Computer Simulation of Singlet Fission in Single Crystalline

Article pubs.acs.org/JPCC

Computer Simulation of Singlet Fission in Single Crystalline Pentacene by Functional Mode Vibronic Theory Justin E. Elenewski,† Ulyana S. Cubeta,† Edward Ko,‡ and Hanning Chen*,† †

Department of Chemistry, The George Washington University, Washington, District of Columbia 20052, United States of America Department of Mechanical Engineering, Columbia University in the City of New York, New York, New York 10027, United States of America

‡

S Supporting Information *

ABSTRACT: We have applied our functional mode framework for singlet fission to pentacene, a prototypical organic material for multiple exciton generation. It was found that singlet fission in pentacene occurs predominantly through a coherent process mediated by a virtual charge-transfer (CT) intermediate, which lies slightly above the photoexcited S1S0 state. This energetic near-degeneracy facilitates a substantial vibronic superposition, leading to a rapid transition rate of 25.1 ps−1. By contrast, the direct S1S0 → T1T1 path constitutes a much more sluggish route with a rate of 2.6 ps−1, largely due to the weak diabatic coupling between participant states. These data collectively afford an experimentally consistent rate of 27.7 ps−1 for the entire singlet fission process. The presence of this lowlying CT intermediate suggests that enhanced electronic coupling between S1S0 and T1T1 states may collude with coherent vibrational mixing to expedite the formation of triplet pairs. The knowledge gleaned from our investigations heralds a new approach to charge transfer-mediated singlet fission, a rapidly growing research field that holds great promise to circumvent the Shockley−Queisser thermodynamic limit for solar energy conversion.

1. INTRODUCTION Singlet fission1 is a spin-allowed transition that converts a singlet exciton into a correlated triplet pair. In the case of organic photovoltaics, this processes initiated when a chromophore is promoted to its first excited singlet state (S1) through optical excitation. A correlated pair of triplet excitons (T1T1) is then formed through spin exchange between the frontier orbitals of the photoexcited sensitizer and an adjacent ground state chromophore.2 Decoherence ultimately erases entanglement between these triplets, leading to a pair of independent triplet excitons (T1 + T1) that are located at distinct sites within the photovoltaic.3 This doubling raises the optimal incident photon-to-current conversion efficiency (IPCE) to 200%, making singlet fission a promising technique to overcome the long-standing Shockley−Queisser limit4 of 34% for solar energy conversion. Detailed balance analysis5 suggests that the thermodynamic efficiency of a single-junction solar cell can reach 44% in the presence of multiple excition generation (MEG), assuming an optimal bandgap of 0.7 eV. The practical utility of this conclusion is bolstered by the observation of a 200% triplet yield via singlet fission in thin films of 1,3-diphenylisobenzofuran,6 confirming an earlier theoretical prediction7 for sensitizer candidates that possess a favorable ordering of adiabatic energy levels (E(S1) > 2E(T1)). Specific physical features may be engineered to promote this route, including the design of tightly stacking polymorphs8 to enhance the electronic coupling between chromophores, and the selection of low-lying charge transfer (CT) intermediates9 © XXXX American Chemical Society

to maximize superposition of the bright S1 state and the dark T1T1 state. Underscoring the efficacy of this strategy, a triplet yield of 170% has been achieved under ambient conditions using a copolymer with alternating units of strong electron donors and acceptors.10 The singlet fission rate of 1,3diphenylisobenzofuran11 is likewise 1 order of magnitude greater before the α → β phase transition, suggesting that singlet fission efficiency may be tailored through chemical functionalization or physical alterations. Singlet fission was first discovered in 1965,12 however, its detailed mechanism remains controversial even for prototypical polyacenes.13−16 The current theoretical consensus indicates two possible routes for singlet fission (Figure 1), and the relative contribution of each is delineated by the energetics of participating states as well as their electronic coupling strengths. The first pathway is direct, affording the correlated triplet T1T1 through the diabatic coupling to the photoexcited S0S1 singlet. Conversely, the second pathway relies on a charge transfer (CT) intermediate, and may be subdivided into two distinct cases. A CT state that is energetically lower than S0S1 but higher than T1T1, may participate as a true intermediate through sequential S1S0 → CT and CT → T1T1 transitions. This process is assumed to be incoherent due to strong environmental coupling. In contrast, a virtual CT intermediate with a higher energy than S0S1 and T1T1 may still contribute to Received: April 1, 2017 Published: May 1, 2017 A

DOI: 10.1021/acs.jpcc.7b03107 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

important role played the CT state is also underscored when comparing the singlet fission yield of three regioisomers, namely the o-, m-, and p-phenylene-linked pentacene dimers.32 It was found that the meta-isomer exhibits a much higher singlet fission yield than its counterparts due to its thermodynamically accessible CT state, which is only 0.03 eV above the lowest lying singlet excited state.32 This small energy difference highlights the significance of wavefuncton mixing, which may help overcome the activation barrier for singlet fission in pentacene29 and its derivatives.33,34 The remainder of this paper is organized as follows. In section 2, our FMSF theory18 is briefly reviewed and adopted to derive the transition rate formulas corresponding to the direct, sequential and mediated pathways of singlet fission. Numerical simulations are then presented in section 3 for a pentacene supercell, and the resultant data are employed to determine the singlet fission rates for each competing mechanism. A detailed analysis is presented in section 4 to reveal the most probable route for singlet fission in crystalline pentacene.

2. THEORETICAL METHODS As detailed in our previous contributions,18,35,36 the reaction coordinate, V⃗ MCM, of an electron-transfer or spin-transition process in a given system can be considered as a linear combination of all vibrational normal modes:

Figure 1. Energy diagrams for three proposed singlet fission pathways: (a) direct mechanism; (b) sequential mechanism; (c) mediated mechanism.

Nvib

singlet fission through a coherent superposition of their vibronic states. The relative contribution of each pathway to singlet fission is difficult to assess using existing theoretical methods. A unified framework that can treat all transition pathways on equal footing is thus highly desired for the computer-aided design of singlet fission sensitizers and thus the refinement of low-cost, third-generation solar cells.17 To this end, we have recently developed a functional mode singlet fission (FMSF) theory18 based on Fermi’s golden rule,19 constrained density functional theory (CDFT),20 and functional mode analysis (FMA).21 This method has previously identified the direct mechanism as the primary route for singlet fission in single-crystal tetracene, with an overall transition rate of 0.02 ps−1,18 which is firmly consistent with the experimental value of 0.01 ps−1.15 In the present study, we apply our FMSF theory18 to model singlet fission in single-crystal pentacene−a material that exhibits a higher fission rate22 (12.5 ps−1) than tetracene despite having significant structural similarity. The enhanced singlet fission rate in pentacene has been partially ascribed to a stronger CT admixture into its lowest excited state23−25 and thus a stronger electronic coupling.26 This material is also characterized by a tighter aromatic stacking, which serves to lower the energy of the CT state. More specifically, experimental data indicate that the CT state lies a mere 0.18 eV above the S1S0 in single-crystal pentacene,27 which is half of the value for single-crystal tetracene. The intermediate CT state usually expedites singlet fission either through a coherent, superexchange-like interaction28,29 or via enhanced vibronic coupling.10,30 A recent study31 on a series of covalent terrylenediimide dimers demonstrated that singlet fission can be enabled or disabled through a deliberate shift of the CT state’s energy, which may be tuned by varying solvent polarity. If the S1S0 and T1T1 states are nearly energetically degenerate, as in compound 2 of ref 31, then the CT state stabilized by a comparatively polar dichloromethane solvent system serves as an exciton trap that effectively blocks singlet fission. The

⃗ VMCM =

∑ ciVi⃗

(1)

i=1

where the expansion coefficient ci indicates the relative contribution of the i th normal mode V⃗ i to weighted vibronic energy underlying a nonradiative transition. The vibrational normal modes can be determined either by diagonalizing the mass-weighted cross-correlation matrix of atomic displacements extracted from a molecular dynamics trajectory, as in the present study, or by diagonalzing the mass-weighted Hessian matrix of an optimized structure as in ref 36. The functional mode analysis method21 may be employed to determine V⃗ MCM by maximizing Pearson linear correlation coefficient R, defined as R=

cov(pV⃗

MCM

(t ), ΔEDA(t ))

σ VMCM ⃗ σΔEDA

(2)

where pV⃗ MCM (t) = (r(⃗ t) − ⟨r⟩⃗ )•V⃗ MCM is the projection of instantaneous atomic displacement r(⃗ t) − ⟨r⟩⃗ on V⃗ MCM, ΔEDA(t) is the instantaneous diabatic energy gap between the reactant ΨD and product ΨA states, cov(pV⃗ MCM (t),ΔEDA (t)) is the covariance function between pV⃗ MCM (t) and ΔEDA(t), and their standard deviations are denoted as σV⃗ MCM and σΔEDA, respectively. The most numerically efficient way to deduce the maximum value of R is to solve a set of linearly coupled equations: Nvib

∑ cicov(pV⃗ (t) , pV⃗ (t) ) = cov(ΔEDA(t ), pV⃗ (t) ) j = 1, ..., Nvib i=1

i

j

j

(3)

In singlet fission, the product T1T1 state has strong multiconfigurational character, and may be constructed using spin-adapted CDFT.20,37 To do so, a spin-polarized and position-dependent Hartree potential was added to the Kohn−Sham Hamiltonian of an adjacent pair of sensitizer B


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The Journal of Physical Chemistry C molecules to achieve a net spin difference of +4 (or −4) between chromophores as quantified using Becke population analysis for heteronuclear systems.38,39 The resulting molecular orbitals are then used as building blocks for the six constituent spin configurations of the T1T1 state (Supporting Information, section S1). Molecular orbitals of the intermediate CT state were formulated through a restricted open-shell Kohn−Sham (ROKS) approach40 in conjunction with CDFT.20 In a CT dimer (Supporting Information, section S1), a given sensitizer carries a net charge of +1 while its counterpart holds the excess electron. Finally, the reactant S1S0 state is defined using linearresponse time-dependent density functional theory (LRTDDFT) through collective excitations of Kohn−Sham electron−hole pairs.41 According to nonradiative transition theory,42 the energy profiles of reactant and product states in a chemical reaction delineate their vibrational wave function overlap. In the present study, the free energy driving force ΔG0 associated with a transition is calculated via thermodynamic integration (TI) using a linearly mixed Hamiltonian, H(η):43 Ĥ (η) = (1 − η)HR̂ + ηHP̂

the S1S0 and T1T1 states. If both stages of the two-step S1S0 → CT → T1T1 transition are assumed to be first-order, the population of the product T1T1 state [T1T1] grows monotonically: CT SC ⎛ SC −kSF CT −kSF kSF e t − kSF e t ⎞⎟ ⎜ = + [TT] 1 1 1 SC CT ⎜ ⎟[S1S0]0 − kSF kSF ⎝ ⎠

where [S1S0]0 is the initial population of the initial S1S0 state, CT and kSC SF (or kSF ) is the transition rate for S1S0 → CT (or CT → T1T1). Once the reaction coordinate for each successive step has been ascertained using functional mode analysis, kSC SF and ̂ kCT SF can be calculated using eq 8. with JSC = ⟨S1S0|H0|CT⟩ and JCT = ⟨CT|Ĥ 0|T1T1⟩. Detailed expressions for these matrix elements are presented in Supporting Information, sections S3 and S4. Under a steady-state approximation that assumes a constant population of [CT], [T1T1] reduces to (1 − SC −kCT CT SF t e−kSF t)[S1S0]0 when kSC )[S1S0]0 when SF ≪ kSF or to (1 − e SC CT S kSF ≫ kSF . As such, the overall reaction rate kSF for the S1S0 → CT → T1T1 transition is determined by its rate-limiting step. 2c. Mediated Mechanism. In the mediated mechanism, a manifold of CT states is needed to facilitate the coherent coupling between a pair of energetically degenerate S1S0 and T1T1 vibronic states. If the number of vibrational quanta within the reactant S1S0 state is νk, the corresponding number in the resonant T1T1 state νi is given by

(4)

where Ĥ R and Ĥ P are the Hamiltonians of the reactant and product states, respectively. After sampling the gradient of H(η) along η, ΔG0 is given by Nη

ΔG0 =

⎛ ∂Hmix(η) ⎞ ⎟ Δη ⎝ ∂η ⎠η

∑⎜ i=1

vi = vk +

(5)

The associated reorganization energy λ is readily deduced from these free energy differences: λ = ⟨Hmix(η = 1) − Hmix(η = 0)⟩η= 0 + ΔG0

3 ⎛ 2EST − SST (ES + ET ) ⎞ ⎟ ⎜ 2⎝ 2(1 − SST 2) ⎠

=

2π JD 2 ℏ2ωF

(6)

M = kSF

λ ℏ

(7)

j=0

⟨vS1S0 , k|vCT , j⟩2 ⟨vTT , i|vCT , j⟩2 (ECT − ET + (vj − vi)ℏωMCM )2

3. SIMULATION RESULTS Our simulation system is an 8 × 10 × 3 pentacene supercell containing a total of 17,280 atoms (Figure 2). The initial cell was constructed from experimental X-ray crystallographic data45 and equilibrated at 300 K using a 1 ns molecular dynamics (MD) simulation with a time step of 1.0 fs. The generalized AMBER force field (GAFF)46 was adopted to treat simulation physics, in conjunction with RESP partial charge fitting scheme.47 Starting from the equilibrated system, a 10 ns MD production run was carried out to provide 100,000 randomly selected snapshots for normal-mode analysis on an

(8)

Nvib

i=1

ℏ2ωMCM

∞

∑

where νs are the vibrational eigenstates of the corresponding functional modes. In this notation, νCT,j denotes the jth vibrational state on the CT potential energy surface. These vibrational eigenstates are reasonably approximated by analytic solutions for one-dimensional displaced harmonic oscillators. Since energetic degeneracy is not required between the CT state and the resonant reactant/product pair, a complete CT manifold should be considered to fully account for quantum coherence. Nonetheless, a diminished contribution from higher-lying CT vibronic states is expected due to energy mismatch as indicated by the denominator of eq 12.

⎛ ΔG0 + λ ⎞2 ⎟ −1⎜ 2 e ⎝ ℏωF ⎠

∑ ci 2ωi

2πJSC 2 JCT 2

(12)

where ωF is the functional-weighted vibrational angular frequency defined as ωF 2 =

(11)

N

where the definitions of EST, SST, ES, and ET are summarized in the Supporting Information, section S2. Once the reaction coordinate of the direct S1S0 → T1T1 has been determined using functional mode analysis, kDSF is given by D kSF

ES − ET ℏωMCM

where ωMCM = (∑i =vib1 ci 2ωi−1)−1 is the angular frequency of the reaction coordinate associated with the S1S0 → T1T1 transition. By following second-order time-dependent perturbation theory, as shown in the Supporting Information, section S5, the singlet fission rate kM SF mediated by the CT manifold is given by

Another important parameter for diabatic electron transfer44 is the electronic coupling strength J = ⟨φR|Ĥ 0|φP⟩, the representation of which depends strongly on the multiexciton characteristics of the reactant and product states. For notational simplicity, we will treat each J individually when calculating the rate of singlet fission kSF for each constituent pathway (Figure 1). 2a. Direct Mechanism. In the direct mechanism, the reactant and product are S1S0 and T1T1, respectively. As shown in ref 18 JD = ⟨S1S0|Ĥ 0|TT 1 1⟩ =

(10)

(9)

2b. Sequential Mechanism. The sequential mechanism contains an intermediate CT state whose energy lies between C


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nearest-neighbor shell of the designated dimer are included into our QM subsystem. Therefore, the dimer model should be sufficient to delineate the CT-admixture into S1S0. Aiming to reduce the computational burden associated with TI, a hybrid quantum mechanics/molecular mechanics (QM/MM) approach50 was exploited to model the chosen reaction center and its first neighbor shell using CDFT,20 while the rest of the system was described using GAFF.46 Unless otherwise specified, all QM/MM simulations were performed using the CP2K software51 with the Goedecker−Tetter−Hutter (GTH) pseudopotential,52 Perdew−Burke−Ernzehof hybrid (PBE0) exchange-correlation functional,53 polarized-valence-double-ζ (PVDZ) basis set,54 electrostatic QM/MM coupling scheme,55 and a wavelet-based Poisson solver.56 Since singlet fission may proceed through three pathways, namely the direct, sequential and mediated mechanisms (Figure 1), the results for each route are presented individually in the following three subsections. 3a. Direct Mechanism. The collective motion driving a direct S1S0 → T1T1 transition was determined by projecting the diabatic energy gaps, ΔE(r)⃗ = ES1S0(r)⃗ − ET1T1(r)⃗ , of 5,000 randomly selected nuclear configurations onto all vibrational normal modes of the designated reaction center for singlet fission. As shown in Figure 3, a concerted ring-deformation

Figure 2. An 8 × 10 × 3 supercell of a pentacene single crystal. The molecular pair designated as the reaction center for singlet fission is highlighted in red, while all eight molecules within its nearest-neighbor shell in the ab crystalline plane are highlighted in blue.

adjacent pair of pentacene molecules (Figure 2). This pair constitutes our reaction center for singlet fission. In order to assess the extent of delocalization of the S1S0 state upon system expansion, we calculated the photodisplaced charge, ∫ dr |ρ (r ) − ρ (r )|

e h , where ρe and ρh are the associated CDP = 2 electron and hole densities, respectively, for both dimer and decamer (Figure 2). Since DFT is known to suffer from orbital overdelocalization issue due to self-interaction error and incorrect asymptotic behavior,48 the long-range-corrected hybrid function, LRC-ωPBEh,49 was adopted for the bigger decamer system. It was found that CDP only slightly increases from 0.28 to 0.30 (Table 2) when all eight molecules within the

Figure 3. Calculated driving vibrational mode V⃗ MCM for the direct S1S0 → T1T1 transition within an adjacent pair of pentacene molecules.

motion with ωMCM = 1590 cm−1 predominates over other modes. Interestingly, among the 210 available normal modes, only two are critical for the direct mechanism as indicated by the profile of V⃗ MCM coefficients with c2i > 0.10. The two essential modes at 1800 and 1813 cm−1 contribute a statistical weight of 55.3%, outweighing all other contributors. To ensure the statistical quality of V⃗ MCM, the coefficients of its constituent modes were projected onto another 3,000 nuclear configurations that were extracted from a separate 3 ns MD trajectory. As shown in Figure 4, the cross-validated Pearson’s coefficient, RC, is 0.77, which is only 0.03 units smaller than the training coefficient, R. This trivial deviation suggests that our sampling for FMA of V⃗ MCM is sufficient. The energy level diagram calculated using our TI approach and LR-TDDFT is presented in Figure 1a. LR-TDDFT was initially employed to obtain a histogram of S0S0 → S1S0 vertical excitation energies for 5,000 MD snapshots. Since the histogram shows a peak centered at 2.03 eV, the relaxed S1S0 state lies above its S0S0 counterpart by 1.88 eV after accounting for a thermal relaxation energy ΔEER of 0.15 eV in the S1S0 state. Using this data, the driving force for the direct S1S0 →

Table 1. Singlet Fission Rates through the Direct, Sequential and Mediated Mechanisms, as Well as Their Relative Significances mechanism

rate (ps−1)

relative significance (%)

2.6 0 25.1 27.7

9 0 91 100

kDSF

direct, sequential, kSSF mediated, kM SF overall, kDSF + kSSF + kM SF

Table 2. Charge-Transfer Character of the S0S1 State Quantified by the Photo-Displaced Charge for Pentacene Dimer and Decamer Using PBE0 and LRC-ωPBEh Functionals system/functional

CDP

dimer/PBE0 dimer/LRC-ωPBEh decamer/LRC-ωPBEh

0.28 0.25 0.30

D


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Figure 5. Dependence of the singlet fission rate kM SF on the number vibrational quanta within the CT intermediate state νj when the reaction proceeds through the mediated mechanism.

Figure 4. Scatter plot for cross-validation of MCM coefficients ci2 through projection of the diabatic energy gap ΔES1S0 → T1T1 onto vibrational normal modes.

of νk = 1 → νj = 0, 1, 2 → νi = 4 afford a cumulative rate of 23.4 ps−1 due to the energetic near-degeneracy of intermediate and product vibronic states ECT + νj ℏωMCM ≈ ET + νk ℏωMCM. −1 When νj = 3 and 4, we find that kM SF drops to a modest 0.8 ps , −4 −1 followed by a more drastic decline to 5.0 × 10 ps as νj −1 reaches 5. Although kM when νj = 6, it SF rebounds to 0.016 ps decays persistently and rapidly at νj = 7 and beyond. As a result, the overall kSSM is given by 25.1 ps−1, which is nearly ten times faster than kDSF. The significant acceleration can be ascribed to the substantially enhanced electronic coupling strengths (JSC = 84.9 meV and JCT = 387 meV) mediated by the stabilized CT state.

T1T1 transition − ΔGD = ΔES0S0 → S1S1 − ΔEER − ΔGS0S0 → T1T1 was found to be 0.67 eV, exceeding the observed reorganization energy λD of 0.48 eV. As a result, the direct transition falls into the Marcus inverted region44 with a highly stabilized product state. While the corresponding electronic coupling strength JD = ⟨φS1S0|Ĥ 0|φT1T1⟩ fluctuates with structural changes, a mean value of 15.9 meV was determined using eq 7 after sampling 100 MD snapshots. Encouragingly, an independent CDFT study57 achieved a comparable value of 19.0 meV when using the same PBE0 functional.53 Using this data, the rate for direct singlet fission kDSF is found to be 2.6 ps−1 as determined using eq 8 (ΔGD = −0.67 eV, λD = 0.48 eV, ωF = 2513 cm−1, and JD = 15.9 meV). 3b. Sequential Mechanism. If the energy of the mediating CT state lies between S1S0 and T1T1, singlet fission may proceed through a two-step sequential mechanism that is associated with a true CT intermediate (Figure 1b). However, this condition is not fulfilled in single-crystal pentacene since the calculated CT state lies above S1S0 by 0.24 eV, which is consistent with the experimental value of 0.18 eV.27 Moreover, an energy gap of 0.24 eV is not surmountable by thermal fluctuations at room temperature, effectively disabling the sequential route for singlet fission. Therefore, kSSF = 0 due to energy level misalignment. 3c. Mediated Mechanism. Vibrational decoherence is negligible within the vibronic manifold for the CT state under conditions associated with coherent mediation, (Figure 1c), permitting a coherent superposition with S1S0. If we assume that all processes preserve the vibrational wave function’s phase, the thermal relaxation of S1S0 is also negligible. Our assumption is based on the experimental findings58−60 that the vibrational relaxation of photoexcited pentacene in the crystalline phase occurs on the picosecond time scale, which is notably slower than its observed singlet fission rate of 12.5 ps−1.22 The relaxation energy of the S1S0 state ΔEER = 0.15 eV is comparable to ℏωMCM ≈ 0.20 eV. Under these conditions, S1S0 contains νk = 1 vibrational quanta while its resonant T1T1 counterpart contains νi = 4 (Figure 1c). This difference is associated with a free energy gap of 0.67 eV between νk and νi in the absence of vibronic coupling. For the sake of completeness, our calculations should include all vibronic states in the CT manifold. Nonetheless, only a few contribute meaningfully to singlet fission. As shown through the dependence of kM SF on νj (Figure 5), the three mediated routes

4. DISCUSSION AND CONCLUSIONS Pentacene and its derivatives have been employed as electron donors in bulk-heterojunction solar cells due to their suitable bandgaps,61 high hole mobilies62 and long exciton diffusion lengths.63 In addition, pentacene is known to undergo ultrafast singlet fission with a characteristic time scale of 80 fs.22 This rate is several orders of magnitude faster than tetracene15 despite only modest differences in molecular structure. In practice, the additional aromatic ring in pentacene shifts its bandgap to absorb at longer wavelengths while simultaneously diminishing the energy gap between its S1S0 and CT states and thereby facilitating intermolecular electron transfer. As shown through our thermodynamic calculations (Figure 1b), the CT state is only 0.24 eV higher in energy than the S1S0 state. This establishes the CT state as an active virtual entity, in superposition with both the reactant S1S0 state and the product T1T1 state, leading to a rapid mediated rate of 25.1 ps−1. By contrast, the feeble electronic coupling strength between the S1S0 and T1T1 states relegates the direct mechanism to a sluggish rate of 2.6 ps−1, affording a relatively small contribution on the order of 10% as shown in Table 1. While the calculated overall singlet fission rate of 27.7 ps−1 overestimates experiment by a factor of 2,22 it is still within the limit of uncertainties associated with morphological distortions and defect-induced strain.64 In the course of these calculations, we have quantitatively dissected the contribution of each possible mechanism to singlet fission in pentacene. These data underscore the importance of the CT mediation and constitute a simulation protocol that is readily adopted to other materials exhibiting singlet fission. We should note that our simulations were performed on a pentacene dimer, which is unlikely to be able to completely delineate the pronounced admixture of the E


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CT state into S1S0 as revealed by many earlier experimental and theoretical studies.9 Therefore, without compromising physical accuracy, a more appropriate treatment is highly desired to afford the computational cost associated with fully periodic simulation systems. Moreover, the current framework of our functional mode vibronic theory35 ignores the collective contribution from different vibrational modes by only accounting for their linear correlation with diabatic energy gap. A further improvement can be possibly made by introducing the nonlinear and even higher-order corrections through measuring the interdependence between diabatic energy gap and vibrational normal modes using optimized mutual information.21

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b03107. Section S1, spin−orbital structure of singlet fission intermediates; section S2, parameters for calculation of JD = ⟨T1T1|Ĥ |S0S1⟩, section S3, calculation of coupling element JSC = ⟨S0S1|Ĥ |CT⟩; section S4, calculation of coupling element JCT = ⟨T1T1|Ĥ |CT⟩; section S5, M derivation of the mediated singlet fission rate kSF (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(H.C.) Fax: (202) 994-5873. Telephone: (202) 994-4492.Email: [email protected]. ORCID

Hanning Chen: 0000-0003-3568-8039 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research was supported by a start-up grant and the Columbian College Facilitating Fund of the George Washington University. Computational resources utilized in this research were provided by the Argonne Leadership Computing Facility (ALCF) at Argonne National Laboratory under Department of Energy contract DE-AC02-06CH11357 and by the Extreme Science and Engineering Discovery Environment (XSEDE) at Texas Advanced Computing Center under National Science Foundation contract TG-CHE130008.

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REFERENCES

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Computer Simulation of Singlet Fission in Single Crystalline

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