Ultrafast Energy Transfer in Excitonically Coupled Molecules Induced

Subscriber access provided by Washington University | Libraries

Spectroscopy and Photochemistry; General Theory

Ultrafast Energy Transfer in Excitonically-Coupled Molecules Induced by a Nonlocal Peierls Phonon Hong-Guang Duan, Peter Nalbach, R. J. Dwayne Miller, and Michael Thorwart J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00242 • Publication Date (Web): 25 Feb 2019 Downloaded from http://pubs.acs.org on February 26, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 17 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

The Journal of Physical Chemistry Letters

Ultrafast Energy Transfer in Excitonically-Coupled Molecules Induced by a Nonlocal Peierls Phonon Hong-Guang Duan,†,‡ Peter Nalbach,¶ R. J. Dwayne Miller,‡,§ and Michael Thorwart∗,† †I. Institut f¨ ur Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany ‡Max Planck-Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany ¶Westfälische Hochschule, M¨ unsterstr. 265, 46397 Bocholt, Germany §The Departments of Chemistry and Physics, University of Toronto, 80 St. George Street, Toronto Canada M5S 3H6 E-mail: [email protected]

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry Letters 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

Abstract Molecular vibration can influence exciton transfer either via a local (intramolecular) Holstein or a nonlocal (intermolecular) Peierls mode. We show that a strong vibronic coupling to a nonlocal mode dramatically speeds up the transfer by opening an additional transfer channel. This Peierls channel is rooted in the formation of a conical intersection of the excitonic potential energy surfaces. For increasing Peierls coupling, the electronically coherent transfer for weak coupling turns into an incoherent transfer of a localized exciton through the intersection for strong coupling. The interpretation in terms of a conical intersection intuitively explains recent experiments of ultrafast energy transfer in photosynthetic and photovoltaic molecular systems.

Graphical TOC Entry

2


Page 2 of 17

Page 3 of 17 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


Nonadiabatic quantum dynamics is at the heart of energy transfer processes in photoactive molecules and is characterized by comparable time scales of the electronic and nuclear dynamics. 1–3 The Born-Oppenheimer approximation of separated electronic and nuclear time scales is no longer valid. The strong influence of the electron-vibrational (vibronic) coupling is known to alter the electronic dynamics qualitatively. Such a situation, e.g., occurs when electronic potential energy surfaces form a conical intersection in molecular systems, which induces an ultrafast radiationless energy transfer between different electronic states. 4,5 Strong nonadiabatic effects are induced by a sizable electron-vibrational interaction. Commonly, the electron couples to a local molecular phonon mode. This concept is well established for the conventional electron-transfer and the Holstein model of a molecular crystal. 6,7 More recently, evidence is growing that the interaction of the electron with a nonlocal phonon is important. 8–10 Actually, this not only occurs in molecules, but also in organic molecular crystals where nonlocal intermolecular low-frequency vibrations are identified. 8,9 Moreover, nonlocal vibrations strongly influence the energy transfer in photosynthetic systems. An extended Holstein model has been used to show that nonlocal exciton-phonon interactions 10 strongly suppress the vibronic enhancement of energy transfer, which otherwise prevails due to a strong interaction of the excitons with a local phonon. Nonadiabatic dynamics is also relevant during singlet fission 11–13 in organic semiconductors, which is a spin-allowed exciton multiplication process that converts one spin-singlet exciton to two spin-triplet excitons. It offers the potential to enhance solar energy conversion by circumventing the (adiabatic) Shockley-Queisser limit on efficiency. Its function depends on a subtle interplay between electronic and electron-phonon interaction. 14–16 In molecular crystals, two major electron-phonon interactions prevail: intramolecular local Holstein modes modulate the electronic site energies, and intermolecular non-local Peierls modes induce fluctuating electronic couplings between molecules. Specifically, singlet fission in tetracene is driven by vibronic coherence. 17 Based on a local Holstein model, the calculated time scales of the fission agree with the experimental observations. 18,19 Furthermore,

3



a strong vibronic coupling to local and nonlocal vibrations significantly reduces the energy gap between the singlet excited state and the triplet-pair state. 20 In addition, the calculated nonlocal vibrational mode works as bridge to link two electronic states. Moreover, the interaction of the exciton with nonlocal intermolecular vibrational modes is highly sensitive to the stacking geometry of the crystal and can dramatically speed up singlet fission. 21,22 In fact, the underlying mechanism of the primary singlet fission in pentacene and its derivatives has been uncovered experimentally using ultrafast nonlinear spectroscopy. 23 In this paper, we develop an intuitive picture of how nonlocal vibrations modulate the nonadiabatic transfer in concert with local vibrations. We show that their influence can be understood in terms of a novel Peierls transfer channel formed by a conical intersection. In a two-bath exciton model, where a dimer is linearly coupled to two harmonic baths representing the damped local and the nonlocal vibration, two potential energy surfaces for the exciton form a conical intersection 4,5 for strong nonlocal (Peierls) coupling. The emergence of this Peierls channel then permits an ultrafast exciton transfer. 24,25 The interplay of the Holstein and the Peierls modes induce exciton localization during the transfer when the Peierls coupling is strong enough and exciton transfer shifts from a delocalized (coherent) to a localized (incoherent) transfer. The conical intersection helps to intuitively interpret the ultrafast transfer in photosynthetic complexes and organic photovoltaics 26–28 and eventually applies to optimize organic conductors. Recently, this concept has been applied to explain the experimental results on non-adiabatic singlet fission in a pentacene film. 23

Theoretical Model. - The basic principle is schematically illustrated in Fig. 1. We study a dimer of two excitonically coupled monomers each of which are coupled to both an intramolecular local Holstein phonon and an intermolecular non-local Peierls phonon. Both modes are additionally coupled to their respective dissipative environment, formed, e.g., by a phonon bath. The total Hamiltonian is H = Hmol + Henv . The molecular part is given by

4


Page 4 of 17

Page 5 of 17 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


Figure 1: Schematic view of the model. The intramolecular Holstein modes act as tuning modes, while the intermolecular Peierls mode is an effective coupling mode, such that a conical intersection is formed between the two-dimensional potential energy surfaces.

the Holstein-Peierls dimer model

Hmol = |Ai hA hA| + |Bi hB hB| + (|Ai V hB| + h.c.) ,

(1)

with hA = A + hg − κQH , hB = B + hg + κQH and the electronic coupling strength V = V0 + ΛQP . Here, |j = A, Bi denotes the single excitation state of the monomer j. P Moreover, hg = 12 i=H,P Ωi (Pi2 +Q2i ) is the sum of the local (Holstein) and nonlocal (Peierls) linear phonon mode. Here, PH/P and QH/P indicate the momentum and the coordinate of the modes with the vibrational frequency ΩH/P , respectively. The local phonon is coupled to the electronic state of each monomer, the latter having the site energy A/B , via the vibronic coupling strength κ. Both monomers are electronically coupled via a Coulomb interaction with a static part V0 and a dynamic Peierls part due to their coupling to the non-local phonon with strength Λ. Both phonon modes are coupled to their own environment each of which is modeled by a harmonic bath with the Hamiltonian " Henv =

X X i=H,P α

2 p2i,α mi,α ωi,α + 2mi,α 2

2 # ci,α Qi . xi,α + 2 mi,α ωi,α

Here, the momenta of the bath oscillators are denoted as pi,α while their coordinates, masses and frequencies are denoted by xi,α , mi,α and ωi,α . The respective coupling constants are

5



Page 6 of 17

Figure 2: Time-dependent population ρA (t) of monomer A for the avoided crossing (AC, blue line) and the conical intersection (CI, red line) case. The magenta and the black lines show the global fit to one or two exponentials, see text.

ci,α . The baths are characterized by the spectral densities JP/H (ω) = ωP/H,α ) = ηP/H ω exp(−ω/ωc ).

π 2

P

c2P/H,α

α mP/H,α ωP/H,α δ(ω −

For simplicity, we assume two Ohmic spectral densities

throughout this work, i.e., with the vibrational damping strengths ηH/P and the cutoff frequency ωc (chosen identical for both baths). To describe exciton transport in organic molecular crystals, relevant model parameters are the energy gap = A − B between the two monomers, and the vibrational mode frequencies. Typically, out of the various local and nonlocal modes, only a specific single mode of both classes dominates and, typically, ωH ωP . 9,29 To be specific, we assign the vibrational frequencies ΩH = 1000 cm−1 and ΩP = 150 cm−1 , respectively. For the vibronic interaction, the electronic potential energy surfaces of the two monomers A and B are shifted relative to each other, which is characterized by the Holstein parameter ∆H . Thus, we obtain κ=

∆√ H ΩH . 2

We are interested in the dynamics of the population of the two monomers when

the Coulomb interaction V0 and the vibronic coupling strengths of the local (κ) and the nonlocal (Λ) modes are varied. To model the damping of the local and nonlocal modes, we set ηP/H = 0.5 and ωc = 100 cm−1 and the baths are both held at room temperature (300 K).

6


Page 7 of 17 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


Exciton transfer dynamics. - To investigate the transfer dynamics between the two monomers, we numerically calculate the reduced density matrix of the electronic subsystem using the time-nonlocal (TNL) quantum master equation. 30,31 Convergence with respect to the restricted dimension of the vibrational Hilbert spaces is ensured. We determine the monomer populations ρj (t) for = 400 cm−1 , V0 = 200 cm−1 , ∆H = 0.3, thus, κ = 212 cm−1 . 27,32,33 At initial time t = 0, monomer A is fully occupied and starts out from the vibrational ground state, ρA (0) = 1. We compare two cases, first, when the nonlocal Peierls mode couples strongly, i.e., Λ = 200 cm−1 , and, second, when no Peierls mode is present, i.e., Λ = 0 cm−1 . As detailed below, we refer to the first case as the conical intersection (CI) case, and to the second as the avoided crossing (AC) case. Fig. 2 illustrates the time-dependent population ρA (t) (ρB (t) = 1 − ρA (t)). We observe a striking difference between the AC and the CI case. In the CI case with a strongly coupled Peierls phonon, the population (red line) rapidly decreases within the first 100 fs, and a slow decay follows, with small coherent oscillations superposed. In contrast, in the AC case in absence of the Peierls phonon, the population of A (blue line) decreases much slower also with coherent oscillations. The decay happens on a time scale similar to the slow decay in the CI case. To quantify the transfer time scales, we fit an exponential to the global dynamics without the oscillations. We obtain a transfer time of 3.2 ps (magenta line) for the AC case. It turns out that two exponentials are required for the fit of the CI case. We find two transfer times, 30 fs and 850 fs (black line). Clearly, the striking fast decrease of the transfer time is a result of the nonlocal Peierls phonon. We note that a faster transfer could also be achieved by simply increasing the local Holstein coupling, but a realistic coupling strength is quantitatively not sufficient to induce such a dramatic decrease by roughly two orders of magnitude. It is essential to observe that the Holstein-Peierls model established here is identical to the two-state twomode model of a conical intersection of two electronic potential energy surfaces. Here, the strong vibronic coupling of the exciton to a nonlocal Peierls phonon leads to the formation of

7



−1

10

NTR (fs−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

Page 8 of 17

−2

10

−3

10

−4

10

0

50

100

150

200

250

300

−1

Λ (cm )

Figure 3: Normalized transfer rate NTR as a function of the Peierls vibronic coupling Λ. The kinetics for Λ > 100 cm−1 (blue dashed line) is fitted by two exponential functions one of which is the newly emerged ultrafast transfer channel.

a conical intersection. This is further illustrated by the two potential energy surfaces of the monomers A and B shown in the Supporting Information. The potential landscape shows that in the CI case a degenerate singular point emerges through which the wave packet can pass very efficiently from one to the other electronic state. Hence, we can conclude that a strong coupling of the exciton to a Peierls phonon opens a new ultrafast transfer channel (the ”Peierls channel”) for the exciton.

Normalized transfer rate. - The conclusion is further supported when the Peierls couP pling Λ is varied. We fit the population dynamics to the function 2j=1 Aj exp(−t/τj ) using either one (if possible) or two exponentials. The resulting transfer times are listed in the Supporting Information. In the bi-exponential case, we define the normalized transfer rate NTR = (A1 /τ1 + A2 /τ2 )/(A1 + A2 ). Fig. 3 shows the normalized transfer rate versus Λ. It turns out that a single exponential is sufficient to fit the kinetics in the regime Λ ≤ 100 cm−1 (see Supporting Information), but for Λ > 100 cm−1 two exponentials are necessary, one of which represents the ultrafast transfer. A jump from 0.002 to 0.01 fs−1 in the NTR is observed once Λ exceeds 100 cm−1 . The additional ultrafast transfer component is the evidence of the additional transfer channel. Analyzing the potential energy surfaces, we observe that the potential barrier for the transfer of a wave packet from its initial position to the 8


Page 9 of 17 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


degenerate point is lowered with increasing Λ (see Supporting Information). The ultrafast deactivation is induced by the non-local Peierls phonon and can yield a transfer time as short as 12 fs for Λ = 300 cm−1 .

Ultrafast transfer via the Peierls channel. - Subtracting the exponential kinetics and performing a Fourier transformation of the residual yields the frequencies and the transition channels involved. We show representative cases with the resolved frequencies in Fig. 4 (see Supporting Information for more data). For Λ = 0 cm−1 (black line), three peaks appear, a dominating one at 570 cm−1 and two further peaks around 1000 cm−1 . The frequency of 570 cm−1 (black dashed vertical line) fits the energy gap between the two monomers in the adiabatic basis and is, thus, the evidence of electronic coherence due to the direct Coulomb coupling of two monomers. The peak at ∼ 1000 cm−1 results from the coupling to the local mode and is a signature of vibrational coherence. No evidence of the vibrational frequency of 150 cm−1 (nonlocal mode, light blue dashed line in Fig. 4) is present. For increasing Peierls coupling Λ, the excitonic peak (570 cm−1 ) diminishes and is reduced by a factor of 5 for Λ & 180 cm−1 . Instead, various peaks with small amplitudes appear around 570 cm−1 , which depend strongly on Λ. These are hallmarks of the complicated transfer path of the wave packet when a CI is present (see the potential energy surfaces in Fig. S1 in the Supporting Information). The vibrational peak of the local mode decreases and is strongly suppressed as well when 40 cm−1 . Λ . 180 cm−1 . However, it interestingly revives for Λ = 240 cm−1 . Furthermore, a new peak around the vibrational frequency of the nonlocal mode, 150 cm−1 emerges with increasing of Λ. A more refined analysis shown in the Supporting Information shows that, in fact, the local and nonlocal modes no longer act as the tuning and coupling modes separately. Instead, the finite electronic coupling V0 = 200 cm−1 shifts the CI between the two potential energy surfaces in the negative QP direction (see Supporting Information), such that the nonlocal vibration acts, in addition to being a coupling mode, also as a tuning mode in the vicinity of the conical intersection.

9



7000 150 cm−1

6000

Amplitude (arb. units)

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

Page 10 of 17

570 cm−1 1000 cm−1

5000

−1

Λ = 0 cm

−1

Λ = 40 cm

4000

−1

Λ = 180 cm

−1

Λ = 240 cm 3000 2000 1000 0 0

500

1000 ω (cm−1)

1500

Figure 4: Fourier transform of the residuals for the Peierls coupling Λ = 0 cm−1 to 240 cm−1 . The frequencies of the electronic and the vibrational transitions (local and nonlocal modes) are highlighted by the dashed lines at 150 cm−1 , 570 cm−1 and 1000 cm−1 , respectively.

Wavelet analysis. - A wavelet analysis elucidates more details (see Supporting Information for a brief summary of the technique). In Fig. 5(a), we show the result for Λ = 0. The transfer is dominated by the pure excitonic coupling at 570 cm1 with an electronic coherence lifetime of ∼350 fs. The vibrational coherence of the local mode at 1000 cm−1 is relatively weak which demonstrates the electronic nature of the coherent transfer. No evidence of the Peierls channel at 150 cm−1 exists. For Λ = 40 cm−1 , see Fig. 5(b), the electronic coherence is dephased more rapidly and the oscillations of the local mode are still present with a rather long lifetime. Again, no evidence of the low-frequency mode is present at 150 cm−1 , i.e., the contribution of the CI still is weak. For Λ = 180 cm−1 , the vibrational coherence of the local mode at 1000 cm−1 has disappeared, see Fig. 5(c). Interestingly, new oscillations appear beyond 100 fs, with the frequency of the nonlocal mode at 150 cm−1 . They show the lifetime of ∼400 fs. Thus, the nonlocal vibrations play the partial role of a tuning mode in the transfer (see the CI of the potential surfaces in the Supporting Information). Moreover, the electronic coherence at 570 cm−1 and its lifetime are further damped and its frequency 10


Page 11 of 17 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


is shifted to 1000 cm−1 with fast decay after 200 fs, which indicates that the transfer of the wave packet is complete within 200 fs (see the dashed curve) and oscillates along the tuning mode at the lower electronically excited state. Hence, the CI dominates the energy transfer. The oscillations are more complicated when Λ is increased further. In Fig. 5(d), the oscillations for Λ = 240 cm−1 are rather small. First, electronic coherence is initially absent, but increases over time with a frequency of 570 cm−1 . It reaches a maximum at ∼120 fs, diminishes afterwards, and then shifts to the new frequency. Second, the vibrational coherence of the local mode is generated after 120 fs and survives for long times. Third, the low-frequency mode at 150 cm−1 is also present after t ≥ 100 fs. Based on the tracking of electronic and vibrational coherence, the wave-packet dynamics can be clearly illustrated. Electronic quantum coherence is completely destroyed in the configuration of the CI with a strong coupling to the nonlocal vibration, Λ = 240 cm−1 . During the energy-transfer process, the transient electronic coherence is generated (∼120 fs) by strong nonadiabatic interaction, when the wave packet approaches the degenerate point between two PESs. After passing the CI, the wave packet further oscillates along the tuning and coupling modes in the lower excited state due to the strong mixing of these two modes at Λ = 240 cm−1 . Thus, the energy-transfer process for Λ = 240 cm−1 is entirely driven by the CI.

Figure 5: Wavelet analysis of the residual obtained after subtracting the kinetics by the global fitting approach, for (a) Λ = 0 cm−1 , (b) Λ = 40 cm−1 , (c) Λ = 180 cm−1 , and, (d) Λ = 240 cm−1 .

11



The impact of the local Holstein coupling strength κ is further analyzed in the Supporting Information. For κ = 0, no CI is present and a single peak at 570 cm−1 reflects full electronic coherence. Increasing κ again suppresses the peak at 570 cm−1 and, accordingly, the electronic coherence. At the same time, the peaks associated with the vibrational frequencies at ∼ 150 cm−1 and ∼ 1000 cm−1 emerge. Overall, the formation of the CI is mainly responsible for the fast electronic decoherence. A strong local mode induces localization and incoherent transfer which is in agreement with Ref. 34

Conclusions. - We have shown that a strong coupling of the exciton to a nonlocal Peierls vibrational mode can generate a novel channel (Peierls channel) allowing an ultrafast transfer. This occurs via the formation of a conical intersection of two potential energy surfaces when the dimer simultaneously couples to a local (Holstein) and a nonlocal (Peierls) vibrational mode. The local and nonlocal modes jointly induce exciton localization during the transfer when the Peierls coupling is strong enough 10,35,36 and exciton transfer shifts from a delocalized (coherent) to a localized (incoherent) transfer, in line with previous studies. 29 The model helps to interpret the ultrafast energy and charge transport in photosynthetic complexes 10 or organic photovoltaics 26–28 and eventually applies to optimize organic conductors. Moreover, the modeling relates recent findings 37–40 with the emergence of a conical intersection and explains the delocalized phase for weak coupling of the nonlocal (or, alternatively, the local) mode. For strong coupling, a conical intersection is formed and the transfer is incoherent, with vibrational oscillations superposed.

Acknowledgments. - This work was supported by the Max Planck Society and the Deutsche Forschungsgemeinschaft. PN acknowledges financial support by the DFG project NA394/2-1.

Supporting Information. - Extracted transfer time constants for varying Peierls coupling. Explicit configurations of the potential energy surfaces. Fourier transforms of the

12


Page 12 of 17

Page 13 of 17 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


residuals in view of the electronic and vibrational coherence properties. Basics of the wavelet transform. Residuals and the wavelet analysis of the energy transfer for varying Holstein coupling.

References (1) Nakamura H. Nonadiabatic Transition; World Scientific: Singapore, 2012. (2) Blankenship R. E. Molecular Mechanisms of Photosynthesis; Blackwell Science: Oxford/Malden, 2002. (3) Schröter, M.; Ivanov, D.; Schulze, J.; Polyutov, S.P.; Yan, Y.; Pullerits, T.; K¨ uhn, O. Exciton-vibrational coupling in the dynamics and spectroscopy of Frenkel excitons in molecular aggregates. Phys. Rep. 2015, 568, 1-78. (4) Domcke, W.; Yarkony, D. R.; Köppel, H. Conical Intersection: Electronic Structure, Dynamics and Spectroscopy; World Scientific: Singapore, 2004. (5) Domcke, W.; Yarkony, D. R.; Köppel, H. Conical Intersection: Theory, Computation and Experiment; World Scientific: Singapore, 2011. (6) Holstein, T. Studies of Polaron Motion: Part I. The Molecular-Crystal Model. Ann. Phys. 1959, 8, 325-342. (7) Holstein, T. Studies of Polaron Motion: Part II. The “Small” Polaron. Ann. Phys. 1959, 8, 343-389. (8) Girlando, A.; Grisanti, L.; Masino, M.; Brillante, A.; Valle, R.G.D.; Venuti, E. Interaction of Charge Carriers with Lattice and Molecular Phonons in Crystalline Pentacene. J. Chem. Phys. 2011, 135, 084701. (9) Aragó, J.; Troisi, A. Dynamics of the Excitonic Coupling in Organic Crystals. Phys. Rev. Lett. 2015, 114, 026402. 13



(10) Lee, M. H.; Troisi, A. Vibronic Enhancement of Excitation Energy Transport: Interplay Between Local and Non-local Exciton-phonon Interactions. J. Chem. Phys. 2017, 146, 075101. (11) Smith, M.B.; Michl, J. Singlet Fission. Chem. Rev. 2010, 110, 6891-6936. (12) Smith, M. B.; Michl, J. Recent Advances in Singlet Fission. Annu. Rev. Phys. Chem. 2013, 64, 361-386. (13) Chan, W.-L.; Berkelbach, T.C.; Provorse, M.R.; Monahan, N.R.; Tritsch, J.R.; Hybertsen, M.S.; Reichman, D.R.; Gao, J.; Zhu, X.-Y. The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory. Acc. Chem. Res. 2013, 46, 1321-1329. (14) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Microscopic Theory of Singlet Exciton Fission. I. General Formulation. J. Chem. Phys. 2013, 138, 114102. (15) Tempelaar, R.; Reichman, D.R. Vibronic Exciton Theory of Singlet Fission. II. TwoDimensional Spectroscopic Detection of the Correlated Triplet Pair State. J. Chem. Phys. 2017, 146, 174704. (16) Tempelaar, R.; Reichman, D.R. Vibronic Exciton Theory of Singlet Fission. III. How Vibronic Coupling and Thermodynamics Promote Rapid Triplet Generation in Pentacene Crystals. J. Chem. Phys. 2018, 148, 244701. (17) Morrison, A. F.; Herbert, J. M. Evidence for Singlet Fission Driven by Vibronic Coherence in Crystalline Tetracene. J. Phys. Chem. Lett. 2017, 8, 1442 -1448. (18) Tamura, H.; Huix-Rotllant, M.; Burghardt, I.; Olivier, Y.; Beljonne, D. First-Principles Quantum Dynamics of Singlet Fission: Coherent versus Thermally Activated Mechanisms Governed by Molecular π Stacking. Phys. Rev. Lett. 2015, 115, 107401.

14


Page 14 of 17

Page 15 of 17 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


(19) Beljonne, D.; Yamagata, H.; Brédas, J.L.; Spano, F.C.; Oliver, Y. Charge-Transfer Excitations Steer the Davydov Splitting and Mediate Singlet Exciton Fission in Pentacene. Phys. Rev. Lett. 2013, 110, 226402. (20) Ito, S.; Nagami, T.; Nakano, M. Density Analysis of Intra- and Intermolecular Vibronic Couplings toward Bath Engineering for Singlet Fission.J. Phys. Chem. Lett. 2015, 6, 4972-4977. (21) Renaud, N.; Grozema, F. C. Intermolecular Vibrational Modes Speed Up Singlet Fission in Perylenediimide Crystals. J. Phys. Chem. Lett. 2015, 6 360-365. (22) Yao, Y. Coherent Dynamics of Singlet Fission Controlled by Nonlocal Electron-phonon Coupling. Phys. Rev. B 2016, 93, 115426. (23) Duan, H.-G.; Jha, A.; Li, X.; Tiwari, V.; Ye, H.; Nayak, P.K.; Zhu, X.-L.; Li, Z.; Martinez, T.J.; Thorwart, M.; Miller, R.J.D. Intermolecular Vibrations Drive Ultrafast Singlet Fission, submitted (2018). (24) Polli, D.; Altoé, P.; Weingart, O.; Spillane, K. M.; Manzoni, C.; Brida, D.; Tomasello, G.; Orlandi, G.; Kukura, P.; Mathies, R. A.; Garavelli, M.; Cerullo, G. Conical Intersection Dynamics of the Primary Photoisomerization Event in Vision. Nature 2010, 467 440-443. (25) Hamm, P.; Stock, G. Vibrational Conical Intersections as a Mechanism of Ultrafast Vibrational Relaxation. Phys. Rev. Lett. 2012, 109, 173201 (26) Eggeman, A. S.; Illig, S.; Troisi, A.; Sirringhaus, H.; Midgley, P. A. Measurement of Molecular Motion in Organic Semiconductors by Thermal Diffuse Electron Scattering. Nat. Mat. 2013, 12, 1045-1049. (27) Tamura, H.; Ramon, J. G. S.; Bittner, E. R.; Burghardt, I. Phonon-Driven Ultrafast Ex-

15



citon Dissociation at Donor-Acceptor Polymer Heterojunctions. Phys. Rev. Lett. 2008, 100, 107402. (28) Bakulin, A. A.; Lovrincic, R.; Yu, X.; Selig, O.; Bakker, H. J.; Rezus, Y. L. A.; Nayak, P. K.; Fonari, A.; Coropceanu, V.; Brédas, J. L.; Cahen, D. Mode-selective Vibrational Modulation of Charge Transport in Organic Electronic Devices. Nat. Comm. 2015, 6, 7880. (29) Aragó,J.; Troisi, A. Regimes of Exciton Transport in Molecular Crystals in the Presence of Dynamic Disorder. Adv. Func. Mat. 2016, 26, 2316-2325. (30) Meier, C.; Tannor, D. J. Non-Markovian Evolution of the Density Operator in the Presence of Strong Laser Fields. J. Chem. Phys. 1999, 111, 3365-3376. (31) Kleinekathöfer, U. Non-Markovian Theories Based on a Decomposition of the Spectral Density. J. Chem. Phys. 2004, 121, 2505-2514. (32) Qi, G.; Govind, N.; Ratner, M. A.; Carmer, C. J.; Gagliardi, L. Influence of Coherent Tunneling and Incoherent Hopping on the Charge Transfer Mechanism in Linear DonorBridgeAcceptor Systems. J. Phys. Chem. Lett. 2015, 6, 4889-4897. (33) Savoie, B. M.; Rao, A.; Bakulin, A. A.; Gelinas, S.; Movaghar, B.; Friend, R. H.; Marks, T. J.; Ratner, M. A. Unequal Partnership: Asymmetric Roles of Polymeric Donor and Fullerene Acceptor in Generating Free Charge. J. Am. Chem. Soc. 2014, 136, 2876-2884. (34) Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Brˇedas, J. L. Charge Transport in Organic Semiconductors. Chem. Rev. 2007, 107, 926-952. (35) Trionfi, A.; Wang, D. H.; Jacobs,J. D.; Tan, L.-S.; Vaia, R. A.; Hsu, J. W. P. Direct Measurement of the Percolation Probability in Carbon Nanofiber-Polyimide Nanocomposites. Phys. Rev. Lett. 2009, 102, 116601. 16


Page 16 of 17

Page 17 of 17 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


(36) Troisi, A.; Orlandi, G. Dynamics of the Intermolecular Transfer Integral in Crystalline Organic Semiconductors. J. Phys. Chem. A 2006, 110, 4065-4070. (37) Yao, Y.; Zhou, N.; Prior, J.; Zhao, Y. Competition Between Diagonal and Off-diagonal Coupling Gives Rise to Charge-transfer States in Polymeric Solar Cells. Scient. Rep. 2015, 5, 14555. (38) Guo, C.; Weichselbaum, A.; von Delft, J.; Vojta, M. Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models. Phys. Rev. Lett. 2012, 108, 160401. (39) Bruognolo, B.; Weichselbaum, A.; Guo, C.; von Delft, J.; Schneider, I.; Vojta, M. TwoBath Spin-Boson Model: Phase Diagram and Critical Properties. Phys. Rev. B 2014, 90, 245130. (40) Zhou, N.; Chen, L.; Xu, D.; Chernyak, V.; Zhao, Y. Symmetry and the Critical Phase of the Two-Bath Spin-Boson Model: Ground-State Properties. Phys. Rev. B 2015, 91, 195129.

17


Ultrafast Energy Transfer in Excitonically Coupled Molecules Induced

Recommend Documents