Singlet Fission in Perylenediimide Dimers - The Journal of Physical

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C: Energy Conversion and Storage; Energy and Charge Transport

Singlet Fission in Perylenediimide Dimers Marwa H. Farag, and Anna I. Krylov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05309 • Publication Date (Web): 18 Oct 2018 Downloaded from http://pubs.acs.org on October 18, 2018

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The Journal of Physical Chemistry

Singlet Fission in Perylenediimide Dimers Marwa H. Farag and Anna I. Krylov∗ Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, United States E-mail: [email protected]

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Abstract Singlet fission is a process in which one singlet exciton is converted to two triplets. By using transient absorption and time-resolved emission spectroscopy, recent experimental study (J. Am. Chem. Soc. 2017, 140, 814) investigated how different crystal packing of perylenediimide (PDI) molecules modulates their singlet fission rates and yields. It was observed that the rates of the PDIs vary between 0.33 and 4.3 ns−1 . By employing a simple three-state kinetic model and restricted active-space configuration interaction method with double spin-flip, we study the electronic factors (excitation energies and coupling between relevant states) responsible for the variation of singlet fission rates in these PDI derivatives. Our approach reproduces the trends in singlet fission rates and provides explanations for the experimental findings. Our analysis reveals that the electronic energies and the coupling play significant roles in controlling the speed of the singlet fission rates. The wave-function analysis of the adiabatic electronic states shows that in many model PDI structures the multi-exciton character is spread over several states, in contrast to previously studied systems. This different nature of the multi-exciton state poses interesting mechanistic questions. By mapping the relation between the stacking geometries of PDIs and the rates of the singlet multiexciton formation and the binding energies, we suggest favorable PDI structures that should not lead to exciton trapping.

Introduction Widespread applications of photovoltaic devices are hindered by lower power conversion efficiency. 1 The efficiency can be enhanced by incorporating materials capable of multi-exciton generation, such as singlet fission. 2,3 Singlet fission occurs in organic molecules in which the lowest singlet excitation energy is approximately twice the lowest triplet excitation energy. 2,4,5 Singlet fission has been documented in tetracene, 6,7 pentacene, 8,9 perylene-3,4:9,10bis-dicarboximide (PDI), 10–12,12,13 1,3-diphenylisobenzofuran (DPIBF), 14 their derivatives, 11,15–17

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and other conjugated organic molecules. 2,4,5 In organic photovoltaic materials, the absorption of light generates a singlet excitonic state (S1 ). In singlet fission materials, S1 can be converted to two triplet states with an overall singlet character. Such a correlated triplet pair is called singlet multi-exciton state (1 ME). 18–20 The two triplet states of the 1 ME can separate into two uncoupled triplets via electron decoherence and Dexter energy transfer 9,21–23 to nearby molecules. The singlet fission process is illustrated in Figure 1.

S0

S1

1

ME

T1 + T 1

Figure 1: Schematic representation of the singlet fission process in a molecular crystal (each circle represents a molecule). S0 denotes the ground state (GS, grey circles) and S1 (red circle) denotes the initially excited singlet state. 1 ME (purple) and T1 + T1 (green) refer to the multi-exciton state and the two uncoupled triplet-exciton states, respectively.

Singlet fission materials produce two charge carriers per absorbed photon, which allows to overcome the Shockley-Queisser efficiency limit for single-junction devices. 1,24,25 For example, an organic solar cell based on pentacene as a singlet fission material can produce external quantum efficiency of (109±1) %. 26 However, this cell operates under conditions that are not suitable for commercial applications. The main obstacle in the advancement of solar cells based on singlet fission materials is our limited understanding of factors controlling singlet fission rates and yields. Because singlet fission involves energy transfer between neighboring molecules, the singlet fission rates and yields depend not only on the properties of the individual chromophores but also on their relative arrangement in the molecular solid. It is unclear how different crystal packing affects the energetic driving force and the coupling between the states. In singlet fission materials with optimal conversion efficiency, the energy of S1 should be 3



approximately equal to the sum of the two triplets 2E(T1 ), i.e., the reaction should be nearly isoergic. Singlet fission in strongly exothermic systems such as in hexacene 27 was found to be slower relative to less exothermic systems, 16,28 by virtue of energy gap law (akin to the inverse region in the Marcus theory 29 ). Even more importantly, large exothermicity leads to energy losses to vibrations, which is detrimental to the overall device efficiency. On the other hand, singlet fission in slightly endothermic systems can be very efficient. 2,4 For example, singlet fission in tetracene is endothermic by about 200 meV, 30 yet, singlet fission at room temperature is sufficiently fast, so it can outcompete other relaxation pathways. 30–33 The conversion of S1 to 1 ME has been extensively studied both experimentally, 28,34–42 using time-resolved spectroscopy, and theoretically. 6,43–57 A recent review, 5 which describes advances in theoretical modeling of singlet fission, provides a concise description of the contemporary mechanistic picture of this process. A minimal model of singlet fission comprises two interacting chromophores whose electronic states can be approximately classified as locally excited, charge-resonance or charge-transfer, and multi-excitonic. The transition between the states of predominantly locally excited (S1 ) and multi-excitonic (1 ME) character is facilitated by the contributions from the charge-resonance configurations present in the respective adiabatic wave-functions; this is sometimes described as charge-transfer mediated singlet fission. 41 One can also imagine a mechanism involving transitions via a real physical intermediate state of charge-transfer character, 2 however, no unambiguous examples of such mechanism have been reported so far. The involvement of charge-resonance configurations manifests itself by the strong dependence of the singlet-fission kinetics on solvent’s dielectric constant 58 and on electron-donating ability of side groups. 40 The effect of the relative arrangements of the chromophores on the singlet fission yields and rates has been extensively investigated. 16,45,58,59 By varying the slip-stack displacements of the chromophores, the singlet fission rate and, consequently, yield can be either enhanced or suppressed. One particularly fascinating illustration of this phenomenon is provided by the studies of polymorphs which reported vastly different rates of singlet fission in different

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crystal forms of the same molecular solid. 60–63 Although the connection between the molecular solid morphology and efficiency of singlet fission is indisputable, it is not entirely clear how to search for optimal local arrangements of the chromophores because different packing affects multiple factors contributing to the overall rate. Experimentally, the control of the morphology is also far from trivial. One practical approach is to use covalently linked dimers, in which the relative orientation of the two chromophore moieties can be controlled by the rational design of the linker. 7,38,42,48,64–71 Alternatively, packing can be tweaked by augmenting the chromophores by strategically chosen bulky substituents. One class of chromophores where packing can be easily controlled by substituents is PDIs, which form ordered π-stacked structures. The crystal structures and their optical properties can be modulated by adding different functional groups at the imide positions. 72–78 In addition to this tunability, PDIs possess several other attractive features. PDI chromophore is thermally and photochemically stable 79 and strongly absorbs visible light. 73 The excitation energy of the lowest triplet state in PDI is approximately half the excitation energy of the lowest singlet state. 80 Thus, PDI is a promising organic chromophore for singlet fission photovoltaic cells. The effect of packing on singlet fission rates has been systematically investigated in a recent experimental study. 11 In this work, transient absorption and time-resolved emission spectroscopy were used to compare the singlet fission rates and yields in six different PDI derivatives. The singlet fission rates varied between 0.33 and 4.3 ns−1 and the triplet yields varied between 80%-178%. This experiment was in part motivated by an earlier theoretical study, which predicted the singlet fission rates for various PDI arrangements computed using Redfield theory parameterized by DFT calculations. 59 The trends in experimentally measured rates 11 agreed reasonably well with the theoretical predictions, 59 despite discrepancies in the absolute rates. In this paper, we investigate the effect of different intermolecular PDI structures on the energetic driving force and on the electronic coupling between the relevant states and provide

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an explanation of the experimental trends. 11 We employ the restricted active-space configuration interaction method with double spin-flip (RAS-2SF) to calculate the adiabatic excitation energies for different PDI dimers and the coupling between the adiabatic electronic states. Using adiabatic framework is a distinguishing feature of our protocol, which does not assume that the relevant electronic states have simple diabatic character and allow for multiple electronic configurations to couple and interact, as dictated by the many-body Hamiltonian. As explained below, this feature is particularly important in the case of PDIs, owing to an unusual electronic structure of their multi-excitonic states. We use a simple three-state kinetic model based on electronic energies and coupling to compute the relative rates and compare the results with the experimental trends. 11 By using wave-function analysis tools, we investigate the character of the excited states. This analysis reveals an interesting feature that distinguishes PDIs from other singlet fission systems: in many PDIs the multi-exciton character is spread over several electronic states, which might have significant mechanistic consequences. We note that a very recent experimental study 81 of terrylene-based dimers has reported spectroscopic evidence of the multi-exciton state containing large contributions of the charge-resonance configurations. The paper is organized as follows. First, we present the theoretical framework and explain computational details of the protocol employed. We then discuss the energies and the character of the relevant excited states. This is followed by the discussion of the computed rates and comparison with the experimentally measured rates. Finally, we map the relation between singlet fission rates and the PDI stacking geometries and predict the PDI geometries with optimal singlet fission rates and yields.

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Theoretical framework Adiabatic wave-functions of two coupled chromophores can be described in terms of the locally excited (LE), charge resonance (CR), and multi-exciton (1 TT) configurations 82

|Ψi = cLE |ΨLE i + cCR |ΨCR i + c

Here, cLE , cCR , and c

1 TT

1 TT

|Ψ1 TT i.

(1)

are the collective amplitudes of the LE, CR, and 1 TT configurations,

which give rise to the collective weights ω LE , ω CR , and ω

1 = |cLE |2 + |cCR |2 + |c

1 TT

1 TT

:

|2 = ω LE + ω CR + ω

1 TT

.

(2)

Figure 2 shows these configurations expressed in terms of localized molecular orbitals of a dimer. By calculating ω LE , ω CR , and ω

1 TT

, one can assign a character to an adiabatic

S0 (AB) A

B

S1 (AB)

+

+ LE

1

LE

1

TT

CR

CR +

+

+

ME (AB)

1

TT

+ …

+

CR

+ … CR

Figure 2: Electronic configurations for S0 (ground state), S1 (lowest singly excited singlet state), and 1 ME (singlet multi-exciton state) in AB dimer expressed in terms of localized molecular orbitals.

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wave-function. For example, the S1 state is identified by large ω LE , while the 1 ME state is identified by large ω

1 TT

. The S1 and 1 ME states can also contain the CR configurations,

which lend ionic character to the wave-function. In the context of singlet fission, mixing of CR configurations in the S1 and 1 ME states plays an important role by facilitating electronic couplings between these states. Following previous work from our group, 82 we use the chargeand the spin-cumulant analysis to quantify the weights of different contributions in the total wave-function. We calculate singlet fission rates by using adiabatic wave-functions 47 and the three-state kinetic model summarized in Figure 3, as was done in Refs. 48,83,84. The kinetic model describes singlet fission as a two-step process: the first step is a non-

1

ME

r1 ME Esf 1

S1S0 S0S0

5

r2

ME ≈ 2 T1 Eb

hν

Figure 3: Energy diagram for the three-state kinetic model of singlet fission. r1 is the rate for the formation of the 1 ME state (first step) and r2 is the rate for the production of the independent triplets (second step).

adiabatic transition from S1 to 1 ME and the second step is decoupling and separation of the triplets. The respective rates are estimated 48 using Fermi’s Golden Rule and the linear free energy approach (which assumes that the rate of a process is proportional to the free energy difference between the initial and final states):

r1 ∼ (NAC)2 e−αβEsf

(3)

r2 ∼ e−αβEb

(4)

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where r1 denotes the rate of the first step, the formation of the 1 ME state. The rate depends on energy drive Esf , Esf = E(1 ME) − E(S1 ),

(5)

and the non-adiabatic coupling (NAC) between the S1 and the 1 ME states. The norm of the one-particle transition density matrix ||γ|| can be used as a proxy for NAC, 42,47–49 because ||γ|| captures the changes in the adiabatic wave-function, such as variation of CR characters responsible for the derivative coupling. 85 α is a parameter in the free energy relationship (we use α = 0.5 as in our previous work 48,84 ) and β = 1/kT where k is the Boltzmann constant and T is the temperature (we use T =300 K). r2 is the rate of the second step, the production of the independent triplets. The rate of the second step depends on several factors including the multi-exciton binding energy, Eb . Eb is the energy penalty for separating two triplets which requires unmixing the CR contributions from the adiabatic wave-function of the singlet ME state. Eb can be estimated as: Eb = E(5 ME) − E(1 ME),

(6)

where 5 ME is the quintet multi-exciton state, which, in contrast to 1 ME, always has pure diabatic TT character. We also calculate the Estt ,

Estt = E(S1 ) − 2E(T1 )

(7)

where E(S1 ) is the lowest singlet exciton energy of the dimer and E(T1 ) is the triplet excitation energy of the monomer. Estt is an asymptotic value of the overall energy change in the SF process (positive values correspond to exoergic singlet fission). As in previous work, we do not attempt to compute absolute rates, rather, we use Eqs. (3) and (4) to compute relative rates in homologically similar compounds. We stress that the purpose of this simple model is not to achieve a quantitative description of singlet fission

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kinetics, but rather to provide quick estimates of the magnitude of the effect of variations in underlying electronic structure factors on the rate, in the same fashion as in applications of linear free energy approach to rationalize and predict trends in a large variety of processes in organic chemistry 86,87 (in depth discussion of the assumptions behind linear free energy relationships can be found in Ref. 88). Linear free energy approach works the best when applied to series of sufficiently similar compounds/processes. Since different PDIs have very similar structural and physical properties, this is a good case for applying a linear free energy relationship. An important limitation of our model is that it does not account for coupling of electronic dynamics with vibrations, which plays an important role in the singlet fission processes. 50,52,89 Consequently, the model does not predict slow-down of the rate in strongly exothermic situations. This drawback can be remedied by amending Eq. (3) to include reorganization energy, as in Marcus rate expression, 29 at a price of introducing an additional parameter. This type of a kinetic model has been used by Van Voorhis and co-workers. 16 Of course, to elucidate full mechanistic details of singlet fission, i.e., to determine whether the process proceeds through conical intersections and to pinpoint the precise role of intra- or inter-molecular vibrations, much more sophisticated theoretical framework is required 52,89,90 and we hope that our findings will stimulate such studies in the future.

Computational details Figure 4 shows the structure of the PDI monomer. We constructed model dimer structures from the optimized monomer structures by arranging them as in the respective crystal structures. In our model structures, we used R=H and K=H, as the substituents do not affect electronic properties of the PDI chromophores. 11 Below we refer to this structure as PDI(p); p for planar. The structure of the planar PDI monomer was optimized by using RI-MP2/cc-pVDZ. The planar PDI has D2h symmetry. The effect of slightly non-planar structures adopted by PDIs with bulky substituents (as in WD 10 ) is discussed in the SI.

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Figure 4: The structure of perylenediimide (PDI). R and K denote substituents at the imide and the core positions.

Because all electronic properties of the planar and non-planar model structures are very similar, we used planar model structures in all calculations. We calculated singlet and triplet vertical excitation energies of the optimized monomers using ωB97X-D 91 and RAS-SF. 92,93 In the latter, the active space consists of 2-electronsin-2-orbitals and the reference restricted open-shell determinant is a high-spin triplet state (Ms = 1). We constructed model PDI dimers from the optimized monomer structures by stacking two identical monomers at the arrangement shown in Figure 5. All model dimers were constructed from the planar PDI monomer. For the WD dimer, we also considered a structure built from non-planar dimers, WD(np); the results for WD(p) and WD(np) are discussed in the SI. We computed the excitation energies of the dimers using RAS-2SF with 4-electrons-in-4orbitals with a high-spin quintet reference (Ms = 2). 94 We analyzed the resulting electronic wave-functions in terms of weights of the LE, CR, and 1 TT configurations (Eq. (1)), which allowed us to identify the S1 and 1 ME states. We estimated the coupling between the S1 and 1

ME states by computing ||γ||2 . We used the cc-pVDZ basis set in all calculations and kept

the core electrons frozen. To examine the basis set effects, we performed additional calculations with the cc-pVTZ basis; the results are presented in the SI. All electronic structure

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calculations were carried out using the Q-Chem electronic structure program. 95,96

a)

b)

Figure 5: (a) The structure of a PDI dimer showing the displacement along the long (x) and the short (y) axes. (b) Slip-stack displacement dx and dy for PDI dimers from Ref. 11. The plane-to-plane distance (dz) are 3.40 (C1), 3.46 (MO), 3.28 (C8), 3.48 (EP), 3.41 (C3-II), 3.50 (C7), 3.40 (C3-I), and 3.68 (WD) Å.

For each model PDI dimer shown in Figure 5 (b), we computed the key electronic energies, Eqs. (5) and (6). We then computed the relative rates for the formation of the 1 ME state (Eq. (3)) and the production of the triplets (Eq. (4)) using the three-state kinetic model. Here, all rates were calculated relative to the WD dimer. We compared the rates relative to the planar and non-planar PDI and found that they are not significantly different. Thus, we only present the rates relative to the planar WD dimer. The rates relative to the non-planar WD dimer are given in the SI.

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Correction of the RAS-2SF energies Although RAS-2SF provides a balanced description of the electronic configurations shown in Fig. 2, the excitation energies are overestimated due to an incomplete account of dynamic correlation. To account for the missing dynamic correlation, we followed the same procedure as in Ref. 49 and corrected the RAS-SF energies as follows. The energy correction scheme is based on wave-function decomposition of the adiabatic states Ψ in terms of LE, TT, and CR configurations (an additional small term, denoted by ω SS , contains the contributions due to the simultaneous excitations of both chromophores that are not of the ME type 49 ). The corrected energies are computed as

E[Ψ] = E0 [Ψ] + (ω LE + 2ω SS ) Ec (LE) + ω

1 TT

Ec (1 TT) + ω CR Ec (CR),

(8)

where E0 is the uncorrected energy and Ψ is the adiabatic wave-function of the dimer, i.e., S1 or ME. Ec (LE), Ec (1 TT), and Ec (CR) are the energy corrections to the LE, CR, and 1

TT contributions, respectively. These corrections are computed by adjusting the RAS-SF

energies of the respective diabatic configurations to match more accurate reference values. Here, we use ωB97X-D/cc-pVDZ energies as a reference, as was done in Ref. 49. Recent studies of PDIs by Engels and coworkers 97,98 have also employed ωB97X-D as the best available method. As discussed below, although the resulting excitation energies are not in a perfect agreement with the experimental values, the discrepancy is much smaller than for RAS-SF. We note that a recent benchmark study focusing on the excited-state wavefunctions 99 have shown that among various popular functionals, the range separated density function theory hybrid functionals ωB97X-D and CAM-B3LYP yields the best description of the exciton properties, as compared to the high-level correlated wave-function methods. 100 For the PDI monomer and dimers, the the excitation energies computed with ωB97X-D and CAM-B3LYP are close, within less than 0.1 eV from each other. Here we employ the ωB97XD energies as a reference for correcting RAS-2SF excitation energies. Below we present the

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results for the corrected energies; the results for the uncorrected energies are given in the SI. Table 1 reports the correction associated with the LE, 1 TT, and CR contributions for each PDI dimer. The magnitude of the correction for the LE contribution is larger than for the 1 TT contribution. This is expected because the dynamical correlation affects singlet states more than triplet states. We also note that the magnitude of Ec (LE) for the EP dimer is larger than those obtained for other PDI dimers. This is due to the significant CR contribution in the S1 state in the EP dimer (see Table 3 in the results and discussion section). Table 1: The energy correction (in eV) for the LE, 1 TT, and CR contributions. Dimer C1 C3-I C3-II C7 C8 WD(p) MO EP

Ec (LE) Ec (1 TT) Ec (CR) -0.5854 -0.2755 -0.0535 -0.7262 -0.2672 -0.1653 -0.6211 -0.2595 0.1281 -0.7083 -0.2658 0.0068 -0.7032 -0.2749 -0.1249 -0.7554 -0.2590 -0.0988 -0.7991 -0.5777 -0.1100 -1.1382 -0.2628 -0.2179

Results and discussion Excited-state analysis of the PDI monomers The experimental linear absorption spectra of different PDI derivatives in solution are very similar, 10,11 meaning that the functional groups at the imide and core positions have a relatively small effect on the electronic states of the PDI chromophore. Thus, in our calculations we used model structures in which the substituents were replaced by hydrogens. Table 2 shows singlet and triplet excitation energies of the PDI monomers. Within the experimental resolution, E(S1 ) of the PDI p and np derivatives is indistinguishable, 10,11 suggesting that the slight distortion of the PDI core does not significantly perturb its electronic structure. 14


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The triplet excitation energy of the planar PDI is red-shifted by 0.09 eV relative to the nonplanar PDI. The absolute values of the ωB97X-D excitation energies are red-shifted by ∼0.5 eV relative to the RAS-SF values. The calculated E(S1 ) for the planar and the non-planar geometries are similar (Table 2), in agreement with the experiment. The calculations do not reproduce a small red shift of the triplet state in the non-planar structure, but the magnitude of the shift is very small and could be due different experimental conditions. 101 Table 2: The S1 and T1 vertical excitation energies (eV) in the PDI monomer. State Exp. (p)a S1 2.34 T1 1.19 Estt -0.04

Exp. (np)b 2.34 1.28 -0.22

ωB97X-Dc (p) ωB97X-Dc (np) RAS-SFc (p) RAS-SFc (np) 2.903 2.865 3.524 3.470 1.553 1.510 1.885 1.803 -0.203 -0.155 -0.246 -0.136 a Refs. 80 and 11 b Ref. 10 c cc-pVDZ basis.

As expected, the RAS-SF excitation energies are strongly blue-shifted relative to the experimental values. The errors in the ωB97X-D/cc-pVDZ energies are smaller: 0.56 and 0.36 eV for S1 and T1 , respectively. This functional has been also used by Engels and coworkers. 97,98 Additional calculations with CAM-B3LYP yielded excitation energies that are very close to the ωB97X-D values; the corresponding Estt is -0.118 eV, which is close to 0.203 eV obtained by ωB97X-D. In contrast to popular functionals that might yield excitation energies in better agreement with experimental values, 102 ωB97X-D was shown to better reproduce the character of excited-state wave functions. 99 These reference values determine the accuracy of the absolute excitation energies of the dimers. While the magnitude of the error is large, we expect similar magnitude of errors in model dimers, such that the variations of excitation energies due to different chromophore packing can be reproduced due to error cancellation. Using the monomer values, the estimated errors in the gap between the S1 and 1

ME states is 0.56-2×0.36=-0.16 eV; the 1 ME state would be computed at higher energies

relative to S1 .

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Excited-state analysis of the PDI dimers Table 3: The S1 , 1 ME, and 5 ME vertical excitation energiesa (eV) and the wave-functions composition of the S1 and 1 ME states of the PDI dimers. Dimer State C1 S1 1 ME 5 ME C3-I S1 1 ME 1 ME0 5 ME C3-II S1 1 ME 5 ME C7 S1 1 ME 1 ME0 5 ME C8 S1 1 ME 1 ME0 5 ME WD S1 1 ME 1 ME0 5 ME MO S1 1 ME 1 ME0 5 ME EP S1 1 ME 1 ME0 5 ME a

a Eex ω LE ω CR 2.6054 72.54 23.15 3.5755 2.33 15.54 3.4911 2.6813 54.43 34.40 2.9704 39.74 36.98 3.6298 7.22 30.43 3.5045 2.6567 80.49 15.37 3.5211 1.41 15.88 3.4577 2.6242 67.52 27.87 3.2482 21.83 40.68 3.5904 6.44 32.06 3.4893 2.5461 55.75 38.23 3.1569 34.46 34.14 3.5896 6.40 29.45 3.4963 2.6932 58.26 30.88 3.1541 31.48 24.62 3.7443 8.94 44.58 3.4910 2.5947 56.34 32.95 2.8495 36.80 32.87 3.3886 5.16 41.06 3.4771 2.6717 34.37 62.79 2.8752 16.02 56.16 3.5446 14.52 35.09 3.5110 Corrected energy.

1

ω TT 1.39 82.02 8.46 21.11 62.04 0.29 82.63 1.19 36.25 61.14 3.33 29.48 63.82 7.66 42.02 46.00 8.24 28.34 53.52 0.13 26.83 49.84

Table 3 lists the excitation energies and the properties of the S1 , 1 ME, and 5 ME states for the model PDI dimers. It also lists the weights of the LE, CR, and 1 TT configurations in the RAS-SF wave-functions (Figure 2). The adiabatic wave-functions of the singlet states of 16


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the model PDI dimers show considerable configuration mixing. The S1 state has significant contributions from the LE and CR configurations, which is common for Frenkel excitonic states 56,97,103–105 and agrees with findings of Engels and co-workers. 97,106 We also note a small contribution from the 1 TT configuration in the S1 state in some dimers. In striking contrast to other systems studied by our group (various acenes, 1,3-diphenylisobenzofuran, 1,6diphenyl-1,3,5-hexatriene), 42,47,49,84 in most model PDI structures, the multi-exciton character is spread over several singlet adiabatic states. For example, in C3-I, C7, C8, MO, EP, and WD dimers (Table 3), the largest 1 TT contribution varies between 46-64%. This is different from, for example, model tetracene dimers in which the 1 TT character of the 1 ME state is always large (ω

1 TT

≥ 80%). 47,48 This is an interesting aspect of the electronic structure of

PDI dimers, which likely affects the mechanism of singlet fission in these systems. As one can see from Table 3, the states that we labeled as the 1 ME states contain contributions from all singlet configurations, including a large but not always dominant 1 TT contribution. In the C3-I, C7, C8, MO, and EP dimers, there are two adiabatic states that have a large contribution from the 1 TT configuration (varying between 21 and 64%). These states are denoted by the 1 ME and 1 ME0 labels. The wave-function analysis shows that the lower 1 ME state has comparable weights of the LE, CR, and 1 TT configurations, with the weight of the 1 TT configuration being less than 40%. The upper 1 ME state (denoted 1 ME0 ) shows a larger contribution from the 1 TT configuration, more than 50%. In the WD dimer, there are two adiabatic 1 ME states with comparable weights of the 1 TT configuration. In the C1 and C3-II dimers, however, the 1 ME state has a dominant contribution from the 1 TT character (ω

1 TT

≈80%). In contrast to the 1 ME state, 5 ME preserves pure diabatic 5 TT character;

this is similar to other singlet-fission systems. 47,49 To test the robustness of this finding, we performed additional calculations using an entirely different approach, ab initio FrenkelDavydov exciton model (AIFDEM) 107 for the MO dimer, following the same protocol as used in Ref. 52. We found that although the exact weights of the configurations are slightly different, the AIFDEM calculations confirm strongly mixed character of the multi-exciton

17



Page 18 of 39

states. We also extended the set of model systems to the trimers of PDI chromophores. The RAS-3SF calculations of all model trimers show that the characters of the relevant states remain qualitatively similar to the dimers (Table S7 in the SI). Thus, although the precise state composition of the electronic states in bulk crystal needs to be elucidated by more sophisticated calculations, the mixed nature of the multi-excitonic states appears to persist at different levels of theory and in larger systems. With an exception of a very recent experimental study of terrylene-based dimers, 81 such strongly mixed singlet multi-exciton states have not been reported before. We note that this type of electronic structure highlights the benefit of the adiabatic framework and poses new questions regarding the mechanism of triplet production from 1 ME. Table 4 lists relevant electronic energies (Esf and Eb ) and coupling ||γ||2 between the multi-exciton and S1 states. These quantities depend strongly on the relative orientations of the individual chromophores and are likely to be responsible for the observed differences in the singlet fission rates of various PDIs. Table 4: dimersb .

Relevant electronic energiesa (eV) and coupling for different PDI

Dimer Esf C1 0.9701 C3-I 0.2891 0.9485 C3-II 0.8644 C7 0.6240 0.9662 C8 0.6108 1.0435 WD 0.4609 1.0511 MO 0.2548 0.7939 EP 0.2035 0.8729 a

Eb -0.0844 0.5341 -0.1253 -0.0634 0.2411 -0.1011 0.3394 -0.0933 0.3369 -0.2533 0.6276 0.0885 0.6358 -0.0336

||γ||2 , S1 -1 ME 0.10 0.07 0.12 -0.4493 0.09 -0.4818 0.11 0.11 -0.5599 0.08 0.11 -0.4128 0.10 0.10 -0.5113 0.09 0.12 -0.4343 0.30 0.29 Estt -0.5006 -0.4247

||γ||2 , S0 -1 ME 0.06 0.30 0.13 0.07 0.25 0.16 0.27 0.14 0.22 0.22 0.27 0.18 0.29 0.15

||γ||2 , 1 ME-1 ME0 0.19 0.26 0.27 0.27 0.24 0.15

Corrected energies. b When two sets of values are given, the upper and lower rows correspond to 1 ME and 1 ME0 , respectively.

Esf , the energy gap between 1 ME and the S1 , is endoergic in all dimers. Likewise, the computed Estt is also endoergic. We note that our protocol, which is based on ωB97X-D 18


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energy correction, probably overestimates the extent of the endothermicity by as much as 0.16 eV. For the lower 1 ME state, the thermodynamic drive for the formation of the 1 ME state is less endothermic. This state has positive multi-exciton binding energy Eb (i.e., the energy of this state is lower than the 5 ME state). The energy of the upper 1 ME state is above 5 ME, which means that the triplet separation step for this state is exothermic. We estimated the coupling between the S1 and 1 ME states by using the norm of the oneparticle transition density matrix. The values of the coupling are similar (∼0.1) in all dimers except the EP dimer in which the coupling is ∼0.3. This is due to the large contribution of the CR configuration in the S1 and 1 ME/1 ME0 states (Table 3). Figure 6 shows the computed rates of the multi-exciton formation and compares the results with the experimental rates. 11 The experimental rates were determined from the transient absorption spectroscopy and time-resolved photoluminescence and it is not clear whether one can distinguish between the rates of formation of the 1 ME state or the production of the two independent T1 states. All rates are computed relative to the reference system, the planar WD dimer. Figures 6 (a) and (b) show the computed rate for the formation of the lowest state with more than 20% multi-exciton character and for the state with the largest multi-exciton character (this is an upper state, 1 ME0 , in all dimers except C1 and C3-II). In the discussion below we refer to these two cases as “lower ME” and “upper ME” states. In the reference WD dimer, there are two 1 ME states with a similar contribution from the 1 TT character (Table 3). For consistency, in Figure 6 (a), the rates are calculated relative to the lower 1 ME state in the WD dimer and in Figure 6 (b) the rates are calculated relative to the upper 1 ME state (the choice of the 1 ME state (lower or upper) for the reference system does not affect the trend in the rates, but changes the relative magnitude of the rates). The fitting of the data points in Figure 6 to a linear equation shows that the rate of the higher 1 ME state formation agrees better with the experiment than the rate of the lower 1 ME state formation. Calculations using uncorrected energies (shown in SI) lead to similar conclusion (better correlation for the upper ME state), but the energy correction

19



C7

(a) Experimental relative rate (log(r1/r01))

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

Page 20 of 39

C8 C3-I EP

C1

MO

(b)

Figure 6: Relative rates of singlet fission: theory versus experiment. (a) The log of the relative rate is calculated for the lowest state with significant (more than 20%) multi-exciton character (1 ME state). (b) The log of the relative rate is computed for the state with the largest multi-exciton character (this is the upper ME state, 1 ME0 , in all dimers except C1 and C3-II). Relative rates are computed relative to the planar WD rate (a) relative to the 1 ME state and (b) relative to the 1 ME0 state. Solid black lines show a linear fit for all data points (R2 =0.32 for the 1 ME state and 0.77 for the 1 ME0 state). The standard deviation for each data point from the fitted line is shown in the SI. The experimental rates are from Ref. 11.

clearly improves the agreement with the experimental rates, i.e., the R2 value for the rates computed using uncorrected energies is 0.49, whereas the correction brings it up to 0.77. However, the calculations overestimate the magnitude of the differences in rates for different dimers. To illustrate this point, consider the slowest and the fastest dimers from Figure 6, MO and C8. The experimental rate in C8 is ∼ 1.1 orders of magnitude slower than in MO, whereas in our calculations the C8 rate is slower by ∼ 2.4 orders of magnitude than the MO

20


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rate. Previous theoretical work, which used entirely different protocols based on DFT-parameterized Redfield theory, investigated the singlet fission rates for different stacking PDI arrangements. 59 Their calculated rates correlate with the experimental rates slightly better (R2 is 0.87), but the absolute values of the computed rates are three orders of magnitude faster than the experimental ones. 11 In this study, the calculated rate in C8 was slower by ∼ 1.2 orders of magnitude than in MO, which agrees well with the experiment. The analysis of the data in Table 4 allows us to identify the key factors controlling the singlet fission rates in these dimers. Table 4 shows that Esf is 0.79 and 1.04 eV in the MO and C8 dimers, respectively and that the coupling is approximately 0.1 for both dimers. Thus, we attribute faster singlet fission rate in the MO dimer to more favorable Esf . The experimental study 11 reported that the 1 ME state produced in thin films relaxes to the ground state within 28 ns in EP and 10 ns in the MO, C3-I, C7, and C8, and that the overall yield of the free triplets was small. To fully understand the singlet fission process, we need to consider the triplet separation step. While rigorous theoretical modeling of this process would require much more advanced approaches 52,89,90 that are beyond the scope of this work, here we discuss various factors controlling competing relaxation pathways of the multi-exciton state and possible mechanistic implication of its strongly mixed character. We begin by analyzing the couplings between the lower and the upper multi-exciton states, the coupling between the ground state and the two singlet multi-exciton states, and the multi-exciton binding energy. The data in Table 4 shows that the coupling between the lower and the upper 1 ME states is large. Moreover, the coupling between the lower 1 ME state and the ground state is larger than the coupling between the lower 1 ME state and S1 . The multi-exciton binding energy for the upper 1 ME states varies between -101 meV (unbound biexciton) and 88 meV, while for the lower 1 ME state, it varies between 636 meV and 241 meV (the biexciton is bound). There are at least two competing channels for the decay of the initially produced multi-exciton state: a separation into two uncorrelated

21



triplets and radiationless relaxation to the ground state. The structures where an upper 1 ME state is produced in the first step, the production of triplets is likely to be affected by the radiationless relaxation into the lowest 1 ME state, which results in exciton trapping because of large multi-exciton binding energy, and consequent relaxation to the ground state. Large values of ||γ|| for the 1 ME-1 ME0 transition suggest that this channel might be completive with triplet separation. If this is the case, one possible route to improving efficiency of singlet fission in PDIs is to find arrangements at which the lowest 1 ME state is nearly degenerate with 5 ME.

(a) 1TT

ω

(b)

(c)

(d)

(e)

||γ||2

Esf

rate

Eb

Short axis, dy (Å)

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

Page 22 of 39

Long axis, dx (Å) Figure 7: Mapping the relation between the stacking geometries of the PDI dimers and (a) the weight of the 1 TT configuration in the 1 ME state (b) the coupling ||γ||2 between S1 and 1 ME (c) Esf , eV, (d) the log of the relative singlet fission rate (relative to the lower 1 ME in the WD dimer) (e) the multi-exciton binding energy, Eb , eV. Lower panel corresponds to the lowest-energy state with significant multi-exciton character (ω TT > 20), while the upper panel refers to the state with the largest 1 ME character (ω TT > 40). See text for explanation.

To find the PDI geometries that maximize the rate of 1 ME formation while not leading to exciton trapping, we map the relation between the singlet fission rates for the 1 ME formation and binding energies and the stacking arrangements of the PDI dimers along the long axis (x) and the short axis (y). We begin by quantifying the weight of the multi-exciton 1 TT character in the 1 ME state. The weight of the 1 TT configuration for the lowest 1 ME state

22


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(identified as the lowest state with noticeable ω TT ) is shown in the lower panel of Figure 7 (a). As one can see, in this range of displacements, ω TT varies from 98% to 28%. The red domains correspond to the structures in which the lowest state has large ME character (>80%). The blue domains correspond to the structures in which the lowest state has small ω TT (less than 40%); these are the structures where the “real” multi-exciton state is the upper one (ω TT > 40). Green domains correspond to the structures in which multi-exciton character is split almost equally between the two (or more) states. Thus, one can expect that in the structures from the blue and green domains singlet fission proceeds via an upper state and that the triplet separation competes with relaxation to the lower ME state. The upper panel of Figure 7 (a) shows the weight of the 1 TT configuration for the upper 1 ME state for the blue and green domains. As one can see, the blue domain for the upper 1 ME state has 1 TT character more than 40%. Figure 7 (d) shows the rate of the multi-exciton formation. The rate is fast when Esf (Figure 7 (c)) is less endothermic and the coupling (Figure 7 (b)) is large. The red domain in Figure 7 (d) corresponds to the PDI structures for which the singlet fission rate is fast. As one can see, the red domain region for the lower and upper 1 ME state is different. Moreover, we find that in the regions where Esf is more endothermic and the coupling is large, the rates are slow. Therefore, to optimize the singlet fission rate for the formation of 1 ME, both electronic energy and coupling should be favorable. This conclusion differs from that of a previous study 59 of PDIs, which proposed that in order to optimize the singlet fission rate, one needs to find the geometry that optimizes the coupling between the two chromophores. Figure 7 (e) shows Eb which provides information about the energy needed for the separation of the singlet multi-exciton into two triplets. For the lower 1 ME, the comparison of the rates with Eb suggests that despite favorable energetics and couplings of structures in red domain in Figure 7 (d), the efficiency of triplet production is impeded by the exciton trapping (see Figure 7 (e)). For the upper 1 ME state, however, the region where the formation of the multi-exciton is fast is the same region where Eb is favorable even if the

23



system decays to the lower 1 ME state. On the basis of these maps, we conclude that singlet fission would be more favorable in the PDI structures where both dx and dy are ≥ 2.5 Å. We note that varying the intermolecular crossing angle, which was zero in the crystal structures studied in this paper and in Ref. 12, can provide an additional degree of freedom for tuning the couplings and energetics. Of course, higher level of theory that includes intra- and inter-molecular vibrational motions and non-adiabatic transitions is needed for developing a quantitative picture of this step.

Conclusion By employing a simple three-state kinetic model, we investigated the singlet fission rates for different PDI geometries and provided explanation for the measured rates in PDI derivatives. Unlike many other singlet fission systems, the singlet multi-exciton state in PDIs does not always have dominant 1 TT character and contains large contributions from the LE and CR configurations. This leads to the splitting of the 1 ME state into the lower and upper multiexciton states with different 1 TT contribution. The calculated rates reproduce the trend of the measured rates better when using the 1 ME state with the largest 1 TT character, which often corresponds to the upper 1 ME state. When singlet fission proceeds via the upper 1 ME state, the triplet separation might compete with radiationless relaxation to the lowest 1 ME state for which the triplet separation is strongly endothermic and inefficient. This radiationless relaxation to the lower 1 ME state may result in the trapping of the multi-exciton and might be the reason for the experimentally observed decay of the produced multi-excitons to the ground state. By mapping the relation between the stacking geometry of the PDIs and the rate for the formation of the 1 ME, we posit that PDI geometries with displacement ≥ 2.5 Å along x and y coordinates are favorable. Experimental work to synthesize and investigate singlet fission in these PDI structures is needed to check this proposal. The reported findings warrant more detailed theoretical studies and an extension of the current mechanistic picture

24


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of singlet fission to incorporate the case of strongly mixed multi-exciton states.

Supporting Information Available Supporting information is available: the excitation energies of the PDI monomer obtained with cc-pVTZ; electronic energies and couplings for the the non-planar WD dimer; the rates relative to the non-planar WD dimer; raw energies; results for model trimers; rates computed from uncorrected energies; relevant Cartesian geometries.

Notes A.I.K. is a part owner and a board member of Q-Chem, Inc.

Acknowledgment This work is supported by the Department of Energy through the DE-FG02-05ER15685 grant. We are grateful to Professor Sean Robers and Dr. Jon Bender from UT Austin for stimulating discussions and for valuable feedback about the manuscript. We also would like to thank Professor Anatoly Kolomeisky from Rice University for helpful discussions.

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Singlet Fission in Perylenediimide Dimers - The Journal of Physical

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