Prediction of Oscillator Strength and Transition Dipole Moments with

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Prediction of Oscillator Strength and Transition Dipole Moments with Nuclear Ensemble Approach for Thermally Activated Delayed Fluorescence Emitters Weixuan Zeng, Shaolong Gong, Cheng Zhong, and Chuluo Yang J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 25 Mar 2019 Downloaded from http://pubs.acs.org on March 25, 2019

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The Journal of Physical Chemistry

Prediction of Oscillator Strength and Transition Dipole Moments with Nuclear Ensemble Approach for Thermally Activated Delayed Fluorescence Emitters Weixuan Zengab, Shaolong Gonga, Cheng Zhonga*, and Chuluo Yangab* a Hubei

Key Lab on Organic and Polymeric Optoelectronic Materials,

Department of Chemistry, Wuhan University, Wuhan 430072, P. R. China b Shenzhen

Key Laboratory of Polymer Science and Technology, College of

Materials Science and Engineering, Shenzhen University, Shenzhen 518060, China.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (C. Z.) and [email protected] (C. Y.)

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ABSTRACT: This work incorporates nuclear ensemble approach into emission simulation of thermally activated delayed fluorescence (TADF) emitters with strong charge-transfer (CT). The vibrational distribution of the excited state is described by an ensemble of nuclear geometries with vertical transition properties computed for each point in the ensemble through time-dependent density functional theory (TDDFT) method. Comparing to TDDFT calculation at stationary geometry, this method provides better estimate of oscillator strength and distribution of transition dipole moments (TDMs). Four different types of CT states are explored. For twisted intramolecular CT state, the oscillator strength is promoted strongly and direction distribution of TDMs are concentrated, while it increases less with dispersed TDMs for the through space CT state. The present work provides a feasible calculation method for TADF emitters and will compensate the flaws of traditional stationary point TDDFT method which hamper their application in understanding and predicting the photophysical properties of emitters with strong CT characteristics.

INTRODUCTION Organic molecules with distinct electron-donating and electron-withdrawing units are known to exhibit special optoelectronic properties because of the significant charge-transfer (CT) feature of their lowest excited states and thus enable widely optoelectronics application, such as organic photovoltaics1, organic field-effect transistors2, bio-/chemo-/photo-sensors3, and organic light 2 ACS Paragon Plus Environment

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emitting diodes (OLEDs)4-7. The appearance of electron donor (D) and acceptor (A) moieties result in different degrees of special separation of the frontier molecular orbitals (FMOs) because of the highest occupied molecular orbital (HOMO) is mainly distributed in D unit, while the lowest unoccupied molecular orbital (LUMO) is mainly located in A unit. Minimizing the overlap of FMOs brings about small energy splitting (ΔEST) between the lowest singlet (S1) and triplet (T1) states. When the ΔEST small enough, effective up-conversion from

T1 to S1 can be realized at room temperature, giving thermally activated delayed fluorescence (TADF).8-9 Emitting materials exhibiting TADF are of great interest to researchers for their interesting optoelectronic properties, such as long lifetimes of delayed fluorescence and triplet excitons harvesting in OLED applications via a reverse intersystem crossing (RISC) process. In the past few years, efforts have been made to develop accurately molecular modeling of TADF materials, not only for better understanding, but also for predicting the photophysical properties that helps to obtain a complete mechanistic picture of TADF and design of improved emitters.10 Most efforts have been devote to calculate ΔEST and corresponding RISC process.11-13 However, the oscillator strength (f) and transition dipole moment (TDM) of the radiative transition from S1 to ground state (S0) receives relatively little attention. Modeling f can help analyze the radiative transition process of TADF emitter. And direction of TDM is important for analyzing the orientation of emitting dipoles of TADF emitters, which is proved to have effect on the out-coupling 3 ACS Paragon Plus Environment


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efficiency and regarded as a crucial factor to increase efficiency of OLEDs.14-16 For most TADF molecules, because of extremely low hole-electron overlap caused by separated D-A structure and strong CT character, TDDFT calculation at stationary molecular geometry results in nearly zero f and the radiative transition could almost be asserted as forbidden with such small f, which is usually contrary to the experimental observed good emitting performance. Besides, the corresponding direction of TDM is also inaccurate because the value of each component is too low. To better model the TDM and f of S1 → S0 radiative transition process of TADF emitter with near zero oscillator strength, the contribution of dynamic disorder of molecular structures as the result of molecular vibration should be taken into account.17-18 In a D-A structure, the torsion and rotation around the single bond between the D and A fragment is proved to be the vibrational mode that most significantly affect the electronic structure of excited state. According to this theory, the method base on Boltzmann distribution of torsion angle was developed.19-20 The core concept is rotating the D-A dihedral angle of the D-A structure manually to generate a bunch of geometries, each of which was calculated by TDDFT method. Then, the Boltzmann weighted average of the twisting angle related transition properties were calculated, affording calculated

f and TDM direction with more reliable values. But there are two drawbacks with this method. First, vibration modes besides torsion motion that contribute to the radiative transitions are neglected in this case. Second, the method could only 4 ACS Paragon Plus Environment

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suitable for the single bond connected D-A systems with a dihedral angle that easy to adjust manually. Recently, efforts have been made to develop general methods for spectrum simulation of large molecules.21 Among them, the nuclear ensemble approach22-24 represents one accessible method to deal with the challenge of emission simulation of TADF molecules. The vibrational distribution of the excited state can be described by an ensemble of nuclear geometries in the harmonic approximation with the Wigner distribution. Then, vertical transitions properties are computed for each point in the ensemble through TDDFT method. The final spectrum is obtained as an incoherent sum over all these individual transitions. Thus, the nuclear ensemble approach can well describe dynamic disorder with a relatively low computational cost. In this work, the nuclear ensemble approach is introduced into the calculation of S1 → S0 radiative transition process of TADF emitter, aiming at reliable investigation of both f value and TDM direction. To better evaluate feasibility of the new calculation method, four molecular systems were chosen from previous reports. The top priority of the chosen systems is observed with TADF properties. Considering the great variety for the molecule catalog of TADF emitters, we try to cover different conjugation types that may have effect on the dynamic disorder properties. The calculations are carried out for a series of four molecules whose chemical structures are shown in Figure 1. This series consists of four types of intramolecular D-A systems, namely, a) twisted 5 ACS Paragon Plus Environment


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intramolecular CT (TICT)25 (PXZ-TRZ26); b) spiro-conjugation CT (Spiro-CN27); c) through-space CT (XPT28) and d) homo-conjugation (TPA-PRZ(CN)229). All the chosen systems are reported shown typical TADF features but very low calculated values of f in conventional TDDFT. Additionally, the semiempirical method based on manual adjusted D-A dihedral angle has been also processed with the TICT system.20 To further evaluate the potential of the method in predicting the emission properties, existing experimental data are collated for comparison.

Figure 1. Chemical structure and donor (blue)/acceptor (red) geometry of the molecules investigated in this work. METHODS

S1 state geometries, including vibrational modes for all of the molecules collected in Figure 1 are optimized at the TD-CAM-B3LYP30/def2-SVP31 level. 6 ACS Paragon Plus Environment

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The dispersion correction was conducted by Grimme's D3 version with BJ damping function32 by Gaussian 16 program. Based on the optimized S1 state geometries and vibrational normal modes, the nuclear ensemble approach was performed with the Newton-X program24. 1000 nuclear configurations were sampled according to the finite-temperature uncorrelated Wigner distribution for room temperature (300K). TDDFT calculations for S1 → S0 transitions were then performed at the same level to collect the transition dipole moments, oscillator strength and transition energies of all the configurations. Arithmetic mean of the f and x, y and z components of TDM were then calculated to describe the radiative transition in the dynamic disorder system. The required sample size n of the nuclear configurations for a ± α confidence interval with 95% confidence level were estimated by the values of

f or TDM components, expressed as followed: 1.96𝜎 2 ) 𝛼𝑝

𝑛=(

(1)

where the 𝜎 is the standard deviation and the 𝑝 is the average of f or TDM components. The torsion angle Boltzmann distribution method was processed using the optimized S1 geometry of TICT model PXZ-TRZ. The dihedral angle between D/A fragments (θ) is manually rotated around the C-N single bond with the step size of 0.5 degree from 50 to 89.5 degree, affording 80 structures to represent the torsion angle disorder. Then, TDDFT calculations at the CAM-B3LYP/def27 ACS Paragon Plus Environment


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SVP level were performed on these structures. The average f and TDM direction were calculated by the weighted average where the weight of each structure was obtained by Boltzmann distribution: 𝐹(state) ∝ 𝑒

―

𝐸 𝑘𝑇

(2)

in which, F (state) is the population of state, E is state energy, k is the Boltzmann’s constant and T is the thermodynamic temperate set as 300 K at this case. The values of TDM components and f were then obtained using the same method based on Equation S1 with the average f and TDM direction. RESULTS AND DISCUSSION The stationary point TDDFT data obtained at optimized S1 geometry is summarized in Table 1. The calculated f scaled at the order of < 10-4 to 10-2, generally below the approximated experiment value of 10-3 to 10-1 (while more detailed experimental approach of f can be found in the Supporting Information)33-35. The small calculated f’s were not only unable to match the experiment data, but also unable to satisfy the demand for reliable TDM analysis. It was especially prominent for the case of PXZ-TRZ, in which the value of f was almost zero, accompany with almost zero TDM components and TDM direction could not be determined. The direction of the calculated TDMs of the molecules relative to the coordinate of the molecular structures are shown at Figure S1.


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Table 1. Calculated TDM components and oscillator strengths at CAMB3LYP/def2-SVP level with stationary point calculation and nuclear ensemble approach.

Compound

PXZ-TRZ

Spiro-CN

XPT

Stationary point calculation x 0.000 0 0.025 0 0.075 9

TPA-

0.357

PRZ(CN)2

1

y

z

f

Nuclear ensemble approach 𝑥

𝑦

𝑧

𝑓

0.0000 0.0000 0.0000 0.4098 0.0329 0.0141 0.0167

0.0822 0.0000 0.0005 0.2221 0.2189 0.1060 0.0083

0.0860 0.0666 0.0011 0.1404 0.0984 0.0517 0.0029

0.0749 0.0793 0.0109 0.3677 0.0910 0.3894 0.0331

The average TDM and f for the molecules obtained by the nuclear ensemble approach is summarized in the right part of Table 1. The “trajectory” of key values and the direction of TDMs relative to the coordinate of the molecular structure for each type of model system are discussed below. For the TICT structure PXZ-TRZ, the average TDM vector from nuclear ensemble had large x components (0.4098) and small y, z components (0.0329 and 0.0141). This resulted in almost all the TDM vectors aligning to the D-A direction with little dispersion (Figure 2a, Figure 2b, Figure 2c and Figure 2d), 9 ACS Paragon Plus Environment


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distinguishing from zero TDM in stationary point calculation. The values of f were changing according to the TDMs, and the running average curves are shown in Figure 2e. As the structure number increasing, the average f showed obviously convergence and increased from nearly zero to 0.0167, which had reached the rational order of magnitude in theory. This result indicated the transition forbidden CT state could be made allowed with the molecular vibration introduced by the nuclear ensemble approach, and made the outcome more reasonable and satisfactory.

Figure 2. a) Average TDM vector (green arrow), donor plane (x-y plane, blue) and acceptor plane (x-z plane, red) and b) frontier orbital distributions (HOMO is shown in blue and LUMO is shown in red) with optimized S1 state structure of PXZ-TRZ. c) TDM vectors of each nuclear configuration (blue line) with D-A 10 ACS Paragon Plus Environment

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centers. d) Trajectory of TDM components. e) Running average of TDM components and oscillator strength of PXZ-TRZ.

For the spiro-conjugation CT structure Spiro-CN, the average TDM vector could be aligned to the direction between one diphenylamine group in the donor unit and one cyanobenzene group in the acceptor unit (Figure 3a and Figure 3d). Considering the symmetry of the structure, the TDM vectors were distributed in the direction between the two diphenylamine donors and the two cyanobenzene acceptors with similar x, y-components (0.2221 and 0.2189) and smaller z-component (0.1060). The average f turned out to be 0.0083, which was much larger than f at stationary geometry (0.0005) but smaller than the TICT structure PXZ-TRZ. This result could be attributed to the rigid spiro D-A structure, which did not have the “soft” torsion vibrational modes between donor and acceptor units.



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Figure 3. Average TDM vector (green arrow), donor plane (blue plane) and acceptor plane (red plane) of a) Spiro-CN, b) XPT and c) TPA-PRZ(CN)2. TDM vectors of each nuclear configuration (blue line) with D-A centers of d) SpiroCN, e) XPT and f) TPA-PRZ(CN)2.

For the through-space CT structure XPT, the phenothiazine donor and the diphenyl triazine acceptor were arranged parallel to the y-z plane. The average TDM vector aligned to the direction linking the centers of them (Figure 3b) with major x component (0.1404) and minor y and z components (0.0984 and 0.0517). On the contrary, almost the same x, y and z component in stationary point calculation (0.076, 0.086, and 0.067) did not make much sense. As shown at Figure 3e, the TDM vectors of the nuclear configuration show larger dispersion in y, z direction compared to PXZ-TRZ, which is due to the multiple interaction points between orbitals in stacked donor and acceptor. The average 12 ACS Paragon Plus Environment

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oscillator strength turned out to be 0.0029 for XPT, in which case the promotion comparing to the stationary point calculation (0.0011) was not as significant as the other models. This result indicated that the dynamics disorder has less effect in enhancing the oscillator strength of through-space CT state. For the homo-conjugation structure TPA-PRZ(CN)2, the average TDM was on the direction linking the center of the acceptor unit and one of the diphenylamine donors with similar x and z components (0.3677 and 0.3894) and smaller y component (0.0910), as shown at Figure 3c and Figure 3f.

In

contrast, TDM at stationary point showed large x (0.3571) and small y, z (0.0749, 0.0793) component, which could not reflect the spatial relationship of donor and acceptor. Average f of 0.0331 was about three times as large as the

f at stationary point (0.0109). This result indicated that the partially allowed radiative transition could also be promoted by dynamic disorder, although the magnitude was not large. To evaluate the number of structures needed for obtaining reliable TDM and

f, one-sample proportion in the confidence interval was introduced. First, some statistic quantities were introduced: Sample size is the number of structures calculated. Confidence interval α estimate a range around average value of TDM and f (the range is 1±α times average value) within which the true value is estimated to lie. Confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value. As details can be found in the experimental section, confidence level was set as 13 ACS Paragon Plus Environment


95% and different confidence intervals were plugged into Equation 1 for the four systems. The required sample size versus confidence interval curves are shown at Figure 4. It could be seen that sample sizes varied with different systems, and the oscillator strengths needed bigger sample size than the TDM vectors. And as a corollary, accuracy of the nuclear ensemble approach can be controlled by sample size, and sample size of 1000 in this work could ensure 10% confidence interval of f’s and 5% confidence interval of TDM vectors for all systems.

f

104

Sample Size

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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TDM

PXZ-TRZ Spiro-CN XPT TPA-PRZ(CN)2

103

102

101

0

5

10

15

Confidence Interval (%)

20

Figure 4. Sample size curves of confidence interval of oscillator strengths (solid dot) and TDMs (open dot) for the molecules analyzed in this work.


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The emission spectra of the molecules were obtained according to the literature method21 (Figure S4). All spectra show characteristic CT emission board band expended to long wavelength. The overestimate of the energy levels may be due to the functional and the lack of solvation effects. Further studies including benchmark of functional and incorporation of solvation effects may be necessary to obtain more reliable emission spectra calculation. In the single bond connected D-A molecules, internal rotation around the single bond has dominate effect on electronic structure of excited states. The Boltzmann distribution method had been mentioned and applied to approximate analysis of TICT type TADF emitters by Jan-Michael Mewes based on this concept.20 In this study, with the TICT emitter PXZ-TRZ selected as model, the dihedral angle between D/A fragments (θ) was manually rotated around the C-N single bond with the step size of 0.5 degree from 50 to 89.5 degree (Figure 5a).



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Figure 5. a) Average TDM vector (green arrow), donor plane (y-z plane, blue) and acceptor plane (x-y plane, red) with optimized S1 state structure of PXZTRZ. Dependence of b) TDM components, oscillator strength and c) relative energy and Boltzmann weighted oscillator strength on dihedral angle θ between D-A planes of PXZ-TRZ. As can be seen in Figure 5b, x-component increased with the θ decreasing, while the y and z components kept zero. And the value of f increased with the x-component, showing a significant quadratic curve-shaped. The calculated f’s 16 ACS Paragon Plus Environment

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were than weighted using Boltzmann factors for 300 K with relative energies changing with θ (Figure 5c). The values at nearly vertical region (> 87

°)

contributed little because of very small f. Meanwhile, as the θ decreasing, the relative energy increased, the density of states became very low so that could not contribute to the emission though the f was very big. The Boltzmann weighted values provided average f of 0.0239 with average TDM vector of (0.9005, 0.0000, 0.0000). The value of f was larger than the one at nuclear ensemble approach. And the direction of TDM vector was on the D-A direction without any dispersion in y and z direction. This result could be attributed to all other vibrational modes neglected in this method. Given the above, the Boltzmann distribution method provides a simple and practical solution for approximating the TDM of TICT structure emitter, but the disadvantages of the method is obvious: i) apply only to single bond connected TICT structures; ii) neglect the effects of vibrational modes other than torsion motion. CONCLUSIONS In summary, this work demonstrates that nuclear ensemble approach is a selectable tool for rational assessment or prediction of the radiative transition of an emitter with very small hole-electron overlap (strong CT state). For four types of CT states investigated here, it could be clearly seen that the molecular vibration promoted f and TDM and directions of TDM were along the D-A link directions. The extent of promotion of f and dispersion of TDM were related to 17 ACS Paragon Plus Environment


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the type of conjugation: The direct π conjugation that broken by twisting (TICT) is very sensitive to the vibration and the f is promoted strongly with concentrated TDMs; the through space CT such as spiro-conjugation, π-π stacking, homoconjugation are less sensitive to vibration and the f is promoted relatively weak with dispersed TDMs. As for the former case of TICT, we can also estimate TDM and f with Boltzmann distribution of torsion angle. This work provides a feasible calculation method for TADF emitters and will compensate the flaws of traditional TDDFT method which hamper their application in understanding and predicting

the

photophysical

properties

of

emitters

with

strong

CT

characteristics.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.xxxxxxx. Coordinate transformation of transition dipole moment; correlation of oscillator strength and transition dipole moment; stepwise protocol of the method; experimental approach of oscillator strength; direction of transition dipole moment obtained by TDDFT calculation; hole-electron overlap distribution obtained by TDDFT calculation.


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AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (C. Z.) and [email protected] (C. Y.) Notes The authors declare no competing financial interests. ACKNOWLEDGMENT This research was supported from the National Natural Science foundation of China (Nos. 51873160 and 91833304), the National Basic Research Program of China

(973

Program

2015CB655002),

Shenzhen

Peacock

Plan

(KQTD20170330110107046) and the key Technological Innovation Program of Hubei Province (No. 2018AAA013). The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University. We thank Sobereva (USTB Beijing) for providing the Multiwfn program and providing instruction on how to run the software.

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TOC Graphic


Prediction of Oscillator Strength and Transition Dipole Moments with

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