Triplet Pair States in Singlet Fission - Chemical Reviews (ACS

Review Cite This: Chem. Rev. XXXX, XXX, XXX−XXX

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Triplet Pair States in Singlet Fission Kiyoshi Miyata,†,‡ Felisa S. Conrad-Burton,† Florian L. Geyer,† and X.-Y. Zhu*,† †

Department of Chemistry, Columbia University, New York, New York 10027, United States Department of Chemistry, Kyushu University, Fukuoka 819-0395, Japan

‡

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S Supporting Information *

ABSTRACT: This account aims at providing an understanding of singlet fission, i.e., the photophysical process of a singlet state (S1) splitting into two triplet states (2 × T1) in molecular chromophores. Since its discovery 50 years ago, the field of singlet fission has enjoyed rapid expansion in the past 8 years. However, there have been lingering confusion and debates on the nature of the all-important triplet pair intermediate states and the definition of singlet fission rates. Here we clarify the confusion from both theoretical and experimental perspectives. We distinguish the triplet pair state that maintains electronic coherence between the two constituent triplets, 1(TT), from one which does not, 1(T···T). Only the rate of formation of 1(T···T) is defined as that of singlet fission. We present distinct experimental evidence for 1(TT), whose formation may occur via incoherent and/or vibronic coherent mechanisms. We discuss the challenges in treating singlet fission beyond the dimer approximation, in understanding the often neglected roles of delocalization on singlet fission rates, and in realizing the much lauded goal of increasing solar energy conversion efficiencies with singlet fission chromophores.

CONTENTS 1. Introduction 2. Fundamentals of Singlet Fission 2.1. General Mechanisms 2.2. Diabatic and Adiabatic States 2.3. Sixteen-Spin State Representation of the Triplet Pair: 1(TT) 2.4. Nine-Spin State Representation of the Triplet Pair: 1(T···T) 2.5. Binding Energy of the Triplet Pair State 2.6. Delocalization in the Extended Solid: Exciton Bands 2.7. Delocalization in a Dimer: The Excimer 2.8. Rates of Forming 1(TT) and 1(T···T) 2.9. Triplet Pair Separation, Transport, and Entropic Contributions 3. Computational Studies of Singlet Fission 3.1. 1(TT) State from First Principle Quantum Chemical Calculations 3.2. Electron−Phonon Coupling in Singlet Fission 3.3. Incoherent Rate Equation Calculations of Singlet Fission Kinetics 3.4. Coherent Singlet Fission and Quantum Dynamics 4. Experimental Evidence for Singlet Fission Mechanisms 4.1. Spectroscopic Signatures of 1(TT) 4.2. Vibronic Coherence in Singlet Fission 4.3. Delocalization and Charge-Transfer States © XXXX American Chemical Society

5. Conclusion and Outlook Associated Content Supporting Information Author Information Corresponding Author ORCID Notes Biographies Acknowledgments References

A B B C E F G G I J

Y AA AA AA AA AA AA AA AA AB

1. INTRODUCTION Singlet fission refers to the conversion of a singlet excited state to two triplet excited states in a molecular system. Since its inception 50 years ago, the field of singlet fission has enjoyed a renaissance over the past eight years. The renewed interest has been motivated in large part by the potential of using singlet fission in solar cells to exceed the Shockley−Queisser limit.1 This motivation can be traced to a seminal paper by Hanna and Nozik in 2006 on the concept of carrier multiplication solar cells,2 but Dexter was the first to propose the sensitization of conventional solar cells by singlet fission chromophores in a lesser-known paper in 1979.3 The basic idea is as follows. Solar radiation comes from a white light source. For a solar cell based on a single semiconductor bandgap (Eg), the excess excitation energy from photons with hν > Eg is lost as heat, and

J K K L M N Q R V W

Received: September 17, 2018

A

DOI: 10.1021/acs.chemrev.8b00572 Chem. Rev. XXXX, XXX, XXX−XXX

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2. FUNDAMENTALS OF SINGLET FISSION

this loss is partially responsible for the Shockley−Queisser limit of 33% in power conversion efficiency.4 If a singlet fission chromophore absorbing at ≥2 × Eg is combined with a conventional semiconductor/chromophore absorbing at ≥Eg, part of the excess excitation energy is harvested as increased photocurrent and the theoretical power conversion efficiency is increased to 44%. While the oft-cited reason in recent publications on singlet fission has been solar energy conversion, there are also intrinsic interests from a fundamental perspective. Singlet fission involves the triplet pair state, which, in a simplistic sense, is considered as double excitations in a single or a pair of neighboring molecules. Thus, singlet fission serves as a fascinating model system for the investigation of many-body interactions. While the single-electron view exemplified by molecular orbitals in chemistry and band structures in physics has shaped our understanding of electronic structures and electronic excitations, this view is inadequate when it comes to the triplet pair state. To appreciate the importance of manybody interactions, one only needs to look at the success of quantum chemistry methods based on multireference configuration interactions. Understanding the nature of the triplet pair state, how it couples to the singlet state, and how it separates into two individual triplets provides deep insight into many-body interactions in molecular systems. The 2010 review in this series by Smith and Michl has provided much guidance to the singlet fission renaissance.5 More concise updates by Michl and co-workers6,7 and others8−13 have appeared over the past 5 years. In the present account, we do not attempt a comprehensive summary of relevant literature but rather focus on understanding singlet fission mechanisms using selected examples. The need for understanding will become evident as we discuss the misconceptions and confusion in the singlet fission literature. A major source of confusion in singlet fission literature arises from the different meanings of the triplet pair states, the singlet fission rates, or the experimental observables to different authors. It is therefore our objective to clarify the definitions of the triplet pair state and the rate of singlet fission. We discuss how the triplet pair state differs from two triplets or one singlet, how it is manifested in spectroscopy, and how it couples to the singlet or two triplets. Specifically, we emphasize the necessity of distinguishing a triplet pair state that maintains electronic coherence between the two constituent triplets from one that has lost it. Following recent notations of Scholes and co-workers,14 the former can be labeled 1(TT) and the latter 1 (T···T). We present rationale for defining singlet fission rate as the rate of forming 1(T···T), not that of forming 1(TT) as is assumed by many authors. We show the necessity of considering vibronic, not purely electronic, coupling. In the case of singlet fission in crystalline molecular solids, we discuss the often neglected role of electronic delocalization and band formation,8 the entropic contributions to triplet separation,15 and the dynamic equilibrium among the electronic states involved.16 In the discussion on experimental measurements, we remind the readers that each experiment provides a specific and often limited view of a complex physical process. On the practical side, attempts of implementing singlet fission in solar energy conversion have been addressed by a number of authors17−22 and will not be discussed further here.

2.1. General Mechanisms

We mainly consider the lowest-energy excited singlet, S1, and the lowest-energy triplet, T1, in the singlet and triplet manifolds, respectively; much of the following discussion also applies to higher-energy Sn and Tn (n > 1) states involved in singlet fission.15 The S1 state is higher in energy than the T1 state in molecules, and the difference is twice the quantum exchange energy in simple HOMO/LUMO approximation.9 When the S1 energy is approximately twice that of the T1 energy, the energy conservation condition in the conversion of one S1 state to two T1 states is satisfied with or without the involvement of vibrational quanta. This process, i.e., singlet f ission, also preserves spin angular momentum if the two triplets are properly coupled. Because of spin conservation, singlet fission can be considered a form of internal conversion and can occur on ultrafast time scales. This contrasts the orders-of-magnitude slower process of intersystem crossing, i.e., the conversion of one singlet to one triplet, where angular momentum conservation is achieved only with spin−orbital coupling or scattering with magnetic impurities.23 Singlet fission was discovered in the 1960s in photophysical studies of anthracene24 and tetracene.25 It was proposed as the reverse of triplet annihilation to explain intriguing fluorescence properties, including delayed fluorescence and two-photon excitation.26,27 This proposal was subsequently verified by the magnetic field dependence of the delayed fluorescence.28−30 In the commonly used but inaccurate view of two molecules in a crystalline solid, an S1 on one molecule interacts with the neighboring molecule in the ground state, S0, to form the triplet pair state, 1(TT), assumed to be distributed over the two molecules. Here, 1(TT) most often represents the singlet state from the electronic coupling between two T1 states in homofission involving two identical molecules, but it can also describe that between two different triplets, Tn and Tn′, in hot homofission15 or in heterofission involving two different chromophores.31 The triplet pair subsequently separates into two individual triplets (T1) as they move away from each other. This mechanism is represented by a two-step process:28,29,32 1

2

S0 + S1 V 1(TT) → T1 + T1 ←⎯⎯⎯

(1)

Despite the widespread acceptance of this model, much confusion has appeared in recent years, particularly regarding the nature of the triplet pair state. Rigorously speaking, one should take a dynamic view of the 1(TT) state as it evolves from its initial formation, followed by interaction with the environment (electronic and phononic), and the ultimate formation of two individual triplets. Initially, the 1(TT) state is a many-electron singlet state with the two constituent triplets electronically coupled and maintaining electronic coherence. The loss of electronic coherence between the two triplets plays a decisive role in granting individuality to each triplet and determining its properties in subsequent charge- or energytransfer processes. In contrast, the loss of spin coherence on longer time scales involves triplet pair states with very small energy differences (GHz)10,33 and should have little effect on the electronic or chemical properties of the triplets. In addition, the S1 state is usually delocalized in either crystalline solid or a molecular dimer; as a result, the notation of S0 + S1 is a misnomer. We adapt the proposal of Scholes and coB


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workers14 and write down singlet fission as a three step process: 1 1

2

2.2. Diabatic and Adiabatic States

An essential distinction one must make in discussing singlet fission is adiabatic versus diabatic states.11,51 All states in eq 2 (or 1) can in principle be presented as adiabatic states, i.e., eigenstates of the total Hamiltonian. The time-dependent wave function in the presence of an initial electromagnetic field (optical excitation) can be expressed as a linear combination of adiabatic states. These adiabatic states provide accurate descriptions of the dynamic process and are relevant to timedependent experimental observables. Because of the explicit dependence of the electronic wave function of the adiabatic states on nuclear coordinates along the reaction path, the time evolution of the nuclear wavefuction depends on the so-called derivative couplings between different electronic states (the derivative is with respect to the nuclear coordinate).51 These derivative couplings are difficult to obtain from a computational perspective. An alternative to the adiabatic approaches is to find a basis in which the character of each electronic wave function does not change along a set of nuclear coordinates or reaction path. More precisely speaking, the derivative coupling between each pair of electronic wave functions in this basis is strictly zero. These wave functions constitute a diabatic basis, the advantage of which is that coupling between two states in this basis is through the electronic Hamiltonian only. Although diabatic and adiabatic states can describe a dynamic process with equal accuracy, constructing the former from the latter is far from straightforward.51 This difficulty is partly responsible for the practice in the singlet fission community of using approximate or intuitive diabatic basis,52−57 e.g., constructing model diabatic states from a few single configurations of localized excitations based on the highest occupied molecular orbital (HOMO or H) and the lowest unoccupied molecular orbital (LUMO or L) in fixed nuclear geometries. This is illustrated by the molecular dimer model within the smallest 2 × 2 active space consisting of HOMO and LUMO on each molecule (Figure 1). Here, each gray box represents a diabatic state and each colored arrow is electronic coupling between a pair of diabatic states. The lefthand side shows the four spin−orbital wave functions of the

3

S1 V (TT) ⎯⎯→ (T···T) ⎯⎯→ T1 + T1 1

←⎯⎯⎯⎯⎯⎯

←⎯⎯⎯⎯⎯⎯

(2)

1

where the correlated triplet (T···T) represents the triplet pair that has lost electronic coupling/coherence but retains spin coherence. A similar classification was made earlier by Zhu and co-workers,17 who labeled 1(TT) and 1(T···T) as two distinct multiexciton states, ME and ME′. A great deal of singlet fission research involves finding appropriate theoretical and experimental descriptions of the all-important triplet pair state, which may have different meanings to different authors. While some authors opted not to distinguish the triplet pair state that retains electronic coherence from that which has lost it, such practices have led to major confusion and misconceptions on the definition or calculation of singlet fission rates. From both conceptual and practical perspectives, we argue that the rate of singlet fission should be defined as the rate of forming 1(T···T), not that of 1 (TT). The importance of this distinction is underscored by the discovery that the 1(TT) state with strong intertriplet electronic coupling in pentacene dimers exhibits spectroscopic signatures and chemical properties distinct from those of individual T1 states.34 In aggregates or crystalline solids, the distinction between 1(TT) and 1(T···T) may be attributed to intertriplet distance, with the former being a triplet pair on neighboring molecules and the latter being beyond nearest neighbors. However, we prefer the strict definition of intertriplet electronic coherence as the distinguishing factor. In the presence of excitonic band formation discussed later, the two triplets can be located at distances beyond that of the nearest neighbor and still retain electronic coherence;35 decoherence of such a delocalized 1(TT) state or band occurs via coupling to the phonon bath, leading to the formation of 1 (T···T). Only for sufficiently weak inter-triplet electronic coupling, of the order of or smaller than coupling of 1(TT) to the phonon bath, can one neglect the distinction between 1 (TT) and 1(T···T). The necessary distinction between 1(TT) and 1(T···T) also calls into question the appropriateness of the term “intramolecular singlet fission” used in some of the recent publications on covalently linked and strongly coupled molecular dimers/oligomers.36−42 For example, in an endlinked pentacene dimer without spacer, the photogenerated triplet pair state clearly remains as 1(TT) and exhibits a characteristic 1(TT) → S2 transition in the excited-state absorption spectrum.34,43 This 1(TT) state is more appropriately called a dark singlet state, as is known in the photophysics literature on oligoenes and carotenoids, where an optically dark S1 state of Ag symmetry is located energetically below the bright Bu state (π → π*).40,44−50 This dark Ag state is essentially an intramolecular 1(TT) state with large intertriplet binding energy, up to ∼1 eV, and is believed to facilitate nonradiative recombination. In addition to the dark Ag state, a less tightly bound triplet pair state labeled S* state may result from the twisting of neighboring CC units in carotenoids. This more weakly bound triplet pair state has been proposed to undergo singlet fission, albeit with low yields due to competition from nonradiative recombination of the more strongly bound Ag state.47−50 For a detailed discussion on this subject, interested readers are referred to the recent perspective by Kim and Zimmerman.12

Figure 1. Electronic configurations for a molecular dimer within the 2 × 2 HOMO−LUMO approximation. Photoexcitation from ground state (S0S0) in the left box creates excited states in the right box, within which the gray boxes represent diabatic states: left, singlet excited states; middle, charge-transfer states; right, triplet pair states. Each colored arrow represents electronic coupling between a pair of diabatic states. C


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singlet excited state (S0S1), with one molecule in the S1 state and the other in the S0 state. The middle shows the chargetransfer states (CT) with one molecule in the cationic (D+) state and the other in the anionic (D−) state. The right shows the two triplet pair state configurations with total singlet spin. Here we have neglected double excitations localized on each individual molecule. Each adiabatic state may be formed from mixing of these diabatic states. The colored arrows represent intermolecular one-electron transfer integrals between HOMOs (JHH), LUMOs (JLL), LUMO−HOMO (JLH), and mixed HOMO−LUMO two-electron transfer integrals (JJLH). These couplings are usually non-zero (depending on symmetry) in the presence of intermolecular orbital overlap. The one-electron transfer and two-electron transfer integrals are responsible for CT-mediated and direct triplet pair generation, respectively. While the use of these approximate diabatic states has been common practice and has contributed to the understanding of singlet fission in model systems,52−57 one should be mindful of its limitations. As pointed out by Casanova most recently in this series, the use of these approximate diabatic states “··· might be unable to capture the complexity of the fission process and the subtleties of the states involved” and “··· different approximations to the definition of diabats might result in sensible quantitative discrepancies or even drive to qualitative differences and apparent contradictions regarding the character of the electronic states or the magnitude of the computed couplings”.11 For example, the localized diabats in Figure 1 for the S1 and CT states do not accurately represent those in crystalline solids where both electronic and excitonic delocalization introduces non-negligible band characters (section 2.6). Perhaps no example is more dramatic in illustrating the perils of intuitive diabats than the practice of constructing the 1 (TT) state within the 9-spin state representation, as detailed in sections 2.3 and 2.4. In the following, we use bold italic fonts to represent adiabatic states and normal italic fonts for diabatic states. The “S1” in each of the four diabatic S0S1 states in Figure 1 represents a localized singlet state on one of the two molecules in the dimer. This is to be distinguished from the adiabatic “S1”, which may be formed from a linear combination of all the ÄÅ ÅÅ ⟨S S |Ĥ |S S ⟩ ÅÅ 0 0 0 0 ÅÅ ÅÅ ÅÅ ⟨S1S0|Ĥ |S0S0⟩ ÅÅ ÅÅ ÅÅ ⟨S S |Ĥ |S S ⟩ ÅÅ 0 1 0 0 ÅÅÅ ÅÅ⟨D+D−|Ĥ |S S ⟩ ÅÅ 0 0 ÅÅ ÅÅ − + ÅÅ⟨D D |Ĥ |S0S0⟩ ÅÅ ÅÅ ÅÅ ⟨TT |Ĥ |S S ⟩ 0 0 ÅÅÇ 1 1

diabatic states, due to the interdiabatic electronic couplings (colored arrows). The same argument applies to each CT or TT state, which can be a linear combination of all the diabatic states in Figure 1. Buchanan, Havlas, and Michl provided the hitherto most comprehensive treatment of singlet fission within this 2 × 2 active space.7 The spin−orbitals of the ground (S0S0) and the excited diabats in Figure 1 are given by S0S0 = NS0S0|hAαhAβhBαhBβ|

(3)

S1S0 = NS1S0 2−1/2(|hAαlAβhBαhBβ| − |hAβlAαhBαhBβ|)

(4)

S0S1 = NS0S12−1/2(|hAαhAβhBαlBβ| − |hAαhAβhBβlBα|)

(5)

1 + −

D D = N+−2−1/2(|hAαlBβhBαhBβ| − |hAβlBαhBαhBβ|)

1 − +

D D = N −+2−1/2(|hAαhAβlAαhBβ| − |hAαhAβlAβhB α|)

ÄÅ 1 −1/2 Å ÅÅ|h αl αh βl β|+|h βl βh αl α| TT N 3 = 1 1 TT A A B B ÅÅÅÇ A A B B 1 1 1 − (|hAαlAβhBαlBβ| + |hAαlAβhBβlBα| + 2 ÑÉÑ |hAβlAαhBαlBβ| + |hAβlAαhBβlBα|)ÑÑÑÑ ÑÖ

⟨S0S0|Ĥ |S0S1⟩

⟨S0S0|Ĥ |D+D−⟩

⟨S0S0|Ĥ |D−D+⟩

⟨S1S0|Ĥ |S1S0⟩

⟨S1S0|Ĥ |S0S1⟩

⟨S1S0|Ĥ |D+D−⟩

⟨S1S0|Ĥ |D−D+⟩

⟨S0S1|Ĥ |S1S0⟩

⟨S0S1|Ĥ |S0S1⟩

⟨S0S1|Ĥ |D+D−⟩

⟨S0S1|Ĥ |D−D+⟩

⟨D+D−|Ĥ |S1S0⟩ ⟨D+D−|Ĥ |S0S1⟩ ⟨D+D−|Ĥ |D+D−⟩ ⟨D+D−|Ĥ |D−D+⟩ ⟨D−D+|Ĥ |S1S0⟩ ⟨D−D+|Ĥ |S0S1⟩ ⟨D−D+|Ĥ |D+D−⟩ ⟨D−D+|Ĥ |D−D+⟩ ̂ ⟨TT 1 1|H |S0S1⟩

(7)

(8)

where each |···| is a 4 × 4 Slater determinant ensuring proper antisymmetrization of the four-electron wave function; each spin−orbital is a product of constituent spatial (h or l for the HOMO or LUMO, respectively) and spin (α or β) on the two molecules (A and B). The Ns are normalization constants. The two charge-transfer states are represented by 1D+D− and 1 − + D D states in eqs 6 and 7, respectively. The diabatic state consisting of an electronically coupled triplet pair on the two chromophores is shown in eq 8 and will be detailed in the next section. Buchanan, Havlas, and Michl wrote down the Hamiltonian matrix from this diabatic basis as

⟨S0S0|Ĥ |S1S0⟩

̂ ⟨TT 1 1|H |S1S0⟩

(6)

̂ + − ⟨TT 1 1|H |D D ⟩

where Ĥ is the static electronic Hamiltonian; the electromagnetic field of optical excitation can be added to Ĥ as a time-dependent perturbation to simulate the dynamics. In the latter, the oscillator strength is usually dominated by those of the S0S0 → S0S1 and S0S0 → S1S0 transitions. To the first approximation, one can set the transitions to D+D−, D−D+, and T1T1 states to zero. Adiabatic states that are relevant to experimental observables are obtained from the diagonalization of the Hamiltonian matrix in eq 9.

̂ − + ⟨TT 1 1|H |D D ⟩

ÉÑ Ñ ⟨S0S0|Ĥ |TT ÑÑ 1 1⟩ Ñ ÑÑ ÑÑ Ñ ⟨S1S0|Ĥ |TT ⟩ 1 1 Ñ ÑÑ ÑÑ ÑÑ ⟨S0S1|Ĥ |TT 1 1⟩ Ñ ÑÑ ÑÑ + − ̂ ÑÑ ⟨D D |H |TT ⟩ 1 1 Ñ ÑÑ ÑÑ − + ̂ ÑÑ ⟨D D |H |TT 1 1⟩Ñ ÑÑÑ ÑÑ ̂ Ñ ⟨TT 1 1|H |TT 1 1⟩ Ñ ÑÖ

(9)

The singlet fission process is dynamic and can be represented by the density matrix, ρ̂, which evolves with time. Each diagonal element of the density matrix, ρ̂jj, is the population of the diabatic state j, and each off-diagonal element, ρ̂ij, is the coherence between diabatic states i and j. Within the density matrix formulation, as represented by the Liouville−von Neumann equation, one can use the Hamiltonian matrix such as that in eq 9 with the time-dependence in Ĥ given by an excitation laser pulse included explicitly. D


Chemical Reviews d ρ ̂( t ) = [Ĥ (t ), ρ (̂ t )] − iℏD̂ (10) dt ̂ where D represents interaction with the environment. Using dynamic perturbation theory to the second order and including the Markovian approximation, one turns eq 10 into the Redfield equation with D̂ = R̂ ρ̂, where the Redfield tensor R̂ describes population relaxation and pure dephasing due to interaction with the phonon bath at a finite temperature.54,55,58 We will return to this equation later when we discuss singlet fission rates. 2.3. Sixteen-Spin State Representation of the Triplet Pair: 1 (TT) Within the minimal 2 × 2 active space, eq 8 represents an antisymmetric spin−orbital wave function. While this is presumably the lowest-energy triplet pair diabat of singlet character, there is a manifold of other triplet pair states within the four-electron representation. In the limit of weak spin− orbital coupling, which is the case for most organic molecules of interest, we can write the total wave function as the product of a spatial wave function (ϕi) and a spin wave function (χi). iℏ

Ψi = P[|ϕi⟩·|χi ⟩]

χ1 =

1 3

ÅÄÅ ÑÉ ÅÅθ + θ − 1 (θ + θ + θ + θ )ÑÑÑ ÅÅ 1 6 2 3 4 5 Ñ ÅÇ ÑÑÖ 2

1 = 3

χ3, M = 0 s

1 (θ11 + θ12) − 6

χ3, M =−1 = s

s

(h)

1 [θ2 − θ3 + θ4 − θ5] 2

(i)

χ5, M = 1 =

1 (θ8 − θ9) 2

(j)

χ5, M =−1 =

1 (θ11 − θ12) 2

(k)

1 (θ1 + θ6 + θ2 + θ3 + θ4 + θ5) 6

(l)

1 [θ7 + θ8 + θ9 + θ10] 2

(m)

1 [θ11 + θ12 + θ13 + θ14 ] 2

(n)

|χ6, M = 2 = θ15

(o)

|χ6, M =−2 = θ16

(p)

s

(12)

s

s

χ6, M = 0 = s

χ6, M = 1 =

M=0

s

θ1 θ2 θ3

ααββ αβαβ βααβ M = +1

θ4 θ5 θ6

αββα βαβα ββαα M = −1

θ7 θ8 θ9 θ10

ααβα αβαα βααα αααβ M = +2

θ11 θ12 θ13 θ14

αβββ βαββ ββαβ βββα M = −2

θ15

αααα

θ16

ββββ

(f)

1 3 (θ11 + θ12 + θ13) − θ14 12 2

χ5, M = 0 =

Table 1. Primitive Spin Functions for the Four-Electron System (Reprinted with Permission from Ref 59; Copyright 2015 American Chemical Society)

(e)

(g)

s

χ4, M =−1 =

(d)

1 3 (θ7 + θ8 + θ9) + θ10 12 2

χ4, M = 1 = −

There has been much confusion in the literature on the form of the total spin wave function, χi(1,2,3,4). In principle, one needs 2 × 2 × 2 × 2 = 16 basis functions from four electrons, each with two possibilities (α or β), to construct the orthonormal Hilbert space. Table 1 shows the 16 primitive

2 θ13 3

1 [θ1 − θ6 + θ2 + θ3 − θ4 − θ5] 6

s

where P is antisymmetrizer, i.e., Slater determinant. Within the minimal 2 × 2 active space, Ψi becomes

(b)

2 1 (θ8 + θ9) θ7 − 3 6

s

χ4, M = 0 =

(a)

ÄÅ ÉÑ ÅÅ Ñ ÅÅθ1 − θ6 − 1 (θ2 + θ3 − θ4 − θ5)ÑÑÑ (c) ÅÅÇ ÑÑÖ 2

χ3, M = 1 =

(11)

|Ψ⟩ i ≈ P[|hA (1)lA(2)hB(3)lB(4)⟩·|χi (1, 2, 3, 4)⟩]

1 [θ2 − θ3 − θ4 + θ5] 2

χ2 =

Review

χ6, M =−1 = s

s

s

(13)

where |χ1⟩ and |χ2⟩ are singlet states; |χ3⟩, |χ4⟩, and |χ5⟩ are triplet states; |χ6⟩ is the quintet state. We combine these with the spatial part of the wave function, |ϕi⟩, and apply the antisymmetrizer P to obtain the 16 spin−orbital functions (eqs 11 or 12). These functions are only eigenfunctions of the total Hamiltonian under the approximation that there is no spin− orbital coupling. Including spin−orbital coupling will turn these into diabatic states. The energies corresponding to these spin−orbital eigenfunctions within the four-electron space are

functions used for the basis set.59 The construction of a basis set with multiple spins is not trivial;60 in the case of χi(1,2,3,4), the basis is not unique because there are multiple basis

Ei = ⟨Ψ|i H |Ψ⟩ i

functions that give the same spin quantum numbers (S and MS) in overall singlet or triplet states. Using proper linear

(14)

Including only two-electron exchange integrals (and neglecting three- and four-electron exchanges), we obtain the following energies for the singlets (1−2), the three triplets (3− 5), and the quintet (6). The detailed mathematical steps for obtaining the energies, referenced to the ground-state energy,

combinations to give a diagonalized Hamiltonian matrix, Scholes constructed 16 orthonormal spin eigenfunctions that are adapted here:59 E


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are shown in the Supporting Information. These energies are shown in descending order as

where

1 |hA(2)lA(1)⟩ (a) r12 1 = ⟨hB(1)lB(2)| |hB(2)lB(1)⟩ r12

K 0 = ⟨hA(1)lA(2)|

Khh=⟨hA(1)hB(2)|

1 |lA(2)lB(1)⟩ (b) r12

1 |hA(2)hB(1)⟩ (c) r12

Klh = ⟨lA(1)hB(2)|

1 |lA(2)hB(1)⟩ (d) r12

Khl = ⟨hA(1)lB(2)|

1 |hA(2)lB(1)⟩ (e) r12

the two-electron Coulomb repulsion (in atomic

units). In this representation, E0 + K0 and E0 − K0 are the singlet and triplet excitation energies, respectively, on a single molecule. These two-electron exchange integrals are related to the spatial colocalization of the wave functions involved. Thus, the intramolecular K0 is much larger than the intermolecular Kll, Khh, Klh, and Khl. As pointed out by Scholes,59 the magnitudes of Kij (i, j = l, h) are of the order O(S2K0), where S is the overlap integral for each pair of molecular orbitals involved. When intermolecular wave function overlap is negligible, one can set the intermolecular exchange integrals (Kll, Khh, Klh, and Khl) to zero, and the energies in eq 16 reduce to the results at the weak electronic coupling limit, as reported by Scholes.59 This weak coupling limit is appropriate for the (T···T) state. The very definition of the 1(TT) state requires electronic coupling between the two triplets. This necessitates the inclusion of intermolecular exchange integrals. With the inclusion of intermolecular exchange integrals, we arrive at the following tentative conclusions: (1) The energy of the quintet (Ψ6) could be lower than that of the singlet (Ψ1). While the singlet Ψ1 is commonly assumed to be most relevant to singlet fission, the quintet Ψ6 can also be involved and can in some cases be energetic sinks within the triplet pair manifold. This is contrary to previous beliefs but consistent with recent observations in electron spin resonance (ESR) experiments.61−63 An accurate prediction of energy ordering requires configuration interactions in larger spaces,64,65 beyond the 2 × 2 active space. (2) From eq 15e, we note that E1 > E0 − 2K0. Thus, within the four-electron spin space in the absence of configuration interaction, one can consider the triplet pair state (Ψ1) repulsive as 1(TT) evolves into 1(T···T) and the Kij(s) terms decrease to zero. However, recent experiments on pentacene dimers,34 polymers,66 and crystals67 suggest that the triplet pair binding energy is higher than kBT at room temperature. Again, accurate estimates of triplet pair energies require configuration interactions in larger spaces.64,65 (3) Because of the non-negligible values of intermolecular exchange integrals, the triplet pair states delocalize into bands in crystalline solids, with the bandwidths of the order of Kij(s), depending on symmetry and dimensionality. A many-body GW-PBE calculation on crystalline pentacene suggests a T1 bandwidth of ∼50 meV and an S1 bandwidth of ∼100 meV.68 The 1(TT) bandwidth is expected to be between these values. A recent measurement by Huang and co-workers reported similar diffusivity for 1(TT) and S1 in crystalline tetracene,69 suggesting the two states are similarly delocalized.

Here, E0 is the HOMO−LUMO excitation energy. The K(s) are intramolecular (K0) or intermolecular (Kll, Khh, Klh, and Khl) two-electron exchange integrals (Figure 2),

Kll = ⟨lA(1)lB(2)|

1 is r12

(16)

2.4. Nine-Spin State Representation of the Triplet Pair: 1 (T···T)

Many authors in the singlet fission community have adopted the 9-spin state representation in the construction of the triplet pair wave function. This approach treats the two triplets as electronically isolated entities and the 3 × 3 = 9 primitive spin basis functions are simply |TmA s⟩ | TmB s′⟩, where A and B label the two molecules and ms, ms′ = 1, 0, −1 are the projected spin quantum numbers. With proper linear combinations, one obtains the following orthonormal spin function basis set |S,MS> consisting of singlet |φ1⟩, triplets |φ2⟩, and quintets |φ3⟩:

Figure 2. Schematics of two-electron exchange integrals in the 2 × 2 active space. F


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2.5. Binding Energy of the Triplet Pair State

The 9-spin state representation of the triplet pair diabat, 1(T··· T), assumes zero binding energy between the two triplets, as implicit in the absence of intertriplet electronic interaction. The 16-spin state representation of the triplet pair diabat, 1 (TT), predicts repulsive interaction between the two triplets, because it is higher in energy than 2 × T1 by the two-electron exchange integrals, eq 15e. However, actual experimental observables are related to adiabatic states, not diabatic states. In particular, the 1(TT) adiabatic state results from configuration interaction of 1(TT) with S1 and CT diabats in Figure 1 and with higher-order excitations not included in this minimal basis set. The triplet−triplet binding energy of 1(TT) can be dominated by contributions from configuration interactions. Recent experiments on pentacene dimers,34 polymers,66 and other solid-state acenes67,71 suggest that the binding energies between the two triplets in the 1(TT) state are larger than kBT at room temperature. Trinh et al. showed that the tightly bound 1(TT) state exhibits spectroscopic signatures distinctly different from those of individual triplets.66 More importantly, the large inter-T1 binding energy in 1(TT) inhibits the charge transfer from the constituent T1 state to the electron acceptor.34 Yong et al. reported an interT1 binding energy of ∼30 meV for 1(TT) from singlet fission in thin films of a series of acenes and modified acenes based on the red-shifted fluorescence emission;71 however, this interpretation is debatable as it is difficult to distinguish fluorescence emission from 1(TT) or a self-trapped S1 state due to exciton−phonon coupling. Ginsberg and co-workers found that the two triplets from singlet fission in (triisopropylsilyl)ethynyl (TIPS)-pentacene remain bound for hundreds of picoseconds within each single crystal domain, suggesting triplet pair binding energy larger than kBT at room temperature.67 In contrast to experimental suggestions, computational studies based on high-order CI methods reported little binding energy in the 1(TT) state.43,72 This discrepancy between experiments and computations reflects the limitations of current theoretical methods in treating singlet fission in large molecular systems.11,12

The 9-spin state representation was originally proposed by Merrifield to describe triplet−triplet annihilation, i.e., the reverse of eq 2, in oligoacene crystals.70 In T−T annihilation, the first step is the diffusional encounter of two triplets to form a spin-coupled triplet pair, 1(T···T), with negligible inter-T electronic coupling. Because forming 1(T···T) is most important in triplet−triplet annihilation and subsequent steps of 1(TT) and S1 formation are likely not rate-limiting, one can easily comprehend why the 9-spin state formulism is an excellent approximation in the treatment of triplet−triplet annihilation kinetics. The widely accepted practice of adopting the Merrifield 9spin state formulism to treat the reverse process of singlet fission is problematic. The Merrifield formulism assumes that the spin Hamiltonian is a perturbation and neglects the spatial portion of the electron wave function. This approximation is valid as long as there is no wave function overlap between the two triplets, as in 1(T···T), but breaks down when intertriplet electronic coupling is present, as in 1(TT). To understand this problem, let us examine the total wave function consisting of both the spatial part and the spin part: |Ψ′⟩ i = P′[|ϕA · ϕB⟩·|φi(A , B)⟩]

2.6. Delocalization in the Extended Solid: Exciton Bands

(a)

≈ P′[|hA(1)lA(2)⟩·|hB(3)lB(4)⟩·|φi(A , B)⟩] (b)

While dimer models as exemplified by the minimal 2 × 2 active space discussed earlier have been widely used in theoretical/ computational studies on singlet fission, their applicability to solids is problematic. In extended aggregates and crystalline solids, intermolecular coupling results in delocalization of electronic excitations and the singlet state is extended over a group of molecules. Such delocalized electronic excitations are called excitons. There are two types of intermolecular couplings that result in singlet excitons: (1) dipole−dipole interaction and (2) intermolecular charge transfer. These two interactions lead to the formation of Frenkel and Mott− Wannier (also called charge-transfer) exciton bands, respectively, although these two bands are often mixed in many molecular crystals.73−75 The first type of interaction is the well-known coupling between transition dipoles responsible for resonant energy transfer between chromophores. At intermolecular separations larger than molecular dimensions, this coupling is approximated by point dipole−point dipole Coulomb interactions. However, this approximation breaks down at shorter distance, and more involved quantum mechanical calculations become

(18)

The modified antisymmetrizer P′ performs permutation on electrons only within molecule A or B.59 Although the spin wave functions of the two lowest-energy triplet pair states, |φ1⟩ and |φ3⟩ in the 9-spin state formulism, are equivalent to |χ1⟩ and |χ6⟩ in the 16-spin state formulism, the restricted permutation of the spatial part of the wave function |ϕA·ϕB⟩ in eq 18 is invalid when one applies it to 1(TT). Implicit in this restricted permutation is the absence of inter-triplet electronic coupling, and this amounts to setting all the inter-molecular two-electron exchange integrals, Kij(s) in eq 15, to zeros. This leads to erroneous energies for the 1(TT) diabatic state. In addition to problems with energies, the incorrect antisymmetric wave function (spatial and spin) for 1(TT) should also propagate the errors to all the coupling matrix elements between the triplet pair diabat and each of the other configurations in the Hamiltonian matrix, eq 5. This may lead to problems in understanding the dynamics and in calculating singlet fission rates, as detailed later. G


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usually somewhere in between the two extremes, and this results in the mixing of J- and H-aggregate characteristics. When inequivalent molecules are contained in a unit cell, the eigenenergies shift both higher and lower depending on light polarization. This is called Davydov splitting.79 The second type results from intermolecular CT interactions, the same interaction responsible for the formation of valence band (VB) and conduction band (CB) from HOMOs and LUMOs, respectively. In the Merrifield approach, the Frenkel-only Hamiltonian in eq 19 is now modified within the nearest-neighbor approximation as

necessary. Such pairwise coupling leads to the formation of socalled J- or H-aggregates in ordered molecular assemblies and, more generally, Frenkel exciton (FE) bands in molecular crystals. A Frenkel exciton can be viewed as a coherent superposition of localized excitons (S1) on individual molecules in the assembly. The Frenkel exciton band has been treated with various levels of theory, including the Bethe−Salpeter equation (BSE).74 To understand the delocalization of a localized S1 state into the Frenkel exciton band and the resulting optical properties, we consider a linear chain of ordered molecules with only one excitation. For illustrative purposes, we consider only nearest-neighbor (NN) interaction, which is equivalent to the tight binding model in electronic structure; this is a crude approximation because Coulomb interaction is long-ranged. Following the approach of Merrifield, the Hamiltonian operator is given by76,77 FE HNN

= ES1 ∑

an†an

+ JD ∑

n

(an†an + 1

n

+

an†+ 1an)

HNN =

k

CT HNN (k) = ECT (ck†, +1ck , +1 + ck†, −1ck , −1)

+ h. c.

c†k,+1

(20)

where μ→i is the transition dipole moment on site i and r→12 is the position vector between two neighboring molecules. Within this nearest-neighbor approximation, the Frenkel exciton band has the familiar tight-binding solution: EFE(k) = ES1 + 2JD ·cos k

(23)

c†k,−1

(24)

HCT NN

Here, or in creates from ground state a CT exciton at momentum k with electron transfer to the neighbor. HFE−CT describes the mixing between FE and CT excitons; te NN (th) is the electron (hole) transfer coupling between neighboring LUMOs (HOMOs). The results of the Hamiltonian in eq 22 are mixed FE−CT exciton bands. Experimentally, the CT and FE characters are manifested in electro-absorption spectra; the transition energy of a pure CT exciton and a pure FE exciton scale linearly and quadratically, respectively, with electric field. For example, experimental electro-absorption spectra of crystalline acenes80,81 can only be described by the mixed Frenkel-CT exciton Hamiltonian.82 Similarly, CT-Frenkel mixing reproduces Davydov splitting in linear absorption spectra56,83 and excitonic dispersions84,85 in pentacene. Note that, when we only consider the electronic degrees of freedom in a perfect molecular crystal, both Frenkel and CT excitons are delocalized over the entire crystal. However, electron−phonon coupling and structural disorder lead to the localization of the excitonic wave function spanning a few to a few hundreds of molecules, depending on the competition between delocalization and localization.86 To appreciate the extent of exciton delocalization from intermolecular FE and CT interactions, we show recent computational results of Sharifzadeh et al. (Figure 4)87 on pentacene crystals. These authors applied GW-PBE many-electron theories to crystalline pentacene and found that low-energy singlet excitons are highly delocalized and have significant CT character. The calculated electron−hole correlation functions projected in the ab (left) and ac (right) molecular planes are shown for the (a) S1 and (b) T1 excitons in Figure 4. The S1 state is delocalized over ≥18 molecules with an average electron−hole distance of ≥6 Å, indicating more CT than FE characters, while the T1 exciton is localized to a single pentacene molecule. The major CT character of the lowest bright S1 state is most obvious in the electron−hole correlation functions in Figure 4. Berkelbach et al. employed a system-bath exciton Hamiltonian to obtain the wave functions and electron−hole correlation distributions for the S1 state in crystalline pentacene;58 their results showed somewhat less S1 delocalization than those in Figure 4. In contrast to the well-developed theories of the singlet FE− CT exciton bands, much less is known about delocalization of the 1(TT) state. Because the 1(TT) state is optically dark, its

(μ1⃗ ·μ2⃗ )r12 2 − 3(μ1⃗ · r12⃗ )(μ2⃗ · r12⃗ ) r12 5

(22)

FE − CT HNN (k) = ak†[(te + the−ik)ck , +1 + (te + the−ik)ck , −1]

(19)

where the creation a†n (annihilation, an) operator creates (destroys) an excitonic state at site n: a†n|0⟩ = |n⟩ (an|n⟩ = |0⟩); ES1 is the local exciton energy and JD is the coupling, which in the point dipole approximation is given by JD =

FE CT FE − CT (k) + HNN (k) + HNN (k )] ∑ [HNN

(21)

In typical van der Waals assemblies or crystals of conjugated molecules, the magnitude of JD is on the order of 10−100 meV and its sign can be negative or positive, depending on the relative orientation of the dipoles (Figure 3).74,78 For dipoles

Figure 3. Frenkel excitonic bands. Schematic illustration of (a) Jaggregate and (b) H-aggregate and corresponding Frenkel exciton band dispersions. The color arrows illustrate (a) the red-shifted transition and (b) the blue-shifted transition.

arranged head-to-tail in a line (Figure 3a), JD < 0, and the molecular assembly is called a J-aggregate. For dipoles arranged side-by-side (Figure 3b), JD > 0 and it is an H-aggregate. Compared to S1 in an individual molecule, the optical transition energy is red-shifted and the transition strength is enhanced in a J-aggregate. The opposites hold for an Haggregate. In reality, the relative molecular orientation is H


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weaker than CT-mediated coupling as large as 100 meV.57,92 The strong mixing between FE- and CT-excitons in aggregates or crystalline solids of organic molecules is expected to result in strong S1−1(TT) band-to-band coupling and ultrafast or coherent singlet fission. The rate of singlet fission is a sensitive function of the relative energetic positions of S1 and 1(TT). On the one hand, FE−CT and 1(TT) exciton band formation, along with electronic-vibrational coupling, can easily result in energy-resonant conditions for ultrafast singlet fission.8 On the other hand, the lowering in S1/CT bands much more than that in the 1(TT) or 1(T···T) manifolds may result in an unfavorable free energy landscape that inhibits singlet fission.6 Finally, the presence of band dispersion in both FE−CT68,73,74,78,91 and the 1 (TT)68,93 manifolds may lead to momentum-dependent singlet fission.68 2.7. Delocalization in a Dimer: The Excimer

An all-encompassing but highly confusing term used in the singlet fission literature is “excimer”. The excimer, short for “excited-state dimer”, usually refers to the transient binding of an electronically excited molecule (atom) with a ground-state molecule (atom). If the two species involved in the binding are not identical, the dimer is referred to as an “exciplex”. The excimer concept was originally developed to describe photophysical phenomena in solutions and gas phases and later in solids.94−96 The driving force for excimer formation can be (1) the sharing of the S1 state between the two chromophores; (2) the mixing of charge-transfer states; and (3) transient formation of covalent bonds. Barring the last mechanism, which is of little relevance to singlet fission, the excimer wave function can be expressed as a superposition of singlet and charge-transfer states in the dimer, i.e., eqs 4−7:94−96

Figure 4. Delocalization and CT characters of excitons in pentacene. Two-dimensional electron−hole correlation function in (a, c) the ab molecular plane (averaged over the c axis) and (b, d) the ac molecular plane (averaged over the b axis) for the (a, b) singlet and (c, d) triplet exciton in crystalline pentacene. Reprinted with permission from ref 87. Copyright 2013 American Chemical Society.

delocalization comes from the CT-like inter-molecular charge transfer integrals, te and th, and the two-electron exchange integrals, Kij (i, j = h, l). In the 2 × 2 active space in Figure 1, the matrix element ⟨T1T1|Ĥ |T1T1⟩, eq 9, consists of various Fock integrals and two-electron integrals that are similar to those in ⟨D+D−|Ĥ |D+D−⟩ and related matrix elements.7 Thus, band formation is also expected for the 1(TT) state. Computational and theoretical studies35,68,88 have addressed the role of delocalization in singlet fission, but little is known about the bandwidth of 1(TT). A recent calculation by RefaelyAbramson et al. of crystalline pentacene based on the manybody GW-PBE approach yields a T1 bandwidth of ∼50 meV, as compared to the ∼100 meV bandwidth for the S1 state.68 The difference results from the fact that the wave function of the T1 state is more compact and localized than that of the S1 state, as shown in Figure 4. We may expect the bandwidth of the 1(TT) state to be between those of T1 and S1 but close to that of the former for a weakly bound triplet pair state. This conclusion contrasts the suggestions of dimer localized triplet pair states in calculations on tetracene by Mayhall89 and by Casanova and Krylov. 90 The extend of 1 (TT) state delocalization in crystalline solids deserves further theoretical investigations. Interestingly, a recent measurement on crystalline tetracene by Huang and co-workers reported that 1 (TT) possesses similar diffusivity as S1 does.69 The high diffusion constant of S1 in crystalline tetracene comes from the strong intermolecular electronic and dipole couplings; we hypothesize that 1(TT) is similarly delocalized. Given the extent of delocalization and exciton band formation, with bandwidths on the order of 50−100 meV in crystalline acenes,68,73,74,78,91 it is no longer accurate or even valid to consider S1 as an isolated molecular excitation or 1 (TT) as two localized T1 excitons on neighboring molecules. These delocalized exciton states may affect or dictate singlet fission in the following ways: (1) the dominance of CTmediated singlet fission; (2) energy-resonant conditions for singlet fission; and (3) k-dependent singlet fission. The direct S1−1(TT) coupling is usually on the order of a few meV, much

|Excimer⟩ = C1|S1S0⟩ + C2|S0S1⟩ + C3|1D +D−⟩ + C4|1D −D+⟩

(25)

Absent in this description is the further stabilization of the excimer state by movement in nuclear coordinates, i.e., vibronic coupling, with the extreme example of direct covalent-bond formation. It is evident from eq 25 that excimer formation in a dimer picture is conceptually equivalent to the delocalization of excitons in a crystalline solid detailed in the last section. Likewise, vibronic coupling in an excimer is similar to a selftrapped exciton.97 The latter is sometimes also called a polaron-exciton, a product of the further stabilization and localization of an exciton by deformation and Coulomb potentials in a crystalline lattice. Given the conceptual equivalency of an excimer in a molecular dimer and an exciton band in a crystalline solid, calling a delocalized excitation in a crystalline solid an excimer is a misnomer. In the presence of translational symmetry, delocalization of electronic excitation in a crystal should not be localized only to two neighboring chromophores. The concept of excimer, i.e., excited-state dimer, is applicable to molecular solids only in the presence of disorder, as in polymeric or amorphous films,96 or when the size of the self-trapped exciton in a crystalline solid is smaller than the unit cell dimension.97 In the latter case, the excited-state delocalization is confined to the unit cell containing at least two molecules. In the following, we opt not to use the confusing term excimer to describe exciton delocalization in crystalline solids. I


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Rigorously speaking, only the rate of forming 1(T···T) in the red box in Figure 5 is defined as the rate of singlet fission. There are two pathways to 1(T···T) formation. The first is the two-step process via the intermediate 1(TT), which can be formed coherently and/or incoherently from S1 (see below). Here, 1(T···T) forms from the electronic dephasing of 1(TT), as a result of coupling to the phononic bath, i.e., nuclear reorganization, or to a mixed electronic and phononic bath due to vibronic coupling to neighboring molecules. The latter essentially describes the diffusive separation of the two triplets in an extended molecular structure such as molecular crystals, polymers, or aggregates. The second pathway involves the direct conversion from S1 to 1(T···T). Because there is little electronic coupling between the constituent T1 states within 1 (T···T), each T1 is expected to be highly localized, and we expect the coupling between S1 and 1(T···T) to be weak. Thus, this conversion step is dominated by an incoherent rate process, as exemplified by calculations based on the nonadiabatic transition formulism, such as Fermi’s golden rule.57 While it is not the rate of singlet fission, the rate of 1(TT) formation is of interest as it may be directly relevant to observables in some experiments. The formation of 1(TT) can occur coherently (1) and/or incoherently (1′), as detailed later in section 3.4. Both processes can also occur in the same system, as reported for singlet fission in crystalline hexacene99 and rubrene.100 The last step (3) in Figure 5 describes the spin dephasing to form two individual triplets. This process occurs on the much longer time scale of ns−μs, as a result of the weak spin−orbital coupling or scattering with magnetic impurities. While this dephasing process can be probed in magnetic field dependent experiments or in electron spin resonance (ESR) measurements,61,62 the physical and chemical properties of 1 (T···T) are expected to be similar to those of two T1 states as far as charge or energy transfer is concerned. Most recently, Nagashima and co-workers examined the singlet-to-quintet conversion dynamics in detail using TR-ESR.63 They concluded that entropy and disorder govern the spin dynamics, concurrent with triplet migration.

Note that the wave function in eq 25 does not include dynamic information. When the time-dependent evolution of nuclear parts of the wave functions is taken into account, we may view excimer formation as ultrafast, determined by coherent nuclear wavepacket motions, or on slower time scales as given by incoherent rate processes. These are similar to the coherent and incoherent channels for 1(TT) formation as shown below. Similar to how exciton delocalization may dictate singlet fission in the crystalline solid, excimer formation in a dimer may (1) enhance electronic coupling between S1 and 1(TT) states via the CT component, thus increasing the singlet fission rate, and (2) change the free energy landscape for singlet fission. In the latter, excimer formation is expected to lower the S1 and CT energy more than it does to the 1(TT) state. When the S1 state is lower in energy than that of the 1(TT) or 1(T··· T), singlet fission is inhibited due to endoergicity. Similarly, if CT is lower in energy than 1(TT), the former may serve as a trap state that provides a nonradiative recombination channel.38 2.8. Rates of Forming 1(TT) and 1(T···T)

The 1(TT) and 1(T···T) states are distinct excitonic states, and, as such, their rates of formation have different physical meanings. However, there has been much confusion in the literature on this elementary definition. This confusion permeates throughout both computational and experimental literature, as pointed out by Scholes.59 In experiments, one spectroscopic signature may probe either 1(TT) or 1(T···T) while the other may probe both; the lack of clear assignment of what one is actually seeing in a specific experiment can lead to fruitless debates.98 In computations, the use of eqs 12 or 18 for the triplet pair wave function should yield different rates, for the formation of 1(TT) or 1(T···T), respectively; yet this critical distinction has been overlooked. To clarify these confusions, we recast eq 2 into Figure 5 where each adiabatic state is labeled by its main diabatic

2.9. Triplet Pair Separation, Transport, and Entropic Contributions

The loss of electronic coherence in 1(TT) to form 1(T···T) results from the spatial separation of the two constituent triplets and coupling of the electronic states to the phonon bath. Here, the spatial separation between the two T1 components can be viewed as a triplet energy-transfer process. While singlet exciton transfer in molecular solids occurs efficiently via the long-range dipole−dipole coupling, also called the Förster mechanism,101 triplet exciton transfer is based on the much slower Dexter mechanism.102 The latter involves short-range exchange coupling between neighboring molecules. Scholes and co-workers carried out kinetic analysis for temperature-dependent singlet fission in TIPS-pentacene. This analysis revealed a triplet energy-transfer step in determining triplet pair separation, with an intermolecular triplet hopping time of ∼3.5 ps. Turner and co-workers applied two-dimensional electronic spectroscopy to singlet fission in rubrene single crystals.103 Their measurement showed that the 1 (TT) state is formed within 20 fs, and its separation to form 1 (T···T) occurs on the picosecond time scale via triplet energy transfer. While the energetics of singlet fission is often discussed in the context of enthalpic change (ΔH), any spontaneous

Figure 5. Schematic representation of singlet fission kinetics. The rate of forming 1(T···T) in the red box is defined as the singlet fission rate.

character. For instance, the adiabatic S1 can contain contributions from CT and 1(TT) diabats, and the adiabatic 1 (TT) can be mixed with S1 and CT diabats. While this adiabatic representation includes the CT diabats implicitly, the direct involvement of adiabatic CT as an intermediate is also possible, but this may be a nonradiative recombination pathway that competes with singlet fission.38 J


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triplet pairs, a result of the dynamic equilibrium among these excitonic populations. Such cooperativity suggests that NTT and NT···T may be approximated by the molecular sites sampled by the S1 state; this approximation is supported by Huang and co-workers who showed similar diffusion constants for S1 and triplet pair states.69 Thus, the entropy change for triplet pair separation can be approximated by

process is in fact governed by free energy change. At constant pressure, it is the change in Gibbs free energy, ΔG = ΔH − TΔS, that determines the direction of a process and the resulting equilibrium. In a molecular solid, the separation of 1 (TT) to 1(T···T) involves a large increase in phase space; thus, there is an entropic driving force for singlet fission. This issue was first addressed by Chan, Ligges, and Zhu15 and was later explored by Kolomeisky et al.104 and Nagashima et al.63 Chan et al. discussed entropic driving force in the context of temperature-independent singlet fission rates in crystalline tetracene using transition-state theory.15 They assumed that the transition state (≠) for singlet fission is the electronically dephased multiexciton state ME′, i.e., the 1(T···T) state in the updated representation, and showed that the entropic gain (TΔS) can compensate for enthalpic increase (ΔH) to account for the temperature-independent singlet fission rate in crystalline tetracene.15 Note that Chan et al. used a widely accepted value of ΔH = 0.17 eV for endoergic singlet fission in crystalline tetracene and found that TΔS ≥ ΔH for T ≥ 170 K.15 However, follow-up measurement revealed temperatureindependent singlet fission down to 10 K.105 Thus, ΔH for crystalline tetracene was likely overestimated, and ΔH in this system is close to zero. Now we address the entropic driving force for triplet pair separation 1(TT) → 1(T···T). The ΔH of this step is simply the triplet pair binding energy, and the entropic change is ΔS = k ln(ΩΤ···Τ/ΩΤΤ), where ΩΤ···Τ and ΩΤΤ are the total configurations sampled by the 1(T···T) and 1(TT) states, respectively. Consider an oligoacene crystal where exciton diffusion occurs predominantly in the ab plane with each molecule surrounded by four neighbors in a herringbone packing motif. We assume that each 1(TT) state resides on two nearest-neighbor molecules while each 1(T···T) state is made up of any pair of molecules. Under these approximations, the total configurations of the 1(TT) and 1(T···T) states are ΩTT = γTT ·2NTT

ij γ NS − ji Ω zy ΔSTT → T ··· T = kB lnjjj T ··· T zzz ≈ kB lnjjjj T ··· T · 1 j Ω z 2 k TT { k γTT

(27)

For crystalline tetracene at room temperature, the LD = 10 nm diffusion length corresponds to NS1 ≈ 1.3 × 103.15 We obtain ΔSTT→T···T = 48 J K−1 mol−1 at room temperature, assuming both singlets and quintets are involved in 1(TT) but only singlets in 1(T···T).

3. COMPUTATIONAL STUDIES OF SINGLET FISSION Computational studies have played major roles in advancing our understanding of singlet fission. A number of reviews on computational and theoretical aspects of singlet fission have appeared recently.7,9,11,12,107,108 Interested readers are referred to these recent reviews for details. Here we choose illustrative examples aimed at understanding singlet fission. We focus on the following topics: (1) ab initio calculations on the nature of the electronic states in singlet fission, particularly the 1(TT) state; (2) the role of electron−phonon coupling in coherent and incoherent singlet fission; (3) incoherent rate equation approaches to singlet fission; and (4) quantum dynamics simulations of singlet fission. 3.1. 1(TT) State from First Principle Quantum Chemical Calculations

Among the electronic states involved in singlet fission, the singlet (S1) and the triplet (T1), including their mixing with CT states, are well-known in molecular photophysics. They can be obtained from standard quantum mechanical methods, such as multireference wave function methods or time-dependent density function theory (TDDFT). When molecules are present in a crystalline solid environment, the singly excited states can be obtained from many-body theories, such as Green’s function approaches, to reveal the delocalization of excitations, as discussed in section 2.6. The 1(T···T) state can also be easily constructed from T1, as discussed earlier in the 9spin state formulism in section 2.4. Unlike the standard treatments of the S1 and T1 states, describing a pair of electronically coupled triplets, i.e., 1(TT), is much more difficult. The computational difficulty lies in the inherent multiexcitation nature of the 1(TT) state, which includes four open-shell molecular orbitals. In addition, correction from dynamic correlation is necessary for such piconjugated systems.109,110 Thus, we cannot use conventional calculations such as Hartree−Fock theory or TDDFT that lack the capability of describing multiexcitation or strong correlation. Instead, the 1(TT) state requires higher-level calculations, such as the complete-active space self-consistent field (CASSCF) method, which can describe multireference states, with complete active space perturbation theory (CASPT2) to take into account dynamic correlation effects.111,112 In practice, the system size of these calculations is limited because of computational cost.

(a)

ΩT μ T = γT μ T ·NT μ T ·(NT μ T − 1) (b)

1 yz zz zz {

(26)

where γTT and γT···T are the spin configurations in the two triplet pair states. Within the 16-spin representation for 1(TT), γTT = 6 if the lowest-energy singlet and quintet states are involved, while in the 9-spin representation for 1(T···T), γT···T = 3 for the three singlet configurations. NTT and NT···T are the number of molecular sites sampled by the triplet pair state due to exciton delocalization and diffusion. The factor of 2 in eq 26a comes from the nearest-neighbor constraint of 1(TT) in the ab plane. What are the numbers of molecular sites sampled by the two triplet pair states, 1(TT) and 1(T···T)? For the S1 state, the efficient Förster coupling leads to long-range diffusion. Take crystalline tetracene as an example. The S1 diffusion length is LD = 10 nm at T = 295 K, and it increases with decreasing temperature to reach a plateau of LD = 60 nm for T < 170 K.106 The diffusion of the 1(TT) or 1(T···T) state is nontrivial. In principle, triplet pair diffusion is dictated by the short-range Dexter mechanism, but there are also more efficient channels that can result in long-range diffusion. In an elegant experiment, Huang and co-workers applied transient absorption microscopy to track the spatial and temporal populations of excitonic states in tetracene crystals.13,16 These authors discovered that the diffusion of S1, 1(TT), and 1(T···T) are cooperative among the fast-moving singlet and slow-moving K


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Figure 6. Vibronic coupling for singlet fission in a covalent pentacene dimer, BP-37. (A) Slice of the potential energy surface, as each substituted pentacene monomer relaxes along the 1435 cm−1 mode shown, with each point calculated at the 8o8e level of theory. The 1(TT) state (i.e., ME) is higher than S1 at Q = 0, but the two states are degenerate at Q = (0.33,0.33). (B) Percent diabatic state composition of the three lowest excited adiabatic states of BP-37 as a function of the normal mode coordinate Q. (C) Representation of the 1435 cm−1 mode in pentacene and the postulated structure of T1 in one substituted pentacene chromophore, reached through the avoided crossing. Reprinted with permission from ref 42. Copyright 2016 American Chemical Society.

GW-PBE to probe singlet fission in crystalline pentacene; the triplet pair state they treated was 1(T···T), not 1(TT), because these authors assumed no electronic interaction between the two constituent triplets.68

Zimmerman et al. pioneered CASSCF/CASPT2 calculations on pentacene monomer and CASSCF calculation (without CASPT2) on pentacene dimers in a 12o12e active space.111 They reported a “dark” double-excited state (D) that is slightly lower in energy than the bright singlet state in a single pentacene molecule, suggesting that it may be the 1(TT) state that undergoes ultrafast coupling via a conical intersection into an intermolecular triplet pair. However, this result contradicts the experimental observation that the lowest-energy S1 state is optically bright in a single pentacene molecule. The interpretation of Zimmerman et al. was revised by Zeng et al. in similar calculations using 12 π-orbitals for the active space for both pentacene monomer and dimers.113 The improved active space revealed that the intramolecular D state is higher in energy than S1 is, but the intermolecular 1(TT) is lower, consistent with experimental observations. This conclusion has been recently confirmed by calculations based on the method of density matrix renormalization group (DMRG).114 Restricted active space double spin-flip (RAS-2SF) method115 has been suggested as an alternative for balanced but lower-cost treatment. This has been successful for relatively small systems including intramolecular and intermolecular singlet fission.116−119 However, the neglect of dynamic correlation effect may sacrifice quantitative accuracy. Most recently, an update on spin-flip method to properly consider the effect of dynamic correlation is demonstrated in the dimer case,120 but further study would be needed for practical use on larger systems.114 While CASSCF/CASPT2, RAS-2SF, and other advanced quantum chemical techniques have been successfully used in calculating the 1(TT) state in singlet fission,43,72,111−114,116−119,121 these methods are restricted to small systems, mainly molecular dimers. In an important development, Refaely-Abramson et al. employed many-body

3.2. Electron−Phonon Coupling in Singlet Fission

Dynamic processes in molecules are characterized by strong vibrational-electronic (vibronic) coupling. This is exemplified by the famous reorganization energy in electron-transfer theory. In molecular crystals, one needs to take into account phonon dynamics properly to explain their optoelectronic properties.122,123 Singlet fission is no exception, and the effect of vibrational/phonon degrees of freedom on both energetics and couplings has been discussed since the onset of this research field.124 In the most extensively studied model systems of pentacene dimers, Zimmerman et al. suggested that motion along the intermolecular coordinate modulates the energies of S1 and 1(TT) states, leading to level crossing and a conical intersection for ultrafast singlet fission despite modest electronic coupling.111,116 Ananth and co-workers reexamined the problem in a more refined calculation and found no level crossing along the intermolecular coordinate.42 Instead, it is the intramolecular coordinate, i.e., the aromatic ring breathing mode at 1435 cm−1, which modulates the energies of the S1 and 1(TT) states and leads to avoided level crossing on sub-ps time scales in the end-linked pentacene dimer, despite the small electronic coupling (∼2 meV) (Figure 6). These theoretical predictions are in excellent agreement with the experimental finding of ultrafast 1(TT) formation in pentacene dimers26,32 and suggest the decisive role of energy resonance conditions. The latter is also evident in the simulation of Bakulin et al., who used a vibronic multistate model in pentacene derivatives to interpret coherent phonons in 2Delectronic spectroscopy.125 L


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PES followed by an incoherent rate process to 1(TT) formation (Figure 8e). The other coherent phonon mode (Figure 8b) breaks symmetry, increases VSF = ⟨S0S1|Ĥ |T1T1⟩, and transfers the coherence to the 1(TT) potential energy surface (PES) (Figure 8d). More importantly, the rapid 1(TT) generation is possible because the Franck−Condon region overlaps with the crossing point of the S1 and 1(TT) PES associated with symmetry breaking. This discovery of the coexistence of coherent and incoherent singlet fission in rubrene, along with similar conclusions in pentacene125 and hexacene,99 reconciles the coherent15,17 and incoherent57 singlet fission mechanisms.

Tempelaar and Reichman recently formulated a vibronic exciton theory in which electronic and vibrational degrees of freedom are treated microscopically and nonperturbatively for singlet fission.126 They calculated energetics and oscillator strengths relevant to singlet fission in pentacene.126,127 Note that these authors constructed the triplet pair diabat from two T1 in the 9-spin state formulism discussed above; their triplet pair is the 1(T···T) state. Phonon modes modulate the energetics of all excitonic states in molecular crystals.122,123 An important state that does not appear in eq 2 is the CT state, which may directly participate in singlet fission or mediate the coupling between S1 and 1(TT). Fujihashi and Ishizaki examined the phonon impacts on the energetics of S1, 1(TT), and CT states by model quantum calculations and showed that the relative energetics of the CT states is also modulated by phonons.128 The phonon modes strongly coupled to CT states are different from those coupled to the 1(TT) state.129 Besides energetics, phonon modes also modulate electronic coupling, and this can be understood as the violation of the Condon approximation. Take, as an example, electronic coupling for the direct singlet fission mechanism, ⟨S0S1|Ĥ | T1T1⟩, which is sensitive to the symmetry between the adjacent molecules. For a pair of linear acenes with C2v symmetry, ⟨S0S1 |Ĥ |T1T1⟩ is strictly zero due to the perfect cancellation of molecular orbital overlaps.6 In such a case, intermolecular symmetry breaking modes may have a dominant effect on SF dynamics, as illustrated by calculations on the tetracene dimer in Figure 7.130,131 In a rubrene crystal, the center tetracene

3.3. Incoherent Rate Equation Calculations of Singlet Fission Kinetics

Johnson and Merrifield initially analyzed singlet fission using conventional rate equations to model the ns−μs dynamics observed in time-resolved fluorescence from tetracene crystals in the presence of magnetic field.134,135 More recently, Burdett and Bardeen adapted this rate process in a density-matrix formulation to model the quantum beats due to spin coherence.33,136 In this approach, the triplet pair is the 1(T··· T) constructed from two T1 within the 9-spin formulism. It is well-suited for the description of step (3) in eq 2, i.e., the separation of 1(T···T) into 2 × T1 or the reverse process in the first step of triplet annihilation. What we are mostly concerned with for the singlet fission process is the rate of forming 1(TT) or 1(T···T) on shorter time scales. The rate constant of forming the triplet pair in singlet fission, kSF, has been treated by a number of authors using Fermi’s golden rule formulism, i.e., the incoherent and nonadiabatic transition between two states, kSF =

2π (HSF )2 (FC) ℏ

(28)

Figure 7. Non-Condon effect in singlet fission demonstrated in covalently linked tetracene. With strict C2v symmetry, coupling between S1 and 1(TT) vanishes. Vibrational motions breaking the C2v symmetry turn on the electronic coupling. Reprinted with permission from ref 131. Copyright 2015 American Chemical Society.

where HSF is the electronic coupling and FC is the densityweighted Franck−Condon factor. In the simplest case of direct singlet fission, HSF = ⟨S0S1|Ĥ |T1T1⟩, but more generally HSF may include contributions from other indirect channels, particularly those mediated by CT states. The FC term enforces the energy-conservation condition in the transition. At the high-temperature limit, the density-weighted Franck− Condon factor is reduced to the form best known as the Marcus electron-transfer theory at the weak coupling limit, ÄÅ É ÅÅ (λ + ΔG 0)2 ÑÑÑ 1 Å ÑÑ Å (FC) ≈ expÅÅ− Ñ ÅÅÇ 4πλkBT ÑÑÑÖ 4πλkBT (29)

chromophores are slip-stacked along the long axis with the offset corresponding to zero ⟨S0S1|Ĥ |T1T1⟩ in a frozen geometry. Experiments revealed thermally activated singlet fission with a time constant of ∼10 ps at room temperature.132,133 In addition to the thermally activated singlet fission occurring on ∼10 ps time scale, Miyata et al. discovered a coherent channel that occurs on the ultrafast time scale ≤ 100 fs.100 Evidence for the coexistence of these two channels comes from the observation of two distinct coherent phonon modes, as detailed in the next section on experimental results. These authors calculated the potential energy surfaces using state-ofthe-art multireference second-order perturbation theory. One coherent phonon mode (Figure 8a) does not change intermolecular symmetry and reflects dynamics on the S1

where k B T is thermal energy, λ is the well-known reorganization energy, and ΔG0 is the free energy difference between the final and initial states. Fermi’s golden rule formulism is valid at the weak electronic coupling limit, and the transition is nonadiabatic. Here the word “weak” refers to the electronic coupling being smaller than coupling to the phonon bath, He‑ph. Equations 28 and 29 have been applied to the estimation of singlet fission rates in model systems that are not particularly efficient.137,138 As HSF increases to the intermediate region, i.e., comparable to He‑ph, the rate process evolves to adiabatic transition on a mixed potential energy surface. An empirical and simple formulism to encompass both nonadiabatic and adiabatic electron-transfer rates is the Bixon and Jortner equation,139 which is reduced to Fermi’s golden rule formulism at the low electronic coupling limit but becomes independent of electron coupling at the M


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Figure 8. Symmetry breaking turns on electronic coupling and coherent singlet fission in rubrene. (a) Displacement patterns of the phonon mode coupled to S0−S1 transition (Q74) and (b) the mode coupled to VSF (Q126) calculated with the quantum mechanics/molecular mechanics (QM/ MM) method. Variations in energy of the S1 and TT states along the coordinates are calculated with the CASSCF method. Each curve is shifted so as to be 0 cm−1 at the ground-state equilibrium structure. The displacements of tetracene chromophores are highlighted by red arrows. (c) Potential energy curves of the S0, S1, and TT states along the interpolated reaction path between the minimum points of the S0 and TT states calculated with XMS-CASPT2. The shaded area indicates the spread of a vibrational wave function at the zero-point energy. (d, e) Schematic representations of (d) coherent SF and (e) incoherent SF based on wavepacket dynamics. Reprinted with permission from ref 100. Copyright 2017 Springer Nature.

of the former, not the latter. The experimental time scales for the formation of 1(TT) in all three acenes15,17,99,143 and in rubrene100,103 are much faster than those predicted by the rate equations and can be accounted for by the vibronic coherent model, as is done in Figure 9b−d for hexacene, detailed below.

strong coupling limit. This approach was adapted by Yost et al. to describe singlet fission rates in a range of oligo-acenes and functionalized oligo-acenes with varying strengths of electronic coupling.57 In their work, the triplet pair was constructed from two triplets in the 9-spin state representation and thus was the 1 (T···T) state, not the 1(TT) state. Because the singlet fission rate is defined as that for the formation of 1(T···T), these authors were in fact reporting the rate of singlet fission in step 2′ in Figure 5, i.e., S1 → 1(T···T). Rigorously speaking, such calculation does not give the rate of 1(TT) formation. Only when the inter-T1 electronic coupling is sufficiently small can we ignore the distinction between 1(T···T) and 1(TT). An important task in future research is to quantify this distinction. Monahan et al. calculated the singlet fission rates in tetracene, pentacene, and hexacene using Fermi’s golden rule in the Marcus−Levich−Jortner formulation,140 including coupling to the dominant ring breathing mode at 1450 cm−1.99 The calculated singlet fission time constants (Figure 9a) are τSF = 10 ps, 80 fs, and 160 fs for energetic driving forces of ΔESF = 2E(T1) − E(S1) = 0.11, −0.11, and −0.59 eV for tetracene, pentacene, and hexacene, respectively, in good agreement with experimental time constants (dots) for S1 population decay.17,57,105,141,142 Given the construction of the triplet pair in the form of 1(T···T), not 1(TT), the rate equation approach essentially treats the incoherent formation

3.4. Coherent Singlet Fission and Quantum Dynamics

When inter-T1 electronic coupling is not negligible, as is the case for the end-connected pentacene dimer,34,43 1(T···T) is distinctively different from 1(TT). In this scenario, we are interested in the formation of 1(TT) not only because the dephasing of this intermediate state defines singlet fission but also because it can be directly related to experimental observables.34,43,99,100,125 The formation of 1(TT) can occur both incoherently and coherently. The former is exemplified by the rate equation calculations in the last section, while the latter represents the formation of a coherent superposition of 1 (TT) with S1 and/or CT states and becomes significant when electronic couplings to S1 and/or CT are sufficiently strong and the energy-resonant condition is satisfied with aids from vibronic coupling and electronic band formation.8,92 An accurate treatment of the coherent process would require quantum dynamics simulation from first principles, but this is not computationally practical for a multichromophore system. A realistic compromise is to simulate the dynamics using model Hamiltonians, e.g., within the Liouville−von Neumann N


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Figure 9. Incoherent and coherent rates of singlet fission. (a) Incoherent singlet fission rate as a function of energy driving force in acenes from Marcus−Levich−Jortner electron-transfer theory (circles correspond to experimental data). (b−d) Results from quantum dynamics simulations for hexacene. (b) S1−1(TT)⊗3 coherence, which is particularly large for the first 100 fs. (c) Relative (integrated to one) TT⊗n populations as a function of time; n is the vibrational quantum number. Note the 1(TT)⊗3 state is formed first, with a single-exponential rise time of 40 ± 10 fs. (d) S1 population (gray) as a function of time; the red curve is a single exponential fit (red) with a time constant of 100 fs. The initial conditions correspond to excitation of S1. Reprinted with permission from ref 99. Copyright 2017 Springer Nature.

density-matrix formulism, eq 10.54,92,144 These model simulations have provided insight into singlet fission, particularly the role of charge-transfer states and electron−phonon coupling. Greyson et al. carried out the first quantum dynamics simulation based on the density-matrix approach to understand singlet fission in model dimers.144 This study revealed that ultrafast singlet fission can be realized via charge-transfer mediated coupling, resulting in coherent generation of the triplet pair state on the 10−100 fs time scale (Figure 10). They also pointed out the critical role of energy-resonant conditions in the coherent singlet fission dynamics. Berkelbach et al. extended the quantum dynamics approach using the Redfield equation to account for coupling of the electronic system to the phonon bath.54,55,58,92 These authors revealed the “super-exchange path” where CT states higher in energy than either that of S1 or TT can give rise to similarly efficient singlet fission as the sequential path (Figure 11). The authors applied the scheme to the model pentacene dimer and crystalline systems, further suggesting the importance of CT states and phonon bath modes. In the work of Berkelbach et al., the phonon bath was assumed to consist of continuous spectral density of the Debye form.54,55,58 Tempelaar and Reichman went beyond this assumption and calculated the singlet fission rate from a combination of Redfield theory and the vibronic exciton theory, which includes vibrational modes microscopically. They found that the use of one vibrational mode in the dynamics calculation leads to particular sensitivity of the singlet fission rate to the vibrational energy (Figure 12), as expected from the critical importance of energy-resonant conditions.145 Such sensitivity is relaxed when one includes

Figure 10. Coherent simulation of singlet fission in an isoenergetic system. Time dependence of the sum of all populations (solid) in S1 (blue), CT (red), and TT (green) configurations, the percentage of population that has left via fission (purple dash−dot) or fluorescence (orange dash) routes, and the percentage of population with fluorescence (black solid). The inset shows the first 0.1 ps. Reprinted with permission from ref 141. Copyright 2011 American Chemical Society.

more vibrational modes.146 Note that the quantum coherent mechanism is responsible for triplet pair population on the ultrashort time scale of ≤20 fs, when the energy of the vibrational mode involved is close to the energy-resonant condition (Figure 12). This condition is easily met in a crystalline solid with sizable bandwidths in both electronic and phonon degrees of freedom.8 O


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singlet fission depending on molecular stacking symmetry. The slip-stacked structure in TIPS-pentacene is shown to promote a coherent mechanism for ultrafast singlet fission, while the C2h stacking symmetric in rubrene necessitates symmetry-breaking intermolecular vibrations to enable thermally activated singlet fission. As discussed above in section 3.2 and Figure 8, Miyata et al. discovered that, in addition to thermally activated singlet fission, symmetry-breaking vibrational modes can also promote a coherent singlet fission channel in rubrene.100 The success of the quantum dynamics approaches requires the judicious construction of model Hamiltonians, an issue carefully addressed by Shiozaki and co-workers for model dimers.64 However, the most challenging problem is limitation of the dimer model.8,35 Carrying out quantum dynamics simulations for singlet fission in extended solids, with proper treatment of electronic and excitonic delocalization, and with inclusion of vibronic coupling effects on both energetics and electronic coupling, is a formidable challenge at the present time and requires major efforts from the theoretical and computational community. While the quantum coherent mechanism for singlet fission was initially suggested as originating from electronic coupling,17,144 follow-up studies revealed a mixed electronic and vibrational origin, i.e., vibronic coherence.99,100,125,143,145,152 Within the adiabatic picture, the rate of forming 1(TT) from S1 in the vibronic coherent mechanism can be represented as12

Figure 11. Super exchange coupling in singlet fission. Singlet fission yield after the four periods of time indicated for the [1/2 1/2] pentacene dimer. The dashed line qualitatively separates the superexchange (SX) regime, E(CT) > E(S1), from the sequential (SEQ) regime, E(S1) > E(CT). Estimated energy levels for the pentacene dimer are denoted by the white circle. Reprinted with permission from ref 55. Copyright 2013 AIP Publishing.

1

dQ ⟨(S1)i |∇Q H | (TT)j ⟩ d[ 1(TT)] ∝ · dt dt E(S1)i − E1(TT)j

(30)

where Q is the relevant nuclear coordinate; i and j represent vibrational quanta on the potential energy surfaces of S1 and 1 (TT), respectively. The first term on the right-hand side of eq 30 is the velocity of classical nuclear motion, i.e., vibrational wavepacket motion. Given the typical vibrational frequencies involved in singlet fission, such vibrational wavepacket motion can lead to the ultrafast formation of the superposition of S1 and 1(TT) on time scales of a few to a few 10s fs. The second term is the derivative coupling between the two adiabatic states, S1 and 1(TT). While the coupling matrix element (numerator) can be small, the energy-resonant condition (denominator) can give rise to large effective coupling to drive the vibronic coherence. Monahan et al. simulated the vibronic quantum coherent dynamics of singlet fission in crystalline hexacene.99 In such a highly exoergic system (ΔESF = −0.59 eV), the ultrafast experimental 1(TT) formation time is consistent with a vibronic quantum coherent model involving three vibrational quanta of the dominant 1450 cm−1 ring breathing mode, i.e., coherence involving S1⊗n = 0 and 1(TT)⊗n = 3 (⊗n = i indicates a state with quantum state n = i). The experimental τfTT < 50 fs is on the order of the vibrational period of τυ = 23 fs, suggesting that the S1 PES crosses the 1(TT) PES within the Franck−Condon region for the optical excitation (S0 → S1) and vibrational wavepacket motion leads to the formation of the S1 and 1(TT)⊗n = 3 superposition. These authors performed quantum dynamics simulations for a model twoelectronic state system using the multiconfigurational timedependent Hartree (MCTDH) method151,153 and showed that the initially prepared S1⊗n = 0 Franck−Condon state evolves into a coherent superposition (Figure 9b) with the resonant 1 (TT)⊗n = 3 state on the time scale of τυ. This process is

Figure 12. Calculated singlet fission dynamics of pentacene. A 2 × 2 supercell from vibronic pexciton theory with different vibrational energies was used. Curves indicate the total population of diabatic triplet pair states as a function of time. Reprinted with permission from ref 145. Copyright 2018 AIP Publishing.

The bath effect was carefully examined by Tao using symmetrical quasi-classical nonadiabatic dynamics; this work revealed the interesting effect of quantum interference between direct and CT-mediated singlet fission pathways and how such coherent effect is modulated by the phonon bath.147,148 Zang et al. simulated the quantum interference in singlet fission and pointed out how this effect differs from those associated with radiative transitions in J- and H-aggregates.149 Nakano et al. applied the quantum master equation simulation to pentacene dimers and showed how interchromophore excitonic coupling modulates energetics and, consequently, singlet fission dynamics.150 These authors suggested the importance of electron−phonon coupling to CT states,150 but this may not be unique as similar electron−phonon couplings to S1 and triplet pair states are equally important.128 Tamura et al. carried out a comparative study of singlet excitons in TIPS-pentacene and rubrene using first principles nonadiabatic quantum dynamical modeling.151 These authors provide a unified view of coherent versus thermally activated P


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imental results from ultrafast laser spectroscopic measurements, it is helpful to understand what each experiment is actually probing. In the most commonly used experimental technique of TA, the electronic transitions in the probe step are

accompanied by population transfer to the triplet pair states with a rise time of 40 ± 10 fs (Figure 9c), in excellent agreement with experiment. Dephasing and population conversion from S1 to 1(TT) occur with a time constant (Figure 9d) of the same order as that from incoherent rate calculations (Figure 9a).

4. EXPERIMENTAL EVIDENCE FOR SINGLET FISSION MECHANISMS Initial evidence for singlet fission came from photoluminescence (PL) and its dependence on magnetic fields in crystals of linear oligoacene molecules.28,135,154,155 With the advent of ultrafast laser technologies, time-resolved spectroscopies, such as time-resolved photoluminescence (TR-PL) and transient absorption (TA) spectroscopies, have been applied to the study of singlet fission. While TR-PL probes only the bright states (S1), TA can in principle be used to track all states, bright or dark, involved in singlet fission. Since the initial study by Jundt et al.156 on singlet fission in pentacene thin films, TA spectroscopy has become the most commonly employed technique in singlet fission studies.105,132,141 Despite the power and broad applicability of TA spectroscopy, we point out two limitations. The first concerns the mechanistic assumptions often invoked in data interpretation. For example, early studies assumed the process of S1 → 2 × T1, but this assumption ignores the critical intermediate states of 1(TT) and 1(T···T). The second concern pertains to the quantitative interpretation of TA spectra. Many authors are satisfied with the association of each excited-state species with particular spectral components obtained at different time scales or from global/target analysis, without assigning the specific transitions involved for each excited-state species. This common practice can limit mechanistic insight, e.g., the subtle distinction between 1(TT) and 1(T···T).34 Besides TA, complimentary laser spectroscopic techniques have played critical roles in shaping our understanding of singlet fission. These include, among others, timeresolved two-photon photoemission (TR-2PPE), which yields energetics of the excitonic states;15,17,99 ultrafast Raman spectroscopy157−159 and ultrafast infrared spectroscopy160,161 to provide time-dependent structural information; two-dimensional electronic spectroscopy on quantum coherence and coupling;103,125 time-resolved PL to track radiative emission S1 and/or 1(TT);71,162 and time-resolved electron spin resonance61,62 on the long time process of spin dephasing. We focus on understanding singlet fission from spectroscopic measurements and their connections to theories and computations. One should be mindful that each experiment is almost without exception a limited view of a complex and inherently dynamic process. In various laser spectroscopic probes of singlet fission, each optical transition can be more sensitive to one species than to another, e.g., 1(T···T) vs 1(TT), and the cross section of each probe transition can vary with time due to the extent of its electronic coupling with the phonon environment.99 Accepting these complexities, one can clearly see the naivete of insisting that one experiment tells the whole truth and another one does not. This is amply illustrated by the quantum coherent singlet fission mechanism, which was first explored theoretically in the simulation of Greyson et al.53 and demonstrated experimentally by Chan et al.,17 challenged by some researchers in the following few years,98 and now widely accepted as the vibronic coherent mechanism in singlet fission.15,17,99,100,103,125,143,145 Before we present key exper-

S0 + hυ1 → S1

(31)

S1 + hυ2 → S0 + 2hυ2

(32)

S1 + hυ2 → Sn

(33)

1

(TT) + hυ2 → Sn

1

(T1T1) + hυ2 → 1(T1Tn)

1

(34) (35)

(T1 ··· T1) + hυ2 → 1(T1 ··· Tn)

(36)

T1 + hυ2 → Tn

(37)

where hν1 is the pump photon and hν2 is the probe photon. Equation 31 is the excitation step and appears as bleaching in the probe step, while eq 32 is the reverse process of stimulated emission. Equation 33 is the photoinduced absorption (PIA) of S1. The newly discovered eq 34 is the PIA of 1(TT).34 The PIAs in eqs 35 and 36 probe the constituent T1 component in 1 (TT) and 1(T···T) and are hardly distinguishable from the PIA of the individual T1 state in eq 37. In a TR-2PPE experiment, light absorption from a pump laser pulse creates excitons in the molecular material and that from the probe pulse (hν2) ionizes the excited state. The photoelectrons are detected by an electron energy analyzer, and their kinetic energies as a function of time delay measure the dynamics involved.163 The electronic transitions are S0 + hυ2 → h+ + e

(38)

S1 + hυ2 → h+ + e

(39)

T1 + hυ2 → h+ + e

(40)

(T1 ··· T1) + hυ2 → h+ + T1 + e

(41)

(TT) + hυ2 → h+ + e

(42)

(TT) + hυ2 → h+ + T1 + e

(43)

1

1

1

To appreciate the quantitative nature of a 2PPE spectrum, one needs to understand the final state involved. For the ionization of S1 (eq 39) and T1 (eq 40), the kinetic energy of the photoelectron from each excitonic state, referenced to that from S0 (eq 38, i.e., ionization from the HOMO), is a direct measure of the excitation energy of the singlet or triplet excitonic state.163,164 For the ionization of one T1 in 1(T···T) and leaving behind the other T1, the final state is also a hole in a HOMO. However, for the ionization of 1(TT), the final state can be either a hole in HOMO (eq 42) or a hole in HOMO plus a remaining T1 (eq 43). The former can be called the first ionization potential and measures the total excitation energy of 1 (TT). The latter is the second ionization potential and gives the excitation energy close to that of a single T1.15,17,99 An important point to remember when interpreting experimental results is the difference between diabatic and adiabatic states. Experimental observables are inherently adiabatic states, not diabatic states. An adiabatic state can be a superposition of diabats, e.g., S1, CT, and 1(TT). In probing Q


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τ1 = 110 ± 20 fs, can be used to define the kinetics of 1(TT) formation. Note that in the 1(TT) and 1(T···T) spectral region, there is an energy relaxation with a time constant of τ2 = 260 ± 50 fs; this can be interpreted as the conversion of the highenergy 1(TT) state to the lower-energy 1(T···T) state, which appears as the apparent decrease in average energy. Chan et al. also presented indirect evidence from electron transfer to C60 that the triplet pair state behaviors as individual triplets, which we now call 1(T···T), in ∼900 fs. We take τ2 = 260−900 fs as the time window for singlet fission (τSF), not the commonly assumed τ1 ≈ 80−100 fs.141,156 Note that, while the initial work of Chan et al. suggested that the coupling responsible for the coherent formation of 1(TT) was electronic in nature,17 follow-up work over the past 7 years has established electronic plus vibration, i.e., vibronic, coupling as mainly responsible for the coherent singlet fission mechanism.99,100,125,143,145,152 In the case of crystalline pentacene, the identification of 1 (TT) comes from the second ionization potential, which gives a photoelectron with kinetic energy close to that from an individual T1. The photoelectron from the first ionization potential of 1(TT) overlaps in energy with that from S1 and is not resolved in Figure 13. This problem is overcome for crystalline hexacene, in which singlet fission is highly exoergic and 1(TT) lies at 0.59 eV below S1. Thus, the first ionization potential of 1(TT) is well-separated from that of S1 and can serve as a quantitative probe of 1(TT) without interference by photoelectron signal from either S1 or 1(T···T). Indeed, TR2PPE spectra from crystalline hexacene (Figure 14a and b)

such an adiabatic state, a specific spectroscopic transition can emphasize a particular diabatic component of the adiabatic state. 4.1. Spectroscopic Signatures of 1(TT)

As discussed in section 2 and summarized in Figure 5, 1(TT) is distinguished from 1(T···T) by the presence of intertriplet electronic coupling in the former, not the latter. This difference could in principle be established in electronic spectroscopies. In most spectroscopic measurements involving electronic transitions, the 1(T···T) state is indistinguishable from the individual T state, but the 1(TT) state can possess distinct spectroscopic signatures. There have been considerable efforts in singlet fission studies to obtain spectroscopic signatures of the 1(TT) state. Zhu and co-workers reported the first experimental evidence for the 1(TT) state in singlet fission using TR-2PPE spectroscopy on crystalline pentacene;17 this was followed by similar experiments on tetracene15,18 and hexacene.99 Spectroscopic signatures of the 1(TT) state have also come from distinct excited-state absorption in TA spectroscopy,34,165,166 emission in PL spectroscopy,71,162 and coherent features in two-dimensional electronic spectroscopy.103,125,127 The 2D pseudocolor plot of TR-2PPE spectra from a polycrystalline pentacene, Figure 13, reveals the short-lived S1

Figure 13. TR-2PPE spectra showing 1(TT) and its evolution to 1 (T···T) in crystalline pentacene. Pseudocolor plots of TR-2PPE spectra of a pentacene thin film (≥15 nm) excited at hν1 = 2.15 eV and probed with hν2 = 4.65 eV. The energetic positions of the S1, 1 (TT), and 1(T···T) states are indicated.17 The feature “X” at short time scale and extending to negative time delay is a charge-transfer exciton specific to the organic semiconductor surface.164 Shown on the right is a schematic of the 2PPE process. Adapted with permission from ref 17. Copyright 2011 AAAS.

Figure 14. Energetic positions and dynamics of S1 and 1(TT) in crystalline hexacene from TR-2PPE spectra. (a) Pseudocolor plot of TR-2PPE spectra collected with hν1 = 1.48 eV and hν2 = 4.20 eV (a) and with hν1 = 1.48 eV and hν2 = 4.70 eV (b). (c) Comparison of experimental cross-correlations (black dots) and associated fits for S1 (top), 1(TT) (center), and T1 (bottom). The colored lines in (c) are kinetic fits as detailed in the text. The black dashed curve is the laser pump−probe cross-correlation (CC, full width at half maxiumum (fwhm) = 99 ± 5 fs). Reprinted with permission from ref 99. Copyright 2017 Springer Nature.

state upon photoexcitation and long-lived 1(T···T) and/or 2 × T1 from singlet fission.17 Evidence for the 1(TT) state comes from the ultrafast (≤20 fs) appearance of a photoelectron signal in the spectral region for 1(T···T) and/or 2 × T1, but with kinetic energy higher by as much as 0.11 eV, which is the known exoergicity for singlet fission in pentacene. This highenergy feature is taken as evidence for the 1(TT) state, which is initially resonant with the S1 state and which relaxes to the 1 (T···T) state at lower energy. The ultrafast (≤20 fs) appearance of the 1(TT)-like spectroscopic feature prompted Chan et al. to propose a quantum coherent mechanism in which initial photoexcitation leads to the formation of a quantum superposition of S1 and 1(TT) states,17 most likely with the mixing of CT.8,92 Dephasing of this quantum superposition, as reflected in the S1 population decay time of

clearly reveal the transient 1(TT) state (from the first ionization potential) at 0.59 eV below S1. For comparison, the photoelectron signal from the second ionization potential of 1(TT) (1.04 eV below S1) is at similar positions as the signal from 1(T···T), and this spectral region consists of contributions from both. Remarkably, the dynamics of S1 and 1(TT) (Figure 14c) show very different time scales. While the population of S1 decays with a time constant of τdS1 = 180 ± 10 fs, the R


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Figure 15. Probing singlet fission in pentacene thin films and aggregates by transient absorption. (a) Transient absorption from a polycrystalline thin film of pentacene in the range from 500 to 1000 nm. Note that even with 20 fs time resolution, discrimination between 1(TT) and 1(T···T) in the spectral window is not clear. Reprinted with permission from ref 141. Copyright 2011 American Chemical Society. (b) Spectral components resolved from TA in an aggregate of pentacene derivative in near-infrared region (900−1400 nm). A flat absorption in 1100−1400 nm region is assigned to an interacting triplet pair, 1(TT). Reprinted with permission from ref 14. Copyright 2016 American Chemical Society.

Figure 16. Transient absorption from bipentacene reveals the spectroscopic signature for the 1(TT) state. (a) TA spectra of bipentacene (BP0) for the S1 (light red) and 1(TT) (blue) states from global analysis, which gives an S1 → 1(TT) conversion time constant of τ = 0.7 ps. The inset shows the molecular structure of BP0 where the side group (gray) is (triisopropylsilyl)ethynyl (TIPS). The transitions, along with vibronic progressions, are shown on each spectrum. (b) Estimated potential energy surface for S1 → 1(TT) conversion and near-IR PIA transitions. Adapted with permission from ref 34. Copyright 2017 AAAS.

population of 1(TT) rises with a time constant of τfTT = 45 ± 20 fs and decays with τdTT = 270 ± 10 fs. On the basis of these different time constants, Monahan et al. proposed the coexistence of both coherent and incoherent singlet fission (Figure 5). The former is attributed to the ultrafast (τfTT) formation of coherence due to wavepacket motion from the Franck−Condon region on the S1 PES to resonant vibronic levels on the 1(TT) PES. The incoherent rate process (τdS1) occurs from a vibronically relaxed S1 to 1(TT). The two constituent triplets in 1(TT) lose electronic coherence and form 1(T···T) on the time scale of τdTT. These conclusions are supported by transient absorption spectroscopy and quantum dynamics simulations (Figure 9b−d). Detecting 1(TT) from singlet fission using TA spectroscopy has been most difficult and may have in part contributed to the prevailing confusion in the literature on the distinction between 1(TT) and 1(T···T). This difficulty arises from the often lack of assignment of the specific transitions involved in a TA spectrum, the overlapping nature of excited-state absorption (photoinduced absorption (PIA)), ground-state bleaching (GSB) and stimulated emission (SE), the closeness

of 1(T1T1) → 1(T1Tn) and 1(T1···T1) → 1(T1···Tn) occurring in the spectral region as that of T1 → Tn, and the ultrafast time scales involved in 1(TT) dynamics. These difficulties are illustrated by the TA spectra of Wilson et al.141 on singlet fission in a pentacene polycrystalline thin film (Figure 15a), where a broad spectral region between 725 and 950 nm was assigned to triplet PIA with no evidence for distinguishing an early triplet pair from a later triplet pair we now know as 1(TT) and 1(T···T). Pensack et al. examined TA spectra from singlet fission in (triisopropylsilyl)ethynyl (TIPS) pentacenes and discovered that the difference between 1(TT) and 1(T···T) lies in the near-infrared (NIR) region.14 They assigned a flat PIA feature in this spectral region (Figure 15b) to the electronically coupled triplet pair, i.e., 1(TT), which loses electronic coupling on the time scale of ∼1 ps to form 1(T···T). This time constant is close to the τ2 = 260−900 fs discovered in the TR-2PPE study of crystalline pentacene. A similar conclusion was reached by Herz et al. using pump-depletion-probe spectroscopy on TIPS-pentacene and its N-containing derivatives.166 These authors discovered a near-IR PIA feature exclusively to the short-lived 1(TT) state and assigned it to the 1(T1T1) → S


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Figure 17. Comparison of TA spectra in the visible region for TIPS-pentacene dimers with (a) 0, (b) 1, and (c) 2 phenyl spacers. The arrows point to the assigned the 1(TT) → S3 transitions. Reprinted with permission from ref 36. Copyright 2015 American Chemical Society.

Figure 18. Theoretical confirmation of the spectroscopic signature for 1(TT) in bipentacene. Calculated PIA spectra of the pentacene dimer (BP0) for dihedral angles of θ = 0° and 30° in the (a) near-IR and (b) visible regions. Also shown in (a) is the calculated PIA spectrum of BP1, i.e., pentacene dimer with a phenyl linker. Reprinted with permission from ref 43. Copyright 2017 American Chemical Society.

0−1 vibronic peaks. In addition to the 1(TT) → S2u transition, there is also another transition at ∼1.8 eV that is specific to the 1 (TT) state and assigned similarly to the 1(TT) → S3 transition.34 The oscillator strength of the distinct 1(TT) → S2u or 1(TT) → S3 transition diminishes with decreasing intertriplet electronic coupling, as shown for pentacene dimers with a phenylene spacer or larger dihedral angle between the two pentacene chromophores.34,36 In contrast, the 1(T1T1) → 1 (T1T3) transition probes the individual triplet character and is independent of the intertriplet electronic coupling. This point is also evident in the comparison of TA spectra of pentacene dimers with zero, one, and two phenyl linkers (Figure 17). With decreasing intertriplet electronic coupling, as evidenced by both 1(TT) formation and decay rates, the 1(TT) → S3 transition (∼680 nm, labeled by arrows in parts a and b of Figure 17) diminishes, but the 1(T1T1) → 1(T1T3) (∼470− 550 nm) transition remains constant and nearly identical to that of T1 → T3 of an individual triplet from sensitization.36 The experimental work of Trinh et al. on the signature of the 1 (TT) state has been verified independently by Khan and Mazumdar, who calculated the excited-state ordering with high-level configuration interactions.43 The calculation predicted that the 1(TT) state possesses a strong transition in the near-infrared region to the S2 state and this transition is absent for T1 (Figure 18a). In agreement with experiments, the computational work showed that the oscillator strength of the distinct 1(TT) → S2 transition, not the 1(T1T1) → 1(T1T3)

1

(T1Tn) transition. This assignment needs to be revised in view of the experimental work of Trinh et al.34 and the theoretical work of Khan and Mazumdar43 on pentacene dimers. While the reason for the broadness of the 1(TT) PIA in Figure 15b was not addressed in detail by Pensack et al., we suggest that this broad feature may be related to the delocalization of the 1 (TT) state, as it narrows to a well-defined transition with vibronic signature in the pentacene dimer (see below). Trinh et al. applied TA spectroscopy to end-connected pentacene dimer (BP0, see inset in Figure 16a for molecular structure).34 This molecular system represents an ideal model for the investigation of the 1(TT) state, as the triplet pair is confined in the dimer and likely remains as 1(TT) without undergoing further conversion to 1(T···T). TA studies show the simple kinetics of S1 to 1(TT) conversion with a time constant of τS1 = 0.7 ps; the latter decays on the time scale of τTT1 = 450 ps.36 Figure 16a shows TA spectra of the 1(TT) state (blue) and the S1 state (light red) obtained from global analysis. The 1(TT) state features a distinct PIA peak at ∼1 eV, assigned to the 1(TT) → S2u transition with vibronic progression of the pentacene ring breathing mode; this is similar to the S1u → S2g transition from the S1 state. Here “2g” and “2u”, respectively, refer to the symmetric and antisymmetric combination of constituent S2 states localized on each pentacene unit in the dimer molecule. Figure 16b shows the optical transitions on potential energy surfaces from the Huang−Rhys factors or the relative amplitudes of the 0−0 and T


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Figure 19. Spectroscopic evidence for the triplet pair state from polarization resolved transient absorption. Correlation of packing scheme and optical response along all three crystal axes. Schematic perfluoropentacene (PFP) stacking patterns along (A) the b⃗- and c⃗-axes on NaF(100) substrates and (B) a⃗- and b⃗-axes on KCl(100) substrates (top view). Linear absorption (open circles) along (C) b⃗-axis, (D) c⃗-axis, (E) a⃗-axis and corresponding differential absorption spectra at time delays of 300 fs (dashed) and 1 ps (solid) for the b⃗- and c⃗-axes. For the a⃗-axis, differential absorption spectra at 1 ps (dashed) and 90 ps (solid) are shown together with a Fano fit (red) of the induced absorption. False-color plots: time evolution of the differential absorption spectra along (F) the b⃗-axis, (G) the c⃗-axis, and (H) the a⃗-axis shown on a nonlinear time scale. Reprinted with permission from ref 165. Copyright 2014 American Chemical Society.

contrast, the shape of the broad PIA peak in the 1.5−1.7 eV region observed in the b direction (Figure 19C and F) but not in the c direction (Figure 19D and G) remains constant throughout the probe time window. While the authors assigned this broad PIA to an excimer state, it may be appropriately attributed to excited-state absorption from the S1 band in the crystalline lattice. The formation of 1(TT) was faster than that of an excimer-like state, suggesting that these two states are distinctly different. The critical information on transition dipole orientation and the resulting polarization-dependent TA spectra of 1(TT) is lost in experiments on polycrystalline films. This point is demonstrated for singlet fission in crystalline hexacene. While initial TA spectroscopy measurement on a polycrystalline hexacene thin film reported a singlet fission time constant of 500 fs,167 this time turned out to be an average for TA evolution along different crystalline directions in polarizationresolved TA on the single crystal.99 The latter is attributed to different orientations and magnitudes of transition dipole moments for 1(TT) vs 1(T···T).99 Similar discrepancies have been observed for singlet fission in polycrystalline thin films versus single crystals of pentacene and tetracene.105,168,169 There have been reports on vibrational signatures of 1(TT) and other transient states in singlet fission. Frontiera and coworkers applied femtosecond Raman spectroscopy to singlet fission; these authors attributed time-resolved shifts in

transition, diminishes with decreasing intertriplet electronic coupling, as shown for pentacene dimers with a phenylene spacer (BP1) or with larger dihedral angle between the two pentacene chromophores.43 Both the experimental finding34 and the computational confirmation43 unambiguously distinguish 1(TT) from 1(T···T) or 2 × T1 and call for quantitative spectroscopic assignments as a necessity in understanding singlet fission. Kolata et al. employed TA microscopy to investigate polarization-dependent spectroscopic signatures of 1(TT) from singlet fission in a perfluoropentacene single crystal (Figure 19).165 These authors found that the transition dipole moment of 1(TT) is highly oriented to the crystal axis due to the electronic coupling between adjacent triplets, unlike that of noncoupled triplets in 1(T···T). These authors found evidence for 1(TT) in the strong PIA peak at 2.28 eV along the a crystalline axis (Figures 19E and 18H). This feature is assigned to the 1(T1T1) → 1(T1T4) transition. Evidence for 1(TT) comes from the spectral line shape evolution with time and with its ultrafast formation within the experimental time resolution. Initially, the asymmetric peak (gray filled-in area in Figure 19E, Δt = 1 ps) resembles a Fano-type resonance, consistent with the strong coupling of 1(TT) to the S1/CT exciton bands. The Fano-resonance evolves into a symmetric peak shape (black curve in Figure 19E, Δt = 90 ps), consistent with the dephasing of 1(TT) to the more isolated 1(T···T). In U


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aromatic ring modes to the transient 1(TT) state in rubrene158

crystalline hexacene (Figures 9 and 14) where the coherent coupling is between S1 and excited vibrational levels on the 1 (TT) PES.99 Using TA spectroscopy with high time resolution, Musser et al. found that, for singlet fission in TIPS-pentacene films, vibrational coherence in the initially photogenerated singlet state is transferred to the triplet state.152 This finding contradicts the incoherent rate model and is evidence for the vibronic coherent formation of the 1(TT) state. These authors suggest the presence of a conical intersection for the efficient transition from the S1 state to the 1(TT) state. Bakulin et al. studied the formation of the 1(TT) state in films of pentacene derivatives using two-dimensional electronic spectroscopy.125 As shown in coherent vibrational frequency resolved 2D spectra (Figure 21), the involvement of high-frequency modes (1170 and 1360 cm−1) leads to resonant conditions. This is responsible for the coherent formation of the 1(TT) state, as evidenced by the appearance of the lower-energy diagonal peak, as well as the off-diagonal coupling (coherence). The vibronic resonant condition is responsible for ultrafast 1(TT) generation with modest electronic coupling (∼30 meV). The vibronic coherent mechanism is also observed in the endoergic singlet fission in tetracene. Stern et al. reported evidence for the 1(TT) state from singlet fission in TIPStetracene based on ultrafast formation ( 0.4 ps. (d) 2D plot of the probe wavelength dependence of Fourier amplitude spectra of coherent phonon signals. The signals in 480−550 nm are multiplied by a factor of 0.3. The modes at ∼80 and 124 cm−1 are denoted as modes α and β, respectively. Reprinted with permission from ref 100. Copyright 2017 Springer Nature.

might be a direct measure of the dynamic relaxation. Interestingly, they discovered a negative effect of delocalization on singlet fission: aggregation of the polymers in poor solvents and in thin films significantly lowers the S1 energy, making singlet fission endoergic, thus suppressing singlet fission. Aryanpour et al. carried out theoretical analysis of singlet fission using a model polyene with atomic site energies that mimic the donor−acceptor interactions.175 This work revealed that strong electron correlations and broken symmetry were essential to describe the photophysics leading to singlet fission. Remarkably, these authors discovered that the 1(TT) became optically bright, and direct transition to 1(TT) peaks with particular charge-transfer character represented by the site energy ϵB (Figure 26). The simple dipole selection rule dictates that the 1(TT) state is optically dark, but this selection rule is clearly broken when multiconfiguration interaction is taken into account. The nonzero optical oscillator strength of the 1 (TT) state may also partially explain the observation that that the 1(TT) state possesses similar diffusivity as that of S1.69 This theoretical work supports two major experimental observations: (1) the ultrafast formation of 1(TT) states in a number of experimental systems (see last section on the coherent singlet fission mechanism)15,17,99,100,103,125,143 and (2) the suppression of singlet fission rate when charge-transfer character in the donor−acceptor system is too large.37

In addition to affecting electronic coupling and energetics for singlet fission, delocalization is essential to triplet pair separation. Both experiments34,66 and theoretical analysis175 reveal that the 1(TT) state in an oligomer or polymer remains bound and does not separate into individual triplets. This is in contrast to a crystalline lattice, where formation of 1(T···T) is facilitated by triplet energy transfer14 as the two triplets separate into a larger phase space. The latter is essentially an entropic driving force in singlet fission, as first proposed by Chan et al.15 and most recently summarized by Casanova in this series.11 The need for intermolecular triplet energy transfer to separate an intramolecular 1(TT) is experimentally demonstrated by Trinh et al. 40 in the model system of diphenyldicyano-oligoene (DPDCn), where n is the number of double bonds in the oligoene (Figure 27d). These authors demonstrated that photoexcitation of DPDCn to the S2 state in the solution leads to the ultrafast (0.3 ps) formation of intramolecular 1(TT), in competition with decay into the dark S1 state on the same 0.3 ps time scale. The 1(TT) subsequently decays back into the Sn manifold and recombines in 40 ps without undergoing triplet separation. In the solid film, the TA spectra (Figure 27a) reveal the presence of a long-lived T1 state, which yields phosphorescence (Figure 27b) with a lifetime of 15 μs. The long-lived T1 originates from the intermolecular separation of the intramolecular 1(TT) state on X


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However, in the solid state, a significant T1 population is formed, indicating again the importance of intermolecular interactions for the separation of the intramolecularly formed 1 (TT) state.

5. CONCLUSION AND OUTLOOK The above is the authors’ attempt at providing a state-of-theart understanding of singlet fission mechanisms, with a focus on the nature of triplet pair states. We point out the necessity of distinguishing two distinct triplet pair states, 1(TT) and 1 (T···T). Failing to do so in the past has led to confusion in the literature on the construction of the triplet pair wave function (9- vs 16-spin state formalisms), on the definition of singlet fission rates, on the interpretation of experimental observables, and on the debate of coherent versus incoherent singlet fission mechanisms. Moving forward, we see major challenges in theory, experiment, and applications. On the theory/ computation front, it is highly desirable to carry out firstprinciples calculations and quantum dynamics simulations of singlet fission beyond the dimer approximation. On the experimental front, there is need to provide quantitative assignment of spectroscopic transitions of all states involved in singlet fission, with necessary time and energy resolutions. While the alleged reason for studying singlet fission in nearly every paper published recently has been to increase the efficiency of solar energy conversion, achieving this in practice has remained challenging and elusive. The demonstration of quantum efficiency in singlet fission solar cells above 100% in narrow wavelength windows has been an exciting step,22 and efforts are underway to use singlet fission chromophores to sensitize conventional solar cells.177−179 However, the power conversion efficiencies of all solar cells incorporating singlet fission reported to date have been far below those of conventional single junction solar cells. Among the many challenges in implementing singlet fission for solar energy conversion is the limited choice of molecules exhibiting high

Figure 23. Controlling singlet fission by the CT states in TDI derivatives. (a) Molecular/cystalline structure of TDI derivatives. (b) Schematic illustration of how CT mediates singlet fission when it is energetically higher than or isoenergetic with S1 and 1(TT) and how it becomes a nonradiate trap when it falls below S1 and 1(TT). Reprinted with permission from ref 174. Copyright 2017 American Chemical Society.

the 30 ps time scale, as evident from the growth of additional bleaching signal in Figure 27c. Pun and co-workers164 studied a fully conjugated tetracene polymer (PolyTc) and found similar results. While in solution, photoexcitation of PolyTc leads to the formation of 1(TT), but not individual triplets, before decaying back to the S1 state.

Figure 24. Structures and absorption spectra of singlet fission exhibiting and control materials. (A) Strong electron acceptor units based on thiophene dioxide (TDO) are shown in red, and the donor units are shown in blue. (B) Materials without TDO units. (C) Absorption spectra of compounds. Reprinted with permission from ref 37. Copyright 2015 Nature Springer. Y


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Figure 25. Singlet fission in an isoindigo-based donor−acceptor polymer. Structures and absorption spectra of (a) isoindigo-based low-bandgap polymer IIDDT-Me. (b) Aromatic and quinoidal resonance structures of IIDDT-Me. (c) Natural transition orbital analysis of the electron and hole states for the singlet and the triplet. (d) Normalized UV−vis absorption spectra of IIDDTMe in the TCE, DCB, and CHCl3 solutions and films. Reprinted with permission from ref 162. Copyright 2018 Nature Springer.

Figure 26. Theoretical optical transitions in polyene with different charge-transfer characters. HOMO and LUMO of the D and A segments of the monomer in the model polyene (a) for zero site energies and (b) for nonzero site energies with ϵB = 2.75 eV. (c) Calculated ground-state absorption spectra for a range of ϵB. The 1(TT) state continues to remain optically allowed up to ϵB ≈ 2.75 eV. Reprinted with permission from ref 175. Copyright 2015 American Physical Society.

singlet fission yields, the limited approaches for the efficient harvesting of resulting triplets or triplet pairs, the lack of understanding or control of how charge separation or triplet energy transfer occurs across material interfaces, and the mismatch in the long-term stability of singlet fission molecules

and conventional solar cell materials. While this account focuses narrowly on key mechanistic questions on singlet fission, we hope such understanding will serve to guide the design and development of a much larger singlet fission tool Z


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Figure 27. Intra- to intermolecular singlet fission in oligoenes. (a) Transient absorption pumped at 450 nm and (b) phosphorescence spectrum pumped at 335 nm for a DPDC7 film. (c) Comparison of PIA at 765 nm (red circles, left axis) and GB at 555 nm (blue circles, right axis) as a function of pump−probe delay. The solid gray curve is a biexponential fit to the excited state absorption (ESA) signal, and the dashed black curve is from the kinetic model for intra- to intermolecular singlet fission. (d) DPDCn molecules for inter- to intramolecular singlet fission. Reprinted with permission from ref 40. Copyright 2015 American Chemical Society.

box to achieve the much lauded goal of increasing solar energy efficiency with singlet fission chromophores.

physics/chemistry and electron−phonon coupling in molecular and hybrid semiconductors. Felisa Shai Conrad-Burton received her B.S. in Chemistry from George Mason University in 2016. She entered the graduate school at Columbia University and joined the XYZ group in 2016. She is currently working towards her Ph.D. in Chemical Physics. She studies singlet fission in molecular systems using a variety of ultrafast laser spectroscopic techniques.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemrev.8b00572.

Florian L. Geyer studied chemistry at Heidelberg University and carried out research at the Karlsruhe Institute of Technology and the Lawrence Berkeley National Lab. He completed his Ph.D. in 2016 with Prof. U. H. F. Bunz at Heidelberg. He received a Feodor Lynen Fellowship of the Alexander von Humboldt Society and carried out postdoctoral research at Columbia University in the groups of XYZ and Colin Nuckolls. He is currently a research scientist at BASF.

16-Spin state triplet pair eigenenergies calculations (PDF)

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Xiaoyang Zhu (a.k.a. XYZ) is the Howard Family Professor of Nanoscience and a Professor of Chemistry at Columbia University. He received a B.S. degree from Fudan University and a Ph.D. from the University of Texas at Austin. He did postdoctoral research at the Fritz-Haber-Institute. His current research interests include photophysics, light-matter interaction, and ultrafast spectroscopy of molecular, nano, and hybrid materials and interfaces.

X.-Y. Zhu: 0000-0002-2090-8484 Notes

The authors declare no competing financial interest. Biographies Kiyoshi Miyata received his Ph.D. in Chemistry in 2015 from Kyoto University under the supervision of Prof. Yoshiyasu Matsumoto. He did postdoctoral research in the XYZ group at Columbia University (2015−2018) with Fellowship support from the Japan Society for the Promotion of Science (JSPS). He is currently an assistant professor of chemistry at Kyushu University. His research focuses on photo-

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Grant DE-SC0014563. X.-Y.Z. acknowledges collaborations and discussions with, among others, Colin Nuckolls, Josef AA


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Michl, David Reichman, David Beljonne, Hiroyuki Tamura, Troy van Voorhis, Timothy Berkelbach, Matthew Sfeir, Luis Campos, and Nandini Ananth. X.-Y.Z. thanks his group members, past and present, who worked on the fascinating problem of singlet fission. K.M. acknowledges the Japan Society for the Promotion of Science for fellowship support. F.L.G. was supported by a Feodor Lynen Fellowship of the Alexander von Humboldt Society.

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