Fluctuations in Electronic Energy Affecting Singlet Fission Dynamics

Letter pubs.acs.org/JPCL

Fluctuations in Electronic Energy Affecting Singlet Fission Dynamics and Mixing with Charge-Transfer State: Quantum Dynamics Study Yuta Fujihashi and Akihito Ishizaki* Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan S Supporting Information *

ABSTRACT: Singlet fission is a spin-allowed process by which a singlet excited state is converted to two triplet states. To understand mechanisms of the ultrafast fission via a charge transfer (CT) state, one has investigated the dynamics through quantum-dynamical calculations with the uncorrelated fluctuation model; however, the electronic states are expected to experience the same fluctuations induced by the surrounding molecules because the electronic structure of the triplet pair state is similar to that of the singlet state except for the spin configuration. Therefore, the fluctuations in the electronic energies could be correlated, and the 1D reaction coordinate model may adequately describe the fission dynamics. In this work we develop a model for describing the fission dynamics to explain the experimentally observed behaviors. We also explore impacts of fluctuations in the energy of the CT state on the fission dynamics and the mixing with the CT state. The overall behavior of the dynamics is insensitive to values of the reorganization energy associated with the transition from the singlet state to the CT state, although the coherent oscillation is affected by the fluctuations. This result indicates that the mixing with the CT state is rather robust under the fluctuations in the energy of the CT state as well as the high-lying CT state. experimental fission rate.10 They showed that the measured fission rates in the pentacene derivatives exhibit a transition from a nonadiabatic regime to an adiabatic one, as predicted by the theoretical investigation.32 Many theoretical studies based on electronic structure calculations24,25,29,30,39 and quantum dynamics calculations27,31−37 have suggested two possible pathways for the formation of TT, as shown in Figure 1. The first process

inglet exciton fission is a spin-allowed process by which two triplet states are generated from a singlet excited state.1,2 This process was first observed in anthracene crystals in 1965.3 Current interest in this process is driven by the potential to dramatically increase the efficiency of organic photovoltaics, as it is possible to inject two charges into an external circuit per absorbed photon.4,5 Fission in organic photovoltaics must outcompete singlet exciton dissociation at the donor−acceptor interface. In the case of a slow fission material, the singlet state can diffuse to the acceptor interface and generate charge before fission occurs. Therefore, ultrafast fission is essential for organic photovoltaic applications. Singlet fission has been observed in a number of molecular systems2 such as pentacene6 and tetracene.7 Time-resolved spectroscopy experiments have revealed that fission in pentacene can occur very quickly on time scales of 80−110 fs.8−10 Despite a significant number of experimental7,8,10−19,21,22 and theoretical studies23−40 on a variety of molecular materials, the molecular mechanism underlying the process has not yet been fully elucidated. In a conventional four-electron view, the singlet state of a molecule, S1, transforms into a correlated triplet pair state, TT, on adjacent molecules, which subsequently dissociates into two separate triplet states. The TT state, often called the multiexciton state, is considered an optically dark state.24 Recently, Chan et al.8,12 directly observed the TT state using a time-resolved two-photon photoemission (TR-2PPE) spectroscopy experiment in pentacene and tetracene and suggested that an initial photoexcitation creates a quantum superposition between S1 and TT. Yost et al. measured the fission rates in acene derivatives with vastly different structures using ultrafast transient absorption spectroscopy and presented a kinetic model based on a fission rate expression by a resummation technique that successfully reproduces the

S

© XXXX American Chemical Society

Figure 1. Schematic representation of two possible pathways within the four-electron view of the singlet fission process in two molecules.

is a mediated mechanism in which the singlet state converts to a triplet pair state via a charge-transfer (CT) state; here the two neighboring molecules are positively and negatively charged. The second process is a direct mechanism involving simultaneous two-electron transfer without the CT state. Quantum-chemical Received: December 2, 2015 Accepted: January 5, 2016

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DOI: 10.1021/acs.jpclett.5b02678 J. Phys. Chem. Lett. 2016, 7, 363−369

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The Journal of Physical Chemistry Letters

CT and the other states.16,40 It is natural to question whether the fluctuations in the energy of the CT state destroy the mixing with the CT state. Therefore, we explore the impact of fluctuations in the CT state energy on the fission dynamics and mixing with the CT state. We also consider the comparison between the population dynamics and the corresponding free energy surfaces to clarify the relation between the fission dynamics and the fluctuations in the energy of the CT state. Model for Singlet Fission Dynamics. In this study, we focus on a model dimer for singlet fission based on the four-electron four-orbital HOMO−LUMO basis, as shown in Figure 1. We consider a simple scheme of the singlet fission process, |g⟩ → |S1⟩ → |CT⟩ → |TT⟩. The electronically diabatic Hamiltonian for the electronic ground state |g⟩, singlet state |S1⟩, charge-transfer state |CT⟩, and correlated triplet pair state |TT⟩ is given by

calculations of pentacene clusters showed that the energy of the CT state is ∼300 meV above that of the singlet states; this indicates that singlet fission in pentacene should occur via the direct pathway.25 On the contrary, Berkelbach et al. investigated the role of the high-lying CT state in the fission process by applying the Redfield theory and showed that this state does not necessary imply the direct mechanism.32 Tao calculated the dynamics of a pentacene dimer by a quasi-classical approach based on the Meyer−Miller−Stock−Thoss representation to investigate the applicability of the Redfield theory in the typical parameter region for describing the fission dynamics. This calculation showed that the result obtained by the quasi-classical approach is in good agreement with those obtained using the Redfield theory;36 however, the fission dynamics obtained by these theories does not generate a quantum superposition of S1 and TT immediately after photoexcitation, and it is not consistent with the suggestion by the TR-2PPE experiment.8 In molecular crystals in which fission occurs, fluctuations in the electronic energy are mainly induced by intermolecular motions in the crystal and intramolecular vibrational motions of the molecules that constitute the crystal.41−43 Time-resolved transient absorption spectroscopy experiments on 6,13bis(triisopropyl-silylethynyl)-pentacene (TIPS-pentacene) demonstrated that high-frequency vibrational modes play a role in efficient singlet fission.17 Berkelbach et al.13,31,32 and Tao37 explored the impacts of fluctuations on the fission dynamics and demonstrated that the fission dynamics show temperatureindependent behavior in a wide range of temperatures, in keeping with the spectroscopy measurements.8,12,15 In previous theoretical studies of fission dynamics, it has often been assumed that each electronic state is coupled to an independent vibrational mode; that is, that fluctuations in electronic energy are uncorrelated.31−38 The uncorrelated fluctuation model is typically used for describing electronic energy transfer between pigments in a photosynthetic light-harvesting system.44 The moving electronic excitation energy in a light-harvesting system is described as a single excitation on a pigment embedded in the protein. In this framework, the different electronic excited states represent excitations of different pigments in the protein. Therefore, the excited electronic state of each pigment experiences a different dynamical effect from its local protein environment, and the uncorrelated model is appropriate for describing the electronic energy transfer in the protein. Indeed, molecular dynamical simulations have shown that uncorrelated fluctuations are valid for an Fenna−Mathews−Olson protein environment;45,46 however, the fission process is different from this situation. S1 and TT exist within the same dimer. The electronic structure of TT is similar to that of S1 except for the spin configuration,10 and these electronic states could experience the same fluctuation induced by the surrounding molecules. Therefore, the fluctuations in the electronic energies could be correlated, and a 1D coordinate model, which is usually employed to describe electron-transfer reactions in condensed phases, might adequately describe the fission dynamics.47−49 Indeed, Yost et al.10 showed that the rate formula based on the 1D coordinate model successfully reproduced the experimental fission rate in acene derivatives. In this work, we develop a model for describing the fission dynamics to explain the behavior of the fission dynamics observed by experimental spectroscopy. We apply the 1D coordinate model to describe the fission dynamics in a pentacene dimer. Recent theoretical works have presented the value of the reorganization energy larger than the electronic coupling between

Ĥ =

Ĥ m(x) m⟩⟨m +

∑ m = g,S1,CT,TT

∑

∑ Jmn m⟩⟨n

m = S1,CT,TT n ≠ m

(1)

where Ĥ m(x) (m = g,S1,CT,TT) represents the diabatic Hamiltonian for the vibrational degrees of freedom (DOFs), x, when the system is in state |m⟩. For simplicity, we assume that the interstate coupling Jmn is independent of the vibrational DOFs. The electronic energy of each diabatic state experiences fluctuations caused by the intramolecular vibrational motion and surrounding crystal vibrational motion. Such fluctuations in electronic energies can be characterized by the collective energy gap coordinate defined as49,50 umn ̂ = Ĥ m(x) − Ĥ n(x) − ⟨Ĥ m(x) − Ĥ n(x)⟩n

where the canonical average, ⟨Ô ⟩n =

(2)

−βĤ n ̂ ) with ρ̂eq tr(Ô ρneq /tr n =e

̂ e−βHn, has been introduced for any operator Ô . Here β is the inverse temperature, 1/kBT. The reorganization energy associated with the transition from |m⟩ to |n⟩ is given by49

λmn = ⟨Ĥ m(x) − Ĥ n(x)⟩n − (ϵ°m − ϵ°n)

(3)

where ϵm° denotes the equilibrium energy of diabatic state |m⟩, namely, ϵm° = ⟨Ĥ m(x)⟩m. In the case of photoinduced singlet fission, the reaction coordinate associated with the initial photoexcitation, ûS1,g, and those for the subsequent singlet fission, ûCT,S1 and ûTT,S1, are required. The coordinates ûS1,g, ûCT,S1, and ûTT,S1 may generally involve statically orthogonal components of fluctuations.47,49 The electronic structure of TT is similar to that of S1 except for the spin configurations,10 and these electronic states could experience the same fluctuations induced by the surrounding molecules. Therefore, the fluctuations in the energies of the electronic states could be correlated; that is, they could be described by the same components of fluctuations. We describe the singlet fission dynamics using the 1D reaction coordinate model,47,49 which is used to investigate electrontransfer reactions. The reaction coordinates, ûS1,g, ûCT,S1, and ûTT,S1, are described by the same components of fluctuations, as follows49 uŜ 1,g = − λS1,g ·,̂

(4)

uCT,S ̂ 1 = −ηCT,S λCT,S1 ·,̂

(5)

u TT,S ̂ 1 = −ηTT,S λ TT,S1 ·,̂

(6)

1

1

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The Journal of Physical Chemistry Letters with ηCT,S1= ±1 and ηTT,S1= ±1. The case of ηCT,S1 = 1 (ηTT,S1 = 1) and that of ηCT,S1 = −1 (ηTT,S1 = −1) correspond to positive and negative correlations between S1 and CT (TT), respectively. In general, the electronic structure of the CT state is distinct from those of the S1 and TT states. Therefore, ûCT,S1 involves components orthogonal to ,̂ , and the environmental reorganization may take place along coordinates orthogonal to ,̂ on a multidimensional reaction coordinate space. In this work, we assume that the electronic energy of the CT state is much higher than those of the S1 and TT states, and hence the CT state is hardly populated. Namely, we consider a case that the fission dynamics proceed through virtual charge-transfer state. Therefore, we may ignore the reorganization along coordinates orthogonal to ,̂ and employ the 1D reaction coordinate model. The values of ηCT,S1 and ηTT,S1 also control the relative positions of the free energy surfaces of |g⟩, |S1⟩, |CT⟩, and |TT⟩ regarding the reaction coordinate, as shown in eqs 13−15. In this work, we assume that the vibrational DOFs, x, can be modeled as a set of harmonic oscillators. Under this assumption, the reaction coordinates ûmn and ,̂ can be modeled as Gaussian fluctuations; therefore, the dynamical properties can be characterized by several types of two-body correlation functions ̂ ̂ ̂ ̂ of ,̂ (t ) = e iHgt / ℏ,̂ e−iHgt / ℏ or ûmn (t) = eiHgt/ℏûmn e−iHgt/ℏ, as shown later. In addition, the free energies of the diabatic states are quadratic with respect to ,̂ .47−49 The reorganization dynamics51,52 is described by the relaxation function, Ψmg(t), defined as the canonical correlation of ûmg (t)53 Ψmg(t ) = β⟨um̂ g(0); um̂ g(t )⟩g

GS1(,) = ϵ°S1 +

GCT(,) = ϵ°CT +

GTT(,) = ϵ°TT +

∑

+ (JS ,CT S1⟩⟨CT + JCT,TT CT⟩⟨TT + h.c .) 1

In general, the relaxation function, ψ(t), may have complicated forms involving various components. To focus on the effects of the time scales of the fluctuations in the electronic energy induced by the intermolecular modes in the crystal, we model the relaxation function by an exponential decay form (overdamped Brownian oscillator model) with ψ (t ) = 2 exp( −γt )

where γ corresponds to the time scale of the fluctuations. Equation 9 gives the corresponding spectral density as j(ω) = 2γω/(ω2 + γ2). The expression of an equation of motion describing the fission dynamics is given in the Supporting Information (SI). Numerical Results. We present and discuss numerical results regarding fission dynamics in the pentacene dimer. We consider a three-site model consisting of S1, CT, and TT for the numerical calculations of the fission dynamics, as shown in Figure 1. To model pentacene, we take the values of the electronic coupling from ref 32, JS1,CT = 116 meV and JCT,TT = −152 meV. A direct electronic coupling, JS1,TT, is found to be very small, only a few millielectronvolts, compared with the electronic coupling between CT and other states.25,32 We neglect the direct coupling, JS1,TT; that is, we fit JS1,TT = 0 and thus consider only the mediated mechanism in our study. From the absorption spectra, the energy of S1 lies at 1.83 eV.22 The energy of TT is expected to be simply two times the energy of the lowest triplet state, ϵTT ° ≈ 2ϵT° = 1.72 eV, for which the lowest triplet energy has previously been measured to be 0.86 eV.2 Therefore, the reaction driving force from S1 to TT, ΔGS1,TT = ϵ°TT − ϵ°S1, is set to be ΔGS1,TT = −110 meV. Although it is difficult to accurately determine the CT state energy experimentally and theoretically, recent theoretical studies have reported that the energy gap between CT and TT, ϵCT ° − ϵTT ° , lies between 300 and 600 meV.25,32 In this study, therefore, we consider two cases of CT state energy in the range of the theoretically predicted values, that is, ΔGS1,CT = ϵ°CT − ϵ°S1 = 390 meV and ΔGS1,CT= 640 meV. Tamura et al.’s DFT calculation40 of a TIPS-pentacene crystal found λS1,g = 150 meV and λTT,g = 450 meV. When the correlation coefficient, ηTT,S1, is set to be ηTT,S1 = 1, the reorganization energy for the fission reaction in TIPS-pentacene, λTT,S1, is evaluated as 80.4 meV using eq 11. If ηTT,S1 was set to be ηTT,S1 = −1, the reorganization energy for the fission reaction would be λTT,S1 = 1120 meV and be too large to reproduce the fission process with a time scale of 100−200 fs. In our study, therefore, the correlation coefficient and the reorganization energy associated

(9)

Equation 8 and the equality uCT,g = uCT,S ̂ ̂ 1 + uŜ 1,g = −( λCT,g + ηCT,S λCT,S1 ), 1

yield a relational expression among the reorganization energies, λCT,g, λCT,S1, and λS1,g as57,58 1

(10)

In the same manner, the relation among the reorganization energies λTT,g, λTT,S1, and λS1,g can be expressed as λ TT,g = λS1,g + λ TT,S1 + 2ηTT,S λS1,g λ TT,S1 1

(11)

As a result, the free energy surfaces for the four diabatic states with respect to the coordinates , can be expressed as47,49

Gg (,) =

1 2 , 4

(16)

−1

∞

λCT,g = λS1,g + λCT,S1 + 2ηCT,S λS1,g λCT,S1

Gm(,) m⟩⟨m

m = S1,CT,TT

(8)

dt ψ (t ) cos ωt

1 [, − 2( λS1,g + ηTT,S λ TT,S1 )]2 1 4 (15)

with the normalization of ⟨,̂ ; ,̂ ⟩g = 2kBT . The corresponding spectral density j(ω) is defined as53

∫0

1 [, − 2( λS1,g + ηCT,S λCT,S1 )]2 1 4

where , has been treated as a classical variable. The adiabatic free energy surfaces are given by diagonalizing

The relaxation function is independent of temperature in the linear response region. Under this assumption, the Stokes shift magnitude is given as two times the reorganization energy, 2λmg; therefore, the relaxation function satisfies Ψmg(0) = 2λmg. In the same fashion as in eq 7, the relaxation function of ,̂ (t ) is introduced as

j(ω) = ω

(13)

(14)

(7)

ψ (t ) = β⟨,̂ (0); ,̂ (t )⟩g

1 [, − 2 λS1,g ]2 4

(12) 365


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The Journal of Physical Chemistry Letters with photoexcitation and the fission reaction are chosen to be ηTT,S1 = 1, λS1,g = 150 meV, and λTT,S1 = 80 meV, respectively. Comparatively less information is available for the reorganization energy for the transition from S1 to CT, λCT,S1. The reorganization energy for the transition from S1 to CT is set to be λCT,S1 = 50 meV. The λCT,S1 dependence of the fission dynamics is discussed later, and it is demonstrated that the overall behavior of the fission process in pentacene is insensitive to the value of λCT,S1. In Figure 2, we present the population dynamics for various values of the reaction driving force from S1 to CT, ΔGS1,CT, and

Figure 3. Panels a−d show the free energy surface corresponding to the free energy arrangement of Figure 2a−d, respectively. The solid and dashed lines indicate the diabatic and adiabatic free energy surfaces, respectively. ,* is the position of the intersection between GS1(,) and GTT(,).

and adiabatic free energy surfaces of S1, CT, and TT with respect to , . Figure 3a,b shows the free energy surfaces corresponding to the parameters employed in Figure 2a,b, respectively. The strong electronic couplings between CT and the other states create quantum mixing between S1 and TT near the crossing point of the diabatic free energy surfaces of S1 and TT, leading to large splitting around the crossing point, as shown in Figure 3a. The splitting between the adiabatic free energy surfaces around the crossing point owing to the mixing with the CT state reduces the energy barrier between the free energies of S1 and TT, as shown in Figure 3a. The vertical Franck−Condon transition from the origin to the adiabatic state marked as 1 lies above the crossing point. Therefore, the electronic state is delocalized immediately after photoexcitation, and a fission process occurs from 1 to 2 to 4, as shown in Figure 3a. In the case of the high-lying CT state, the creation of quantum mixing between S1 and TT can be understood as the effective electronic coupling between S1 and TT owing to CT mixing, Jeff ≈ JS1,CTJCT,TT/ (ϵ°CT − ϵ°TT), as discussed in refs 31 and 32. The effective coupling induces the transition near the crossing point between the diabatic free energies of S1 and TT. The adiabatic free energy in the case of ΔGS1,CT = 640 meV shows small splitting near the crossing point compared with that of ΔGS1,CT = 390 meV because the large energy gap between CT and TT leads to small effective coupling, Jeff. The weak quantum mixing between S1 and TT, that is, the small effective coupling, decreases the lifetime of the superposition between S1 and TT and the fission rate, as shown in Figure 2b. We address impacts of fluctuations in the CT state energy on the fission dynamics and the mixing with the CT state.

Figure 2. Population dynamics of S1, CT, and TT for various energy gaps, ΔGS1,CT, and correlation coefficient, ηCT,S1, with the relaxation time fixed at γ−1 = 100 fs. The employed parameters are ΔGS1,TT = −110 meV, JS1,CT = 116 meV, JCT,TT = −152 meV, λS1,g = 150 meV, λCT,S1 = 50 meV, λTT,S1 = 80 meV, T = 300 K, and ηTT,S1 = 1.

the correlation coefficient, ηCT,S1, with the relaxation time fixed at γ−1 = 100 fs. The relaxation time used here is based on the Yost et al.’s kinetic model, which adopted γ−1 = 70 fs as the relaxation time and successfully reproduced the experimental fission rates in acene derivatives.10 In Figure 2a,b, we fix ηCT,S1 = 1. The quantum dynamics calculations in Figure 2a,b point to a singlet fission process with a time scale of 100−200 fs, which shows qualitatively good agreement with the experimental rate of 80−110 fs.8−10 Figure 2b shows the fast fission dynamics despite the high-lying CT state. The population dynamics of the TT state in Figure 2a involve two oscillating components in the short-term region (t < 100 fs). The faster oscillatory component with small amplitude reflects the superposition between CT and TT. The slower oscillatory component with large amplitude indicates the superposition between S1 and TT, and this behavior is consistent with the suggestion by the TR-2PPE spectroscopic measurement.8 The recent 2D electronic spectra of pentacene obtained by Bakulin et al.22 showed that there is no evidence of longlasting electronic coherence between S1 and TT. The behavior in Figure 2a does not contradict the results of 2D electronic spectra because the oscillations by the superposition between S1 and TT in Figure 2a disappear by 100 fs. To analyze the fission dynamics driven by the mixing with the CT state, we consider the diabatic 366


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The Journal of Physical Chemistry Letters Figure 2c,d shows the population dynamics for various ΔGS1,CT with the correlation coefficient fixed at ηCT,S1 = −1 to investigate the effects of the correlation coefficient, ηCT,S1, on the fission dynamics. Figure 3c,d show the free energy surfaces corresponding to the parameters employed in Figure 2c,d, respectively. Figure 2c,d show the long-lasting oscillations compared to Figure 2a,b, respectively. This behavior can be explained by comparing the free energy surfaces in the case of ηCT,S1 = −1 with those in the case of ηCT,S1 = 1. The diabatic free energy of CT in the case of ηCT,S1 = −1 is close to the crossing point between the diabatic free energy surfaces of S1 and TT compared with that of ηCT,S1 = 1. As a result, the adiabatic free energy surface in the case of ηCT,S1 = −1 shows large splitting near the crossing point between the diabatic free energies of S1 and TT compared with that of ηCT,S1 = 1. Figure 4a shows the population dynamics of S1

Figure 5. Population dynamics of S1 for various temperatures. The other parameters are the same as those in Figure 2a.

is located around the Franck−Condon point of the free energy surface of S1, , = 0, marked as 1 in Figure 3a. A higher temperature causes a wider distribution of , , which spreads the wave packet farther from the Franck−Condon point. As a result, a large portion of the , distribution lies below the intersection at higher temperature. This portion requires thermal activation from 3 to 2 for the fission process to proceed. On the contrary, the fission in the case of fast fluctuations (γ = 180 meV) show temperature-independent dynamics, as shown in Figure S2. The impact of the fluctuation time scales on the fission dynamics and the details of temperature-independent dynamics in the case of γ = 180 meV are discussed in the SI. In conclusion, we have developed a model of fission dynamics to explain some experimental results. For this purpose, we applied the 1D coordinate model to the description of the fission dynamics in the pentacene dimer. The population dynamics of the TT state as calculated using the 1D coordinate model involves two oscillating components in the short term region, and it shows a fission process with a time scale of 100−200 fs, which is in qualitatively good agreement with the experimental rate. This behavior reflects both the superposition between S1 and TT and that between CT and TT, and it is consistent with the suggestion by the TR-2PPE spectroscopy experiment. We also investigated the influence of the fluctuations in the energy of CT state on the fission dynamics and the mixing with the CT state. The overall behavior of the fission process is insensitive to the value of λS1,CT and the correlation coefficient, ηCT,S1, although the coherent oscillation due to the superposition between S1 and TT is affected by the fluctuations. This result indicates that the mixing with the CT state is rather robust under fluctuations in the energy of the CT state as well as the high-lying CT state. We mention the limitation of the overdamped Brownian oscillator model approach to the singlet fission process. The term “spectral density” may be misleading unless we draw attention to its definition. The spectral density is defined with the relaxation function as53 j(ω) = ω ∫ ∞ 0 dt ψ(t) cos ωt, rather than the distribution of vibrational modes itself. The spectral density of the overdamped Brownian oscillator model j(ω) = 2γω/(ω2 + γ2) is obtained from the relaxation function, ψ(t) = 2 exp(−γt). In this modeling, the time scale of the fluctuations of the electronic energy, τ, is simply τ = γ−1. Although this model is useful for focusing on the time scale of the fluctuations in the electronic energy, it cannot include discrete high-frequency vibrational modes explicitly. Bakulin et al.’s recent observation of 2D electronic spectra in pentacene derivatives showed that some discrete vibrational modes induce vibronic coherence persisting

Figure 4. (a) Population dynamics of S1 for various reorganization energies, λS1,CT. The other parameters are the same as those in Figure 2a. (b) Free energy surface corresponding to the case of λS1,CT = 350 meV. The solid and dashed lines indicate the diabatic and adiabatic freeenergy surfaces, respectively.

for various values of λS1,CT. The other parameters in Figure 4a are the same as those in Figure 2a. Figure 4a shows that the overall behavior of the fission process is insensitive to the value of λS1,CT. To analyze the cause of this insensitivity, we show the free energy surfaces corresponding to the case of λS1,CT = 350 meV in Figure 4b. The equilibrium position of the diabatic free energy surface of the CT state significantly moves from the origin toward the positive direction of , axis because of the large value of λS1,CT. The distance between the diabatic free energy of CT in Figure 4b and the crossing point between the diabatic free energy surfaces of S1 and TT is of the same degree as that shown in Figure 3b. As a result, the cases of Figure 4b and Figure 3b show a similar degree of splitting around the crossing point and lead to the similar behavior of the fission reaction. Therefore, Figure 2c,d and Figure 4a indicate that the mixing with the CT state is rather robust under fluctuations in the energy of the CT state, although the coherent oscillation due to the superposition between S1 and TT is affected by the fluctuations. To study the temperature dependence of the fission dynamics, we present the time evolution of S1 population for various temperatures in Figure 5. The overall behaviors of fission dynamics show slightly anomalous temperature dependence, where the fission dynamics speed up as temperature decreases. As discussed in ref 49, this temperature dependence of the initial behavior can be explained as follows: By definition in eq 8, the variance 2 of the distribution of , is given by ⟨,̂ ⟩ ≃ ⟨,̂ ; ,̂ ⟩ = 2k T . g

g

B

Immediately after the initial photoexcitation, a wave packet of , 367


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The Journal of Physical Chemistry Letters for 1500 fs.22 Their analysis also demonstrated that the cooling process observed with the 2PPE8 might correspond to the relaxation process between the vibronic states of TT. Their observation and modeling indicated that the Huang−Rhys factors of these discrete modes are large and that these modes might play a key role in the fission dynamics. Therefore, the overdamped Brownian oscillator model might be inadequate for the fission dynamics in the acene crystal, although this model has often been used to model the spectral density in recent theoretical studies of fission dynamics. The underdamped Brownian oscillator model can straightforwardly include the discrete high-frequency vibrational modes with long dephasing time. This model was successfully employed for interpreting the vibronic coherence of the nonlinear spectra caused by discrete vibrational modes in photosynthetic light harvesting systems.54,59−62 An investigation of the fission dynamics by the underdamped model is currently in progress.

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Ensuing Ultrafast Multielectron Transfer. Science 2011, 334, 1541− 1545. (9) Wilson, M. W.; Rao, A.; Clark, J.; Kumar, R. S. S.; Brida, D.; Cerullo, G.; Friend, R. H. Ultrafast Dynamics of Exciton Fission in Polycrystalline Pentacene. J. Am. Chem. Soc. 2011, 133, 11830−11833. (10) Yost, S. R.; Lee, J.; Wilson, M. W.; Wu, T.; McMahon, D. P.; Parkhurst, R. R.; Thompson, N. J.; Congreve, D. N.; Rao, A.; Johnson, K.; et al. A Transferable Model for Singlet-Fission Kinetics. Nat. Chem. 2014, 6, 492−497. (11) Johnson, J. C.; Nozik, A. J.; Michl, J. High Triplet Yield from Singlet Fission in a Thin Film of 1, 3-diphenylisobenzofuran. J. Am. Chem. Soc. 2010, 132, 16302−16303. (12) Chan, W.-L.; Ligges, M.; Zhu, X. The Energy Barrier in Singlet Fission Can Be Overcome through Coherent Coupling and Entropic Gain. Nat. Chem. 2012, 4, 840−845. (13) Chan, W.-L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. R.; Tritsch, J. R.; Hybertsen, M. S.; Reichman, D. R.; Gao, J.; Zhu, X.-Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321−1329. (14) Walker, B. J.; Musser, A. J.; Beljonne, D.; Friend, R. H. Singlet Exciton Fission in Solution. Nat. Chem. 2013, 5, 1019−1024. (15) Wilson, M. W.; Rao, A.; Johnson, K.; Gélinas, S.; di Pietro, R.; Clark, J.; Friend, R. H. Temperature-Independent Singlet Exciton Fission in Tetracene. J. Am. Chem. Soc. 2013, 135, 16680−16688. (16) Busby, E.; Berkelbach, T. C.; Kumar, B.; Chernikov, A.; Zhong, Y.; Hlaing, H.; Zhu, X.-Y.; Heinz, T. F.; Hybertsen, M. S.; Sfeir, M. Y.; et al. Multiphonon Relaxation Slows Singlet Fission in Crystalline Hexacene. J. Am. Chem. Soc. 2014, 136, 10654−10660. (17) Musser, A. J.; Liebel, M.; Schnedermann, C.; Wende, T.; Kehoe, T. B.; Rao, A.; Kukura, P. Evidence for Conical Intersection Dynamics Mediating Ultrafast Singlet Exciton Fission. Nat. Phys. 2015, 11, 352− 357. (18) Musser, A. J.; Maiuri, M.; Brida, D.; Cerullo, G.; Friend, R. H.; Clark, J. The Nature of Singlet Exciton Fission in Carotenoid Aggregates. J. Am. Chem. Soc. 2015, 137, 5130−5139. (19) Pensack, R. D.; Tilley, A. J.; Parkin, S. R.; Lee, T. S.; Payne, M. M.; Gao, D.; Jahnke, A. A.; Oblinsky, D.; Li, P.-F.; Anthony, J. E.; et al. Exciton Delocalization Drives Rapid Singlet Fission in Nanoparticles of Acene Derivatives. J. Am. Chem. Soc. 2015, 137, 6790−6803. (20) Stern, H. L.; Musser, A. J.; Gelinas, S.; Parkinson, P.; Herz, L. M.; Bruzek, M. J.; Anthony, J.; Friend, R. H.; Walker, B. J. Identification of a Triplet pPair Intermediate in Singlet Exciton Fission in Solution. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 7656−7661. (21) Miyata, K.; Tanaka, S.; Sugimoto, T.; Watanabe, K.; Uemura, T.; Takeya, J.; Matsumoto, Y. Coherent Phonon Dynamics in Singlet Fission of Rubrene Single Crystal. In Ultrafast Phenomena XIX; Springer, 2015; pp 218−221. (22) Bakulin, A. A.; Morgan, S. E.; Kehoe, T. B.; Wilson, M. W.; Chin, A. W.; Zigmantas, D.; Egorova, D.; Rao, A. Real-Time Observation of Multiexcitonic States in Ultrafast Singlet Fission Using Coherent 2D Electronic Spectroscopy. Nat. Chem. 2015, 8, 16−23. (23) Greyson, E. C.; Vura-Weis, J.; Michl, J.; Ratner, M. A. Maximizing Singlet Fission in Organic Dimers: Theoretical Investigation of Triplet Yield in the Regime of Localized Excitation and Fast Coherent Electron Transfer. J. Phys. Chem. B 2010, 114, 14168−14177. (24) Zimmerman, P. M.; Zhang, Z.; Musgrave, C. B. Singlet Fission in Pentacene through Multi-Exciton Quantum States. Nat. Chem. 2010, 2, 648−652. (25) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M. Mechanism for Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets. J. Am. Chem. Soc. 2011, 133, 19944−19952. (26) Minami, T.; Nakano, M. Diradical Character View of Singlet Fission. J. Phys. Chem. Lett. 2012, 3, 145−150. (27) Teichen, P. E.; Eaves, J. D. A Microscopic Model of Singlet Fission. J. Phys. Chem. B 2012, 116, 11473−11481. (28) Renaud, N.; Sherratt, P. A.; Ratner, M. A. Mapping the Relation between Stacking Geometries and Singlet Fission Yield in a Class of Organic Crystals. J. Phys. Chem. Lett. 2013, 4, 1065−1069.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02678. Expression of an equation of motion describing the fission dynamics; influence of the fluctuation time scales on the fission dynamics. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 (0)564 55 7310. Fax: +81 (0)564 53 4660. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Prof. Takeshi Yanai and Prof. Yuki Kurashige for the inspiring discussions. This work is supported by a Grantin-Aid for Scientific Research (No. 25708003) from the Japan Society for the Promotion of Science.

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REFERENCES

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Fluctuations in Electronic Energy Affecting Singlet Fission Dynamics

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