Correlated Pair States Formed by Singlet Fission and Exciton–Exciton

Article pubs.acs.org/JPCA

Correlated Pair States Formed by Singlet Fission and Exciton− Exciton Annihilation Gregory D. Scholes* Department of Chemistry, Princeton University, Washington Road, Princeton, New Jersey 08544, United States ABSTRACT: Singlet fission to form a pair of triplet excitations on two neighboring molecules and the reverse process, triplet−triplet annihilation to upconvert excitation, have been extensively studied. Comparatively little work has sought to examine the properties of the intermediate state in both of these processesthe bimolecular pair state. Here, the eigenstates constituting the manifold of 16 bimolecular pair excitations and their relative energies in the weak-coupling regime are reported. The lowest-energy states obtained from the branching diagram method are the triplet pairs with overall singlet spin |X1⟩ ≈ 1[TT] and quintet spin |Q⟩ ≈ 5[TT]. It is shown that triplet pair states can be separated by a triplet−triplet energy-transfer mechanism to give a separated, yet entangled triplet pair 1[T···T]. Independent triplets are produced by decoherence of the separated triplet pair. Recombination of independent triplets by exciton−exciton annihilation to form the correlated triplet pair (i.e., nongeminate recombination) happens with 1/3 of the rate of either triplet migration or recombination of the separated correlated triplet pair (geminate recombination). denoted SD**, where the spin state is S = 0, 3, or 5. At first, all of the possible pair excitations will be compared by elucidating their spin eigenstates. The latter part of the paper focuses on the properties of the lowest-energy bimolecular pair excitations, referred to as the correlated triplet pair and here labeled |X1⟩. |X1⟩ is the immediate product of singlet fission and can be a key intermediate state during triplet−triplet annihilation.

1. INTRODUCTION Singlet fission is the process whereby a pair of electronic excitations is produced from a single photoexcitation. A wellstudied example is the conversion of a singlet excited state either localized on one molecule or, more likely, delocalized as a molecular excitonto a special “correlated” pair of triplet excitations shared by two molecules and having overall singlet spin.1−3 That state may subsequently transition to two independent triplet excitations. A particular application foreseen for this process is to enhance the power conversion efficiency of organic or mixed organic−inorganic solar cells.4−6 Incoherent triplet upconversion7−10 is a related phenomenon, where two independently photoexcited triplet states annihilate to yield a higher-energy singlet excitation localized on one molecule. Recently, researchers have focused on elucidating the mechanism and dynamics of the first step in singlet fission.11−36 However, there remain a number of fundamental questions regarding the nature of the correlated excitation pair state and its dissociation that require clarification. For example, (i) what precisely are the correlated triplet pair states, and how are they related to other pair excitation states such as a correlated singlet pair or singlet−triplet pairs? (ii) How do excitations separate, and does that affect their correlation? (iii) How are independent pairs of excitations formed from the correlated pair state? This paper contributes in particular to questions i and ii. In addition, the results of this work are relevant to the historical assumption in molecular photochemistry37 that triplet−triplet annihilation yields a mixture of one overall singlet bimolecular pair excitation, three overall triplet pairs, and five quintet pairs in the ratio 1:3:5. The first part of this paper considers the general problem of the electronic structure of bimolecular pair states that I define as a collective electronic state that mainly comprises a pair of electronic excitationsone on each molecule in the pair. In the photochemistry literature,37 these pair excitations are often © 2015 American Chemical Society

2. PAIR EXCITATION STATES The electronic structures of both the initially excited state and the bimolecular pair excitation states are described with balanced correlation in terms of the configurations I−VI shown in Figure 1. Configurations III−V represent classes of charge-transfer configurations that enable intermolecular orbital overlap effects to be accounted for in the classical valence bond model.38 In Figure 1a, the four-orbital basis is shown, where orbitals occupied in the ground state are denoted a and c, while those unoccupied in the ground state are labeled b and d. The two molecules are more generally represented as two electron groups, R and S. The electron groups are a useful way of dividing up electrons conceptually,39,40 and a physical division is not necessary to define a group (e.g., the antisymmetrizer can operate to permute electrons between groups). The configuration I is the primary configuration of the correlated pair states when intermolecular orbital overlap is moderate to weak, and this is a focus of this report. Calculations of the correlated triplet pair state for molecules positioned at geometries representative of crystal structures show that configuration I typically contributes to the wave function with a weight of ∼90%30 and will approach 100% as the excitations Received: October 5, 2015 Revised: November 19, 2015 Published: November 23, 2015 12699

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

Article

The Journal of Physical Chemistry A

excitation state |I⟩. There are two singlet states, |X1⟩ and |X2⟩, three triplet manifolds, |Z1⟩, |Z2⟩, and |Z3⟩, and one quintet manifold |Q⟩. Bardeen and co-workers have also calculated these 16 spin eigenstates but reported only the two singlet states.45 The zero-field splitting and mixing of the lowest states have been reported by Benk and Sixl.46 The result of 16 eigenstates contrasts discussions in the literature where it has been supposed that the pair excitation of the dimer can be one of only 9 states: singlet, triplet, or quintet.37,47−51 Moreover, it has often been assumed that those nine states are formed with equal probability, whereas it is found below that the lowest pair state, that is, an overall triplet state, lies significantly above the singlet and quintet states (relative to thermal energy). These assumptions have carried through to recent literature.3 Fundamentally, there are 16 states because there are 16 unique ways of distributing four α and/or β spin electrons among four orbitals (Table 1). The branching diagram, Figure 2, explains the Table 1. Primitive Functions Used to Construct the Spin Eigenstatesa

Figure 1. (a) Four-orbital basis. Orbitals occupied in the ground state are denoted a and c, while those unoccupied in the ground state are labeled b and d. The two molecules are represented as two electron groups, R and S. (b) Examples of valence-bond configurations for describing a correlated pair excitation in this basis. I is the primary configuration.

M=0

separate to nonadjacent molecule sites in the crystal. The admixture of configurations III−V is strongly dependent on the degree of interchromophore orbital overlap.2,41−43 These configurations are decisive in promoting singlet fission and, as I will report here, separation of the triplet pair state. The spatial part of the wave function for the correlated triplet pair is given by eq 1 Ψ(r ) ≈ N ( I +

∑ λiII II + ∑ λiIII III i

i

a

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

M labels eigenvalues of Sz.

+ ...) (1)

states. There is only one way to form a quintet state with the four electrons (|Q⟩). Singlet states can be formed either by adding two α electrons to group R followed by two β electrons to group S and then spin pairing across groups (the triplet pair state |X1⟩), or pairs can be added to each group in turn (the singlet pair state | X2⟩). There are three ways of adding three α electrons and one β electron to form the triplet states. Evidently, each of |Z1⟩, |Z2⟩, and |X3⟩ is a singlet−triplet pair, but the precise assignment is not resolved until we construct the spin eigenstates below. The spin eigenstates are constructed from primitive functions θi, with eigenvalues M of Sz, Table 1. Linear combinations of θi associated with each path in the S2 branching diagram are obtained for each M using the procedure described by Pauncz52 to obtain the spin eigenstates (simultaneous eigenstates of Sz and S2). Specifically, I used the genealogical construction of spin eigenfunctions. As an example, consider the correlated triplet pair state produced by singlet fission or triplet−triplet annihilation, |X1⟩. It is written as a product of space and spin wave functions as follows

where N normalizes the wave function and the coefficients λ depend on orbital overlap squared and hence are often small compared to 1. In the remainder of the paper, I will examine the spin eigenstates of |I⟩. Using the branching diagram method,44 Figure 2, it is evident that there are a total of six eigenvectors of S2 for the pair

X1 = P ΦR ΦS × = P ΦR ΦS ×

Figure 2. Branching diagram for calculating the spin states as a function of the number of electrons added to the system, N. For the four electrons in the pair states, there are six eigenstates of S2 (eigenvalues S shown on the plot): two overall singlet states X1 and X2, three overall triplet states, Z1, Z2, and Z3, and one quintet state Q.

⎤ 1 ⎡ 1 ⎢θ1 + θ6 − (θ2 + θ3 + θ4 + θ5)⎥⎦ 3⎣ 2 ⎤ 1 ⎡ 1 1 (αβ + βα) (αβ + βα)⎥ ⎢⎣θ1 + θ6 − ⎦ 3 2 2

(2)

where the electron configurations are written in the respective electron groups R and S for the two molecules so that ΦR = ab and ΦS = cd. The electron groups are distinguishable in the 12700

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

Article

The Journal of Physical Chemistry A example considered here because they are associated with the two molecules that are only weakly interacting (compared, for example, to a chemical bond). P is the antisymmetrizer that synchronously permutes space and spin functions (P is the permutation operator, and p keeps track of the number of permutations from 0 to N) P=

1 N!

∑ (−1)p P (3)

P

In the second line of eq 2, the sum of primitive spin functions + θ3 + θ4 + θ5) is factorized to show explicitly that it is a product of M = 0 functions. By further restricting the permutations to electrons within each group, specified by the modified antisymmetrizer P′, eq 2 is simplified to eq 4. This restriction is strictly valid only when orbital overlap between the groups is negligible (otherwise the wave function is not properly antisymmetrized). Equation 4 gives a reasonable zeroth-order wave function for the correlated triplet pair in cases where orbital overlap is not too strong, for instance, the crystalline forms of pentacene derivatives that we labeled “type I” in a recent report.11 It is an excellent approximation when the excitations separate to nonadjacent molecules. 1 (θ 2 2

X1 ≈ P′ΦR ΦS ×

1 [T+R1T−S 1 + T−R1T+S 1 − T0R TS0] 3

Figure 3. Schematic diagram showing the relative energies of the bimolecular pair states predicted by the branching diagram method, in units of the exchange energy J0. It is assumed that ES = 2ET.

The results for all of the pair states are collected in Table 2. Note that asymmetric terms like T0RS0S should be interpreted as 1 (T0R SS0 + S0R TS0) because the wave function as written is 2 indistinguishable from one where we permute the electron groups ΦRΦS → ΦSΦR (there are an even number of electrons in each group; therefore, a negative sign is not acquired from the permutation). For the M = 0 wave functions, the decomposition into spin pairs localized on the electron groups is provided. In other cases, that decomposition seems to be apparent after applying the antisymmetrizer. For example, by factorizing θ8 + θ9 1 to 2 2 (αβ + βα)αα and applying the antisymmetrizer to group S so that θ7 expands to ααβα − αααβ, it can be seen that

(4)

T0R

where indicates the localized state S = 1, M = 0 on molecule (electron group) R and so forth. Similarly S0R will mean S = 0, M = 0. Equation 4 shows that even for negligible orbital overlap such that the electron groups in |X1⟩ are distinguishable, the triplet excitations associated with each group are not independent because |X1⟩ is a superposition state of the three possible triplet pairs. The excitation energy of each state is estimated by evaluating the matrix element, for example, EX1 = ⟨X1|H − Eg|X1⟩, where Eg is the ground state energy, H is the Hamiltonian operator and EX1 is the excitation energy of state |X1⟩. The triplet excitation energy of one molecule is set to equal to E0 − J0, where J0 is the exchange energy (half of the singlet−triplet splitting). E0 is defined as half of the singlet and triplet transition energies for one molecule. The excitation energy of |X1⟩ is thereby found to be EX1 ≈ 2E0 − 2J0 + O(S2RSJ0). The main corrections to this estimation, apart from the excitonic interactions that I exclude here for clarity, are dependent on orbital overlap; the leading terms are of order overlap squared times the exchange energy, indicated by O(S2RSJ0). Aside from these corrections, the |X1⟩ state has an excitation energy equivalent to two triplet excitations, as expected. The relative energies of the pair excitation states are shown in Figure 3. A notable result is that |X1⟩ and |Q⟩ are predicted to be degenerate (to a first approximation), but the three |Zi⟩ all lie higher in energy. These states comprise singlet− triplet pairs to various degrees; therefore, their energies are raised relative to the triplet pair states because the exchange lowering is less. However, it might be possible to construct linear combinations of |Z1⟩ and |Z2⟩ that produce an overall triplet state triplet pair. This is an issue to examine later. In recent work, Burdett and Bardeen53 express the spin eigenstates of |X1⟩ in terms of the basis functions54 |x⟩ = i 1 1 (ββ − αα), |y⟩ = 2 (ββ + αα), and |z⟩ = 2 (αβ + βα), to 2

⎤ ⎡ 2 1 Z1, M =+1 = P ΦR ΦS × ⎢ (θ8 + θ9)⎥ θ7 − ⎦ ⎣ 3 6 ⎡ 2 +1 0 1 0 +1⎤ T S − TT ⎥ ≈ ΦR ΦS × ⎢ ⎣ 3 ⎦ 3

(5)

The physical interpretation of each of the pair excitation states is more-or-less evident from Table 2. |X2⟩ is the correlated singlet pair. |X1⟩ is the correlated triplet pair. |Z3⟩ is a correlated triplet− singlet pair. The |Z1⟩ and |Z2⟩ states comprise an explicit singlet− triplet pair contribution as well as a linear combination of triplets that is an overall triplet state. The singlet−triplet decomposition of this latter term is evident as spin pairing between the groups that can be deduced from a Rumer diagram drawn from the relative branching diagrams of Figure 1. That is, instead of spin pairing electrons within group R or S, the spin pairs span groups (e.g., electrons in orbitals a with c and b with d).

3. DISSOCIATION OF THE TRIPLET PAIR In a molecular crystal, the correlated excitations can also separate physically but retain entanglement until decoherence thermalizes the triplet population. It turns out that the separation occurs by triplet−triplet energy transfer (the matrix element is elucidated below), and it is a spin-allowed process. Lanzani and corworkers have previously hypothesized that triplet−triplet energy transfer is at play in the spatial dissociation of triplet pairs.55 The initial and final states have the same spin components as |X1⟩, and only the spatial part of the wave function changes. For example, the excitation labeled group S moves to a neighboring molecule labeled U, ΦRΦS → ΦRΦU. What is particularly interesting about this process is that the entanglement among the triplets is

1

yield |X1B⟩ = 3 (|xx⟩ + |yy⟩ + |zz⟩). By multiplying out the product functions and collecting terms, it can be found that |XB1 ⟩ is equivalent to |X1⟩ within an overall phase, that is, |XB1 ⟩ ≡ −|X1⟩. 12701

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

Article

The Journal of Physical Chemistry A Table 2. Bimolecular Pair States and Their Approximate Excitation Energies X1 = P ΦR ΦS × ≈ P′ΦR ΦS × X 2 = P ΦR ΦS ×

1 3

⎡ ⎤ 1 ⎢⎣θ1 + θ6 − (θ2 + θ3 + θ4 + θ5)⎥⎦ 2

E X1 ≈ 2E0 − 2J0

1 [T+R1T−S 1 + T−R1T+S 1 − T0R TS0] 3 E X2 ≈ 2E0 + 2J0

1 [θ2 − θ3 − θ4 + θ5] 2

≈ P′ΦR ΦS × S0R SS0 Z1, M = 0 = P ΦR ΦS ×

⎤ 1 ⎡ 1 ⎢θ1 − θ6 − (θ2 + θ3 − θ4 − θ5)⎥⎦ 3⎣ 2

EZ1 ≈ 2E0 −

4 J 30

EZ2 ≈ 2E0 −

2 J 30

1 [T+R1T−S 1 − TR−1T+S 1 − T0R SS0] 3

≈ P′ΦR ΦS ×

⎡ 2 ⎤ 1 Z1, M =+1 = P ΦR ΦS × ⎢ θ − (θ8 + θ9)⎥ ⎣ 3 7 ⎦ 6

⎡ 1 Z1, M =−1 = P ΦR ΦS × ⎢ (θ + θ12) − ⎣ 6 11

2 ⎤ θ13⎥ 3 ⎦

Z 2, M = 0 = P ΦR ΦS ×

1 [θ1 − θ6 + θ2 + θ3 − θ4 − θ5] 6

≈ P′ΦR ΦS ×

1 [T+R1T−S 1 − T−R1T+S 1 + 2T0R SS0] 6

⎡ −1 3 ⎤ Z 2, M =+1 = P ΦR ΦS × ⎢ (θ + θ8 + θ9) + θ10⎥ ⎣2 3 7 ⎦ 2

⎡ 1 3 ⎤ Z 2, M =−1 = P ΦR ΦS × ⎢ (θ + θ12 + θ13) − θ14 ⎥ ⎣ 2 3 11 ⎦ 2 Z3 = P ΦR ΦS ×

EZ3 ≈ 2E0

1 [θ2 − θ3 + θ4 − θ5] 2

≈ P′ΦR ΦS × S0R TS0 Z3, M =+1 = P ΦR ΦS ×

1 (θ8 − θ9) 2

Z3, M =−1 = P ΦR ΦS ×

1 (θ11 − θ12) 2 EQ ≈ 2E0 − 2J0

Q M = 0 = P ΦR ΦS ×

1 [θ1 + θ6 + θ2 + θ3 + θ4 + θ5] 6

≈ P′ΦR ΦS ×

1 [T+R1T−S 1 + T−R1T+S 1 + 2T0R TS0] 6

1 (θ7 + θ8 + θ9 + θ10) 2 1 = P ΦR ΦS × (θ11 + θ12 + θ13 + θ14) 2

Q M =+1 = P ΦR ΦS × Q M =−1

Q M =+2 = P ΦR ΦS × θ15 Q M =−2 = P ΦR ΦS × θ16

retained, now even in the absence of orbital overlap. It is retained for the same reason that it is present in the first place; we do not know which triplet pair is present, and because triplet−triplet energy transfer preserves the spin state, we do not make a measurement that can distinguish the three possibilities. To derive the matrix element for separation of the correlated triplet pair, we need to include charge-transfer configurations in the reactant and product wave functions. Specifically, they are those that account for orbital overlap between ΦS and ΦU. The derivation follows exactly that given in refs 41 and 43 where we derived the matrix element, eq 7, from reactant and product wave functions (ΨR and ΨP) for the M = 0 triplet pair analogous to those given in eq 6

ΨR = [P ΦR ΦSΞ U + λP ΦR ΦS +Ξ U − + μP ΦR ΦS −Ξ U +] ⎛ 1⎞ 1 (αβ − βα) × ⎜ − ⎟(θ2 + θ3 + θ4 + θ5) ⎝ 2⎠ (6a) 2 ΨP = [P ΦR ΦUΞS + μP ΦR ΦU +ΞS − + λP ΦR ΦU −ΞS +] ⎛ 1⎞ 1 (αβ − βα) × ⎜ − ⎟(θ2 + θ3 + θ4 + θ5) ⎝ 2⎠ (6b) 2

where ΞS indicates the closed-shell ground state of molecule S and the product functions ΦU+ΞS− and so forth are the relevant charge-transfer configurations. λ and μ are small mixing coefficients.41 After some manipulation, we obtain56 12702

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

Article

The Journal of Physical Chemistry A

consistent with experiments,37,47 the microscopic origin of the result does not seem to have been elucidated. Spin restriction in the final state makes the triplet−triplet annihilation rate different from dissociation of the triplet pairs. Hence, the correlated triplet pair will naturally tend to be dissociated at equilibrium. Once the mixed state of thermalized triplets is formed, the recombination to form the correlated triplet pair, triplet−triplet annihilation, is only 1/3 as likely as an equivalent energy-transfer jump that keeps the triplets separated. In recent work, Greenham and co-workers have used optically detected magnetic resonance experiments to detect two distinct recombination pathways,57 one of which is assigned to nongeminate recombination, controlled by bimolecular spin statistics.

V 00 = ΨP H ΨR ≈

2βETβHT A

=V

+1 + 1

= V −1 − 1

(7)

where A is the energy difference between charge-transfer configurations and the locally excited donor configuration. βET and βHT are electron- and hole-transfer matrix elements, respectively. The three contributions to the total matrix element are labeled with superscripts pertaining to M. For example, V+1+1 means the matrix element for triplet−triplet energy transfer of an M = +1 triplet state. These matrix elements diminish exponentially with the donor−acceptor separation.56 The total matrix element for separation of the correlated triplet pair is given by eq 8 V sep ≈

(V 00 + V +1 + 1 + V −1 − 1) 3

5. ENTANGLEMENT AND DECOHERENCE OF THE CORRELATED TRIPLET PAIR The pair excitation states are quantum mechanical superpositions. To understand why the correlated pair states in Table 2 are distinct from independent excitation pairs and to see how the latter are formed by decoherence, we need to consider the density matrix for each state. The relevant density matrix is a 9 × 9 matrix containing all combinations of triplet pairs. The first −1 −1 +1 0 0 three row/column indices denote T+1 R TS , TR TS , and TRTS, respectively. For example, in the case of |X1⟩, the initial state prior to deoherence (a pure state) has a density matrix

(8)

and hence the rate of separation is given by the result that we would compute classically: ksep = (k00 + k+1+1 + k−1−1)/3. (To obtain this result using the Fermi Golden Rule, note that all six interference terms need to be accounted for in addition to the three classical terms.) Therefore, the excitations comprising a correlated triplet pair state can separate with a rate constant approximately equal to the rate of triplet−triplet energy transfer.

4. NONGEMINATE RECOMBINATION OR ANNIHILATION OF TRIPLET PAIRS Now let us consider the reverse process, combination of two independent triplets by annihilation (nongeminate recombination). The two triplet excitations can be generated by completely independent photoexcitations, or they may be a geminate pair produced by singlet fission that has subsequently dephased (thermalized), as discussed below. When the triplets come together to occupy adjacent sites in a crystal, the valence electron wave functions overlap. The spin eigenstate |X1⟩ captures the fact that in the orbital overlap region, we cannot associate the electrons with certainty to particular electron groups (or molecules). This is associated with a weak bonding interaction, represented by the charge-transfer configurations mixed into the wave function, that binds the triplet pair. If, for instance, it were possible to know for sure that molecule U is in the M = +1 triplet state and molecule R is in the M = −1 triplet state, then excitation could jump from molecule U to S to form that correlated triplet pair with wave function |X1⟩. The matrix element for this energy-transfer hop is V+1+1/ 3 . The 1 corresponding rate of an excitation jump k+1 + 1 ∝ 3 |V+1 + 1|2 contrasts with a normal triplet−triplet energy transfer jump that would be 3-fold faster. To understand this prediction, we have to recall that there are no spin restrictions on triplet−triplet energy transfer, but there are in the case of triplet−triplet annihilation; the final state cannot involve any pair of triplets.37 Further, it is not possible to know which triplet pairs are located at sites R and U; therefore, there are nine possibilities, of which three can combine to form the correlated triplet pair. Hence, the ensemble rate constant is 1/3 of the energy-transfer rate constant when there is solely a single excitation involved (here there is no constraint on the initial and final spin states). The phenomenon is a kind of Pauli blocking. While this basic result has been conjectured in the past and is well-accepted and generally

X1

⎡1 ⎢ ⎢1 1 ⎢− 1 X1 = ⎢ 3⎢ ⎢ ⎢⎣

1 −1 1 −1 −1 1 0

⎤ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⋱ ⎥ 0 ⎥⎦ 0

(9)

In contrast, the density matrix for the fully mixed state of independent triplet pairs is completely diagonal (all elements 0 except the nine diagonal values each of 1/9). That is because any pair of excitations is written as a product of localized spin eigenstates, but overall, the bimolecular state is not an eigenstate of the spin operator. Decoherence58,59 converts |X1⟩⟨X1| to independent triplet pairs. This happens by random spin flips on +1 0 +1 either molecule, for example T−1 R TS → TRTS . These spin−flips occur by coupling to nuclear spins and/or radiationless transitions.46 According to interpretations of quantum mechanics, we cannot know which triplet excitation pair is formed until we perform a suitable measurement; therefore, all possibilities coexist as a quantum superposition. That is indicated by nonzero off-diagonal elements in the density matrix |X1⟩⟨X1|, showing that this is an entangled state. |X1⟩ is an example of a so-called W state,60 that is, an important kind of state in quantum information because of the way entanglement is distributed among the three wave functions. The entanglement of each pair m and n of the −1 −1 +1 three triplet pairs, T+1 R TS with TR TS and so forth, can be quantified by the concurrence Cmn(t) = 2|ρmn(t)|. Entanglement implies correlations that exceed those allowed by classical physics. It is not obvious what significance those correlations might have for singlet fission, but there is an important physical reason why |X1⟩ comprises a superposition of the three different triplet pairs that ensure M = 0. Classically, we know that the there is 1/3 chance for any of these pairs to be formed via singlet fission 12703

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

The Journal of Physical Chemistry A

■

−1 −1 +1 0 0 (i.e., T+1 R TS , TR TS , or TRTS) and that information is encoded in the superposition by the 1/ 3 prefactor. Subject to decoherence, the pure state |X1⟩⟨X1| gradually transitions to a mixed state with all nine triplet pair combinations equally populated as the off-diagonal density matrix elements are lost in the ensemble average time-evolving density matrix ρ(t), where ρ(0) = |X1⟩⟨X1|. The concurrences will tend to zero, for instance, C10(0) = 2/3 → C10(∞) = 0. Decoherence converts the −1 −1 +1 0 0 triplet pair from certainly being one of T+1 R TS , TR TS , or TRTS to a statistical mixture of all possible triplet pairs. As discussed in the previous section of the paper, decoherence thereby hinders recombination of the triplet pair.

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by Princeton University through the Innovation Fund for New Ideas in the Natural Sciences. I thank Ryan Pensack for discussions, particularly concerning Scheme 1. Prof. Tao Zeng (Carlton University) is thanked for insightful comments on the paper.

■

6. CONCLUSIONS To summarize, the spin eigenstates and relative energies of the manifold of bimolecular pair excitations have been derived. The lowest-energy states are the overall singlet spin triplet pair |X1⟩ ≈ 1 [TT] and the overall quintet spin triplet pair |Q⟩ ≈ 5[TT]. Three pair states that are overall triplet eigenstates were identified, all higher in energy than |X1⟩ and |Q⟩. |Z1⟩ and |Z2⟩ are mixed in character, part triplet pair and part singlet−triplet. | Z3⟩ ≈ 3[ST] is a singlet−triplet, while |X1⟩ ≈ 1[SS] is a correlated singlet pair. The correlated triplet pair state |X1⟩ is a superposition of the three triplet pair combinations that ensure overall S = 0 and M = 0. Two manifestations of this entangled triplet pair are noted, that is, two forms of |X1⟩. One, labeled 1[TT] in Scheme 1, is a

REFERENCES

(1) Smith, M.; Michl, J. Recent Advances in Singlet Fission. Annu. Rev. Phys. Chem. 2013, 64, 361−386. (2) Smith, M.; Michl, J. Singlet Fission. Chem. Rev. 2010, 110, 6891− 6936. (3) Smith, M. B.; Michl, J. Recent Advances in Singlet Fission. Annu. Rev. Phys. Chem. 2013, 64, 361−386. (4) Beard, M.; Johnson, J.; Luther, J.; Nozik, A. Multiple Exciton Generation in Quantum Dots Versus Singlet Fission in Molecular Chromophores for Solar Photon Conversion. Philos. Trans. R. Soc., A 2015, 373, 20140412. (5) Lee, J.; Jadhav, P.; Reusswig, P.; Yost, S.; Thompson, N.; Congreve, D.; Hontz, E.; Van Voorhis, T.; Baldo, M. Singlet Exciton Fission Photovoltaics. Acc. Chem. Res. 2013, 46, 1300−1311. (6) Yang, L.; Tabachnyk, M.; Bayliss, S.; Bohm, M.; Broch, K.; Greenham, N.; Friend, R.; Ehrler, B. Solution-Processable Singlet Fission Photovoltaic Devices. Nano Lett. 2015, 15, 354−358. (7) Singh-Rachford, T.; Castellano, F. Photon Upconversion Based on Sensitized Triplet-Triplet Annihilation. Coord. Chem. Rev. 2010, 254, 2560−2573. (8) Schmidt, T.; Castellano, F. Photochemical Upconversion: The Primacy of Kinetics. J. Phys. Chem. Lett. 2014, 5, 4062−4072. (9) Monguzzi, A.; Tubino, R.; Hoseinkhani, S.; Campione, M.; Meinardi, F. Low Power, Non-Coherent Sensitized Photon upConversion: Modelling and Perspectives. Phys. Chem. Chem. Phys. 2012, 14, 4322−4332. (10) Sugunan, S.; Tripathy, U.; Brunet, S.; Paige, M.; Steer, R. Mechanisms of Low-Power Noncoherent Photon Upconversion in Metalloporphyrin-Organic Blue Emitter Systems in Solution. J. Phys. Chem. A 2009, 113, 8548−8556. (11) Pensack, R.; Tilley, A.; Parkin, S.; Lee, T.; Payne, M.; Gao, D.; Jahnke, A.; Oblinsky, D.; Li, P.; Anthony, J.; et al. Exciton Delocalization Drives Rapid Singlet Fission in Nanoparticles of Acene Derivatives. J. Am. Chem. Soc. 2015, 137, 6790−6803. (12) Yang, C.-H.; Hsu, C.-P. First-Principle Characterization for Singlet Fission Couplings. J. Phys. Chem. Lett. 2015, 6, 1925−1929. (13) Zirzlmeier, J.; Lehnherr, D.; Coto, P.; Chernick, E.; Casillas, R.; Basel, B.; Thoss, M.; Tykwinski, R.; Guldi, D. Singlet Fission in Pentacene Dimers. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 5325−5330. (14) Musser, A.; Maiuri, M.; Brida, D.; Cerullo, G.; Friend, R.; Clark, J. The Nature of Singlet Exciton Fission in Carotenoid Aggregates. J. Am. Chem. Soc. 2015, 137, 5130−5139. (15) Aryanpour, K.; Shukla, A.; Mazumdar, S. Theory of Singlet Fission in Polyenes, Acene Crystals, and Covalently Linked Acene Dimers. J. Phys. Chem. C 2015, 119, 6966−6979. (16) Musser, A.; Liebel, M.; Schnedermann, C.; Wende, T.; Kehoe, T.; Rao, A.; Kukura, P. Evidence for Conical Intersection Dynamics Mediating Ultrafast Singlet Exciton Fission. Nat. Phys. 2015, 11, 352− 357. (17) Monahan, N.; Zhu, X.-Y. Charge Transfer-Mediated Singlet Fission. Annu. Rev. Phys. Chem. 2015, 66, 601−618. (18) Renaud, N.; Grozema, F. Intermolecular Vibrational Modes Speed up Singlet Fission in Perylenediimide Crystals. J. Phys. Chem. Lett. 2015, 6, 360−365.

Scheme 1

bound triplet pair where orbital overlap effects are present. The wave function of the primary pair configuration in 1[TT] is given by eq 2 because the entire wave function should be antisymmetrized to account for orbital overlap. This state is the initial product of singlet fission as well as triplet−triplet annihilation. The second entangled triplet pair, the separated pair 1 [T···T], is formed by separation of the triplets, process (ii) in Scheme 1. I showed here how the excitation separation is promoted by a triplet−triplet energy-transfer mechanism. It is striking that the entanglement among the triplet pair configurations is retained as the excitations physically separate. Either of these states can reversibly form the original monomeric singlet state, as evidenced, for example, by coherent oscillations in delayed fluorescence decay data.53 They can also decohere to produce a mixed state of independent triplet excitations (process (iii). Once the mixed state of thermalized triplets is formed, the recombination to form the correlated triplet pair, triplet−triplet annihilation, is only 1/3 as likely as an equivalent energy-transfer jump that keeps the triplets separated. The recombination labeled (ii) in Scheme 1 is geminate recombination, and process (iii) is an example of what Greenham and co-workers describe as nongeminate recombination. The latter is predicted to be limited by bimolecular spin statistics. In recent work, Pensack and coworkers have been able to distinguish 1[TT] from 1[T···T] in ultrafast transient absorption studies by noting the evolution of photoinduced absorption line shapes.61 While qualitatively similar, the 1[TT] spectrum is perturbed by interactions between the proximate excitations, while the 1[T···T] spectrum resembles that of a lone triplet excitation. 12704

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705

Article

The Journal of Physical Chemistry A

(42) Scholes, G. D.; Harcourt, R. D. Configuration Interaction and the Theory of Electronic Factors in Energy Transfer and Molecular Exciton Interactions. J. Chem. Phys. 1996, 104, 5054−5061. (43) Scholes, G. D.; Harcourt, R. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer. Iii. An Ab Initio Study of Electronic Factors in Excitation Transfer and Exciton Resonance Interactions. J. Chem. Phys. 1995, 102, 9574−9581. (44) Grabenstetter, J.; Tseng, T.; Grein, F. Generation of Genealogical Spin Eigenfunctions. Int. J. Quantum Chem. 1976, 10, 143−149. (45) Burdett, J. J.; Piland, G. B.; Bardeen, C. Magnetic Field Effects and the Role of Spin States in Singlet Fission. Chem. Phys. Lett. 2013, 585, 1− 10. (46) Benk, H.; Sixl, H. Theory of Two Coupled Triplet States. Mol. Phys. 1981, 42, 779−801. (47) Birks, J. The Quintet State of the Pyrene Excimer. Phys. Lett. A 1967, 24, 479−480. (48) Birks, J. Triplet-Triplet Interaction of Aromatic Molecules in Solution. Chem. Phys. Lett. 1968, 2, 417−419. (49) Razi Naqvi, K. The Mechanism of P-Type Delayed Fluorecence from Fluid Solutions. Chem. Phys. Lett. 1968, 1, 561−562. (50) Stevens, B. Solute Re-Encounter Probabilities from Delayed Excimer Fluorescence. Chem. Phys. Lett. 1969, 3, 233−236. (51) Groff, R.; Merrifield, R.; Avakian, P. Singlet and Triplet Channels for Triplet-Exciton Fusion in Anthracene Crystals. Chem. Phys. Lett. 1970, 5, 168−170. (52) Pauncz, R. Spin Eigenfunctions: Construction and Use; Springer: New York, 1979. (53) Burdett, J.; Bardeen, C. Quantum Beats in Crystalline Tetracene Delayed Fluorescence Due to Triplet Pair Coherences Produced by Direct Singlet Fission. J. Am. Chem. Soc. 2012, 134, 8597−8607. (54) Hameka, H.; Oosterhoff, L. The Probabilities of Triplet-Singlet Transitions in Aromatic Hydrocarbons and Ketones. Mol. Phys. 1958, 1, 358−371. (55) Srimath Kandada, A. R.; Petrozza, A.; Lanzani, G. Ultrafast Dissociation of Triplets in Pentacene Induced by an External Electric Field. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 075310. (56) Olaya-Castro, A.; Scholes, G. D. Energy Transfer from FörsterDexter Theory to Quantum Coherent Light-Harvesting. Int. Rev. Phys. Chem. 2011, 30, 49−77. (57) Bayliss, S.; Chepelianskii, A.; Sepe, A.; Walker, B.; Ehrler, B.; Bruzek, M.; Anthony, J.; Greenham, N. Gemninate and Nongeminate Recombination of Triplet Excitons Formed by Singlet Fission. Phys. Rev. Lett. 2014, 112, 238701. (58) Zurek, W. Decoherence, Einselection, and the Quantum Origins of the Classical. Rev. Mod. Phys. 2003, 75, 715−775. (59) Zurek, W. Decoherence and the Transition from Quantum to Classical. Phys. Today 1991, 44, 36−44. (60) Coffman, V.; Kundu, J.; Wootters, W. Distributed Entanglement. Phys. Rev. A: At., Mol., Opt. Phys. 2000, 61, 052306. (61) Pensack, R.; Ostroumov, E.; Tilley, A.; Mazza, S.; Grieco, C.; Thorley, K.; Asbury, J.; Seferos, D.; Anthony, J.; Scholes, G. D. Direct Observation of Two Triplet Pair Intermediates in Singlet Exciton Fission. 2015, submitted.

(19) Alguire, E.; Subotnik, J.; Damrauer, N. Exploring Non-Condon Effects in a Covalent Tetracene Dimer: How Important Are Vibrations in Determining the Electronic Coupling for Singlet Fission? J. Phys. Chem. A 2015, 119, 299−311. (20) Zhang, B.; Zhang, C.; Xu, Y.; Wang, R.; He, B.; Liu, Y.; Zhang, S.; Wang, X.; Xiao, M. Polarization-Dependent Exciton Dynamics in Tetracene Single Crystals. J. Chem. Phys. 2014, 141, 244303. (21) Wu, Y.; Liu, K.; Liu, H.; Zhang, Y.; Zhang, H.; Yao, J.; Fu, H. Impact of Intermolecular Distance on Singlet Fission in a Series of Tips Pentacene Compounds. J. Phys. Chem. Lett. 2014, 5, 3451−3455. (22) Wang, L.; Olivier, Y.; Prezhdo, O.; Beljonne, D. Maximizing Singlet Fission by Intermolecular Packing. J. Phys. Chem. Lett. 2014, 5, 3345−3353. (23) Feng, X.; Kolomeisky, A.; Krylov, A. Dissecting the Effect of Morphology on the Rates of Singlet Fission: Insights from Theory. J. Phys. Chem. C 2014, 118, 19608−19617. (24) Berkelbach, T.; Hybertsen, M.; Reichman, D. Microscopic Theory of Singlet Exciton Fission. Iii. Crystalline Pentacene. J. Chem. Phys. 2014, 141, 074705. (25) Piland, G.; Burdett, J.; Dillon, R.; Bardeen, C. Singlet Fission: From Coherences to Kinetics. J. Phys. Chem. Lett. 2014, 5, 2312−2319. (26) Parker, S.; Seideman, T.; Ratner, M.; Shiozaki, T. Model Hamiltonian Analysis of Singlet Fission from First Principles. J. Phys. Chem. C 2014, 118, 12700−12705. (27) Yost, S.; Lee, J.; Wilson, M.; Wu, T.; McMahon, D.; Parkhurst, R.; Thompson, N.; Congreve, D.; Rao, A.; Johnson, K.; et al. A Transferable Model for Singlet-Fission Kinetics. Nat. Chem. 2014, 6, 492−497. (28) Zeng, T.; Hoffmann, R.; Ananth, N. The Low-Lying Electronic States of Pentacene and Their Roles in Singlet Fission. J. Am. Chem. Soc. 2014, 136, 5755−5764. (29) Casanova, D. Electronic Structure Study of Singlet Fission in Tetracene Derivatives. J. Chem. Theory Comput. 2014, 10, 324−334. (30) Feng, X.; Luzanov, A.; Krylov, A. Fission of Entangled Spins: An Electronic Structure Perspective. J. Phys. Chem. Lett. 2013, 4, 3845− 3852. (31) Chan, W.; Ligges, M.; Jailaubekov, A.; Kaake, L.; Miaja-Avila, L.; Zhu, X. Observing the Multiexciton State in Singlet Fission and Ensuing Ultrafast Multielectron Transfer. Science 2011, 334, 1541−1545. (32) Zimmerman, P.; Zhang, Z.; Musgrave, C. B. Singlet Fission in Pentacene through Multi-Exciton Quantum States. Nat. Chem. 2010, 2, 648−652. (33) Zimmerman, P.; Bell, F.; Casanova, D.; Head-Gordon, M. Mechanism for Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets. J. Am. Chem. Soc. 2011, 133, 19944−19952. (34) Wilson, M.; Rao, A.; Clark, J.; Kumar, R.; Brida, D.; Cerullo, G.; Friend, R. Ultrafast Dynamics of Exciton Fission in Polycrystalline Pentacene. J. Am. Chem. Soc. 2011, 133, 11830−11833. (35) Chan, W.-L.; Berkelbach, T.; Provorse, M.; Monahan, N.; Tritsch, J.; Hybertsen, M.; Reichman, D.; Gao, J.; Zhu, X.-Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321−1329. (36) Berkelbach, T.; Hybertsen, M.; Reichman, D. Microscopic Theory of Singlet Exciton Fission. I. General Formulation. J. Chem. Phys. 2013, 138, 114102. (37) Saltiel, J.; Atwater, B. W. Spin-Statistical Factors in DiffusionControlled Reactions. Adv. Photochem. 1988, 14, 1−90. (38) McWeeny, R.; Coulson, C. Coulson’s Valence, 3rd ed.; Oxford University Press: Oxford, U.K., 1980. (39) McWeeny, R. Methods of Molecular Quantum Mechanics, 2nd ed.; Academic Press: New York, 1992. (40) McWeeny, R. The Density Matrix in Many-Electron Quantum Mechanics. I. Generalized Product Functions. Factorization and Physical Interpretation of the Density Matrices. Proc. R. Soc. London, Ser. A 1959, 253, 242−259. (41) Harcourt, R. D.; Scholes, G. D.; Ghiggino, K. P. Rate Expressions for Excitation Transfer 0.2. Electronic Considerations of Direct and through-Configuration Exciton Resonance Interactions. J. Chem. Phys. 1994, 101, 10521−10525. 12705

DOI: 10.1021/acs.jpca.5b09725 J. Phys. Chem. A 2015, 119, 12699−12705