and its derivatives are independent of solvent polarity in the range of low polarities. Such phosphorescence thus originates from the triplet state of LE nature. In DMAC-TRZ, its maximum near 484 nm correlates well with the TDDFT calculated T 2 -S 0 transition energy involving orbitals localized on the phenyl-s- S5 etriazine fragment. Such 3 LE-state energy determined experimentally as onset of phosphorescence spectrum in methylcyclohexane equals 2.82 eV (440 nm). The ΔE 1CT-3LE values in various solvents were thus estimated using the E 1CT values form the solvatofluorochromic measurements and the E 3LE value mentioned above. The emission from 3 CT-state was not registered experimentally in these investigations. .226 *V 3CT-1CT -SOCME value for the 3 CT→ 1 CT transition; μ -statistical weight of a rotational isomer at room temperature calculated using relative energies of isomers in the 3 CT state listed in Table S5.  *χ 3CT and χ 3LE -molar fractions of molecules coexisting in 3 CT and 3 LE states at 298.15 K, calculated using Boltzmann distribution law and energies of the respective triplet states listed in Table S1.

S7
k 1CT →3LE and k 3LE→1CT calculated using λ solv = 0.3 eV (Table S9 and S10) Prediction of the rate of radiative deactivation (k r ). The rate of radiative deactivation was obtained as a statistical sum of respective values for each rotational isomer calculated using Strickler-Berg law: 2,3 where λ S1-S0 (θ) and f(θ) are wavelength in nanometer and oscillator strength of S 1 -S 0 transition for each rotational isomer, respectively; n is refractive index of o-dichlorobenzene (n = 1.551). Table S4. TD-DFT predicted energy (λ S1-S0 ) and oscillator strengths (f) of S 1 -S 0 transition, and calculated radiative deactivation constants (k r ) for various dihedral angles*  Table S5. Figure S1. Dependence of the TD-DFT predicted energy gaps between singlet and triplet states on the dihedral angle between donor and acceptor fragments

Theoretical model including rotational isomers and vibrationally-enhanced SOC
Corrections for various polarity. Alignment of the potential curves of 1 CT, 3 CT, and 3 LE states predicted on the TD-TDF/B3LYP level of theory is in excellent correlation with the experimental data in the media of relatively high polarity (like o-dichlorobenzene and 75% mixture of acetone in hexane). To mimic less and more polar medium, the energies of 1 CT and 3 CT states were corrected for each media according to the procedure below.
Unconstrained geometry optimizations of DMAC-TRZ were conducted for various excited states. The nature of each excited state was established by the analysis of molecular orbitals involved in respective transitions. The ΔE 1CT-3CT and ΔE 1CT-3LE values were calculated using electronic energies (E(TD-DFT)) of respective states in their energetic minima (Table S5), verified by the absence of imaginary (negative) vibrational frequencies. Next, starting with the optimized geometry in each excited state, the value of dihedral angle between DMAC and phenyl ring of acceptor fragment was scanned with a 1 degree step from 89° to 60°. Energies of respective states were calculated by single-point calculations. The ΔE 1CT-3CT and ΔE  values for each rotational isomer were calculated using thus obtained energies.  (Table S1) for each medium: where E 3LE 3LE and E 3LE 3LE are the energies of respective states in their optimized geometries from The example of such calculations for hexane is listed in Table S5.
The example of such calculations for hexane is listed in Table S5.
Reorganization energies of the 3 LE→ 1 CT and 1 CT→ 3 LE transitions (Figure 3) were calculated as follows where E 1CT 3LE (θ) -TD-DFT energy of the 1 CT state at the 3 LE geometry at given dihedral angle value θ; E 3LE 1CT (θ) -TD-DFT energy of the 3 LE state at the 1 CT geometry at given dihedral angle value θ. These values are listed in Table S8. were obtained by extrapolation ( Figure 2H in the main text). Reconstructed k rISC values for all media are shown in Table S3.

S12
Prediction of rISC rates. 3 LE→ 1 CT transition. Examples of calculations of 3 LE→ 1 CT transition rates for various rotational isomers using Marcus-Hush equation are presented in Table S9.  Table S5.
Prediction of ISC rates. 1 CT→ 3 CT transition. Rates of the 1 CT→ 3 CT transition in various media were calculated as statistical sums for various rotational isomers using Marcus-Hush equation. The used ΔE 1CT-3CT values were calculated using equation S11 and are listed in Table S1. SOCME values are listed in Table S2. The λ 1CT-3CT values were assumed equal to the ΔE 1CT-3CT ones. Thus obtained k 1CT→3CT values for all media are listed in Table S3.

Prediction of ISC rates. 1 CT→ 3 LE transition.
Examples of calculations of the 1 CT→ 3 LE transition rates for various rotational isomers using Marcus-Hush equation are presented in Table S10.