Quantum Interference in Singlet Fission: J- and H-Aggregate Behavior

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Quantum Interference in Singlet Fission: J- and H-Aggregate Behavior Hang Zang, Yi Zhao, and WanZhen Liang J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b01996 • Publication Date (Web): 29 Sep 2017 Downloaded from http://pubs.acs.org on September 30, 2017

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The Journal of Physical Chemistry Letters

Quantum Interference in Singlet Fission: J- and H-Aggregate Behavior Hang Zang,† Yi Zhao,∗,‡ and WanZhen Liang∗,‡ †Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China ‡State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China E-mail: [email protected]; [email protected] Phone: +86-592-2189197 (Y.Z.); +86-592-2184300 (W.L.)

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Abstract The quantum interference in singlet fission (SF) among the multiple pathways from singlet excited states to correlated triplet pair states is comprehensively investigated. The analytical analysis reveals that this interference is strongly affected by the excitonexciton coupling and is closely related to the property of J- and H-type of aggregates. Different from the interference in the spectra of aggregates, which depends only on the sign of exciton-exciton coupling, however, the interference in SF is additionally related to the signs of couplings between singlet excited states and triplet pair states. The interference dynamics is further demonstrated numerically by a time-dependent wavepacket diffusion method with electron-phonon interactions incorporated. Finally, we take a pentacene dimer as a concrete example to show how to adjust the constructive and destructive interferences in SF dynamics in terms of J-/H-aggregate behaviors. The results presented here may provide guiding principles for designing efficient SF materials through directly turning the quantum interference via morphology engineering.

Graphical TOC Entry


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Singlet fission (SF) is a spin-conserving process in which one singlet excited (SE) state produces two triplet states via a spin-correlated two triplet (TT) state. 1,2 As the SF process potentially allows single-junction photovoltaic devices to surmount the Shockley-Queisser limit, 3–5 it has attracted many attentions recently. 6–17 Numerous investigations have shown that the coupling between SE state and TT state includes a direct interaction and an indirect superexchange interaction mediated by charge-transfer (CT) states. 18–34 Naturally, the quantum interference may occur between direct and indirect pathways. Several works have demonstrated the existence of such effect via quantum dynamic simulations. 35,36 For some kinds of organic aggregates, the indirect CT mediated SF process may become more crucial than the direct one since the CT states can serve as bridges to interact with the SE and TT states by one-electron processes whereas the direct SF mechanism is an unfavorable simultaneous two-electron process. 22,32 For example, for a certain dimer AB the SF process can go through two pathways via A∗ B→A+ B− →AT BT and AB∗ →A− B+ →AT BT , where A∗ B (AB∗ ) is the localized SE state with the chromophore A (B) excited, A+ B− (A− B+ ) is the CT states with cation or anion on either chromophore, and AT BT represents the spin-correlated TT state. By tuning the property of CT states through molecular arrangements, such as energies and couplings with SE and TT states, the quantum interference between two pathways can be controlled, and the resulting SF rates can vary by nearly two orders of magnitude. 37,38 Actually, there may be more pathways from multiple SE states to TT states for the SF process in aggregates. Several experiments have already observed the significant effect of these pathways on SF dynamics. 39–43 In a dimer model, for instance, there are two localized SE states A∗ B and AB∗ . The SF can go through A∗ B→AT BT and AB∗ →AT BT , and the interference between these pathways should be indispensably considered, which, however, is nontrivial because of the exciton-exciton coupling between A∗ B and AB∗ . It is noticed that an analog process in aggregate spectra has been authoritatively investigated, 44–47 and the sign of exciton-exciton coupling can lead to a dramatically different



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interference behavior. The coupling with a positive sign results in a spectral blue-shift and suppressed radiative decay rate, whereas the coupling with a negative sign causes a spectral redshift and an enhanced radiative decay rate, the corresponding aggregates are named as Htype and J-type aggregates, respectively. However, when the intermolecular distance is short, the intermolecular CT states may couple with the Frenkel excited states, the interference between those two types of excitons can affect the crystallochromy and exciton transport. 48,49 In these cases, the effective coupling between the initial and final states was introduced to distinguish the molecular packing in aggregates. In this work, we will demonstrate that SF rates can be also dramatically affected by H- and J-aggregates via quantum interference, and subsequently show how to manipulate SF dynamics through tuning the packing of aggregates. The effect of electron-phonon interaction on the interference is further investigated by using time-dependent wavepacket diffusion (TDWPD) method. 50–52 Finally, we choose a pentacene dimer as a concrete example to control the quantum interferences in SF dynamics by using different packing arrangements. CA VLL

S1S0

VLH VHH

TT

Vex VHH

S0S1

VHL VLL

AC

Figure 1: Electronic states and the couplings among them in SF process. In present model simulations, the signs of couplings with red characters are changeable, while the signs of the others with black characters are fixed to be positive. As mentioned above, five electronic states are concerned for a dimer in the CT-mediated SF process: two localized SE states A∗ B and AB∗ , two CT states A+ B− and A− B+ , and one TT state AT BT . In the following, we simply denote them as S1 S0 and S0 S1 , CA and AC, and TT, respectively. The coupling between those diabatic electronic states are denoted as Vex , VHH , VLL , and VHL as shown in Figure 1. In this case, the pure electronic Hamiltonian 4 ACS Paragon Plus Environment

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in a matrix form is written as 



 ES1 S0    V  ex   He =   VLL    V  HH  

0

Vex

VLL

VHH

0

ES0 S1 VHH

VLL

0

0

VLH

VHH

ECA

VLL

0

EAC VHL

0

VLH

VHL ET T

       .       

(1)

The interference among the pathways can be numerically revealed by changing the signs of couplings. Interestingly, although there are many different choices for the signs of couplings, it is shown that these different choices can be categorized into a few different groups (see Supporting Information). As the energies of CT states are high enough, the SF pathways can be further simplified by introducing the effective coupling between a SE state and TT state based on the superexchange mechanism. The SF process is thus mapped to the two pathways starting from two correlated SE states to the TT state, i.e., S1 S0 to TT and S0 S1 to TT. This map can be performed by using a partitioning technique 53,54 to the original Hamiltonian Eq. (1), which leads to the effective energies of three mapped states (for the detail, see Supporting Information) EeS1 S0 = ES1 S0 +

2 VLL ES1 S0 −ECA

+

2 VHH , ES1 S0 −EAC

EeS0 S1 = ES0 S1 +

2 VHH ES0 S1 −ECA

+

2 VLL , ES0 S1 −EAC

EeT T = ET T +

2 VLH ET T −ECA

+

2 VHL , ET T −EAC

(2)

and the effective couplings Je = Vex +

2VLL VHH ES1 S0 +ES0 S1 −2ECA

+

2VHH VLL , ES1 S0 +ES0 S1 −2EAC

Ve1 =

2VLL VLH ES1 S0 +ET T −2ECA

+

2VHH VHL , ES1 S0 +ET T −2EAC

Ve2 =

2VHH VLH ES0 S1 +ET T −2ECA

+

2VLL VHL . ES0 S1 +ET T −2EAC


(3)


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The interference in the single pathway from S1 S0 state to TT state can be understood from the effective coupling Ve1 , as already demonstrated in several previous works. 37,38,55 It is seen that Ve1 has two terms corresponding to two pathways S1 S0 → CA → TT and S1 S0 → AC → TT, and a constructive (destructive) interference is thus corresponding to the case that terms VLL VLH and VHH VHL have the same (opposite) signs. To describe the SF dynamics, the electron-phonon coupling should be incorporated. In organic molecular aggregates, the vibrational modes can be described by a collection of harmonic oscillators with Hamiltonian Nn

Hph =

ph 5 ∑ ∑

1 ωnj (a†nj anj + ), 2 n=1 j=1

(4)

where a†nj and anj represent the creation and annihilation operators of the j-th phonon mode of the n-th electronic state with frequency ωnj . Those vibrational modes in each electronic state are commonly assumed be the same as those of dimer in the ground state. The Hamiltonian of electron-phonon interactions is given by Nn

He−ph =

ph 5 ∑ ∑

Cnj (a†nj + anj )|n⟩⟨n|,

(5)

n=1 j=1

where Cnj represents the mode-specific electronic-phonon coupling strength, and it is determined by the spectral density Jn (ω) = π

∑ j

C nj 2 δ(ω − ωnj ). The total Hamiltonian now

reads H = He + Hph + He−ph , and the SF dynamics is readily calculated from quantum dynamic methods. Here, we use the TDWPD method where the effect of phonons is incorporated by introducing time-dependent fluctuation energies Fn (t) on electronic states and SF dynamics can be obtained by solving the following differential equation 50–52 ∫ t ∂|ψ(t)⟩ i = (He + Fn (t) − iL dτ αn (τ )e−iHe τ LeiHe τ )|ψ(t)⟩, ∂t 0

where αn (t) =

∑ j

(6)

2 −iωnj t e is the zero temperature correlation function of electronic-phonon Cnj


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couplings and L =

∑N nm

|n⟩⟨m|. The stochastic force Fn (t) is generated by the following

expression

Fn (t) =

∑ k

√

√ Jn (ωk )∆ω √ [ A(ωk )cos(ωk t + ϕk ) + i B(ωk )sin(ωk t + ϕk )], π

(7)

where A(ωk ) = coth(ωk /2kB T ) + csch(ωk /2kB T ), B(ωk ) = coth(ωk /2kB T ) − csch(ωk /2kB T ), ϕk is uniformly distributed in [0,2π], kB is the Boltzmann constant, and T is the temperature. As the electron-phonon couplings are obviously incorporated, however, it is not quite clear how the phase fluctuations diminish the interference. Especially, when the energies of CT states are close to those of SE and TT states, the definition of effective coupling from superexchange mechanism becomes invalid and the population evolution even hops from SE to CT and then to TT states. In this case, it is not easy to analytically reveal the interference effect, and we thus adopt numerical simulations. The parameters of electronic states in the model are set to be ES1 S0 − ET T = 200 meV, VLL = VLH = VHL = 50 meV and VHH = ±50 meV 20,35,56,57 with ± generating the constructive and destructive interferences in the pure electronic states. For simplicity, the electron-phonon interaction in each state is chosen to be identity and described by a Debye spectral density with λ = 50 meV and ωc = 1450 cm−1 , 22 and the temperature T is set to be 300 K. With those parameters, we simulate the SF dynamics by TDWPD method which is capable for the description of carrier dynamics from hopping to coherence regions. 58,59 By tuning the energies of CT states, which has been performed experimentally, 17 we have observed the coherent interference in high CT energies and incoherent hopping dynamics at low CT energies obviously. Figures 2 (a) and (b) display constructive and destructive population evolution at ECA (EAC ) − ET T = 400 meV. Fitting TT population evolutions with an exponential function P (t) = P∞ + Ae−t/τ ,

(8)

we find that τ /τc ≈ 2.6 and τ /τd ≈ 0.5, where τ, τc and τd are time constants for one single 7 ACS Paragon Plus Environment


1.0

1.0

(a)

(b)

0.8 Population

Population

0.8 SE CT TT TT’

0.6 0.4 0.2

0.6 0.4 0.2

0.0

0.0 0

500

1000

1500

2000

0

500

Time (fs)

10.0

(c)

1000

1500

2000

Time (fs) 0.10

Single way Constructive Destructive

CT population

100.0 Rate constant (ps−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.0

0.1 200 250 300 350 400 450 500 ECT−ETT (meV)

(d)

0.08 0.06 0.04 0.02 0.00 200 250 300 350 400 450 500 ECT−ETT (meV)

Figure 2: Population evolution on SE (S1 S0 ), CT(CA+AC) and TT states at ECT − ET T = 400 meV with constructive (a) and destructive (b) interferences, as well as TT state population with only one CT (CA or AC) state pathway (dashed line). SF rate constants (c) and equilibrium population on CT (CA+AC) state (d) in terms of ECT − ET T . CT pathway, two CT pathways with constructive and destructive interferences, respectively. Although the interference is not perfect (the constructive rate should be 4 times larger than that of the single pathway), it indeed dramatically affects SF dynamics. Figures 2 (c) and (d) display the rate constants and equilibrium CT population in terms of different CT energies. With the decrease of CT state energies, even though the CT states are populated and the interference tends to be diminished, the coherent property of the SF wavepacket is still partially preserved because of the difference of SF dynamics from the constructive and destructive cases. As the exciton-exciton coupling Je plays a role, one cannot deal with the SF process only starting from a single SE state, and the two SE states (S1 S0 and S0 S1 ) as well as their correlation have to be taken into account. Interestingly, we find that the SF dynamics in such a case is very similar to the aggregate emission process 44–46 by the analog of the TT state to the ground state and Ve1 and Ve2 to the transition dipoles of the two monomers. Thus the property of J- and H-type aggregate might be useful to explain the interference of SF 8 ACS Paragon Plus Environment

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dynamics although the fluorescence and SF belong to radiation and radiationless processes, respectively. We employ the similar technique used in the aggregate emission process, and partially diagonalize the two-by-two matrix including two SE states in Eq.(9), resulting in 

e  ES1 S0    Je    Ve 1

Je





Ve1 

EeS0 S1

Ve2

Ve2

EeT T



 ES+     → 0      

V+

0

V+ 

ES−

V−

V−

EeT T

  ,   

(9)

e e where ES+ = EeS1 S0 sin2 θ + EeS0 S1 cos2 θ + Jsin2θ, ES− = EeS1 S0 cos2 θ + EeS0 S1 sin2 θ − Jsin2θ,

. The schematic V+ = Ve1 sinθ + Ve2 cosθ and V− = Ve1 cosθ − Ve2 sinθ, with tan2θ = Ee 2−JeEe S0 S1 S1 S0 process is shown in Figure 3. Under a condition of EeS1 S0 = EeS0 S1 = E, we get ES± = E ± Je and |V± | =

e √1 |V 2 1

± Ve2 |,

and we further assume Ve1 = Ve2 , then exactly the same consequence of J- and H-aggregates in aggregate spectra 44–46 can be used to explain SF dynamics, i.e., the H-aggregates with positive couplings Je open the high energy pathway and suppress the SF rate whereas the J-aggregates with negative couplings Je is favorable to the low energy pathway and enhance the SF rate. In a general case, it is easy from Eq. (9) to find that the condition is switched from the sign of exciton coupling to that of JeVe1 Ve2 . No matter whether the aggregates are J-type or H-type, the JeVe1 Ve2 < 0 is favorable for the low energy SF pathway and JeVe1 Ve2 > 0 is favorable for the high energy one. Figure 3 also displays these processes. It should be addressed that unlike the transition dipole of monomer between the excited state and ground state, the couplings Ve1 and Ve2 in SF can be effectively tuned, for instance, by controlling the symmetry of covalent dimer, 37 breaking local crystal structures 55 and substituting either elements or side chains. 60 To numerically demonstrate the interferences behind J- and H-aggregates, we still use the five-state Hamiltonian of Eq.(1), which allows us to investigate detailed effects of excitonic state properties on SF process. The same parameters in the above four-state model are adopted specifically with VHH = +50 meV, ECA (EAC ) − ET T = 400 meV and the coupling 9 ACS Paragon Plus Environment


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S1S0

~ J

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S+ S0S1

S1S0

S0S1 S-

~ V1

~ V2

V+

TT

TT

E S+

E S-

E S-

E S+

ETT

~ ~ V1 = V2

V-

ETT

~ ~ V1 = - V2

~ J > 0 H-aggregate

~ ~ V1 = V2

~ ~ V1 = - V2

~ J < 0 J-aggregate

Figure 3: Schematic SF processes in J- and H-aggregates in terms of Eq. (9). Vex as a variable to convert the aggregates between J- and H-type. Under these parameters, Ve1 and Ve2 equal each other and their values are relatively large because they come from the constructive interference between two CT pathways. The initial wavefunction is assumed to be

√1 N

∑ i∈SE

|i⟩.

It is noticed that the interference should disappear at Je = 0 because two SE states are effectively decoupled in this null-aggregate, 47,61 and the SF process is thus similar to that in the single pathway from one SE state to TT state. By using Eq.(3), the estimated Vex for the null-aggregate is about 20 meV. As expected, the population dynamics at this value of Vex , shown in Figure 4(a), are consistent with those from a single pathway (dashed line). Figures 4 (b)-(c) also display the population evolutions at Vex = −40 meV, and 90 meV. Obviously, the SF processes are sped up and suppressed with the negative and e respectively. By fitting the TT population evolutions with equation positive values of J,

(8), we draw SF rates k = 1/τ in Figure 4 (d) in terms of exciton-exciton coupling Vex . In the region of Vex < 20 meV (the case of Ve1 = Ve2 ), the SF rates are obviously larger than those in the region of Vex > 20 meV, which can be well explained by the property of Jand H-aggregates. Interestingly, as the destructive interference (Vex > 20 meV) applies, the 10 ACS Paragon Plus Environment

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1.0

1.0

(a)

(b)

0.8 Population

Population

0.8 SE CT TT TT’

0.6 0.4 0.2

0.6 0.4 0.2

0.0

0.0 0

1.0

500

1000 1500 Time (fs)

2000

0

Rate constant (ps−1)

(c)

0.8 Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0.0 0

500

1000

1500

2000

Time (fs)

500

16.0 (d) 14.0 12.0 ~ ~ V1=V2 10.0 8.0 6.0 4.0 2.0 0.0 −100 −60

1000 1500 Time (fs)

2000

~ ~ V1=−V2

−20 20 Vex (meV)

60

100

Figure 4: Population evolution of SF with (a) Vex = 20 meV, (b) Vex = −40 meV and (c) Vex = 90 meV, as well as TT state population from only one SE state pathway (dashed line). (d) The SF rates in terms of Vex in dimer (solid line) and ten monomer (dashed line) models. e although the SF rates first increase and then decrease with increasing Vex . At a small J,

pathway from the S− state with a lower energy to the TT state is destructively interfered, the high energy state S+ with the constructively interfered property is not too high and is still reachable, which may enhance SF rate compared to the null aggregate case. However, the initial population may dominantly relax into S− state at a large value of Vex , resulting in small rates. In the above example, the J-aggregate with Je < 0 favors both the nonradiative SF and radiative emission pathways. As two processes for certain kinds of molecular aggregate can have the similar time scales, 27 one may make one of the pathways decay by tuning the values of Ve1 and Ve2 . For instance, by designing VLL → −VLL , VLH → −VLH to make Ve1 = −Ve2 , the effect of J- and H-aggregates on SF rates is totally reversed, as drawn by a purple line in Figure 4 (d). In such a case, H-aggregates with Je > 0 accelerate SF, but they still suppress the spectral emission. Although a dimer model is adopted in the above analytical analysis and numerical sim-



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ulations, the conclusion should be qualitatively applied to molecular aggregates. To confirm it numerically, we construct an one-dimensional molecular chain which includes ten monomers,, 62 and the same electronic structure parameters and vibrational modes with those in the dimer model are used to simulate the SF dynamics. The corresponding SF rates in terms of Vex are shown in Figure 4 (d) with dashed lines. It is observed that the tendency of SF rates with respect to the coupling is exactly same in both models except the values of SF rates with ten monomers are larger because of exciton delocalization. 41,62 To demonstrate how the molecular packing affects the SF dynamics, we take different pentacene dimers in the molecular crystal of pentacene 63 as a concrete example to simulate the SF dynamics. The dimers are labeled as AB, AC and AD in Figure 5. The SF dynamics in the molecular crystal of pentacene has been intensively investigated by the experiments and theories. It is well recognized that the SF process in pentacene crystal is mediated by CT states. 22,23,27,64 Therefore, in our simulation, we consider five diabatic electronic states for each dimer. Then the electronic structure theory will be employed to build the model Hamiltonian and calculate the reorganization energies. We obtain the following effective model Hamiltonian for the dimer AB, AC and AD (in meV) in Eqs. 10, 11 and 12, respectively. The diagonal elements are the energies of diabatic electronic states which incorporate the polarization effect of environmental molecules. The nondiagonal elements are the electronic couplings between five diabatic states which are calculated by the four-electron four-orbital model at theoretical level of HF/6-31G**. The detailed description about the computational methods can be found in Supporting Information. To account for the polarization effect of environmental molecules on the electronic couplings, in SF dynamics simulation, we will scale all the calculated couplings as shown in the effective Hamiltonian by a screening factor f , 65 the values are 0.79, 0.81 and 0.77, for dimer AB, AC and AD, respectively.


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

HedimerAB



 1802.0    −24.1    =  −130.8    72.5   

0.5

−24.1

−130.8

72.5

1802.0

73.3

−124.9

73.3

2183.0

0.0

−124.9

0.0

2228.0

0.3

101.4

86.0

0.5

   0.3     101.4     86.0    

1554.0





HedimerAC

       =       

1802.0

−25.0

−25.0

1802.0 −116.3

127.9

120.3

−116.3 2170.0

0.1

120.3

−117.9

−117.9

127.9

0.1

2187.0

0.2

−0.2

−110.2

−87.8

0.2

   −0.2     −110.2     −87.8    

(11)

1554.0



HedimerAD

(10)



91.6 −61.8 44.1  1802.0    91.6 1802.0 44.1 −61.8    = 44.1 2322.0 0.1  −61.8    44.1 −61.8 0.1 2322.0    0.1

0.1

60.9

60.9

0.1 0.1 60.9 60.9 1554.0

              

(12)

The calculated couplings show that the interference between two CT pathways is destructive for all the three dimers, although the magnitude may be overestimated, 66 the interference feature is consistent with a recent high accuracy calculation. 67 In dimer AD, however, VLH equals to VHL coming from its high symmetry parallel packing, which is obviously different from the situations in dimers AB and AC. This high symmetry parallel packing leads to e it is clear that the low-energy excitonic state of |Ve1 | ≈ |Ve2 |. Together with the sign of J,

dimer AD is not favorable for SF. The similar phenomenon with |Ve1 | ≈ |Ve2 | has been observed in many molecular crystals with a parallel stacking pattern, 11,12,16,33,57 resulting in a completely vanished V+ or V− . In this case, the SF rate could be obviously suppressed upon large excitonic coupling Je under a condition JeVe1 Ve2 > 0.



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A B

D

A C

A

Figure 5: (a) The geometric arrangements of dimer pentacene. Population dynamics for SE (S1 S0 +S0 S1 ), CT (CA+AC) and TT state for pentacene dimer (b) AB, (c) AC and (d) AD at crystal structures. Figures 5 (b)-(d) display SF dynamics for dimer AB, AC and AD with the initial population equally located on S1 S0 and S0 S1 states. In the calculations, the mode-specific electronphonon interactions are used (see Supporting Information). As expected, the SF processes in dimer AB and AC are much faster than that in dimer AD. The time constants for dimer AB and AC, obtained by fitting the TT population evolution with Eq.(8), are τAB = 59 fs and τAC = 93 fs (kAB = 17 ps−1 and kAC = 11 ps−1 ), which are close to the experimental result of 80 fs. 8,15,27 For dimer AD, the effective coupling between S1 S0 (S0 S1 ) and TT state is weak because of its high symmetry. Furthermore, the destructive interference makes Ve1 and e which leads to an S-shape increase of TT population. 62 Ve2 to be quite small relative to J,

In previous investigations of SF dynamics in pentacene dimers, 22,28,68 the transient population on CT states can reach to 30% ∼ 50%, higher than the present results. This may be caused by the energy gaps between the diabatic states. Indeed, the CT state energies predicted by CDFT in the present work are about 100 ∼ 200 meV higher than reported values, 22,69 whereas they are similar to some other reported values. 20,25,27,70 In spite of high


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CT-state energies, the calculated SF rates for dimer AB and AC are relatively large. The reasons can be ascribed to the overestimated electronic couplings calculated by HF method. 66,67 The other reason may come from the vibronic coherence effect. 14,15 It is observed that there exist obvious oscillations of population on SE and TT states with a period of about 25 fs. This period well matches the frequency (1400 cm−1 ) of vibrational mode with a large reorganization energy (see Supporting Information), manifesting the vibronic coherence in SF dynamics. To qualitatively demonstrate how to control SF dynamics with aggregate packing, we focus on dimer AB and artificially change the arrangements between A and B by rotating and translating molecule A along its long axis at its equilibrium geometry. Positive or negative angle corresponds to clockwise or anticlockwise rotation of molecule A along its long axis as shown in Figure 5 (a). Positive or negative displacement corresponds to upward or downward translation of molecule A along its long axis, as shown in Figure 5 (b). The slight change of arrangement does not significantly affect the energies of CT states in crystal (Figure S3), but it dramatically affects the electronic couplings. Figures 6 (a) and (b) display the electronic couplings of dimer AB with respect to different rotational angles and translation distances of molecule A, respectively, (hereinafter, we focus on dimer AB, the terms “AC” and “CA” represent A− B+ and A+ B− , respectively) and the other couplings are shown in Figure S4. S1S0−S0S1

Coupling (meV)

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S1S0−CA

S1S0−AC

400 (a) 300 200 100 0 −100 −200 −300 −400 −40−30−20−10 0 10 20 30 40 Rotation (degree)

200 150 100 50 0 −50 −100 −150 −200

CA−TT

AC−TT

(b)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Translation (Å)

Figure 6: The calculated electronic couplings between diabatic electronic states for the dimer AB in the gas phase with respect to rotating (a) and translating (b) molecule A along its long axis. It is observed that both the sign and the magnitude of electronic couplings are sensitive 15 ACS Paragon Plus Environment


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to geometric arrangement. At large negative angles, the values of coupling are large because the planes of molecule A and B tend to become parallel, and the two monomers are getting close. As we have discussed, however, SF dynamics is not only determined by the absolute values of those couplings but also the sign. For instance, although the absolute values of couplings are large, the interference between two CT pathways is destructive. As the two monomers are getting parallel, |Ve1 | and |Ve2 | have nearly the same value. Since the effective couplings satisfy JeVe1 Ve2 > 0, the relatively large excitonic coupling Je further suppresses the SF rate. For the translation of molecule A, the couplings change periodically due to the nodes of electronic distributions of HOMO and LUMO, 48,49,61 and the different periodical behavior of these couplings can make SF from two CT pathways constructively interfered at some small ranges of translation displacement as shown in Figure 7 (b). The interference between two CT pathways is constructive at −0.8 and 2.0 ˚ A. The direct exciton-exciton coupling nearly keeps a constant negative value, however, the corresponding contribution from CT states can possibility change the total sign of effective coupling. 49,61 It can be judged that JeVe1 Ve2 < 0 is satisfied at −0.8 and 2.0 ˚ A. It should be emphasized that when the two CT pathways are constructively interfered, the condition JeVe1 Ve2 < 0 will always be satisfied at Vex = 0 according to the mathematical relationship of Eq. 3. To numerically reveal the geometrical dependence of SF rate, we calculate the population evolution with TDWPD method with above parameters (the screening factor 0.79 is adopted to scale the calculated couplings) and get the SF rates by exponential fitting with Eq. (8). The results are shown in Figure 7. As expected, the SF process is quite slow at large negative angles, but it is very fast at 2.0 ˚ A. Although the two CT pathways are constructively interfered at −0.8 ˚ A, the SF rate is not large since the absolute values of electronic couplings are small. To design aggregate with a fast SF rate, therefore, one has to consider both electronic coupling strengths and interference among all pathways. In summary, we have rationalized the effect of J- and H-aggregates on quantum interfer-


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25 Rate constant (ps−1)

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25

(a)

20

(b)

20

15

15

10

10

5

5

0 −40−30−20−10 0 10 20 30 40

0

Rotation (degree)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Translation (Å)

Figure 7: SF rate constant of dimer AB in terms of rotating angle (a) and translating distance (b) of molecule A along its long axis. ence in SF dynamics among the multiple pathways from singlet excited states to correlated triplet pair states. Different from the consequence of J- and H-aggregate on spectral characteristics, which is determined solely based on the sign of exciton-exciton couplings, the sign of JeVe1 Ve2 determines to enhance/suppress SF rate in aggregates. Our finding has been further confirmed numerically with a model system and a realistic pentacene dimer by quantum dynamic simulations and electronic structure calculations. The present results may be helpful to understand the SF dynamics in aggregates and provide a new principle for designing the state-of-the-art SF materials.

AUTHOR INFORMATION Corresponding Authors Yi Zhao, E-mail [email protected]; phone +86-592-2189197 (Y.Z.). WanZhen Liang, E-mail [email protected]; phone +86-592-2184300 (W.L.).

Acknowledgement This work is partially supported by the National Natural Science Foundation of China (NSFC) (Grants No. 21290193, No. 21373163 and No. 21573177). Y.Z. acknowledges financial support from the NSFC (Grants No. 21573175 and No. 21773191).



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Supporting Information Available Categorization of the model SF Hamiltonian with different coupling sign conditions; derivation of the orbital phase effect in four-electron four-orbital SF Hamiltonian; derivation of the effective Hamiltonian; detailed description of the electronic structure parameters of pentacene dimer; pentacene monomer reorganization energy and the spectral density form. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Quantum Interference in Singlet Fission: J- and H-Aggregate Behavior

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