Effect of Underdamped Vibration on Excitation Energy Transfer: Direct

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Effect of Underdamped Vibration on Excitation Energy Transfer: Direct Comparison between Two Different Partitioning Schemes Chang Woo Kim, Weon-Gyu Lee, Inkoo Kim, and Young Min Rhee J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b10977 • Publication Date (Web): 08 Jan 2019 Downloaded from http://pubs.acs.org on January 17, 2019

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The Journal of Physical Chemistry

Effect of Underdamped Vibration on Excitation Energy Transfer: Direct Comparison between Two Different Partitioning Schemes Chang Woo Kim†, Weon-Gyu Lee‡, Inkoo Kim§, and Young Min Rhee†*

†Department

of Chemistry, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Korea

‡Department §Samsung

of Chemistry, Sungkyunkwan University, Suwon 16419, Korea

Advanced Institute of Technology, Samsung Electronics, 130 Samsung-ro, Yeongtong-gu, Suwon 16678, Korea

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Abstracts We study excitation energy transfer (EET) in a model three-site system with a mixed-quantum classical dynamics method, by focusing on the effect of an underdamped vibration. We construct two types of models where the underdamped vibration mode is included either in the quantum subsystem or in the classical bath. We show that the two models yield practically equivalent results despite the different depictions of the vibration. In particular, both models consistently demonstrate accelerations of population relaxation induced by quasi-resonant vibration. This indicates that intricate features of EET dynamics that have been frequently ascribed to the quantal nature of vibrations, such as vibronic mixing, can be successfully reproduced by using physically equivalent but classically described bath modes. The mechanism behind the observed quantumclassical correspondence is proposed. We also systematically examine how the structure of the spectating continuum phonon modes affect the vibronic resonance, and observe that phonon modes with different timescales influence the resonance in different manners.


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1. Introduction The excitation energy transfer (EET) processes in multichromophoric systems have been continuously studied due to their importance in biological and materials science.1,2 An important element of EET dynamics is the interaction between electronic and nuclear degrees of freedom (DOFs), which drives nonadiabatic relaxations.3,4 Treating the entire system with full quantum details is generally not feasible, and theoretical approaches often divide the total system into subsystem and bath DOFs. With this, one can study the system dynamics by propagating the reduced density matrix of the subsystem together with the nuclear perturbation that influences the dynamics. In the case of a multichromophoric system, the effects of the nuclear bath on the subsystem can be decomposed into intermolecular and intramolecular parts,5,6 which respectively correspond roughly to electrostatic interactions and chromophore vibrations. The two different types of interactions manifest themselves as distinct features in the spectral density, namely a frequency-dependent profile of the coupling between the subsystem and the bath. Generally, the intermolecular electrostatic interactions show up as a continuum of phonon modes in the low frequency region while the chromophore vibrations display as a series of relatively sharp peaks spread over a wide frequency range.5,7 While earlier works on EET usually relied only on the intermolecular part of the spectral density,8-10 recent studies started to focus on the intramolecular interactions primarily due to the growing attention on the effects of chromophore vibrations. Underdamped chromophore vibrations with slow dephasing time have been shown to induce mixing between electronic and vibrational DOFs, and this mixing has been suggested as an explanation for the long-lasting beating signals observed in multidimensional spectroscopy experiments.11-19 It has also been 3 ACS Paragon Plus Environment


demonstrated that an underdamped vibration mode that is quasi-resonant with an electronic energy gap can accelerate the energy transfer process associated with the electronic states.12,20-24 Based on these findings, biological relevance of the coherent signatures from spectroscopic measurements are often discussed, although some debates regarding this issue is still on-going.25,26 Theoretical studies regarding the effect of underdamped vibrations on EET dynamics often included the modes of interest directly into the subsystem,14-17,20,27-30 mainly because such an approach allows direct accesses to the involved vibrational quantum states.31 Explicitly including the vibration mode into the subsystem is also necessary when the adopted simulation method does not fully account for non-Markovian dynamics of the bath modes, as in Lindblad master equation32 and Redfield relaxation equation.33,34 Nevertheless, methods for treating the non-Markovianity of the bath without considering them as a part of the subsystem have been still continuously advanced.35-39 In any case, ideally, a quantum dynamics tool should be able to handle the system dynamics properly regardless of whether it treats a vibration mode as a part of the subsystem or the bath. Indeed, important insights with respect to the role of underdamped vibrations have been gained from studies that treated such modes as a part of the bath.12,18,19,21,22,40 In this regard, understanding the differences that may be caused by the classical treatment of the bath mode, in comparison with the results by rigorous quantum mechanical treatment, will be an important task. It will be even more so when we have to handle realistically complex systems, where the classical treatment is the only amenable option. Recent progresses in modeling techniques5,41,42 have allowed one to extract the intramolecular portion of the spectral density nearly quantitatively, and a few studies have already employed such techniques in simulating quantum dynamics of realistic systems.40,42-44 Including all vibration modes in the subsystem makes the simulation prohibitively burdensome. Therefore, one eventually 4 ACS Paragon Plus Environment

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needs to rely on a method that can appropriately treat intramolecular vibrations in an efficient manner. Such a method should be able to describe the non-equilibrium nature of the modes as well as to handle arbitrarily shaped spectral densities with the inclusion of inevitable anharmonic nature. As a candidate for fulfilling these ends, mixed quantum-classical methods,45-51 path integral-based methods,52-57 and other related developments58-61 are conceivable. The authors’ group has also recently developed a mixed quantum-classical method62 that can handle these requirements at least to some degree. Here, we will use it in dealing with the effects of an underdamped vibration especially toward inspecting the differences caused by adopting classical and quantum mechanical treatments on the mode. By considering both of the two dissimilar descriptions of the vibration, we will show that basically the same dynamics is obtained regardless of which description is employed for the underdamped vibration. We will also propose a mechanism underlying this quantum-classical correspondence. Specifically, we will consider an electronic Hamiltonian coupled to a relatively simple model bath and simultaneously modulate both the intermolecular and the intramolecular components of the spectral density. We will thoroughly examine how the intermolecular and the intramolecular components interplay with each other in affecting the EET dynamics. We will demonstrate that the detailed structure of the continuum part of the spectral density, which is out of resonance with the electronic energy gap, still substantially influences the vibronic resonance. Ultimately, we aim to show that simulations that employ classical description of nuclei, such as all-atom molecular dynamics potential63,64 or spectral densities extracted from atomistic models,5,44,65-68 can successfully capture detailed aspects of vibronic dynamics in EET.



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2. Methods 2.1. Hamiltonian Models. The Hamiltonian for a quantum subsystem coupled to a nuclear bath is generally expressed as Hˆ  Hˆ s  Hˆ b  Hˆ sb

(2)

Of course, the three terms on the right hand side respectively describe the subsystem, the bath, and their coupling. Let us first consider the situation where the subsystem only contains electronic DOFs. In the scope of Frenkel-Davydov exciton model69,70 in the single excitation manifold, Hˆ s is expressed as Hˆ s   E A A A   VAB A B

(3)

A B

A

where EA is the electronic transition energy for a localized excitation on the A-th chromophore site, defined with its vertical transition from the ground state minimum. Indeed, if we define the S0 minimum to S1 minimum energy as E A,0 , we can easily see that the relationship E A  E A,0  

(4)

holds with the reorganization energy . VAB is the electronic coupling between chromophore sites A and B. From now on, we will use the terms “Site A” and “site energy” to denote a specific pigment site and its excitation energy. For simplicity, the bath is often treated as a collection of harmonic oscillators linearly coupled to the electronic DOFs. We also invoke the Condon approximation71 to eliminate the environmental effect on electronic coupling. Then the two remaining terms of the Hamiltonian become


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  Hˆ b    hi aî† aî  1ˆ  A iA 

(5)

  Hˆ sb     hi si  aî†  aî   A A A  iA 

(6)

where aî† and aî are respectively the creation and annihilation operators for the i-th bath mode. Here we assume that each pigment site is coupled to its own collection of bath modes, as implied by i  A in the summation over the bath modes. The extent of coupling between the electronic and the bath DOFs are determined by the Huang-Rhys (H-R) factors

si  , which is related to the

site spectral density J A ( ) by

J A ( )  h i2 si (  i )

(7)

iA

As the subsystem Hamiltonian in this case covers the spaces spanned by the electronic DOFs, we will denote this depiction of the system as the electronic model. Let us now consider the case where some selected vibration modes are instead assigned as a part of the subsystem. The basis of the subsystem now becomes a combination of the electronic and the vibrational quantum states

A, m  A

 m k

k A

We should note that

A

lA

l

(8)

designates a direct product electronic state composed of Site A in its

electronically excited state with the other sites in their ground states. Thus, in the right-hand side of this equation, we used Greek and Roman symbols to denote individual vibrational quantum states on the excited and the ground electronic states, respectively. Besides the selected vibrations, 7 ACS Paragon Plus Environment


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the other modes will remain in the bath. Of course, the bold-faced symbol m represents the collective subsystem vibrational DOFs on all sites. With this basis, the subsystem Hamiltonian becomes

Hˆ s   E A,m A, m A, m    VAB FC( A, m | B, n) A, m B, n A

(9)

A  B m ,n

m

Here, E A,m is the energy of the localized electronic-vibrational state on A-th chromophore in reference to the minimum of the electronic ground state, which is the sum of the electronic energy and the vibrational energies of all modes in the subsystem. From eq 4, we can easily infer

1 1    E A,m  E A    k   hk  k     ml   hl 2 2 kA    lA 

(10)

where k  hk sk is the reorganization energy associated with the k-th mode. Note that the effect of vibrational reorganization must be accounted only for the modes attached to the electronically excited pigment site. The electronic coupling is also scaled by Franck-Condon (FC) factors72

FC( A, m | B, n)   k nk kA

 lB

ml  l

 pA pB

mpnp

(11)

Here we ignore Duschinsky rotation and assume that the vibration frequencies in the ground and excited state are identical. Diagonalizing the Hamiltonian in eq 9 yields mixed electronicvibrational, or vibronic states. This model will henceforth be termed as the vibronic model. Certain upper limits are usually imposed on the vibrational quantum numbers to make the dimension of the subsystem Hamiltonian small enough for feasible simulations. For the remaining bath modes in the spectral density besides the ones in the subsystem, we assume that they still participate by following eqs 5 and 6. With this construction, the vibrational quantum states sharing the same 8 ACS Paragon Plus Environment

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pigment site are not directly coupled and intramolecular vibrational relaxation does not readily take place. This is not an issue for the purpose of this work, as the vibrations in the electronic model have a similar nature and we are aiming to build a vibronic model in a physically equivalent manner to the electronic model. In case where intramolecular vibrational relaxation is needed, one may impose damping mechanisms on the quantized vibrations by using, for example, the dualbath approach.23,73,74

2.2. Simulation Details. For the electronic subsystem defined with eq 3, we consider a three-site model inspired by Sites 1 to 3 in the Fenna-Matthews-Olson (FMO) complex75 of the green sulfur bacterium Chlorobium Tepidum. These three pigment sites are known to form a major energy transfer pathway within the complex, with Sites 1 and 3 respectively acting as the entrance and the exit channels of EET.76 The Hamiltonian parameters for this model are

 12445 112.1 5.0    Hˆ s   112.1 12520 48.2   5.0 48.2 12205  

(12)

where the site energies and the electronic couplings were taken from refs 64 and 77. As we mentioned in Introduction, we adopt a relatively simple bath model for each site with one continuous distribution of phonon modes and one additional sharp and discrete mode. These two contributions respectively represent the overdamped intermolecular interactions and the underdamped intramolecular vibration mode. Therefore, this simple bath model may serve to represent a realistic spectral density from atomistic simulations. The continuous part is modeled by the Drude-Lorentz (DL) spectral density 9 ACS Paragon Plus Environment


J DL ( ) 

2ph

 ph    2 2 ph

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(13)

and will be simply denoted as the phonon band hereafter. The parameters ph and ph are respectively the reorganization energy and the dephasing constant of this phonon band. We attach underdamped vibration modes to Sites 1 and 3, but not on Site 2. As we explained in Section 2.1, these underdamped modes are described as a part of the bath Hamiltonian with the electronic model and as subsystem terms with the vibronic model. The absence of an underdamped vibration at Site 2 is for keeping the total dimension of the vibronic model low enough for feasible simulations. To be concise, when we include m and n vibrational quantum states with Sites 1 and 3 within the vibronic model, we will denote it as VBM-(m,n). Thus, the dimensionality of VBM(m,n) will be 3mn. In parallel, for brevity, we will denote the electronic model as ELM. For the continuum phonon bands on the three sites, we adopt the same parameter sets. The reduced density matrix of the quantum subsystem was propagated by the non-Hamiltonian variant of Poisson bracket mapping equation formalism (PBME-nH).62 PBME-nH is a mixed quantum-classical dynamics method derived from quantum classical Liouville equation, and treats the bath variables as classical DOFs after Wigner transforms. Although PBME-nH involves approximations in treating the effect of bath on the subsystem, it was shown to yield reliable results for systems with moderate site energy differences such as the FMO complex.62 The advantage of PBME-nH is that it is a non-perturbative method and it also naturally describes the nonequilibrium dynamics of the bath modes. Interested readers may refer to a brief introduction about PBME-nH given in Supporting Information (SI) and references therein. PBME-nH can be straightforwardly extended for treating the vibronic model, and the related derivations can also be 10 ACS Paragon Plus Environment

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found in SI. One caveat for this extension is that the site population dynamics simulated by using different numbers of vibrational quantum states do not strictly match even for exactly equivalent models, as will be shown later in Section 3.2. Our analysis will show that this is because the forces exerted on the nuclei depend on the dimensionality of the subsystem. Nevertheless, we will demonstrate that this discrepancy does not prevail when the reorganization energy of the phonon band lies in a moderate region. For PBME-nH simulations, the continuous DL spectral density in eq 13 was discretized up to 3000 cm1 with 1000 harmonic oscillators, which recovered 98% of the total reorganization energy. We set the initial condition with a localized excitation at Site 1, and we assumed that the bath modes are at thermal equilibrium in the electronic ground state just before the excitation. For the quantized vibration modes within VBM, we vertically excited thermally populated incoherent mixture of ground state vibrational quantum states, which is physically analogous to the situation with ELM. The phase space variables for each trajectory were randomly sampled as described in ref 62. The system dynamics was followed by simulating 106 PBME-nH trajectories of 1 ps duration at 300 K, although the improvement in convergence was only marginal after averaging 105 trajectories. The time step for numerically integrating the equations of motion was 0.5 fs. We did not assume any correlation between the continuum modes coupled to different pigment sites. Table 1 summarizes the parameters and their numerical ranges adopted in this study.



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Table 1. Adopted parameters and their numerical ranges. Parameter

Description

ph

Reorganization energy of the continuum phonon band Dephasing time of the phonon band

ph

Value 0.5 – 50 cm1 0.01 fs1

v

Frequency of the underdamped vibration mode

0 – 2000 cm1

sv

Huang-Rhys factor of the underdamped mode

0.05

v

Reorganization energy of the underdamped mode: v  hv sv

0 – 100 cm1

3. Results and Discussions 3.1. Energy Level Structures. We start our analysis by showing the energy level structures of the ELM and the VBM models based on direct diagonalization.78 The diagonal elements of the subsystem Hamiltonian defined by eqs 3 and 9 are the energies of the spatially localized excited states, which can be considered to form the diabatic basis. Diagonalization of the subsystem Hamiltonian transforms this into the adiabatic basis. For ELM, the diabatic and the adiabatic bases are often termed as site and exciton bases respectively, while the meaning of exciton states is sometimes extended for describing the adiabatic basis of the vibronic model.20,79 Here, we will strictly limit its usage only to denoting the adiabatic basis in ELM. An exciton state is labeled by following the index of the site from which it inherits the largest contribution. Employing this labeling scheme makes the discussion about exciton-vibration resonance in a later part more succinct compared to the scheme of simply indexing exciton states in the order of increasing energies as adopted in other literatures.23,64 Figure 1 schematically illustrates the energy diagrams of ELM in both site and exciton bases, and how the states are labeled. Excitons 1 and 2 are 12 ACS Paragon Plus Environment

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delocalized over Sites 1 and 2 due to the strong coupling between them, while Exciton 3 is almost localized at Site 3.

Figure 1. Schematic energy diagrams of ELM (a) in the site basis and (b) in the exciton basis. The number under each level is the state energy in cm1. For exciton basis, the relative contribution of each site is represented in gray scale.

Let us now examine the energy level structure of VBM-(1,2). Quite naturally, the state energies in this VBM depend on the oscillation frequency of the mode in the subsystem attached to Site 3. Figure 2 depicts how the diabatic and adiabatic state energies change with this mode frequency. While the energy curves for the diabatic states (Figure 2a) show crossing behaviors, the energy curves for the adiabatic states (Figure 2b) exhibit avoided crossings. Crossing points of diabatic and adiabatic energies also occur at different vibration frequencies. Referring to Figure 1, one can find that the crossings of the diabatic energies and adiabatic energies occur at the frequencies close to the site energy differences and the exciton energy differences, respectively. Strong vibronic mixing actually occurs around the avoided crossing points in Figure 2b, which has been suggested 13 ACS Paragon Plus Environment


as a possible origin of long-lasting oscillations in spectroscopic signals.14,15 In fact, VBM-(1,2) is the simplest possible vibronic model as there is only one vibration mode included in the subsystem with its available quantum states as v  0 and 1. Of course, this restricted degree of freedom on the vibration can distort dynamics. Thus, we will also employ an expanded model, VBM-(4,4), for the simulations.

Figure 2. Energy levels of (a) the diabatic and (b) the adiabatic states with VBM-(1,2) as functions of the frequency of the single underdamped mode. The diabatic states are color-labeled according to the site where the excitation is localized, while the adiabatic states are all colored in gray.

3.2. Reliability of PBME-nH. Before demonstrating the similarity between the quantum dynamics with ELM and VBM, we need to convince ourselves that PBME-nH is an adequate 14 ACS Paragon Plus Environment

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approach for showing the similarity as it is not a perfect method as any other semiclassical method. For this, let us first focus on the population transfer to Site 3, the energy sink. We quantified the EET efficiency to Site 3 by calculating the average transfer rate kET defined as

k ET 

2 T2

T

  P (t )  P (0)  dt 0

3

3

(14)

where T is the length of the simulation and P3 (t ) is the Site 3 population at time t. The explanation on the physical meaning of kET and the related derivations are presented in SI Section S3. We believe that using kET defined in this manner is less ambiguous than just using the site population at an arbitrarily chosen time point, as the integration averages out the population oscillations arising from the initial coherence. For VBM, the site population was calculated by tracing out over the vibrational DOF. We first consider the limit of v  0, at which the reorganization energy from the underdamped mode vanishes ( v  hv sv  0 ). This implies that the influence of the underdamped vibration mode will be completely nullified, and with this condition, the EET dynamics should become exactly identical for ELM and VBM-(m,n) regardless of m and n values. (See SI Section S4 for a mathematical proof.) However, as we described in Section 2.2, the results from different models are not strictly the same because the dynamics from PBME-nH depends on the dimensionality of the subsystem. We therefore tested the reliability of PBME-nH by comparing its results against the ones from the numerically exact hierarchical equations of motion (HEOM) method80 on ELM. In this way, we can find out the range of ph where PBME-nH gives reliable results for all adopted models of Hamiltonian.



Figure 3. EET efficiency as a function of the phonon band reorganization energies ph, obtained by PBME-nH simulations in the limit of v  0. The numerically exact HEOM result is also presented for comparison.

Figure 3 displays the EET efficiencies calculated with the limiting value of v  0 while varying

ph from 0.5 to 1000 cm1. The HEOM simulation results show that the EET efficiency increases with a larger phonon band reorganization energy up to ph  ~100 cm1 and decreases afterward. Such a plateauing region is typically observed with EET in a coupled chromophore system.9,10 The PBME-nH results with all three bath models agree well with the HEOM result up to ph  50 cm1, but they start to deviate for larger ph. Among the three models, VBM-(4,4) deviates most noticeably likely because the PBME-nH artifact arising from its dimensional dependence is the largest with the model. In any case, as PBME-nH is quite reliable with ph ≤ 50 cm1, we will only use this region for our later discussions. Indeed, the reorganization energy contributed by the phonon band of bacteriochlorophyll a lies within this range.81 We also show that PBME-nH is suitable for treating underdamped modes with nonzero v. In 16 ACS Paragon Plus Environment

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this case, including an underdamped mode in HEOM is problematic due to its long dephasing time. We will therefore establish the adequacy of PBME-nH by benchmarking ELM/PBME-nH against more sophisticated VBM-(4,4)/PBME-nH, while maintaining its reliability by keeping ph ≤ 50 cm1. For these simulations, we will compare the population dynamics instead of kET as a more direct test of performance. Even though kET serves as a compact metric for comparing simulation results with multiple adjustable parameters, it does not explicitly provide a time evolution profile. In fact, important information such as a coherence lifetime can be deduced from the time profiles of site populations.

Figure 4. Site population dynamics from ELM (thick lines) and VBM-(4,4) (thin lines), evaluated under different bath conditions. The vibration frequencies are specifically chosen to represent quasi-resonant (v  180 cm1) and off-resonant (v  1000 cm1) cases.



Figure 4 presents the population dynamics from the two models simulated under a few different conditions. Here, we chose v  180 cm1 and v  1000 cm1 as two representative vibrational frequencies. The former is quasi-resonant with the energy difference between Excitons 1 and 3 while the latter is not resonant with any exciton pairs. In the case of VBM, we verified that using four vibrational quantum states was enough by checking the convergence after increasing the number of adopted quantum states. For ph  0.5 cm1, the periods of the population oscillations are nearly identical for the two, but the oscillating amplitude is larger with VBM. Even still, the difference is rather small and it diminishes with a larger ph. With v  1000 cm1, both models exhibit tiny oscillations especially at a later time. This is a typically known feature that can arise from the existence of an underdamped vibration.17 Overall, we can see that the general features from both ELM and VBM are well in accord, and that PBME-nH properly treats the underdamped vibration mode in the spectral density. At this point, it is interesting to see that the quasi-resonant vibration (v  180 cm1) accelerates the energy transfer regardless of its description. Namely, Site 3 population grows faster with this vibration. This phenomenon will be thoroughly discussed in the next section.


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Figure 5. EET efficiency kET calculated for various combinations of ph and v for (a) ELM, (b) VBM-(1,2), and (c) VBM-(4,4).

3.3. Vibronic Resonance in Excitation Energy Transfer. Let us now discuss over the full set of simulation results with the underdamped vibration at nonzero frequencies. Figure 5 displays kET from the three models calculated with various combinations of ph and v. In all cases, we can clearly observe two groups of peaks centered at 174 and 408 cm1 with ph values smaller than 20 cm1. The EET enhancement at these two frequencies is in accord with an earlier study based on a similarly defined three-site model,21 and can be readily explained by using the exciton basis. Namely, from Figure 1, one can see that 174 and 408 cm1 respectively correspond to the energy 19 ACS Paragon Plus Environment


differences between Excitons 1 and 3 and between Excitons 2 and 3. From the same figure, we can observe that the transition between Excitons 1 and 2 induces only a marginal change in the contribution by Site 3, with which we are measuring the population transfer. Thus, a resonance peak at 234 cm1 ( 12604 cm1  12370 cm1) is not observed for kET based on Site 3. With these, we can clearly observe that a vibration mode whose energy is resonant with the energy gap between two exciton states will promote the energy transfer through the exciton pair. Directly observing the exciton population further supports this argument. Namely, the 174 cm1 (408 cm1) mode selectively activates the population relaxation between Excitons 1 and 3 (Excitons 2 and 3) as displayed in Figure S1a and b. In contrast, the exciton populations remain nearly stationary without a resonant vibration (Figure S1c and d). These resonant frequencies match the positions of avoided crossing in the adiabatic energy diagram shown in Figure 2b and this aspect again shows that strong vibronic mixing indeed accelerates the speed of EET.29,30 One additionally noticeable feature in Figure 5 is the existence of two satellite peaks centered near 87 and 204 cm1 with ph ≤ 5 cm1, in the cases of ELM and VBM-(4,4). These frequencies are the halves of the resonance frequencies, and thus the peaks correspond to multiphonon resonances. These double-phonon resonance peaks are not seen with VBM-(1,2) because a doubly excited phonon state is not available in its basis and the associated EET pathway is not accessible. In addition, the 87 cm1 double-phonon peak with VBM-(4,4) is smaller than with ELM, but this difference is not actually from the shortcomings of ELM with a completely classical bath. Indeed, when we added more vibrational quantum states to the state basis in VBM, we could observe a better reproduction of the peak with ELM (Figure S2). With these comparisons, it is quite interesting to see that the completely classical bath 20 ACS Paragon Plus Environment

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description within ELM performed comparably with the vibronic models. Especially, the more elaborate treatment VBM-(4,4) generated almost identical results with both the fundamental and the double-phonon resonance peaks appearing almost in the same manner throughout the range of tested ph values. Although the agreement with VBM-(1,2) is worse, the discrepancies are obviously caused by the limited vibrational excitations possibly attained within the model. While the resonance enhancement by an underdamped vibration with the matching exciton energy gap may be a rather expected event, it is still striking to see that including important underdamped vibrations as a part of the subsystem is not a necessary condition at least with PBME-nH. Namely, we can still properly account for the vibrational effects on EET by treating them completely classically. This aspect will indeed be beneficial when one has to adopt a complex potential model82 where quantum mechanical treatments of vibrations are practically impossible. In addition, even a quantum treatment may have artifacts as exemplified by the lack the double phonon resonance and by the noticeably smaller height of the 174 cm1 peak with VBM-(1,2). We note that VBM-(1,2) has been continuously used as a minimal model for observing the consequence of vibronic interactions,16,17,27 and the double phonon resonance in photosynthetic EET will not be that important as ph is usually large enough to mask it from appearing. Even still, care needs to be taken when a reduced model is adopted for costly quantum mechanical treatments. Regarding the good agreement of ELM with classical vibration against VBM, one might concern that it simply has resulted from the employment of a rather high temperature compared to the exciton gaps of 174 cm1 and 408 cm1 (corresponding to 250 K and 588 K). In order to rule this out, we also performed the same set of simulations at 77 K. Even at this low temperature, the kET profiles with ELM and VBMs were still almost identical (Figure S3), implying again that the 21 ACS Paragon Plus Environment


EET profiles with discrete vibrational quantum states are successfully reproduced with a completely classical bath treatment in ELM.

3.4. The Role of Off-resonant Vibration. In contrast to the v < 500 cm1 region that shows complex structure originating from the resonance, the v > 500 cm1 region appears relatively featureless. This is easily perceivable, as the underdamped mode will not be matching with any exciton gap. Still, in this high frequency region, the overall trend of kET changes interestingly depending on ph. For ELM with relatively small ph, kET minutely increases at higher v. This behavior is reversed for larger ph, and kET substantially decreases with increasing v. Eventually, kET becomes even smaller than the value in the v  0 limit, physically meaning that an underdamped high frequency vibration hinders the energy transfer. The monotonic change in kET with respect to v for a fixed value of ph likely originates from the changing reorganization energy

v contributed by the underdamped vibration mode. Because we fixed sv, v increases linearly at higher v as v  ħvsv. Indeed, the second moment of the site energy fluctuations associated with this underdamped vibration changes in proportion to v. Therefore, the squared site energy fluctuation grows linearly with v under our setting, and this aspect will likely be the reason behind the almost linear variation of kET with respect to v. At this point, the minute increase in kET with small ph in the large v limit with ELM is odd, and its reason is rather unclear. Later, we will argue that it is an artifact from the classical treatment of bath vibrations. With VBM, beyond v  500 cm1 where no resonance is present, kET shows a monotonic behavior as with ELM. The EET dynamics from VBM-(1,2) and VBM-(4,4) are somewhat 22 ACS Paragon Plus Environment

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different in the low frequency region, but they gradually become similar with increasing v to reach a nearly quantitative agreement for v > 1000 cm1. This is natural because the thermal populations of higher-lying vibrational quantum states become gradually smaller with increasing

v, and thus lesser number of vibrational quantum states will be required for convergence toward handling a high frequency vibration. One difference we can notice against the ELM results is that kET with VBM in the high v limit only decreases with increasing v for all ph. The decrease in kET at large ph is also more moderate than with ELM. These discrepancies should arise from the different treatments of the underdamped vibration. Quantum aspects of vibration such as energy level discreteness and zero-point energy (ZPE) will become more important with increasing v. In the case of VBM, such quantum characters can be handled by construction except the fact that only a finite number of vibrational states can be included in the basis. Hence, we suppose that the trend observed in the vibronic model will be closer to the actuality. This tells us that the EET process is universally hindered by an off-resonant high frequency vibration. This aspect can be understood in terms of the characters of the electronic-vibrational states defined by eq 8. As shown in eq 9, the interstate coupling between these states is expressed as the product of the F-C factor and the pure electronic coupling. The F-C factors are always less than unity with nonzero H-R factors, and thus the coupling between any two electronic-vibrational states is always smaller than the electronic coupling. Such reduction in the interstate coupling does not slow down the energy transfer in the presence of vibronic resonance, as the resonance opens an efficient exchange channel between the two involved states. When the vibrational frequency increases and becomes off-resonant, this channel is gradually blocked due to the energetic detuning. This will unmask the effect of the F-C factors, and kET eventually drops below the one at v  0 cm1 where the coupling is purely electronic. 23 ACS Paragon Plus Environment


Our result may appear to be in conflict with an earlier report83 which stated that the EET dynamics is rather insusceptible to the existence of high-frequency vibrations. We believe this disagreement originates from the difference in the size of the employed H-R factors. Indeed, we adopted a fixed H-R factor of 0.05 for the single underdamped vibration, while the earlier study employed frequency-dependent H-R factors which did not exceed 0.01 in the high frequency region.83 Of course, when the H-R factor is reduced in our simulations, the vibration mode will couple to the EET dynamics only weakly, and kET will stay almost constant with increasing v. Physically, although the H-R factor for an individual vibration mode in the high frequency region can be as small as 103,84 the high frequency region can include many individual modes in the case of large pigment molecules. Thus, the cumulative H-R factor over all modes can become comparable to our value of 0.05. There are even cases where the H-R factor of a single individual mode exceeds 0.1.65 Therefore, the potential hindering effect of off-resonant high-frequency vibrations should not be completely ignored. In this regard, we anticipate experimentally probing this aspect will be an interesting task. Perhaps, this may be achieved by adopting a chromophore pair interacting in the Förster regime. If a polar bond can be introduced either to the donor or to the acceptor where the electron distribution changes noticeably with the electronic transition, it will form a strongly coupled high frequency underdamped mode with a large H-R factor. Instead of tuning the vibrational frequency as in our simulations, the donor-acceptor gap may be modulated by adopting solvatochromic shifts, and the transfer rates may be determined as a function of the energy gap. In addition, we believe the discrepancy between ELM and VBMs that we observed in the high

v side likely originated from the improper treatment of the vibrational ZPE in ELM. In our mixed quantum-classical treatments, the phase space variables of the classical oscillators were sampled 24 ACS Paragon Plus Environment

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from the Wigner distribution of a quantum harmonic oscillator at thermal equilibrium,62,85 which guarantees the allocation of ZPE to each vibration. This quantal ZPE is excessive for a thermalized classical oscillator at high frequency, and it will leak to the subsystem and other bath modes. If it leaks to the electronic degree of freedom of Site 3, kET will likely be overestimated. In any case, from Figure 5, we see that the associated artifact with ELM does not change the kET profile to any qualitatively different extent.

3.5. Effect of Continuum Phonon Band. Another feature that we wish to emphasize is the width of resonance peaks found in Figure 5. Even with a modest disorder represented by a relatively small ph, the peaks are already substantially broad. Thus, we can infer that a perfectly matching resonance is not needed for the EET enhancement as long as it is aided by a reasonably coupling continuum phonon band.21,23 Of course, increasing ph further broadens the width of this quasiresonance. The double-phonon peaks centered near 87 and 204 cm1 are the most quickly affected by the broadening, and become buried in the 174 cm1 peak as ph increases beyond 5 cm1. Quite naturally, the non-zero width of the resonance is due to the disorder in the exciton energy gap. Because we invoked the Condon approximation and did not assign any fluctuating bath to intersite couplings, the only source of any disorder will be the fluctuations in site energies. This section will present a detailed analysis on the disorder induced by the continuum phonon band and discuss how its shape affects the EET dynamics.



Figure 6. EET efficiency calculated with ELM by using different combinations of ph and v: when the continuum phonon band follows (a) the ordinary DL spectral density (eq 13), (b) the low frequency part of DL ( < 53 cm1), (c) the high frequency part of DL ( > 53 cm1), and (d) the shifted DL spectral density (eq 15) with 0  1000 cm1. The phonon band shapes are displayed with insets of each panel.

Figure 6a reiterates the ELM simulation results already shown in Figure 5a. We divided the DL spectral density (eq 13) into a slower and a faster parts with identical reorganization energies of

ph/2. When we re-performed the same simulations with the partial baths, kET appeared quite 26 ACS Paragon Plus Environment

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differently for the two cases with the increase in ph as displayed in Figure 6b and c. Of course, kET with the two partial spectral densities are practically the same with small enough ph, which will negligibly affect EET. However, as ph increases, the slower half of the bath majorly contributes to broadening the resonance (Figure 6b) while the faster half more contributes to speeding up EET. The lack of broadening with the faster half is also highlighted with the multiphonon resonance peaks, which are almost unnoticeable for ph > 5 cm1 in Figure 6b but are still visible as shoulders up to ph  20 cm1 in Figure 6c. We stress that the overall reorganization energies contributed by the two halves of the phonon band are identical. This shows that the resonance broadening is not solely determined by the total amount of disorder but by the frequency of the phonon modes. Basically, the lower frequency part of the spectral density appears to more affect the broadening. As a more drastic demonstration, we additionally adopted a shifted DrudeLorentz (SDL) spectral density

J ( ) 

 ph   ph  ph  2  2 2 2     ph  (  0 )  ph  (  0 ) 

(15)

as the continuum phonon band, with the shifting parameter of 0  1000 cm1. Figure 6d displays the dramatic change caused by this shift away from the low frequency region. With the lack of slow phonons, increasing ph now only very weakly affects the resonance width, and the multiphonon resonance peak is observed even at ph  50 cm1. In fact, this aspect comes from the timescale separation. Namely, the broadening mainly arises by the phonon modes whose period of oscillation is comparable to or longer than the timescale of EET. The slow modes perturb the structure of the electronic states defined in eq 3 and such slow perturbations can be actually considered effectively frozen during the EET timescale. Namely, 27 ACS Paragon Plus Environment


from the perspective of EET, the initial phase of the slow phonon mode changes little over time. Thus, these modes play the same role as static disorder toward the transfer. We can easily imagine that increasing ph will lead to a larger deviation of the effective Hamiltonian from eq 3, thus a larger broadening. It is interesting to note that a similar timescale separation has been utilized for the recently developed reduced density matrix hybrid approach60,86 and its related methods.87 By dividing the total spectral density into slow classical DOFs and fast quantum DOFs and treating them separately, this approach efficiently yields reliable results across a broad range of parameters. This implies that a classification of bath modes by their timescales is indeed an effective way toward interpreting the nature of system-bath coupling.

Figure 7. Effect of the shape of the continuum phonon band on EET efficiency kET without the presence of the underdamped vibration (v  0 cm1).

In the above, we mentioned that the faster half of the DL density contributes more to the EET enhancement. This is more straightforwardly seen from Figure 7, which shows the change of kET as a function of the bath coupling strength for the case without a resonant underdamped vibration 28 ACS Paragon Plus Environment

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(v  0). One can clearly see that kET changes most sensitively with the high frequency portion of the DL spectral density. This is simply due to the same resonance enhancement discussed in a previous part. After all, a continuum phonon band is a collection of many individual modes, each of which will likely affect EET in the same framework as an isolated underdamped vibration. Because the faster half of the DL density contains substantial coupling strengths around 174 and 408 cm1, where the resonance takes place, it will drive EET more efficiently with a larger ph. In contrast, the low frequency part does not have any mode directly resonant with the electronic energy gap and EET must rely on a multiphonon resonance, which is less efficient due to a smaller Franck-Condon factor. With the SDL density, even a multiphonon resonance is not feasible and kET is quite small even with relatively large bath coupling.

3.6. Classical Correspondence. The initial coherence created by the site-localized and impulsive excitation on Site 1 induces coherent exchange of populations between diabatic site-based states. The population exchange between Site 1 and Site 3 is intense when these two states undergo strong mixing. In VBM with quantum mechanical treatment of the vibration, this strong mixing occurs when the frequency of the underdamped vibration matches the energy difference between a pair of exciton states. With the dissipating effect from the continuum phonon band, the exchanged population eventually accumulates on Site 3. Such a mechanism involving the interplay between coherence and decoherence was shown to be crucial for the effective charge transfer process in the bacterial reaction center.29 In the case of ELM with fully classical vibrations, the dynamics closely resembled the one with VBM even without explicitly including resonantly mixing vibrational states. In a sense, this may 29 ACS Paragon Plus Environment


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appear odd as a classical vibration can take an arbitrary amount of energy regardless of its frequency. This classical correspondence can be reasoned in the following manner. With the classical description, an underdamped vibration with a non-zero H-R factor will induce site energy oscillations. With a harmonic vibration without any Duschinsky rotation,88 this can be roughly described by the site energy autocorrelation function of

C ( )  2ph kT e

  ph

 2v kT e  v cos v

(16)

The first and the second terms on the right-hand side correspond to the actions of the continuum phonon band and the underdamped vibration, respectively, with dephasing timescales of  ph and

 v . Thus, the subsystem is under an influence of a perturbing “energy field” with a well-defined frequency v. When this frequency matches the energy difference between two energy eigenstates, namely exciton states, a first-order transition takes place. This is analogous to the semiclassical description of photon absorption and the associated state switch with oscillating classical electric field.89 Just like the time-dependent perturbation by oscillating classical electric field can yield a transition between energy eigenstates, the time-dependent perturbation by an underdamped vibration can induce a population transfer between two exciton states. We note that this situation greatly resembles the vibronic resonance in a collision-induced nonadiabatic quenching process,90 which could also be accessed by classically treating nuclei. There, the resonance was successfully described with an oscillating electronic energy difference, and our explanation indeed conceptually overlaps with that description. In addition, a very recent study also demonstrated that EET rate can be enhanced by periodic modulation in the site energy with the application of monochromatically oscillating external electric field.91


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As EET proceeds, the electronic energy gradually dissipates into the bath. Because the resonant underdamped mode acts as an efficient EET channel in VBM, we can infer that the dissipated energy in ELM as well will accumulate mostly in the underdamped mode. When the vibrational relaxation is comparable to or slower than the EET timescale, the resulting vibrational excitation will strengthen the site energy oscillation, which in turn will further increase the rate of population relaxation. Handling such positive feedback between the quantum subsystem and a bath mode will clearly be outside the regime of Markovian approximation, although to what extent the nonMarkovianity contributes to the vibronic resonance is quite elusive at this stage. At any rate, the fact that the EET acceleration with vibronic resonance can be naturally explained by a classical vibration apparently suggests that this resonance effect bears a strong classical origin.92 Thus, with the results provided by Figure 5, we conclude that the electronic model that treats the bath completely classically can be employed toward analyzing the effects of underdamped vibrations on EET dynamics in the qualitative and even in the quantitative sense. As mentioned earlier, this is actually important when a complex Hamiltonian model needs to be adopted5,42,63,64 where the simple harmonic oscillator model becomes inapplicable.

4. Conclusion Throughout this paper, we examined EET with a three-site Frenkel exciton model under the presence of an underdamped vibration. We compared the dynamics after considering the underdamped mode either as a part of the quantum subsystem or as a part of the bath. When the vibration was quasi-resonant with the exciton energy difference, EET was promoted in both depictions, in accordance with earlier studies with numerically exact calculations.12,24 While this 31 ACS Paragon Plus Environment


resonance has been ascribed to the quantal nature of the vibration,14,24 we showed that classical description can also excellently reproduce the same resonance behavior. In contrast to the resonant vibration, the presence of an off-resonant mode at high frequency was shown to delay EET. We also studied how the structure of the continuum phonon band affects the vibronic resonance. We observed that resonance peak was broadened primarily by the slow modes in the continuum band. This was due to the fact that a mode that is slower than the EET timescale acts in a static manner to displace the exciton levels. Therefore, the amount of the disorder characterized by the reorganization energy of the phonon band is not enough in characterizing the interplay between electronic states and vibrations, and an attention must be paid to the detailed distribution of phonon modes in the frequency domain. We note that it had already been demonstrated that carefully modeling the shape of the spectral density toward its low frequency limit is crucial for correctly simulating the spectroscopic signals arising from underdamped vibration modes.15 Due to the limitation of PBME-nH, our study was limited to the regime of ph  50 cm1. According to Figure 5, the transfer rate kET monotonically increased with ph in this regime. Moreover, the speed-up due to the vibronic interaction gradually became less prominent with increasing ph. With these two observations, we may wonder the relevance of the vibronic assistance in an energy transfer process, especially because a sufficient amount of ph alone was enough for achieving high kET in our simulations. Even still, according to Figure 3, the transfer rate kET does not monotonically increase with ph and reaches a maximum at some value of ph. It is unlikely that all natural (and also artificial) light-harvesting systems satisfy this optimal condition. For example, the experimentally determined ph of bacteriochlorophyll a, which is abundant in the FMO and the LH2 complexes, is around 15 cm1.64,81 This value is quite smaller 32 ACS Paragon Plus Environment

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than the optimal value reported here and in other theoretical studies,9,10 and accordingly, these complexes lie in a regime where the vibronic interaction may still facilitate the transfer to some extent. Near the maximum kET with optimal ph, the vibronic assistance may become limited. In highly disordered systems with large phonon coupling kET will diminish back as in Figure 3, and the vibrational assistance may reappear. One possible example is the charge transfer process, in which the drastic change in the electron density of the chromophore pair can induce large intermolecular reorganization of neighboring molecules. Interestingly, for strongly disordered systems with ph in the range of 100 – 1000 cm1, there have been reports showing the boost of relaxation by underdamped vibrations.20,93 In these cases, the resonance condition is often well explained in terms of site energies rather than the exciton energies, which seems reasonable when we consider the strong localization effect in that regime. In this regard, it will be fruitful to extend the present analysis into such a strongly disordered condition. Overall, the aspect that classical description is enough for modeling resonant EET dynamics is intriguing. It is even encouraging in that directly simulating EET with a realistic atomistic model often has to rely on classical vibrations. Of course, there exist some sophisticated methods that explicitly account for the nuclear quantum effects even in atomistic models,94,95 but our results indicate that we can stick to using classical nuclei at least when studying EET. Of course, one has to be careful as there are quantum effects for which classical descriptions will fail such as the quantum tunneling.96,97 At least, such an effect does not appear to be important for vibronic resonance.14 In the future, we hope to report more on the vibronic resonance effect with realistic pigment-protein bath model, which is currently under progress.



Associated Content Supporting Information Brief introduction to PBME-nH and its extension to vibronic model (Sections S1 and S2); derivation of the effective EET rate kET (Section S3); mathematical proof of the equivalence between ELM and VBM in the v  0 limit (Section S4); time profile of populations in exciton basis (Figure S1); improvement in reproducing double phonon resonance peak with more vibrational quantum states (Figure S2); and EET rate in ELM and VBM at 77 K (Figure S3). These materials are available free of charge via the Internet at http://pubs.acs.org.

Author Information Corresponding Author *E-mail: [email protected].

Notes The authors declare no competing financial interest.

Acknowledgments This work was financially supported by National Research Foundation of Korea (Grant 2017R1A2B3004946), KAIST HRHRP Research Program (N10180009), and Samsung 34 ACS Paragon Plus Environment

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Electronics (SAIT). Discussions with Prof. Seogjoo Jang (City University of New York) is gratefully acknowledged.

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Biosketch for Virtual Special Issue Young Min Rhee is an Associate Professor of Chemistry at KAIST. After finishing his undergraduate studies at Seoul National University, he received his PhD degree from Stanford University and worked as a postdoctoral researcher in UC Berkeley. Before joining KAIST in 2017, he was an Assistant and then Associate Professor at POSTECH. His main research interests include elucidating photodynamics of complex systems and developing nonadiabatic tools for tackling such systems.


Effect of Underdamped Vibration on Excitation Energy Transfer: Direct

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