Singlet Fission in Perylenediimide DimersMarwa H. Farag and Anna I. Krylov*

Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, United States

*S Supporting Information

ABSTRACT: Singlet fission is a process in which one singletexciton is converted to two triplets. By using transientabsorption and time-resolved emission spectroscopy, recentexperimental study (J. Am. Chem. Soc. 2017, 140, 814)investigated how different crystal packing of perylenediimide(PDI) molecules modulates their singlet fission rates andyields. It was observed that the rates vary between 0.33 and4.3 ns−1. By employing a simple three-state kinetic model andrestricted active-space configuration interaction method withdouble spin-flip, we study the electronic factors (excitationenergies and coupling between relevant states) responsible forthe variation of singlet fission rates in these PDI derivatives. Our approach reproduces the trends in singlet fission rates andprovides explanations for the experimental findings. Our analysis reveals that the electronic energies and the coupling playsignificant roles in controlling the singlet fission rates. The wave function analysis of the adiabatic electronic states shows that inmany model PDI structures, the multiexciton character is spread over several states, in contrast to previously studied systems.This different nature of the multiexciton state poses interesting mechanistic questions. By mapping the relation between thestacking geometries of PDIs and the rates of the singlet multiexciton formation and the binding energies, we suggest favorablePDI structures that should not lead to exciton trapping.

■ INTRODUCTION

Widespread applications of photovoltaic devices are hinderedby lower power conversion efficiency.1 The efficiency can beenhanced by incorporating materials capable of multiexcitongeneration, such as singlet fission.2,3 Singlet fission occurs inorganic molecules in which the lowest singlet excitation energyis approximately twice the lowest triplet excitation energy.2,4,5

Singlet fission has been documented in tetracene,6,7

pentacene ,8 , 9 pery lene-3 ,4 :9 ,10-b is -d icarbox imide(PDI),10−12,13 1,3-diphenylisobenzofuran (DPIBF),14 theirderivatives,11,15−17 and other conjugated organic molecules.2,4,5

In organic photovoltaic materials, the absorption of lightgenerates a singlet excitonic state (S1). In singlet fissionmaterials, S1 can be converted to two triplet states with anoverall singlet character. Such a correlated triplet pair is calleda singlet multiexciton state (1ME).18−20 The two triplet statesof the 1ME can separate into two uncoupled triplets viaelectron decoherence and Dexter energy transfer9,21−23 tonearby molecules. The singlet fission process is illustrated inFigure 1.Singlet fission materials produce two charge carriers per

absorbed photon, which allows for overcoming the Shockley−Queisser efficiency limit for single-junction devices.1,24,25 Forexample, an organic solar cell based on pentacene as a singletfission material can produce external quantum efficiency of(109 ± 1)%.26 However, this cell operates under conditionsthat are not suitable for commercial applications. The mainobstacle in the advancement of solar cells based on singletfission materials is our limited understanding of factors

controlling singlet fission rates and yields. Because singletfission involves energy transfer between neighboring mole-cules, the singlet fission rates and yields depend not only onthe properties of the individual chromophores but also on theirrelative arrangement in the molecular solid. It is unclear howdifferent crystal packing affects the energetic driving force andthe coupling between the states.

Received: June 3, 2018Revised: September 28, 2018Published: October 18, 2018

Figure 1. Schematic representation of the singlet fission process in amolecular crystal (each circle represents a molecule). S0 denotes theGS (gray circles) and S1 (red circle) denotes the initially excitedsinglet state. 1ME (purple) and T1 + T1 (green) refer to themultiexciton state and the two uncoupled triplet exciton states,respectively.

Article

pubs.acs.org/JPCCCite This: J. Phys. Chem. C 2018, 122, 25753−25763

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In singlet fission materials with optimal conversionefficiency, the energy of S1 should be approximately equal tothe sum of the two triplets 2E(T1), that is, the reaction shouldbe nearly isoergic. Singlet fission in strongly exothermicsystems such as in hexacene27 was found to be slower relativeto less exothermic systems,16,28 by virtue of energy gap law(akin to the inverse region in the Marcus theory29). Even moreimportantly, large exothermicity leads to energy losses tovibrations, which is detrimental to the overall device efficiency.On the other hand, singlet fission in slightly endothermicsystems can be very efficient.2,4 For example, singlet fission intetracene is endothermic by about 200 meV,30 yet singletfission at room temperature is sufficiently fast, so it canoutcompete other relaxation pathways.30−33

The conversion of S1 to1ME has been extensively studied

both experimentally,28,34−42 using time-resolved spectroscopy,and theoretically.6,43−57 A recent review,5 which describesadvances in theoretical modeling of singlet fission, provides aconcise description of the contemporary mechanistic picture ofthis process. A minimal model of singlet fission comprises twointeracting chromophores whose electronic states can beapproximately classified as locally excited (LE), charge-resonance (CR) or charge-transfer, and multiexcitonic. Thetransition between the states of predominantly LE (S1) andmultiexcitonic (1ME) character is facilitated by the contribu-tions from the CR configurations present in the respectiveadiabatic wave functions; this is sometimes described ascharge-transfer mediated singlet fission.41 One can alsoimagine a mechanism involving transitions via a real physicalintermediate state of the charge-transfer character;2 however,no unambiguous examples of such mechanism have beenreported so far. The involvement of CR configurationsmanifests itself by the strong dependence of the singlet fissionkinetics on the dielectric constant of the solvent58 and on theelectron-donating ability of side groups.40

The effect of the relative arrangements of the chromophoreson the singlet fission yields and rates has been extensivelyinvestigated.16,45,58,59 By varying the slip-stack displacements ofthe chromophores, the singlet fission rate and, consequently,yield can be either enhanced or suppressed. One particularlyfascinating illustration of this phenomenon is provided by thestudies of polymorphs which reported vastly different rates ofsinglet fission in different crystal forms of the same molecularsolid.60−63

Although the connection between the molecular solidmorphology and efficiency of singlet fission is indisputable, itis not entirely clear how to search for optimal localarrangements of the chromophores because different packingaffects multiple factors contributing to the overall rate.Experimentally, the control of the morphology is also farfrom trivial. One practical approach is to use covalently linkeddimers, in which the relative orientation of the twochromophore moieties can be controlled by the rationaldesign of the linker.7,38,42,48,64−71 Alternatively, packing can betweaked by augmenting the chromophores by strategicallychosen bulky substituents. One class of chromophores wherepacking can be easily controlled by substituents is PDIs, whichform ordered π-stacked structures. The crystal structures andtheir optical properties can be modulated by adding differentfunctional groups at the imide positions.72−78 In addition tothis tunability, PDIs possess several other attractive features. APDI chromophore is thermally and photochemically stable79

and strongly absorbs visible light.73 The excitation energy of

the lowest triplet state in PDI is approximately half theexcitation energy of the lowest singlet state.80 Thus, PDI is apromising organic chromophore for singlet fission photovoltaiccells.The effect of packing on singlet fission rates has been

systematically investigated in a recent experimental study.11 Inthis work, transient absorption and time-resolved emissionspectroscopy were used to compare the singlet fission rates andyields in six different PDI derivatives. The singlet fission ratesvaried between 0.33 and 4.3 ns−1 and the triplet yields variedbetween 80 and 178%. This experiment was in part motivatedby an earlier theoretical study, which predicted the singletfission rates for various PDI arrangements computed using theRedfield theory parameterized by density functional theory(DFT) calculations.59 The trends in experimentally measuredrates11 agreed reasonably well with the theoretical predic-tions,59 despite discrepancies in the absolute rates.In this paper, we investigate the effect of different

intermolecular PDI structures on the energetic driving forceand on the electronic coupling between the relevant states andprovide an explanation of the experimental trends.11 Weemploy the restricted active-space configuration interactionmethod with double spin-flip (RAS-2SF) to calculate theadiabatic excitation energies for different PDI dimers and thecoupling between the adiabatic electronic states. Using anadiabatic framework is a distinguishing feature of our protocol,which does not assume that the relevant electronic states havea simple diabatic character and allows multiple electronicconfigurations to couple and interact, as dictated by the many-body Hamiltonian. As explained below, this feature isparticularly important in the case of PDIs owing to an unusualelectronic structure of their multiexcitonic states. We use asimple three-state kinetic model based on electronic energiesand couplings to compute the relative rates and compare theresults with the experimental trends.11 By using wave functionanalysis tools, we investigate the character of the excited states.This analysis reveals an interesting feature that distinguishesPDIs from other singlet fission systems: in many PDIs, themultiexciton character is spread over several electronic states,which might have significant mechanistic consequences. Wenote that a very recent experimental study81 of terrylene-baseddimers has reported spectroscopic evidence of the multiexcitonstate containing large contributions of the CR configurations.The paper is organized as follows. First, we present the

theoretical framework and explain computational details of theprotocol employed. We then discuss the energies and thecharacter of the relevant excited states. This is followed by thediscussion of the computed rates and comparison with theexperimentally measured rates. Finally, we map the relationbetween singlet fission rates and the PDI stacking geometriesand predict the PDI geometries with optimal singlet fissionrates and yields.

■ THEORETICAL FRAMEWORKAdiabatic wave functions of two coupled chromophores can bedescribed in terms of the LE, CR, and multiexciton (1TT)configurations82

|Ψ⟩ = |Ψ ⟩ + |Ψ ⟩ + |Ψ ⟩c c cLELE

CRCR

TTTT

11 (1)

Here, cLE, cCR, and c1TT are the collective amplitudes of the LE,

CR, and 1TT configurations, which give rise to the collective

weights ωLE, ωCR, and ω1TT

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ω ω ω= | | + | | + | | = + +c c c1 LE 2 CR 2 TT 2 LE CR TT1 1(2)

Figure 2 shows these configurations expressed in terms oflocalized molecular orbitals of a dimer. By calculating ωLE, ωCR,

and ω1TT, one can assign a character to an adiabatic wave

function. For example, the S1 state is identified by large ωLE,whereas the 1ME state is identified by large ω

1TT. The S1 and1ME states can also contain the CR configurations, which lendan ionic character to the wave function. In the context ofsinglet fission, mixing of CR configurations in the S1 and

1MEstates plays an important role by facilitating electroniccouplings between these states. Following previous workfrom our group,82 we use the charge- and the spin-cumulantanalysis to quantify the weights of different contributions in thetotal wave function. We calculate singlet fission rates by usingadiabatic wave functions47 and the three-state kinetic modelsummarized in Figure 3, as was done in refs 48, 83, and 84.The kinetic model describes singlet fission as a two-stepprocess: the first step is a nonadiabatic transition from S1 to1ME and the second step is decoupling and separation of thetriplets. The respective rates are estimated48 using the Fermi’sgolden rule and the linear free-energy approach (whichassumes that the rate of a process is proportional to the free-energy difference between the initial and final states)

≈ −αβr (NAC) e E1

2 sf (3)

≈ −αβr e E2

b (4)

where r1 denotes the rate of the first step, the formation of the1ME state. The rate depends on the energy drive Esf

= −E E E( ME) (S )sf1

1 (5)

and the nonadiabatic coupling (NAC) between the S1 and the1ME states. The norm of the one-particle transition densitymatrix ∥γ∥ can be used as a proxy for NAC42,47−49 because∥γ∥ captures the changes in the adiabatic wave function, suchas variation of CR characters responsible for the derivativecoupling.85 α is a parameter in the free-energy relationship (weuse α = 0.5 as in our previous work48,84) and β = 1/kT, wherek is the Boltzmann constant and T is the temperature (we useT = 300 K). r2 is the rate of the second step, the production ofthe independent triplets. The rate of the second step dependson several factors including the multiexciton binding energy,Eb. Eb is the energy penalty for separating two triplets which

requires unmixing the CR contributions from the adiabaticwavefunction of the singlet ME state. Eb can be estimated as

= −E E E( ME) ( ME)b5 1

(6)

where 5ME is the quintet multiexciton state, which, in contrastto 1ME, always has a pure diabatic TT character. We alsocalculate Estt

= −E E E(S ) 2 (T)stt 1 1 (7)

where E(S1) is the lowest singlet exciton energy of the dimerand E(T1) is the triplet excitation energy of the monomer. Esttis an asymptotic value of the overall energy change in the SFprocess (positive values correspond to exoergic singlet fission).

As in previous work, we do not attempt to compute absoluterates; rather, we use eqs 3 and 4 to compute relative rates inhomologically similar compounds. We stress that the purposeof this simple model is not to achieve a quantitative descriptionof singlet fission kinetics but, rather, to provide quick estimatesof the magnitude of the effect of variations in underlyingelectronic structure factors on the rate, in the same fashion asin applications of the linear free-energy approach to rationalizeand predict trends in a large variety of processes in organicchemistry86,87 (in depth discussion of the assumptions behindlinear free-energy relationships can be found in ref 88). Thelinear free-energy approach works the best when applied toseries of sufficiently similar compounds/processes. Becausedifferent PDIs have very similar structural and physicalproperties, this is a good case for applying a linear free-energyrelationship. An important limitation of our model is that itdoes not account for coupling of electronic dynamics withvibrations, which plays an important role in the singlet fissionprocesses.50,52,89 Consequently, the model does not predictslowdown of the rate in strongly exothermic situations. Thisdrawback can be remedied by amending eq 3 to includereorganization energy, as in the Marcus rate expression,29 at aprice of introducing an additional parameter. This type of akinetic model has been used by Van Voorhis and co-workers.16

Of course, to elucidate full mechanistic details of singlet fission,that is, to determine whether the process proceeds throughconical intersections and to pinpoint the precise role of intra-or intermolecular vibrations, a much more sophisticatedtheoretical framework is required,52,89,90 and we hope thatour findings will stimulate such studies in the future.

■ COMPUTATIONAL DETAILSFigure 4 shows the structure of the PDI monomer. Weconstructed model dimer structures from the optimizedmonomer structures by arranging them as in the respective

Figure 2. Electronic configurations for S0 (GS), S1 (lowest singlyexcited singlet state), and 1ME (singlet multiexciton state) in the ABdimer expressed in terms of localized molecular orbitals.

Figure 3. Energy diagram for the three-state kinetic model of singletfission. r1 is the rate for the formation of the 1ME state (first step) andr2 is the rate for the production of the independent triplets (secondstep).

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crystal structures. In our model structures, we used R = H andK = H, as the substituents do not affect electronic properties ofthe PDI chromophores.11 Below, we refer to this structure asPDI(p); p for planar. The structure of the planar PDImonomer was optimized by using RI-MP2/cc-pVDZ. Theplanar PDI has D2h symmetry. The effect of slightly nonplanarstructures adopted by PDIs with bulky substituents (as inWD10) is discussed in the Supporting Information. Because allelectronic properties of the planar and nonplanar modelstructures are very similar, we used planar model structures inall calculations.We calculated singlet and triplet vertical excitation energies

of the optimized monomers using ωB97X-D91 and RAS-SF.92,93 In the latter, the active space consists of 2-electrons-in-2-orbitals and the reference restricted open-shell determinantis a high-spin triplet state (Ms = 1).We constructed model PDI dimers from the optimized

monomer structures by stacking two identical monomers at thearrangement shown in Figure 5. All model dimers wereconstructed from the planar PDI monomer. For the WDdimer, we also considered a structure built from nonplanardimers, WD(np); the results for WD(p) and WD(np) arediscussed in the Supporting Information.

We computed the excitation energies of the dimers usingRAS-2SF with 4-electrons-in-4-orbitals with a high-spin quintetreference (Ms = 2).94 We analyzed the resulting electronicwave functions in terms of weights of the LE, CR, and 1TTconfigurations (eq 1), which allowed us to identify the S1 and1ME states. We estimated the coupling between the S1 and1ME states by computing ∥γ∥2. We used the cc-pVDZ basis setin all calculations and kept the core electrons frozen. Toexamine the basis set effects, we performed additionalcalculations with the cc-pVTZ basis; the results are presentedin the Supporting Information. All electronic structurecalculations were carried out using the Q-Chem electronicstructure program.95,96

For each model PDI dimer shown in Figure 5b, wecomputed the key electronic energies, eqs 5 and 6. We thencomputed the relative rates for the formation of the 1ME state(eq 3) and the production of the triplets (eq 4) using thethree-state kinetic model. Here, all rates were calculatedrelative to the WD dimer. We compared the rates relative tothe planar and nonplanar PDI and found that they are notsignificantly different. Thus, we only present the rates relativeto the planar WD dimer. The rates relative to the nonplanarWD dimer are given in the Supporting Information.

■ CORRECTION OF THE RAS-2SF ENERGIES

Although RAS-2SF provides a balanced description of theelectronic configurations shown in Figure 2, the excitationenergies are overestimated because of an incomplete accountof dynamic correlation. To account for the missing dynamiccorrelation, we followed the same procedure as in ref 49 andcorrected the RAS-SF energies as follows. The energycorrection scheme is based on wave function decompositionof the adiabatic states Ψ in terms of LE, TT, and CRconfigurations (an additional small term, denoted by ωSS,contains the contributions due to the simultaneous excitationsof both chromophores that are not of the ME type49). Thecorrected energies are computed as

ω ω ω

ω

[Ψ] = [Ψ] + + +

+

E E E E

E

( 2 ) (LE) ( TT)

(CR)0

LE SSc

TTc

1

CRc

1

(8)

where E0 is the uncorrected energy and Ψ is the adiabatic wavefunction of the dimer, that is, S1 or ME. Ec(LE), Ec(

1TT), andEc(CR) are the energy corrections to the LE, CR, and 1TTcontributions, respectively. These corrections are computed byadjusting the RAS-SF energies of the respective diabaticconfigurations to match more accurate reference values. Here,we use ωB97X-D/cc-pVDZ energies as a reference, as wasdone in ref 49. Recent studies of PDIs by Engels and co-workers97,98 have also employed ωB97X-D as the best availablemethod. As discussed below, although the resulting excitationenergies are not in a perfect agreement with the experimentalvalues, the discrepancy is much smaller than for RAS-SF. Wenote that a recent benchmark study focusing on the excited-state wave functions99 has shown that among various popularfunctionals, the range-separated density function theory hybridfunctionals ωB97X-D and CAM-B3LYP yield the bestdescription of the exciton properties, as compared to thehigh-level correlated wave function methods.100 For the PDImonomer and dimers, the excitation energies computed withωB97X-D and CAM-B3LYP are close, within less than 0.1 eVfrom each other. Here, we employ the ωB97X-D energies as a

Figure 4. Structure of perylenediimide (PDI). R and K denotesubstituents at the imide and the core positions.

Figure 5. (a) Structure of a PDI dimer showing the displacementalong the long (x) and the short (y) axes. (b) Slip-stack displacementdx and dy for PDI dimers from ref 11. The plane-to-plane distances(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.

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reference for correcting RAS-2SF excitation energies. Below,we present the results for the corrected energies; the results forthe uncorrected energies are given in the SupportingInformation.

Table 1 reports the correction associated with the LE, 1TT,and CR contributions for each PDI dimer. The magnitude ofthe correction for the LE contribution is larger than for the1TT contribution. This is expected because the dynamicalcorrelation affects singlet states more than triplet states. Wealso note that the magnitude of Ec(LE) for the EP dimer islarger than those obtained for other PDI dimers. This is due tothe significant CR contribution in the S1 state in the EP dimer(see Table 3 in the Results and Discussion section).

■ RESULTS AND DISCUSSIONExcited-State Analysis of the PDI Monomers. The

experimental linear absorption spectra of different PDIderivatives in solution are very similar,10,11 meaning that thefunctional groups at the imide and core positions have arelatively small effect on the electronic states of the PDIchromophore. Thus, in our calculations, we used modelstructures in which the substituents were replaced byhydrogens. Table 2 shows singlet and triplet excitation energiesof 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 notsignificantly perturb its electronic structure. The tripletexcitation energy of the planar PDI is red-shifted by 0.09 eVrelative to the nonplanar PDI. The absolute values of theωB97X-D excitation energies are red-shifted by ∼0.5 eVrelative to the RAS-SF values. The calculated E(S1) values forthe planar and the nonplanar geometries are similar (Table 2),in agreement with the experiment. The calculations do notreproduce a small red shift of the triplet state in the nonplanarstructure, but the magnitude of the shift is very small and couldbe due different experimental conditions.101

As expected, the RAS-SF excitation energies are stronglyblue-shifted relative to the experimental values. The errors in

the ωB97X-D/cc-pVDZ energies are smaller: 0.56 and 0.36 eVfor S1 and T1, respectively. This functional has been also usedby Engels and co-workers.97,98 Additional calculations withCAM-B3LYP yielded excitation energies that are very close tothe ωB97X-D values; the corresponding Estt is −0.118 eV,which is close to −0.203 eV obtained by ωB97X-D. In contrastto popular functionals that might yield excitation energies inbetter agreement with experimental values,102 ωB97X-D wasshown to better reproduce the character of excited-state wavefunctions.99 These reference values determine the accuracy ofthe absolute excitation energies of the dimers. Although themagnitude of the error is large, we expect similar magnitude oferrors in model dimers such that the variations of excitationenergies due to different chromophore packing can bereproduced due to error cancellation. Using the monomervalues, the estimated error in the gap between the S1 and

1MEstates is 0.56−2 × 0.36 = −0.16 eV; the 1ME state would becomputed at higher energies relative to S1.

Excited-State Analysis of the PDI Dimers. Table 3 liststhe excitation energies and the properties of the S1,

1ME, and5ME states for the model PDI dimers. It also lists the weightsof the LE, CR, and 1TT configurations in the RAS-SF wavefunctions (Figure 2). The adiabatic wave functions of thesinglet states of the model PDI dimers show considerableconfiguration mixing. The S1 state has significant contributionsfrom the LE and CR configurations, which is common forFrenkel excitonic states56,97,103−105 and agrees with findings ofEngels and co-workers.97,106 We also note a small contributionfrom the 1TT configuration in the S1 state in some dimers. Instriking contrast to other systems studied by our group(various acenes, DPIBF, and 1,6-diphenyl-1,3,5-hexa-triene),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 (Table3), the largest 1TT contribution varies between 46 and 64%.This is different from, for example, model tetracene dimers in

which the 1TT character of the 1ME state is always large (ω1TT

≥ 80%).47,48 This is an interesting aspect of the electronicstructure of PDI dimers, which likely affects the mechanism ofsinglet fission in these systems. As one can see from Table 3,the states that we labeled as the 1ME states containcontributions from all singlet configurations, including a largebut not always dominant 1TT contribution. In the C3-I, C7,C8, MO, and EP dimers, there are two adiabatic states thathave a large contribution from the 1TT configuration (varyingbetween 21 and 64%). These states are denoted by the 1MEand 1ME′ labels. The wave function analysis shows that thelower 1ME state has comparable weights of the LE, CR, and1TT configurations, with the weight of the 1TT configurationbeing less than 40%. The upper 1ME state (denoted 1ME′)shows a larger contribution from the 1TT configuration, morethan 50%. In the WD dimer, there are two adiabatic 1ME stateswith comparable weights of the 1TT configuration. In the C1and C3-II dimers, however, the 1ME state has a dominant

contribution from the 1TT character (ω1TT ≈ 80%). In contrast

to the 1ME state, 5ME preserves the pure diabatic 5TTcharacter; this is similar to other singlet fission systems.47,49 Totest the robustness of this finding, we performed additionalcalculations using an entirely different approach, ab initioFrenkel−Davydov exciton model (AIFDEM)107 for the MOdimer, following the same protocol as used in ref 52. We foundthat although the exact weights of the configurations are

Table 1. Energy Correction (in eV) for the LE, 1TT, and CRContributions

dimer Ec(LE) Ec(1TT) Ec(CR)

C1 −0.5854 −0.2755 −0.0535C3-I −0.7262 −0.2672 −0.1653C3-II −0.6211 −0.2595 0.1281C7 −0.7083 −0.2658 0.0068C8 −0.7032 −0.2749 −0.1249WD(p) −0.7554 −0.2590 −0.0988MO −0.7991 −0.5777 −0.1100EP −1.1382 −0.2628 −0.2179

Table 2. S1 and T1 Vertical Excitation Energies (eV) in thePDI Monomer

stateexp.(p)a

exp.(np)b

ωB97X-Dc

(p)ωB97X-Dc

(np)RAS-SFc

(p)RAS-SFc

(np)

S1 2.34 2.34 2.903 2.865 3.524 3.470T1 1.19 1.28 1.553 1.510 1.885 1.803Estt −0.04 −0.22 −0.203 −0.155 −0.246 −0.136

aReferences.11,80 bReference 10. ccc-pVDZ basis.

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slightly different, the AIFDEM calculations confirm thestrongly mixed character of the multiexciton states. We alsoextended the set of model systems to the trimers of PDIchromophores. The RAS-3SF calculations of all model trimersshow that the characters of the relevant states remainqualitatively similar to the dimers (Table S7 in the SupportingInformation). Thus, although the precise state composition ofthe electronic states in the bulk crystal needs to be elucidatedby more sophisticated calculations, the mixed nature of themultiexcitonic states appears to persist at different levels oftheory and in larger systems. With an exception of a veryrecent experimental study of terrylene-based dimers,81 suchstrongly mixed singlet multiexciton states have not beenreported before. We note that this type of electronic structurehighlights the benefit of the adiabatic framework and posesnew questions regarding the mechanism of triplet productionfrom 1ME.

Table 4 lists relevant electronic energies (Esf and Eb) andcoupling ∥γ∥2 between the multiexciton and S1 states. Thesequantities depend strongly on the relative orientations of theindividual chromophores and are likely to be responsible forthe observed differences in the singlet fission rates of variousPDIs.Esf, the energy gap between 1ME and the S1, is endoergic in

all dimers. Likewise, the computed Estt is also endoergic. Wenote that our protocol, which is based on the ωB97X-D energycorrection, probably overestimates the extent of theendothermicity by as much as 0.16 eV. For the lower 1MEstate, the thermodynamic drive for the formation of the 1MEstate is less endothermic. This state has a positive multiexcitonbinding energy Eb (i.e., the energy of this state is lower than the5ME state). The energy of the upper 1ME state is above 5ME,which means that the triplet separation step for this state isexothermic.We estimated the coupling between the S1 and

1ME statesby using the norm of the one-particle transition density matrix.The values of the coupling are similar (∼0.1) in all dimersexcept the EP dimer in which the coupling is ∼0.3. This is dueto the large contribution of the CR configuration in the S1 and1ME/1ME′ states (Table 3).Figure 6 shows the computed rates of the multiexciton

formation and compares the results with the experimentalrates.11 The experimental rates were determined from transientabsorption spectroscopy and time-resolved photolumines-cence, and it is not clear whether one can distinguish betweenthe rates of formation of the 1ME state or the production ofthe two independent T1 states. All rates are computed relativeto the reference system, the planar WD dimer. Figure 6a,bshow the computed rate for the formation of the lowest statewith more than 20% multiexciton character and for the statewith the largest multiexciton character (this is an upper state,1ME′, in all dimers except C1 and C3-II). In the discussionbelow, we refer to these two cases as “lower ME” and “upperME” states. In the reference WD dimer, there are two 1MEstates with a similar contribution from the 1TT character(Table 3). For consistency, in Figure 6a, the rates arecalculated relative to the lower 1ME state in the WD dimer andin Figure 6b, the rates are calculated relative to the upper 1MEstate (the choice of the 1ME state (lower or upper) for thereference system does not affect the trend in the rates butchanges the relative magnitude of the rates). The fitting of thedata points in Figure 6 to a linear equation shows that the rateof the higher 1ME state formation agrees better with theexperiment than the rate of the lower 1ME state formation.Calculations using uncorrected energies (shown in SupportingInformation) lead to a similar conclusion (better correlationfor the upper ME state), but the energy correction clearlyimproves the agreement with the experimental rates, that is,the R2 value for the rates computed using uncorrected energiesis 0.49, whereas the correction brings it up to 0.77. However,the calculations overestimate the magnitude of the differencesin rates for different dimers. To illustrate this point, considerthe slowest and the fastest dimers from Figure 6, MO and C8.The experimental rate in C8 is ∼1.1 orders of magnitudeslower than in MO, whereas in our calculations, the C8 rate isslower by ∼2.4 orders of magnitude than the MO rate.Previous theoretical work, which used entirely different

protocols based on the DFT-parameterized Redfield theory,investigated the singlet fission rates for different stacking PDI

Table 3. S1,1ME, and 5ME Vertical Excitation Energiesa

(eV) and the Wave Functions Composition of the S1 and1ME States of the PDI Dimers

dimer state Eexa ωLE ωCR ω

1TT

C1 S1 2.6054 72.54 23.15 1.391ME 3.5755 2.33 15.54 82.025ME 3.4911

C3-I S1 2.6813 54.43 34.40 8.461ME 2.9704 39.74 36.98 21.111ME′ 3.6298 7.22 30.43 62.045ME 3.5045

C3-II S1 2.6567 80.49 15.37 0.291ME 3.5211 1.41 15.88 82.635ME 3.4577

C7 S1 2.6242 67.52 27.87 1.191ME 3.2482 21.83 40.68 36.251ME′ 3.5904 6.44 32.06 61.145ME 3.4893

C8 S1 2.5461 55.75 38.23 3.331ME 3.1569 34.46 34.14 29.481ME′ 3.5896 6.40 29.45 63.825ME 3.4963

WD S1 2.6932 58.26 30.88 7.661ME 3.1541 31.48 24.62 42.021ME′ 3.7443 8.94 44.58 46.005ME 3.4910

MO S1 2.5947 56.34 32.95 8.241ME 2.8495 36.80 32.87 28.341ME′ 3.3886 5.16 41.06 53.525ME 3.4771

EP S1 2.6717 34.37 62.79 0.131ME 2.8752 16.02 56.16 26.831ME′ 3.5446 14.52 35.09 49.845ME 3.5110

aCorrected energy.

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arrangements.59 Their calculated rates correlate with theexperimental rates slightly better (R2 is 0.87), but the absolutevalues of the computed rates are 3 orders of magnitude fasterthan the experimental ones.11 In this study, the calculated ratein 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 C8dimers, respectively, and that the coupling is approximately 0.1

for both dimers. Thus, we attribute a faster singlet fission ratein the MO dimer to more favorable Esf.The experimental study11 reported that the 1ME state

produced in thin films relaxes to the ground state (GS) within28 ns in EP and 10 ns in MO, C3-I, C7, and C8 and that theoverall yield of the free triplets was small. To fully understandthe singlet fission process, we need to consider the tripletseparation step. While rigorous theoretical modeling of thisprocess would require much more advanced approaches52,89,90

that are beyond the scope of this work, here, we discuss variousfactors controlling competing relaxation pathways of themultiexciton state and possible mechanistic implication of itsstrongly mixed character. We begin by analyzing the couplingsbetween the lower and the upper multiexciton states, thecoupling between the GS and the two singlet multiexcitonstates, and the multiexciton binding energy. The data in Table4 show that the coupling between the lower and the upper1ME states is large. Moreover, the coupling between the lower1ME state and the GS is larger than the coupling between thelower 1ME state and S1. The multiexciton binding energy forthe upper 1ME states varies between −101 meV (unboundbiexciton) and 88 meV, whereas for the lower 1ME state, itvaries between 636 and 241 meV (the biexciton is bound).There are at least two competing channels for the decay of theinitially produced multiexciton state: a separation into twouncorrelated triplets and radiationless relaxation to the GS.The structures where an upper 1ME state is produced in thefirst step, the production of triplets is likely to be affected bythe radiationless relaxation into the lowest 1ME state, whichresults in exciton trapping because of the large multiexcitonbinding energy, and consequent relaxation to the GS. Largevalues of ∥γ∥ for the 1ME-1ME′ transition suggest that thischannel might be completive with triplet separation. If this isthe case, one possible route to improving efficiency of singletfission in PDIs is to find arrangements at which the lowest 1MEstate is nearly degenerate with 5ME.To find the PDI geometries that maximize the rate of 1ME

formation while not leading to exciton trapping, we map therelation between the singlet fission rates for the 1ME formationand binding energies and the stacking arrangements of the PDIdimers along the long axis (x) and the short axis (y). We beginby quantifying the weight of the multiexciton 1TT character inthe 1ME state. The weight of the 1TT configuration for thelowest 1ME state (identified as the lowest state with noticeable

Table 4. Relevant Electronic Energiesa (eV) and Coupling for Different PDI Dimersb

dimer Esf Eb Estt ∥γ∥2, S1−1ME ∥γ∥2, S0−1ME ∥γ∥2, 1ME−1ME′C1 0.9701 −0.0844 −0.5006 0.10 0.06C3-I 0.2891 0.5341 −0.4247 0.07 0.30 0.19

0.9485 −0.1253 0.12 0.13C3-II 0.8644 −0.0634 −0.4493 0.09 0.07C7 0.6240 0.2411 −0.4818 0.11 0.25 0.26

0.9662 −0.1011 0.11 0.16C8 0.6108 0.3394 −0.5599 0.08 0.27 0.27

1.0435 −0.0933 0.11 0.14WD 0.4609 0.3369 −0.4128 0.10 0.22 0.27

1.0511 −0.2533 0.10 0.22MO 0.2548 0.6276 −0.5113 0.09 0.27 0.24

0.7939 0.0885 0.12 0.18EP 0.2035 0.6358 −0.4343 0.30 0.29 0.15

0.8729 −0.0336 0.29 0.15aCorrected energies. bWhen two sets of values are given, the upper and lower rows correspond to 1ME and 1ME′, respectively.

Figure 6. Relative rates of singlet fission: theory versus experiment.(a) Log of the relative rate is calculated for the lowest state with asignificant (more than 20%) multiexciton character (1ME state). (b)Log of the relative rate is computed for the state with the largestmultiexciton character (this is the upper ME state, 1ME′, in all dimersexcept C1 and C3-II). The relative rates are computed relative to theplanar WD rate (a) relative to the 1ME state and (b) relative to the1ME′ state. The solid black lines show a linear fit for all data points(R2 = 0.32 for the 1ME state and 0.77 for the 1ME′ state). Thestandard deviation for each data point from the fitted line is shown inthe Supporting Information. The experimental rates are from ref 11.

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ωTT) is shown in the lower panel of Figure 7a. As one can see,in this range of displacements, ωTT varies from 98 to 28%. Thered domains correspond to the structures in which the loweststate has the large ME character (>80%). The blue domainscorrespond to the structures in which the lowest state has smallωTT (less than 40%); these are the structures where the “real”multiexciton state is the upper one (ωTT > 40). The greendomains correspond to the structures in which the multi-exciton character is split almost equally between the two (ormore) states. Thus, one can expect that in the structures fromthe blue and green domains, singlet fission proceeds via anupper state and that the triplet separation competes withrelaxation to the lower ME state. The upper panel of Figure 7ashows the weight of the 1TT configuration for the upper 1MEstate for the blue and green domains. As one can see, the bluedomain for the upper 1ME state has a 1TT character more than40%.Figure 7d shows the rate of the multiexciton formation. The

rate is fast when Esf (Figure 7c) is less endothermic and thecoupling (Figure 7b) is large. The red domain in Figure 7dcorresponds to the PDI structures for which the singlet fissionrate is fast. As one can see, the red domain region for the lowerand upper 1ME state is different. Moreover, we find that in theregions where Esf is more endothermic and the coupling islarge, the rates are slow. Therefore, to optimize the singletfission rate for the formation of 1ME, both electronic energyand coupling should be favorable. This conclusion differs fromthat of a previous study59 of PDIs, which proposed that tooptimize the singlet fission rate, one needs to find thegeometry that optimizes the coupling between the twochromophores.Figure 7e shows Eb which provides information about the

energy needed for the separation of the singlet multiexcitoninto two triplets. For the lower 1ME, the comparison of therates with Eb suggests that despite favorable energetics andcouplings of structures in the red domain in Figure 7d, theefficiency of triplet production is impeded by the excitontrapping (see Figure 7e). For the upper 1ME state, however,the region where the formation of the multiexciton is fast is thesame region where Eb is favorable even if the system decays tothe lower 1ME state. On the basis of these maps, we concludethat singlet fission would be more favorable in the PDIstructures where both dx and dy are ≥2.5 Å. We note thatvarying the intermolecular crossing angle, which was zero in

the crystal structures studied in this paper and in ref 12, canprovide an additional degree of freedom for tuning thecouplings and energetics. Of course, a higher level of theorythat includes intra- and intermolecular vibrational motions andnonadiabatic transitions is needed for developing a quantitativepicture of this step.

■ CONCLUSIONSBy employing a simple three-state kinetic model, weinvestigated the singlet fission rates for different PDIgeometries and provided explanation for the measured ratesin PDI derivatives. Unlike many other singlet fission systems,the singlet multiexciton state in PDIs does not always have adominant 1TT character and contains large contributions fromthe LE and CR configurations. This leads to the splitting of the1ME state into the lower and upper multiexciton states withdifferent 1TT contributions. The calculated rates reproduce thetrend of the measured rates better when using the 1ME statewith the largest 1TT character, which often corresponds to theupper 1ME state. When singlet fission proceeds via the upper1ME state, the triplet separation might compete withradiationless relaxation to the lowest 1ME state for which thetriplet separation is strongly endothermic and inefficient. Thisradiationless relaxation to the lower 1ME state may result inthe trapping of the multiexciton and might be the reason forthe experimentally observed decay of the produced multi-excitons to the GS. By mapping the relation between thestacking geometry of the PDIs and the rate for the formation of1ME, we posit that PDI geometries with displacement ≥2.5 Åalong x and y coordinates are favorable. Experimental work tosynthesize and investigate singlet fission in these PDIstructures is needed to check this proposal. The reportedfindings warrant more detailed theoretical studies and anextension of the current mechanistic picture of singlet fission toincorporate the case of strongly mixed multiexciton states.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcc.8b05309.

Excitation energies of the PDI monomer obtained withcc-pVTZ; electronic energies and couplings for thenonplanar WD dimer; rates relative to the nonplanar

Figure 7. Mapping the relation between the stacking geometries of the PDI dimers and (a) the weight of the 1TT configuration in the 1ME state,(b) the coupling ∥γ∥2 between S1 and

1ME, (c) Esf, eV, (d) the log of the relative singlet fission rate (relative to the lower 1ME in the WD dimer),(e) the multiexciton binding energy, Eb, eV. The lower panel corresponds to the lowest energy state with a significant multiexciton character (ωTT >20), whereas the upper panel refers to the state with the largest 1ME character (ωTT > 40). See text for explanation.

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WD dimer; raw energies; results for model trimers; ratescomputed from uncorrected energies; and relevantCartesian geometries (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: krylov@usc.edu.

ORCIDMarwa H. Farag: 0000-0003-0443-1089Anna I. Krylov: 0000-0001-6788-5016NotesThe authors declare the following competing financialinterest(s): Anna I. Krylov is a part owner and a boardmember of Q-chem, Inc.

■ ACKNOWLEDGMENTSThis work is supported by the Department of Energy throughthe DE-FG02-05ER15685 grant. We are grateful to ProfessorSean Robers and Dr. Jon Bender from UT Austin forstimulating discussions and for valuable feedback about themanuscript. We also would like to thank Professor AnatolyKolomeisky from Rice University for helpful discussions.

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