Article pubs.acs.org/JPCC
Bath Effect in Singlet Fission Dynamics Guohua Tao* Shenzhen Key Laboratory of New Energy Materials by Design, Peking University, Shenzhen, China 518055 School of Advanced Materials, Peking University Shenzhen Graduate School, Shenzhen, China 518055 ABSTRACT: Interactions between electronic states and the environmental phonon bath in singlet fission plays a vital role in understanding the underlying mechanisms. In this work we apply the symmetrical quasiclassical nonadiabatic molecular dynamics simulation method [Cotton, S. J.; Miller, W. H. J. Phys. Chem. A 2013, 117, 7190 and Meyer, H. D.; Miller, W. H. J. Chem. Phys. 1979, 70, 3214] to a model system of singlet fission materials and examine the dependence of fission dynamics on some key factors in the bath modeling, such as, functional form of spectra density, characteristic frequencies, reorganization energy, and bath temperatures. Possible schemes of engineering the phonon spectrum to control fission dynamics have been discussed.
temperature range12 and different bath relaxation functions produce similar results.10 Static quantum mechanical calculations,3 such as density functional theory and many body perturbation theory,20−22 are very powerful to identify energy levels and electronic couplings; however, they may be insufficient by themselves to draw a correct comprehensive conclusion without dynamical treatments. Albeit a popular dynamical method for energy and electron transfer, it is well-known that the incoherent Förster theory cannot describe electronic coherence correctly. Recent time-resolved two-photon photoemission (TR-2PPE) spectroscopy experiments in crystalline pentacene23 and tetracene,24 however, suggested that SF, i.e. the conversion of the singlet (S1) and the triplet pair state (TT), is quantum coherent. Furthermore, Fö rster theory only supports a sequential mechanism of SF in pentacene, in which the energy of the CT state should be between those of the S1 and TT state.9 By contrast, results from Redfield theory, a dynamical method that treats electronic couplings accurately, show that the high-lying CT state does not necessary imply the direct mechanism (as suggested by static calculations3), and the superexchange mediated fission could still be efficient.10,11 However, Redfield theory becomes invalid in the strong electronic-phonon coupling regime. Other methods, such as multiconfiguration time-dependent Hartree (MCTDH) method,25,26 mixed quantum-classical technique based on time domain DFT,27 and surface hopping,28 have been successfully applied to study nonadiabatic exciton dissociation dynamics in organic semiconductors29−31 and SF and subsequent charge transfer dynamics at the pentacene/C60 interface.5 They also have their own disadvantages.6
I. INTRODUCTION Singlet fission (SF) in organic molecular materials has been attractive in the past decade to the ever-growing research community due to its potential implementation in highly efficient but low-cost solar cells.1,2 Singlet fission dynamics involves the electronically nonadiabatic process, in which electronic and nuclear motions are coupled with each other and the electronic relaxation and dephasing are mediated by the environmental phonon bath.3−6 It is therefore essential to model the coupling between molecular vibrations and electronic dynamics accurately, and better understandings of electronic−phonon interactions may provide insights into how to control dynamics and help design promising new materials.1−12 Although energy levels, electronic couplings, and phonon bath are all important in determining SF dynamics, Greyson et al. suggested that energy levels should be paid more attention than electronic couplings in chromophore design based on their phenomenological density matrix model.8 Teichen and Eaves pointed out that energy levels fluctuate induced by solvent, and the environmental fluctuations should also be part of the design principle for maximizing SF yield.9 The phonon-induced fluctuations in the state energies have been observed by Prezhdo and co-workers in their ab initio nonadiabatic MD simulations.4,5 The realistic electronic-phonon interactions in singlet fission materials are truly intricate and depend sensitively on intramolecular motions,7 intermolecular couplings,3,8,13−16 stacking motifs,13−16 crystal environment,16−19 and other factors. It is thus desirable to develop accurate and efficient theoretical tools to study bath effect in singlet fission. Zimmerman and co-workers performed quantum mechanical calculations and proposed that intermolecular interactions leads to SF via nonadiabatic transitions.3 Reichman and co-workers investigated the bath effect in SF using Redfield theory10−12 and found that fission is thermally insensitive over a wide © 2014 American Chemical Society
Received: September 18, 2014 Revised: October 27, 2014 Published: October 27, 2014 27258
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suggested by Cotton and Miller32 is a symmetrical binning window function, i.e.
Recently Cotton and Miller developed an extremely efficient symmetrical quasi-classical (SQC) simulation method32−34 for nonadiabatic dynamics. The SQC method treats the electronicnuclear couplings and coherence accurately in the Meyer− Miller (MM) model,35,36 which has been applied to a variety of electronically nonadiabatic problems, including state-to-state reactive scattering,32 electronically nonadiabatic dynamics in the spin-boson systems,33 electron transfer34 and quantum dynamics and pathway interference in SF.6 It has been shown36 that the classical MM Hamiltonian is actually the exact representation of the quantum electronic nuclear coupled system and has been successfully implemented to many interesting electronically nonadiabatic problems. In particular, Tao and Miller37 applied the MM model (combined with linearized semiclassical initial value representation, LSC-IVR) to excitation energy transfer in a photosynthetic model system and reproduced coherent transfer dynamics in very good agreement with hierarchical equation of motion method.38 Note that the model Hamiltonian for singlet fission in this work is the same as that in ref 37, and the SQC method works even better than LSC-IVR in describing SF dynamics.6 Inspired by previous work done by using Redfield theory,10−12 here we investigate the role of the electronicphonon couplings in SF dynamics by using the SQC method. Section II briefly describes the theoretical method and models. Results and discussions are presented in section III. Section IV concludes.
Wk(nk , N ) =
n ⎡ Pk(t ) = ⎢⟨Wi (ni(0), 1) ∏ Wl (nl(0), 0)Wk(nk(t ), 1) ⎢⎣ l≠i
2
k=1
n
+
⎥⎦
n
⎤
l≠k
⎥⎦
(4)
b. Phonon Bath Models. The property of electronicphonon interactions is determined by the spectra density J(ω) =
π 2
∑ k
ck 2 δ(ω − ωk) ωk
(5)
which contains information on the detailed motions of phonon DOFs. There are many choices for the functional form of the spectra density, and we consider in this work mainly the following several different forms and the mixed form of their combinations (see Figure 1): (i) Debye form:
with Hel = ∑ jN b [(P kj 2 /2) + k⟩⟨k|∑j(−ckjQkj). Here (Qkj, Pkj) is the phase space point of the jth bath mode, which is coupled with the k-th electronic state. ωkj and ckj are the frequency and the coupling constant of the corresponding mode, respectively. n is the number of electronic states, and Nb is the number of the bath modes that are coupled to each single electronic state. The electronically nonadiabatic molecular dynamics simulation is performed based on the classical MM Hamiltonian,35,36 i.e. ⎜
l≠k
∏ Wl (nl(t ), 0)⟩⎥
(1)
n
⎤
n ⎡ n /⎢∑ ⟨Wi (ni(0), 1) ∏ Wl (nl(0), 0)Wk(nk (t ), 1) ⎢⎣ k l≠i
+ ∑l≠k|k⟩Ekl⟨l|, Hph = ∑k n= 1 2 (1/2)ω kj Q kj 2 ] and H el−ph = ∑ k =n 1 |
∑ ⎛⎝ 1 xk 2 +
n
∏ Wl (nl(t ), 0)⟩⎥
∑k n= 1 |k⟩Ek⟨k|
H(x , p, Q, P) =
(3)
where h(z) is the Heaviside function, Δn = 2γ, and N is the electronic quantum number representing the electronic state is occupied (N = 0) or not (N = 1). Without loss of generality, assuming the system starts from the state i, we find the timedependent electronic population of state k from a Monte Carlo averaging procedure:
II. THEORETICAL METHOD AND MODELS a. Nonadiabatic Dynamics Simulation Method. The Frenkel exciton Hamiltonian for singlet fission includes the following three parts: electronic, phonon, and electronic− phonon couplings H = Hel + Hph + Hel − ph
⎞ 1 ⎛⎜ Δn h − |nk − N |⎟ ⎠ Δn ⎝ 2
J(ω) =
2λωωc ω 2 + ωc 2
(6a)
⎞ 1 2 pk − γ ⎟Hkk(Q, P) ⎠ 2
n
∑ ∑ k=1 l=k+1
(xkxl + pk pl )Hkl(Q, P)
(2)
where Hkk and Hkl are the diagonal and off-diagonal Hamiltonian matrix elements. (x, p) and (Q, P) are the phase points for the classical electronic and nuclear degrees of freedom (DOFs), respectively. nk = 1/2(xk2 + pk2) − γ is the quantum number of the electronic state k, and γ is a parameter accounting for the effective zero point energy.32 xkxl + pkpl is the electronic coherence between different states. The properties of an electronic state can be obtained by projecting the ensemble of quasiclassical trajectories onto the corresponding state, and the (classical) projection operator
Figure 1. Different functional forms of phonon bath. (a) Debye and Ohmic type; (b) pseudo local type with three components (see the text); (c) mixed form consisting of a low frequency (ωc = 5 meV) Debye component and a high frequency (ωc = 180 meV) pseudo local term. 27259
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(ii) Ohmic form: J(ω) = ηωe−ω / ωc
(6b)
(iii) Pseudo local form: Nk
J(ω) =
∑ k
λk ω Γk (ω − Ωk)2 + Γk 2
(6c)
(iv) Single frequency form: J(ω) = π ∑ λk ωδ(ω − Ωk) k
(6d)
(v) Mixed form: ex.: J(ω) =
2λoωωc 2
ω + ωc
2
+
λk ω Γk (ω − Ωk)2 + Γk 2
(6e)
Figure 2. Dependence of state populations on different forms of phonon spectra density. (a) Singlet state, S1; (b) charge transfer state, CT; (c) coupled triplet pair, TT. Bath parameters: ωc = 180 meV (Debye and Ohmic); Ωk = 180 meV, Γk = 15 meV (pseudo local, single term Nk = 1); Ωk = 180 meV (single frequency, single mode).
The first one is the Debye form used in previous work, and the second is the widely adopted Ohmic bath. Here η is the coupling strength between the electronic system and the bath, λ is the reorganization energy, and ωc is the characteristic frequency of the bath. The third type of bath is the so-called pseudo local bath introduced in ref 10, which is the sum of individual localized spectra functions. The single frequency form consists of one or multiple modes at a single frequency Ωk, and it can be seen as the Γk → 0 limit of the pseudo local type. The last one is a mixed form combining one broad spectra function, such as the Debye or Ohmic form, and one localized spectra function, such as a term in the pseudo local form or the single frequency form. In this work, we use the following parameters for the pseudo local bath:10 λk = {30,40,30} meV, Ωk = {150,180,200} meV, and ℏΓk = 15 meV. For the mixed form, we include a low frequency (ωc = 5 meV) Debye component and a high frequency (Ωk = 180 meV) term either in the pseudo local form or the single frequency form with an equal reorganization energy for each component, i.e., λD = λk = 50 meV. For the spectra density with multiple peaks and the mixed form, the total reorganization energy is given by λ = ∑k λk or λ = λD + λk. We discretize the continuous spectra density in the way suggested by Wang et al.39 For simplicity, we take the threestate model for pentacene (model a in ref 6) as an example to illustrate the bath effect unless specified otherwise, i.e., E(S1) − E(TT) = 200 meV, E(CT) − E(TT) = 300 meV, and V(S1CT) = V(CT-TT) = −50 meV. 6,10−12
phonon baths except for the single frequency case, in which a much slower fission is observed. Convergence with respect to the number of bath modes has been checked, and no appreciable difference is found. Results for the system coupled with the phonon bath including multiple components are shown in Figure 3, along
Figure 3. Fission dynamics mediated by multiple-component bath. (a) Pseudo local bath; (b) single frequency bath (only S1 population is shown). Bath parameters: λk = 30, 40, 30 meV; Ωk = 150, 180, 200 meV; Γk = 15 meV (pseudo local) for k = 1, 2, 3.
III. RESULTS AND DISCUSSIONS First we examine the dependence of fission dynamics on the functional form of the phonon bath. To make a close comparison, we perform the SQC simulations at the same temperature (T = 300 K) with the characteristic frequency (180 meV) and reorganization energy (100 meV) to be same for all types of baths. As seen in Figure 1, both Debye and Ohmic baths have broad spectra density with the maximum at ωc, while the Debye bath shows a longer high frequency tail. The pseudo local bath used here has a much narrow broadening (Γk = 15 meV) and the single frequency bath represents the lower limit of spectra broadening. Figure 2 depicts the results of time dependent state populations in fission dynamics mediated by a number of different types of phonon baths. With the same settings for major parameters of spectra density, the functional form itself only causes mild differences in fission dynamics for all types of
with the ones for the Debye bath for comparison. With the same temperature and same reorganization energy, the overall fission dynamics for both the pseudo local bath and single frequency bath resemble closely that for the Debye bath, considering the effective characteristic frequencies are slightly different. Figure 3b also presents the individual contributions from each component of single frequency bath, which implies that the interactions among different modes significantly enhance the overall fission dynamics in this case (see also the single frequency bath and single mode case in Figure 2). Next we look into the effect of the characteristic frequency on fission dynamics. In addition to the high frequency (ωc = 180 meV) bath being studied in previous work,6 here we also consider a low frequency bath with a much smaller characteristic frequency (ωc = 5 meV, ∼40 cm−1), which could be a good 27260
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representation for intermolecular couplings.40 As shown in Figure 4, the slow bath allows for longer-lived quantum
Figure 5. Dependence of fission dynamics on reorganization energy. Debye bath, and T = 300 K. Results are shown for λ = 25 meV (dashed green); 50 meV (blue solid); 100 meV (black solid); and 200 meV (red solid). (a) ωc = 180 meV; (b) ωc = 5 meV.
Figure 4. Dependence of fission dynamics on the characteristic frequency of different types of baths. T = 300 K, and λ = 100 meV. (a) Debye and (b) ohmic.
Wj → k(t ) = Vjk 2
coherence, and fission dynamics becomes slow too. The results for the Ohmic bath are similar to those for the Debye bath. Note that the energy differences of electronic states are on the order of a few hundred meVs, which falls far outside the significant region of the spectra density for ωc = 5 meV. However, the fission lifetime is still as fast as about 600 fs (see Table 1). Therefore, it is not well justified that the contribution from the low frequency bath is too small to be neglected.
λ (meV)
low ωc
fit Marcus fit Marcus
25
50
100
200
0.619 8.65 11.2(0.003)b 23.1
0.339 0.415 1.36 1.11
0.223 0.223 0.597 0.597
0.317 0.829 3.85(0.293)b 2.22
a Values are obtained by fitting the SQC results (eq 7) and by using Marcus theory (eq 8) for both high frequency (ωc = 180 meV) and low frequency (ωc = 5 meV) bath cases. bWe take the second time constant which represents fission dynamics of the majority of population since a very short-time decay (shown in parentheses) is involved.
Reorganization energy is directly related to the coupling strength of the electronic−phonon interactions; thus, it may be the most important parameter in the bath modeling. We here investigate the dependence of fission dynamics on reorganization energy by using the SQC nonadiabatic simulation method. Results for the time-dependent population of S1 in the model system with either high or low frequency baths are shown in Figure 5. The trend is complicate: as λ increases or decreases from 100 meV, the fission rate decreases in both directions. We fit the SQC results by a two exponential decay model P(S1) = A exp( −t /τ1) + (1 − A) exp( −t /τ2)
(8)
here Ej − Ek = E(S1) − E(TT) = 200 meV, and the effective electronic coupling Vjk, which can be estimated from perturbation theory,1,10 includes contributions from both direct pathway (VS1‑TT) and intermediated pathway (VS1‑CT and VCT‑TT). To get rid of some uncertainties, we identify the calculated rate for λ = 100 meV to the fitted one and rescale other values for different λ in Table 1. Note that the turnover behavior in the fission rate with respect to the reorganization energy cannot be predicted correctly by Redfield theory, which gives a linear dependence.10 In the high frequency bath case, quantum coherence shows up as λ decreases to about 50 meV since strong electronic− phonon couplings would destroy it. Short-time coherent dynamics can also been seen in the low frequency bath case, in which nuclear motions are slow, making the effective electronic-phonon coupling small. The SQC nonadiabatic simulation method therefore provides rich detailed information on fission dynamics. It is worth noting that the incoherent Förster theory cannot describe coherent dynamics well since the electronic couplings are treated perturbatively. Next we study the temperature dependence of fission dynamics. For the high ωc case, the SQC fission dynamics behaves largely temperature independent in a wide temperature range (10−300 K) for both Debye (Figure 6) and Ohmic (not shown) types of baths, consistent with previous Redfield results.12 By contrast, for the low ωc case, the temperature dependence could be seen at T = 77 K and becomes pronounced at room temperature. Other than the current benchmark model (Figure 6a,b), similar results are found for a different setting of energy diagram in a downhill model (Figure 6c,d), in which the energy of the CT state is in between those of the S1 and TT states, i.e., E(S1) − E(TT) = 200 meV, E(CT) − E(TT) = 100 meV. It thus clearly shows that the temperature independence in fission dynamics is not related to the “energy barrier” in the energy diagram, but a result of thermally inactivated bath modes at temperatures (300 K is about 26 meV) that are considerably small in comparison with the vibrational energy level spacing. For the downhill model, it is interesting to see that fission dynamics speeds up as
Table 1. Rate (in ps) for Different Reorganization Energy λ (in meV)a high ωc
⎡ (E − E + 2λ)2 ⎤ π k j ⎥ exp⎢ − ⎢⎣ ⎥⎦ 8λkBT 2λkBT
(7)
and take the first time constant as the stationary rate, i.e., k = 1/ τ1. The fitted rate constants are in qualitatively good agreement (see Table 1) with the predictions from Macus theory,9 i.e. 27261
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parameters for realistic systems without enough microscopic information. Finally, we investigate the bath effect on pathway interference in the SF dynamics. When a small nonzero direct coupling (|V(S1-TT)| = 5meV) is introduced, the interference between direct and CT-mediated pathway could mediate fission dynamics, sometimes to a significant amount.6 Here we consider a high frequency Ohmic bath, a low frequency Debye bath, and a mixed bath of it and a high frequency pseudo local bath. We fit the SQC results of the time-dependent S1 population by using eq 7, and the fitted SF times are given in Table 2. Table 2. Fitted SF Times (in fs) versus the Direct Electronic Coupling (in meV) Figure 6. Temperature dependence of fission dynamics. Results are shown for the benchmark CT barrier case [(a) and (b)] and the downhill case [(c) and (d)] (see the text).
temperature deceases (see Figure 6d). The enhancement in rate may be due to quantum coherence, which may provide a new route to design efficient singlet fission materials. It is worthy to consider this topic in a future study. Since the low frequency bath may behave considerably different from its high frequency counterpart, we consider next the mixed form bath consisting of a low frequency Debye component and a high frequency pseudo local (single frequency) component (Figure 2c). The results are shown in Figure 7, in comparison with those obtained for the original
V(S1-TT)
Debye low
Debye low + pseudo local
Debye higha
Ohmic high
−5 0 5
367 597 1707
256 386 656
162 223 300
159 221 293
a
Data taken from ref 6.
In the high frequency case (ωc = 180 meV), similar quantum coherence in different pathways is found in the Ohmic bath case (not shown) as that in the Debye bath case.6 In the low frequency case (ωc = 5 meV), the small direct coupling (V(S1TT) = −5 meV) with the same sign as the CT coupling (V(S1CT) = V(CT-TT) = −50 meV) enhances the fission dynamics about 40%, while the antisign direct coupling causes a 186% increase in the decay time (see Figure 8 and Table 2). Changes
Figure 7. Fission dynamics mediated by mixed form bath. Results shown are for (a) a mixing of the Debye bath (ωc = 5 meV) and pseudo local bath (Ωk = 180 meV); (b) a mixing of the Debye bath (ωc = 5 meV) and single frequency bath (Ωk = 180 meV). Results for the original bath are the same as those in Figures 2 (pseudo local and single frequency bath) and 4 (Debye bath, both high and low frequencies). Bath parameters: λD = λk = 50 meV. Only the S1 population is shown.
Figure 8. Modulation of the direct S1-TT coupling on the CT mediated fission dynamics. Results are obtained by the SQC simulation at T = 300 K for (a) Debye bath, ωc = 5 meV and (b) a mixing of Debye bath (ωc = 5 meV, λD = 80 meV) and pseudo local bath (Ωk = 175 meV, λk = 20 meV). Solid lines: V(S1-TT) = 0; dotted lines: V(S1-TT) = −5 meV; and dashed lines: V(S1-TT) = 5 meV.
in fission dynamics due to pathway interference are also
bath with the same total reorganization energy. The mixed bath mediates fission dynamics in such a way somewhat in between what the two original baths do. The numerous combination of possibilities therefore allow us vast freedom to control fission dynamics. On the other hand, it is difficult to identify model
appreciable for the mixed bath consisting of a low frequency Debye bath (ωc = 5 meV, λD = 80 meV) and a high frequency pseudo local bath (Ωk = 175 meV, λk = 20 meV). 27262
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IV. CONCLUSIONS We studied the bath effect in singlet fission for a model system by using the recently developed symmetrical quasiclassical (SQC) nonadiabatic molecular dynamics simulation method. The effects of functional forms of bath spectra density, characteristic frequency, reorganization energy, and bath temperature on fission dynamics are examined. With other major parameters set to be the same, changes in the function forms of spectra density only modulate fission dynamics mildly, except for the single frequency case. The distinction between the single frequency bath and other broad bath could be diminished by the interactions of multiple modes of different frequencies. Fission dynamics slows down in the low frequency bath, in comparison with that in the high frequency bath. However, the increase in decay time may not be too significant; therefore, the low frequency bath might also contribute to fission dynamics to some extent. Reorganization energy plays an important role in modulating fission dynamics, which is responsible for the coherent to incoherent transition in the short time dynamics and the nonlinear change in rate. Fission dynamics is largely temperature independent for the high frequency bath when the phonon bath is not thermally excited. An appreciable temperature dependence is observed for the low frequency bath. Interestingly for the downhill model, lowering the temperature enhances fission dynamics, which sheds light on a new route to design efficient fission materials. The modulation of fission dynamics by the mixed bath is largely determined by the original bath involved. Pathway interference is also observed for different forms of baths and their mixtures, which may provide tremendous possibilities to control fission dynamics by engineering phonon spectrum. As demonstrated in this work, fission dynamics is very complicated. Even though rate theories, such as Marcus theory, are very simple and useful in describing some important overall behaviors of fission dynamics, much detailed information is necessary to understand the underlying mechanisms. For example, if the judgment is merely based on single fission rate, it is hard to distinguish fission dynamics mediated by a mixing of the high frequency pseudo local bath and the low frequency Debye bath with a CT-mediated pathway only from that by a low frequency Debye bath with interference between direct and CT-mediated pathways. Therefore, nonadiabatic molecular dynamics simulations,41,42 such as the SQC method used in this work, would be suitable to provide microscopic pictures of fission dynamics and to help develop further design principles.
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Article
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AUTHOR INFORMATION
Corresponding Author
*
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS G.T. is grateful to Professor William H. Miller for stimulating discussions and suggestions. This work was supported by Peking University Shenzhen Graduate School and Shenzhen Science and Technology Innovation Council. We also acknowledge a generous allocation of supercomputing time from the National Supercomputing Center in Shenzhen (Shenzhen Cloud Computing Center). 27263
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dx.doi.org/10.1021/jp509477j | J. Phys. Chem. C 2014, 118, 27258−27264
