Driving Force and Nonequilibrium Vibronic Dynamics in Charge Separation of Strongly Bound Electron-Hole Pairs

Electron-hole pairs in organic photovoltaics dissociate efficiently despite their Coulomb-binding energy exceeding thermal energy at room temperature. The electronic states involved in charge separation couple to structured vibrational environments containing multiple underdamped modes. The non-perturbative simulations of such large, spatially extended electronic-vibrational (vibronic) systems remains an outstanding challenge. Current methods bypass this difficulty by considering effective one-dimensional Coulomb potentials or unstructured environments. Here we extend and apply a recently developed method for the non-perturbative simulation of open quantum systems to the dynamics of charge separation in one, two and three-dimensional donor-acceptor networks. This allows us to identify the precise conditions in which underdamped vibrational motion induces efficient long-range charge separation. Our analysis provides a comprehensive picture of ultrafast charge separation by showing how different mechanisms driven either by electronic or vibronic couplings are well differentiated for a wide range of driving forces and how entropic effects become apparent in large vibronic systems. These results allow us to quantify the relative importance of electronic and vibronic contributions in organic photovoltaics and provide a toolbox for the design of efficient charge separation pathways in artificial nanostructures.


I. INTRODUCTION
When a solar cell made out of an inorganic semiconductor like silicon is exposed to light, electrons can be readily extracted from the valence band to the conduction band and then captured at the electrodes.If, however, light is absorbed by carbon-based materials, photons produce strongly bound electron-hole pairs called excitons, which are collective optical excitations that may be delocalized across several molecular units [1].Excitons are charge neutral, namely the electron and the hole occupy, respectively, the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) of the same molecular unit, and require dissociation in order to produce a current [2].In contrast, the charge-transfer (CT) states describe partially separated electron-hole pairs where an electron and a hole occupy, respectively, the LUMO and HOMO levels that belong to different molecular sites.The transfer from exciton to CT states is thus suitable to describe the dynamics of electron transfer [3,4].In photosynthetic organisms excitons are split in pigment-protein complexes called reaction centers [5][6][7].In organic photovoltaics (OPV), blends of materials with different electron affinities are used to provide an energetic landscape that is favourable to charge separation at the interface [8].These devices exhibit ultrafast, long-range charge separation with high quantum efficiencies [9][10][11].This means that a large proportion of absorbed photons produces excitons or strongly bound CT states that are successfully dissociated.Some of these electron-hole pairs however thermalize towards the lowest-energy CT state localized at the interface, which is for this reason considered an energetic trap that leads to non-radiative electron-hole recombination [12][13][14][15], as schematically shown in Fig. 1(a).This localization process is predominantly mediated by high-frequency vibrational modes that can bridge the energy gap between high-lying exciton/CT states and the lowest-energy interfacial CT state.The energy loss associated to this process is typically larger than 0.6 eV per photon [16][17][18], leading to a low power conversion efficiency in OPV with respect to their inorganic counterparts that results in a small open circuit voltage [19].
Although energetically costly, dissociation of strongly bound electron-hole pairs [10,11,20] takes place despite the much lower thermal energy at room temperature.The energy of bound CT states is largely dependent on the offset between the LUMO of the acceptor and the HOMO of the donor [21][22][23].Fixing the acceptor and employing different donor materials (or vice versa) is a popular strategy to investigate the energetics at the interface and achieve a high voltage, small energy losses and sufficient photocurrent density [19,[24][25][26][27][28].Surprisingly, some of these blends show ultrafast and efficient exciton dissociation despite having small or no apparent driving force [15,18,[29][30][31][32][33][34][35].The driving force is a crucial parameter in charge separation and refers to the energy difference between exciton and interfacial CT state (see ∆ in Fig. 1(a)).Hybridization between exciton and CT states has been thought to be behind the successful ultrafast charge separation of these promising materials, which are often based on small molecules (oligomers) with acceptordonor-acceptor structures that have reached power conversion efficiencies of up to 17 % [15,31,36,37].This represents an astonishing 50 % increase in the state-of-the-art performance of organic photovoltaics in less than a decade.From finite molecular clusters to periodic molecular solids, ultrafast longrange charge separation has appeared across a wide variety of photovoltaic platforms, but the underlying mechanism has not been understood fully, leading some to advocate for a deeper analysis of charge separation processes [38,39].F Y e f q y n u n d W e B J N g N X P g f j / + V P O q W 9 n H G Q M 6 7 3 s / m T 6 G H E a Y u 2 a R e 3 f 0 h 5 u q Q C 6 p U v 5 J G e 6 F l 7 0 F 6 0 V + 3 t h 6 r F o p x N + j O 0 9 2 9 U S 5 5 q < / l a t e x i t > Acceptor J l (!) < l a t e x i t s h a 1 _ b a s e 6 4 = " B 1 T l E z U t X 1 X + G U n 5 g R j H H 8 w J Z r g     Some experimental studies rule out thermal activation as an important mechanism for charge separation in a large number of photovoltaic devices [40][41][42].In contrast, the vibronic coupling to underdamped vibrational modes is presumed to enable coherent charge separation [43][44][45][46][47][48][49], which requires nonperturbative simulation tools for a reliable description of the vibronic interaction between exciton/CT states and molecular vibrations.However, in many theoretical studies on the charge separation in extended systems, a broad and unstructured environmental spectral density has been considered [50][51][52] to reduce simulation costs, neglecting the ubiquitous presence of underdamped vibrational modes in organic molecules and their role in charge separation.In addition, the non-Markovian vibronic effects proposed to suppress the localization of electron-hole pairs at the interfaces, e.g.suggested in Ref. [52], are found to be well described by a Markovian quantum master equation, as shown in Appendix A, due to weak vibronic coupling strength and no underdamped modes considered in simulations.This indicates that a vibronic mechanism inferred solely on the basis of non-perturbative numerical results without the subsequent formulation of an accurate physical mechanism may lead to ambiguities in the interpretation of the underlying mechanism.Some first-principles numerical methods have been employed to simulate vibronic charge separation [53][54][55][56][57][58][59][60], where underdamped vibrational modes are considered.However, the interpretation of simulated results is a non-trivial issue here.For instance, in Ref. [53], an effective one-dimensional Coulomb potential is considered where electron-hole binding energy is assumed to be reduced by in-stantaneous electron delocalization in three-dimensional acceptor aggregates and as a result the electronic coupling being responsible for a hole transfer becomes larger in magnitude than the detunings in energy levels of the effective potential.In Appendix B, we show how, in this case, deactivating completely the vibrational environment has little impact in charge separation dynamics.This leads us to conclude that the ultrafast long-range charge separation observed in Ref. [53] is not necessarily enhanced by vibronic couplings, but merely induced by the weak Coulomb-binding energy.Other theoretical studies have focused on intermolecular modes as the relevant vibrations behind charge separation [61], while sometimes, intramolecular modes are attributed a hampering role [62].This is, as we will demonstrate, in sharp contrast to our findings, as intramolecular modes can induce both effects.
Given the heterogeneity of donor-acceptor materials and model parameters employed across the literature, we aim to discern the underlying mechanisms of charge separation dynamics as a function of the driving force and the structure of vibrational environments based on non-perturbative simulations and detailed reduced model analysis.We determine under what conditions underdamped vibrational motion induces efficient long-range charge separation in the presence of strong Coulomb-binding energy V ∼ 0.3 eV.To this end we consider one, two and three-dimensional donor-acceptor networks, instead of effective one-dimensional Coulomb potentials, by using our non-perturbative simulation method called dissipation assisted matrix product factorization (DAMPF) [63][64][65][66], to investigate how coherent vibronic couplings pro-mote long-range charge separation in high-dimensional multisite systems.We show that there are two available mechanisms for ultrafast long-range charge separation in donoracceptor interfaces.For low driving forces ∆ ∼ 0.15 eV, the transitions between near-resonant exciton and delocalised CT states occur on a sub-ps time scale even if vibronic couplings are not considered.For high driving forces ∆ ∼ 0.3 eV, the vibronic coupling of underdamped high-frequency vibrational modes with frequencies ω h ∼ 0.15 eV induces the transitions between exciton and CT states delocalised over multiple acceptors.Here a vibrationally cold exciton can interact resonantly with vibrationally hot lower-energy CT states and, subsequently, also with vibrationally cold high-energy CT states.The charge separation process becomes significantly inefficient in this case when vibronic couplings are ignored in simulations, hinting the genuine vibronic effects induced by underdamped vibrational modes.For both low and high driving forces, we demonstrate that the time scale of the charge localization towards the donor-acceptor interfaces is determined by the lifetime of the high-frequency vibrational modes, as strongly damped modes promote the transitions to the lowestenergy interfacial CT state.These results demonstrate that experimentally measured long-lived vibrational and vibronic coherences in OPV [43][44][45][46][47][48][49] may have a functional relevance in charge separation processes.

II. MODEL
To investigate the influence of underdamped vibrational motion on the charge separation of strongly bound electronhole pairs, we consider a one-dimensional chain consisting of N sites, composed of an electron donor in contact with a chain of (N − 1) electron acceptors, as schematically shown in Fig. 1(a).Two and three-dimensional donor-acceptor networks will be considered later.The electronic Hamiltonian is modeled by where h.c.denotes the Hermitian conjugate.Here |0 denotes an exciton state localized at the donor, while |k with k ≥ 1 is a CT state with an electron localized at the k-th acceptor.For simplicity, we assume that the hole is fixed at the donor within the time scale of our simulations due to its lower mobility with respect to the electron [50,52,67].The energylevels of the CT states take into account the Coulomb attraction between electron and hole, modelled by Ω k = −V /k with V = 0.3 eV for k ≥ 1.We take J k,k+1 = 500 cm −1 ≈ 0.06 eV for the electronic coupling being responsible for an electron transfer, a common value found in acceptor aggregates such as fullerene derivatives [53,54,68].The exciton energy Ω 0 depends on the molecular properties of the donor [10,21,22,69], which will be considered a free variable parametrized by the driving force ∆ = Ω 0 − Ω 1 , as shown in Fig. 1(a).
For simplicity, we assume that each electronic state |k is coupled to an independent vibrational environment that is initially in a thermal state at room temperature.The vibrational Hamiltonian is written as with b k,q (b † k,q ) describing the annihilation (creation) operator of a vibrational mode with frequency ω q that is locally coupled to the electronic state |k .The vibronic interaction is modeled by where the vibronic coupling strength is quantified by the Huang-Rhys (HR) factors s q .The vibrational environments are fully characterized by a phonon spectral density J (ω) = q ω 2 q s q δ(ω − ω q ) with δ(ω) denoting the Dirac delta function.According to first-principles calculations of functionalized fullerene electron acceptors, the vibrational environment consists of multiple low-frequency modes, with vibrational frequencies smaller than the thermal energy at room temperature (k B T ≈ 200 cm −1 ≈ 0.025 eV), and a few discrete modes with high vibrational frequencies of the order of ∼ 1000 cm −1 and HR factors 0.1 [70][71][72][73].Motivated by these observations, we consider a phonon spectral density J (ω) = J l (ω) + J h (ω) where J l (ω) = λ l ω l ωe −ω/ω l , with ω l = 80 cm −1 and λ l = 50 cm −1 , describing a low-frequency phonon spectrum (see gray curve in Fig. 1(b)).The highfrequency vibrational modes are modeled by a Lorentzian function J h (ω) = with vibrational frequency ω h = 1200 cm −1 ≈ V /2 = 0.15 eV and HR factor s h = 0.1.Here the reorganization energy of the high-frequency mode, defined by ∞ 0 dωJ h (ω)ω −1 = w h s h , is independent of its vibrational damping rate γ (see red and blue curves in Fig. 1(b)).
In order to tackle the problem of simulating large vibronic systems, we have extended DAMPF [64], where a continuous vibrational environment is described by a finite number of oscillators undergoing Markovian dissipation (pseudomodes) and a tensor network formalism is used.With DAMPF the reduced electronic system dynamics can be simulated in a numerically accurate manner for highly structured phonon spectral densities by fitting the corresponding bath correlation functions via an optimal set of parameters of either coupled or uncoupled pseudomodes [63][64][65][66].The extended DAMPFmethod opens the door to non-perturbative simulations of many body systems consisting of several tens of sites coupled to structured environments in one, two-and three spatial dimensions, as will be demonstrated in this work.More details about the method and the explicit equation of motion in terms of pseudomodes can be found in Appendix C.
B l e i c y q j X w Q 4 P 9 E h P W l 9 7 1 l 6 0 1 2 + q l k p y N u j X 0 N 6 A H e M 5 G P 8 v f 9 q 5 P y w a x 0 X j 9 q h Q O k s e R p p 2 a Y 8 O c P s n V K J r K q N e e e u P 9 E T P 2 k h 7 0 V 6 1 t x + q l k p y d u j P 0 N 6 / A Z i 6 m s 0 = < / l a t e x i t > A H e M 5 G P 8 v f 9 q 5 P y w a x 0 X j 9 q h Q O k s e R p p 2 a Y 8 O c P s n V K J r K q N e e e u P 9 E T P 2 k h 7 0 V 6 1 t x + q l k p y d u j P 0 N 6 / A Z i 6 m s 0 = < / l a t e x i t >    Here the vibrational damping rate of the high-frequency modes with frequency ω h = 1200 cm −1 is taken to be γ = (50 fs) −1 or (500 fs) −1 , shown in red and blue, respectively.(d) Electronic eigenstates in the absence of exciton-CT couplings where probability distributions for finding an electron at the k-th acceptor are vertically shifted depending on electronic energy-levels Eα. (e-f) Electronic eigenstates in the presence of exciton-CT couplings where driving force is taken to be (e) ∆e = 0.15 eV or (f) ∆v = 0.3 eV.In (e), hybrid exciton-CT states contributing to initial charge separation dynamics are colored in blue/green.In (f), the exciton and delocalised CT states, governing initial charge separation via a vibronic mixing, are highlighted in blue and green, respectively.In (d-f), the probabilities for finding an electron at donor/acceptor interface are shown in red.

A. Driving Force and Vibrational Environments
Here we investigate the charge separation dynamics on a sub-ps time scale simulated by DAMPF.For simplicity, we consider a linear chain consisting of a donor and nine acceptors (N = 10).Longer one-dimensional chains and higherdimensional donor/acceptor networks will be considered later.We assume that an exciton state |0 localised at the donor site is created at the initial time t = 0 and then an electron transfer through the acceptors induces the transitions from the exciton to the CT states |k with k ≥ 1.The mean dis-tance between electron and hole is considered a figure of merit for charge separation, defined by x(t) = N −1 k=0 kP k (t) with P k (t) representing the populations of the exciton and CT states |k at time t, with the assumption that the distance between nearby sites is uniform.To investigate how the initial charge separation dynamics depends on the exciton energy Ω 0 and the structure of vibrational environments, we analyse the time-averaged electron-hole distance, defined by x t≤T = 1 T T 0 dt x(t) with T = 400 fs, as a function of the driving force ∆ = Ω 0 + V for various environmental structures.The role of high-frequency vibrational modes and their non-equilibrium dynamics in charge separation processes is identified by considering (i) no environments (J (ω) = 0), (ii)  low-frequency phonon baths (J (ω) = J l (ω), see gray curve in Fig. 1(b)), (iii) high-frequency vibrational modes with controlled damping rates γ ∈ {(50 fs) −1 , (500 fs) −1 } (J (ω) = J h (ω), see red and blue curves in Fig. 1(b)), and (iv) the total vibrational environments including both low-frequency phonon baths and high-frequency vibrational modes (J (ω) = J l (ω) + J h (ω)).
In Fig. 2(a), the time-averaged electron-hole distance is shown as a function of the driving force ∆ when vibrational environments are not considered (J (ω) = 0).In this case, the charge separation dynamics is purely electronic and the mean electron-hole distance shows multiple peaks for ∆ 0.3 eV.When electronic states are only coupled to low-frequency phonon baths (J (ω) = J l (ω)), these peaks are smeared out, resulting in a smooth, broad single peak centered at ∆ e ≈ 0.15 eV.In Fig. 2(b) where the electronic states are coupled to high-frequency vibrational modes (J (ω) = J h (ω)), the time-averaged electron-hole distance is displayed for different vibrational damping rates γ = (50 fs) −1 and γ = (500 fs) −1 , shown in red and blue, respectively.With ω h denoting the vibrational frequency of the high-frequency modes, the electron-hole distance is maximized at ∆ e ≈ 0.15 eV, ∆ e + ω h ≈ 0.3 eV, ∆ e + 2ω h ≈ 0.45 eV, making the charge separation process efficient for a broader range of the driving force ∆ when compared to the cases that the high-frequency modes are ignored (see Fig. 2(a)).It is notable that the electron-hole distance is larger for the lower damping rate γ = (500 fs) −1 of the high-frequency vibrational modes than for the higher damping rate γ = (50 fs) −1 .These results imply that non-equilibrium vibrational dynamics can promote long-range charge separation.This observation still holds even if the low-frequency phonon baths are considered in addition to the high-frequency vibrational modes (J (ω) = J l (ω) + J h (ω)), as shown in Fig. 2(c), where the electron-hole distance is maximized at ∆ e ≈ 0.15 eV and ∆ v = ∆ e + ω h ≈ 0.3 eV.We note that the electron-hole dis-tance at low driving forces ∆ ∼ ∆ e is insensitive to the presence of vibrational environments, while at high driving forces ∆ ∼ ∆ v , the charge separation process becomes significantly inefficient when the high-frequency vibrational modes are ignored.These results suggest that vibrational environments may play an essential role in the long-range charge separation at high driving forces, while the exciton dissociation at low driving forces may be governed by electronic interactions.
So far the time-averaged mean electron-hole distance has been considered to identify under what conditions the charge separation on a sub-ps time scale becomes efficient.However, it does not show how much populations of the CT states with well-separated electron-hole pairs are generated and how quickly the long-range electron-hole separation takes place.In Fig. 3, we show the population dynamics of the CT states where electron and hole are separated more than four molecular units, defined by 9 k=5 P k (t), for the case that electronic states are coupled to the total vibrational environments (J (ω) = J l (ω) + J h (ω)).When the high-frequency vibrational modes are weakly damped with γ = (500 fs) −1 , the electron is transferred to the second half of the acceptor chain within 100 fs and then the long-range electron-hole separation is sustained on a sub-ps time scale for a wide range of the driving forces ∆, as shown in Fig. 3(a).When the high-frequency modes are strongly damped with γ = (50 fs) −1 , for low driving forces around ∆ e ≈ 0.15 eV the long-range charge separation occurs within 100 fs, but the electron is quickly transferred back to the donor-acceptor interface, as shown in Fig. 3(b).For high driving forces around ∆ v ≈ 0.3 eV, the long-range charge separation and subsequent localization towards the interface take place on a slower time scale when compared to the case of the low driving forces.These results demonstrate that underdamped vibrational motion can promote long-range charge separation when the excess energy ∆−V , defined by the energy difference between exciton state and fully separated free charge carriers, is negative or close to zero [33-35, 74, 75].

B. Electronic mixing at low driving forces
The long-range charge separation observed in DAMPF simulations can be rationalized by analysing the energy-levels and delocalization lengths of the exciton and CT states.In Fig. 2(d e H A u i q i 0 5 l i d q B h q 3 U f J x D A l l G B P O W x g q 4 6 b s F a y j K l E i S K F v Q 4 r D x 9 W c / / 3 d n g S T Y D d 4 T n Y P 6 9 / E m n s l c w D w r m 1 X 6 + e J w 8 j D R t 0 h b t 4 P Y P q U g X V E K 9 8 v 4 e 6 J G e t H v t W X v R X r + p W i r J 2 a B f Q 3 v 7 A g z 2 m 7 E = < / l a t e x i t >

t [ps]
< l a t e x i t s h a 1 _ b a s e 6 4 = " < l a t e x i t s h a 1 _ b a s e 6 4 = " H P Y m 0 H u H g 8 m C T 7 9 h F 3 k l I 0 I G e H A u i q i 0 5 l i d q B h q 3 U f J x D A l l G B P O W x g q 4 6 b s F a y j K l E i S K F v Q 4 r D x 9 W c / / 3 d n g S T Y D d 4 T n Y P 6 9 / E m n s l c w D w r m 1 X 6 + e J w 8 j D R t 0 h b t 4 P Y P q U g X V E K 9 8 v 4 e 6 J G e t H v t W X v R X r + p W i r J 2 a B f Q 3 v 7 A g z 2 m 7 E = < / l a t e x i t >

t [ps]
< l a t e x i t s h a 1 _ b a s e 6 4 = " H P Y m 0 H u H g 8 m C T 7 9 h F 3 k l I 0 I G the interface for higher energies E (CT) α , as highlighted in red.
We now consider the full electronic Hamiltonian H e including the exciton state at the low driving force ∆ e ≈ 0.15 eV where efficient long-range charge separation occurs even in the absence of vibrational environments (see Fig. 2(a)).The exciton state |0 is coupled to the eigenstates . This implies that the transition between exciton and CT state |E | 2 at the interface (see red bars in Fig. 2(d)).For ∆ e = Ω 0 + V = 0.15 eV, the exciton state can be strongly mixed with a near-resonant CT state delocalised over multiple acceptor sites (see Fig. 2(d)), leading to two hybrid exciton-CT eigenstates of the total electronic Hamiltonian H e , described by the superpositions of |0 and multiple |k with k ≥ 1 (see Fig. 2(e)).This indicates that the multiple peaks in the timeaveraged electron-hole distance x t≤400 fs shown in Fig. 2(a) originate from the resonances between exciton and CT states |E (CT) α .Here the high-lying CT states with energies E (CT) α + V 0.3 eV do not show long-range electron-hole separation, as the interfacial electronic couplings J 0,1 1|E (CT) α are not strong enough to induce notable transitions between exciton and CT states within the time scale T = 400 fs considered in Fig. 2(a).These high-energy CT states can be populated via a near-resonant exciton state, but the corresponding purely electronic charge separation occurs on a slower ps time scale, as shown in Appendix D, and therefore this process can be significantly affected by low-frequency phonon baths.This is contrary to the charge separation at the low driving force ∆ e ≈ 0.15 eV, which takes place within 100 fs and therefore the early electronic dynamics is weakly affected by vibrational environments.We note that when this analysis is applied to the charge separation model in Ref. [53], it can be shown that an exciton state is strongly mixed with near-resonant CT states delocalised in an effective one-dimensional Coulomb potential and as a result the ultrafast long-range charge separation reported in Ref. [53] can be well described by a purely electronic model where vibrational environments are ignored (see Appendix B).

C. Vibronic mixing at high driving forces
Contrary to the case of ∆ e = 0.15 eV, the eigenstates of the full electronic Hamiltonian H e with ∆ v = 0.3 eV show a weak mixing between exciton and CT states, as displayed in Fig. 2 ) denoting the annihilation (creation) operator of the highfrequency vibrational mode locally coupled to electronic state |k .The other high-lying CT states |E CT near-resonant with the exciton state, E CT ≈ E XT , may have relatively small amplitudes around the interface, so the direct vibronic coupling to the exciton state could be small.However, the transitions from the exciton |E XT , 0 v to the vibrationally hot low-lying CT states |E CT , 1 v can allow subsequent transitions to vibrationally cold high-lying CT states |E CT , 0 v , as the delocalised CT states |E CT and |E CT are spatially overlapped.Such consecutive transitions are mediated by vibrational excitations and can delay the process of charge localization at donor-acceptor interfaces if the damping rate of the high-frequency vibrational modes is sufficiently lower than the transition rates amongst exciton and CT states.This picture is in line with the vibronic eigenstate analysis where the high-frequency modes are included as a part of system Hamiltonian in addition to the electronic states, as summarised in Appendix E.

D. Functional Relevance of Long-lived Vibrational Motion
So far we have discussed the underlying mechanisms behind long-range charge separation on a sub-ps time scale.We now investigate how subsequent charge localization towards the donor-acceptor interface depends on the lifetimes of high-frequency vibrational modes to demonstrate that nonequilibrium vibrational dynamics can maintain long-range electron-hole separation.
In Fig. 4(a) and (b), where the high-frequency modes are strongly and weakly damped, respectively, with γ = (50 fs) −1 and γ = (500 fs) −1 , the population dynamics P k (t) of the exciton |0 and CT states |k with k ≥ 1 is shown as a function of time t up to 1.5 ps in addition to the mean electron-hole distance x(t) .Here we consider the high driving force ∆ v = 0.3 eV where the vibronic transition from exciton |E XT , 0 v to delocalised CT states |E CT , 1 v takes place.When the high-frequency modes are strongly damped, the vibrationally hot CT states |E CT , 1 v quickly dissipate to |E CT , 0 v , leading to subsequent vibronic transitions to vibrationally hot interfacial CT states |E ICT , 1 v (see Fig. 2(f)).After that, the vibrational damping of the high-frequency modes generates the population of the lowest-energy interfacial CT state |E ICT , 0 v and makes the electron-hole pair trapped at the interface, as shown in Fig. 4(a).When the high-frequency vibrational modes are weakly damped, the mean electronhole distance is maximized at ∼ 700 fs, as shown in Fig. 4(b), and then the population P 1 (t) of the CT state |1 localised around the interface starts to be increased.This localized interfacial state |1 has been considered an energetic trap that leads to non-radiative losses [19].In Fig. 4(c), the population dynamics of P 1 (t) is shown in red and blue, respectively, for γ = (50 fs) −1 and γ = (500 fs) −1 .In the strongly damped case, P 1 (t) rapidly increases in time and then saturates at ∼ 0.9 on a picosecond time scale.This is contrary to the weakly damped case where P 1 (t) is quickly saturated at ∼ 0.1 within 100 fs and then does not increase until ∼ 500 fs, demonstrating that the charge localization towards the interface can be delayed by the underdamped nature of the highfrequency vibrational modes.The delayed charge localization makes long-range electron-hole separation maintained on a picosecond time scale, as shown in Fig. 4(d) where 9 k=5 P k (t) is plotted.These results suggest that long-lived vibrational and vibronic coherences observed in nonlinear optical spectra of organic solar cells [44,46,48] may have a functional relevance in long-range charge separation.

E. Large Vibronic Systems
So far we have considered a one-dimensional chain consisting of N = 10 sites.Here we investigate the charge separation dynamics in larger multi-site systems, including longer linear chains, and donor-acceptor networks in two and three spatial dimensions.
For the linear chains consisting of a donor and (N − 1) acceptors, we consider the total vibrational environments including low-frequency phonon baths and high-frequency vibrational modes with γ = (500 fs) −1 (J (ω) = J l (ω) + J h (ω)).The driving force is taken to be ∆ v = 0.3 eV, for which long-range charge separation occurs mediated by vibronic couplings in the case of N = 10 sites.In Fig. 5(a), a longer linear chain is considered with N = 20 and the population dynamics P k (t) of the exciton and CT states |k is shown.It is notable that an electron-hole pair is separated more than ten molecular units within ∼ 200 fs.Interestingly, with a hole fixed at the donor site, the probability distributions P k (t) for finding an electron at the k-th acceptor are strongly delocalised over the entire acceptor chain, which are maximized at k ≈ 6 and locally minimized at k ≈ 3.This implies that an exciton state is vibronically mixed with strongly delocalised CT states, as the detunings Ω k+1 − Ω k = V (k(k + 1)) −1 in donor-acceptor networks in the thermodynamic limit.To corroborate these ideas, we consider a variety of donor-acceptor networks with different sizes and dimensions.In Fig. 5(c), the schematic representations of one-, two-and three-dimensional donor-acceptor networks considered in our simulations are displayed where the size of each network is quantified by the number L of acceptor layers.In the one-dimensional chains, the number of acceptors in each layer is unity, while in the two-dimensional triangular (three-dimensional pyramidal) structures, the number of acceptors in each layer increases linearly (quadratically) as a function of the minimum distance to the donor site.We assume that the distances between nearby sites are uniform and the corresponding nearest-neighbour electron-transfer couplings are taken to be 500 cm −1 .The electronic Hamiltonian is described by the exciton and CT states |k where a hole is fixed at the donor while an electron is localized at the k-th acceptor.The corresponding CT energy is modelled by Ω k = −V /|r 0 − r k | with V = 0.3 eV where r 0 and r k denote, respectively, the positions of the donor and k-th acceptor with the distance between nearby sites taken to be unity and dimensionless.To increase the size of the donor-acceptor networks that can be considered in simulations, we only consider the high-frequency vibrational modes (J (ω) = J h (ω)) with ω h = 1500 cm −1 , s h = 0.1 and γ = (500 fs) −1 .
In Fig. 5(d), the time-averaged electron-hole distance x t≤400 fs simulated by DAMPF is shown as a function of the driving force ∆ for one-and two-dimensional networks with L = 4.Here we consider the minimum distance between donor and each acceptor layer in the computation of the mean electron-hole distance, instead of the distances between donor and individual acceptors.We compare the case that the highfrequency vibrational modes are coupled to electronic states (J (ω) = J h (ω)), shown in blue, with that of no vibrational environments (J (ω) = 0), shown in yellow (slightly darker in the overlapped regions).Note that vibronic couplings make charge separation efficient for a broader range of the driving force ∆ in both one-and two-dimensional networks, and that long-range charge separation is further enhanced in the higher-dimensional network.To simulate larger vibronic systems, in Fig. 5(e), we consider a reduced vibronic model constructed within vibrational subspaces describing up to four vibrational excitations distributed amongst the high-frequency vibrational modes in the polaron basis (see Appendix E for more details).For L = 4, the simulated results obtained by the reduced models of one-and two-dimensional networks are qualitatively similar to the numerically exact DAMPF results shown in Fig. 5(d).The reduced model results demonstrate that long-range charge separation can be enhanced by considering a three-dimensional donor-acceptor network with L = 4, or by increasing the number of layers to L = 9 in the one-and two-dimensional cases.In Fig. 5(f), the dynamics of the mean electron-hole distance x(t) of the one-, two-and three-dimensional systems with L = 4, computed by DAMPF, is shown for a high driving force ∆ = 0.35 eV where the time-averaged electron-hole distance of the threedimensional system shown in Fig. 5(e) is maximized.These results demonstrate that long-range charge separation can be enhanced by considering higher-dimensional multi-site systems with vibronic couplings.

IV. CONCLUSIONS
We have extended the non-perturbative simulation method DAMPF to provide access to charge separation dynamics of a strongly bound electron-hole pair in one-, two-and threedimensional donor-acceptor networks where a donor is coupled to acceptor aggregates.By controlling the driving force and the structure of vibrational environments, we identified two distinct mechanisms for long-range charge separation.The first mechanism, activated at low driving forces, is characterized by hybrid exciton-CT states where long-range exciton dissociation takes place on a sub-100 fs time scale, which is not assisted by underdamped high-frequency vibrational modes.In the second mechanism, dominating charge separation at high driving forces, the exciton-CT hybridization occurs and it is mediated by vibronic interaction with underdamped high-frequency vibrational modes, leading to efficient charge separation for a broad range of driving forces.For both mechanisms, we have demonstrated that long-range charge separation is significantly suppressed when the highfrequency vibrational modes are strongly damped or delocalization lengths of the CT states are reduced by static disorder in the energy-levels of Coulomb potentials.These results suggest that non-equilibrium vibrational motion can promote long-range charge separation in ordered donor-acceptor aggregates.
The formulation and analysis of a reduced model whose validity became accessible to numerical corroboration thanks to the extension of the numerically exact simulation tool DAMPF allows us to identify unambiguously the mechanisms that underlie charge separation dynamics.The methods employed here can be applied to more realistic models where multiple donors are coupled to acceptor aggregates, without introducing effective one-dimensional Coulomb potentials, and vibrational environments are highly structured, which deserves a separate investigation.We expect our findings to help open up the engineering of vibrational environments for efficient long-range charge separation in organic solar cells and the identification of charge separation processes in other systems such as photosynthetic reaction centers and other biological processes driven by electron transfer.ronments (J (ω) = 0) is shown as a function of driving force ∆ in Fig. 7(a), which is maximised at ∆ ≈ 0.09 eV, being close to the driving force considered in numerical simulations of Ref. [53].For ∆ ≈ 0.09 eV, the population dynamics of P 0 (t), P 1 (t) and k=2 P k (t) in the absence of vibrational environments are shown in blue, red and grey lines, respectively, in Fig. 7(b), which are qualitatively similar to the results of Ref. [53] where electronic-vibrational couplings were considered in simulations (see a black line in Fig. 7 O 2 7 A 6 t I y q e K E i j r 0 f F h x + q K e y d 0 Z 4 A k 2 A 3 e I 5 6 D 9 v / x x p 7 i X 1 Q 6 z 2 s 1 + K n c a P o w o b d E 2 p X H 7 R 5 S j K 8 q j X h O 3 8 k h P 9 K w 8 K C / K q / L 2 Q 1 U i Y c 4 G / R n K + z f C 0 Z 5 z < / l a t e x i t > P 5 (t) + • • • + P 9 (t) FIG. 8. Charge separation in the absence of vibrational environments.For a linear chain consisting of N = 10 sites, the probability for separating an electron-hole pair more than four molecular units,

Appendix E: Vibronic eigenstate analysis
To analyse vibronic mixing of exciton and delocalized CT states at high driving force ∆ v = 0.3 eV, we consider a reduced vibronic model where electronic states are coupled to high-frequency vibrational modes.For simplicity, the damping of the high-frequency modes and the vibronic coupling to low-frequency phonon baths are not considered.The reduced model Hamiltonian consists of three parts, H r = H e + H v + H e−v , defined by To minimize the number of vibrational states required to achieve the numerical convergence in reduced electronic dynamics, we consider a displaced vibrational basis defined by unitary displacement operator conditional to electronic states (polaron transformation), In the polaron basis, the reduced vibronic Hamiltonian is expressed as In simulations, we consider a vibrational subspace spanned by up to N v vibrational excitations distributed amongst multiple high-frequency modes b k,h .When N v = 0, only the global ).In Fig. 9, we consider a linear chain consisting of N = 10 sites and ∆ v = 0.3 eV where the vibronic mixing of exciton and delocalized CT states is induced by high-frequency vibrational modes with ω h = 1200 cm −1 and s h = 0.1.In Fig. 9(a), the mean electron-hole distance x(t) computed by the reduced Hamiltonian is displayed where the total number N v of vibrational excitations is increased from 0 to 4. When vibrational excitations are not considered in simulations (N v = 0), the mean electron-hole distance remains below 1 up to 2 ps, as shown in red.When vibrational excitations are considered in simulations (N v ≥ 1), charge separation process is significantly enhanced by vibronic couplings.The electronic dynamics shows convergence when N v ≥ 3 where a blue line for N v = 3 is well overlapped with a black line for N v = 4.
The qualitative features of the numerically converged electronic dynamics can be well reproduced by approximate results for N v = 1 and N v = 2, as shown in orange and green, respectively.
To identify the origin of the vibronic enhancement of charge separation, we consider vibronic eigenstates of the polaron-transformed Hamiltonian U H r U † , represented by The presence of the vibronic mixing can be demonstrated more clearly by analysing both electronic and vibrational states of the vibronic eigenstates |ψ j .For a linear chain consisting of N = 10 sites, the electronic states are coupled to ten high-frequency vibrational modes in total.To simplify analysis, for each vibronic eigenstate |ψ j , we investigate the overlap with electronic eigenstates |E α in the presence of N v vibrational excitations, defined by P j (E α , N v ) = sum( nv)=Nv |ψ j (E α , n v )| 2 where the summation runs over all possible vibrational states with N v excitations distributed amongst the ten high-frequency modes.In Table I, all P j (E α , N v ) being larger than 0.01 are shown, demonstrating that the vibronic eigenstates |ψ j=1,2,3 governing initial charge separation dynamics are well described by the superpositions of vibrationally cold exciton state |E XT , 0 v , vibrationally hot |E CT , 1 v and cold CT states |E CT , 0 v delocalised in the acceptor domain.
OHMIC + HIGH-FREQUENCY MODE WITH OHMIC + HIGH-FREQUENCY MODE WITH= (50 fs) 1 = (500 fs)1   OHMIC BACKGROUND (LOW FREQUENCY) t e x i t s h a 1 _ b a s e 6 4 = " 9 r B t w S S g v l g O I y j y R B r 4 E P y 9 + I 8 = " > A A A D B 3 i c h V L L S s N A F L 3 G V 3 3 X x 8 5 N s A i u S i K i 4 q r g A z e C g t V C W 0 q S T u P Q N A m T a b G W f o B / 4 V Y 3 7 s S t n + F P + A U u P D N G Q U v p h M m 9 c + 6 5 Z + 6 d G T c O e C I t 6 3 3 MG J + Y n J r O z M z O z S 8 s L m W X V 6 6 S q C 0 8 V v S i I B I l 1 0 l Y w E N W l F w G r B Q L 5 r T c g F 2 7 z U M V v + 4 w k f A o v J T d m F V b j h / y B v c c C a i W X a t I d i t 7 R 4 J 3 e O i b J x F k + 7 V s z s p b e p i D j p 0 6 O U r H e Z T 9 o A r V K S K P 2 t Q i R i F J + A E 5 l O A r k 0 0 W x c C q 1 A M m 4 H E d Z 9 S n W e S 2 w W J g O E C b + P t Y l V M 0 x F p p J j r b w y 4 B p k C m S Z u Y J 1 r R B V v t y u A n s J + Y d x r z h + 7 Q 0 8 q q w i 6 s C 8 U Z r X g G X N I N G K M y W y n z p 5 b R m a o r S Q 3 a 1 9 1 w 1 B d r R P X p / e o c I S K A N X X E p G P N 9 K H h 6 n U H J x D C F l G B O u U f B V N 3 X I d 1 t G V a J U w V H e g J W H X 6 q p 7 h 3 b n g K T Y D V z 0 H + / / l D z p X 2 3 l 7 N 2 9 f 7 O Q K B + n D y N A 6 b d A W b n + P C n R K 5 6 j X w 6 0 8 0 C M 9 G f f G s / F i v H 5 T j b E 0 Z 5 X + D O P t C 4 U N o E U = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " d W Q z 8 S 6 8 j u 1 J 6 0 N N G 6 9 3 u O b 7 + Q A = " > A A A C / 3 i c h V L L T s J A F L 3 U F + A L d e m G S E x c k d Y Y d U l U j B s T T O S R A J q 2 D D i h t E 0 7 E J C w 8 C / c 6 s a d c e u n + B N + g Q v P j M V E C W G a 6 b 1 z 7 r l n 7 p 0 Z y 3 d 4 K H T 9 I 6 b N z S 8 s L s U T y e W V 1 b X 1 1 M Z m K f S 6 g c 2 K t u d 4 Q c U y Q + Z w l x U F F w 6 r + A E z O 5 b D y l b 7 V M b L P R a E 3 H O v x c B n 9 Y 7 Z c n m T 2 6 Y A d F M T r C + G + b 7 N h e e O b l M Z P a u r k Z 5 0 j M j J U D Q K X u q T a t Q g j 2 z q U o c Y u S T g O 2 R S i K 9 K B u n k A 6 v T E F g A j 6 s 4 o x E l k d s F i 4 F h A m 3 j 3 8 K q G q E u 1 l I z V N k 2 d n E w A 2 S m a R f z X C l a Y M t d G f w Q 9 g v z X m G t q T s M l b K s c A B r Q T G h F C + B C 7 o D Y 1 Z m J 2 K O a 5 m d K b s S 1 K R j 1 Q 1 H f b 5 C Z J / 2 r 8 4 Z I g G w t o q k K a + Y L W h Y a t 3 D C b i w R V Q g T 3 m s k F Y d N 2 B N Z Z l S c S N F E 3 o B r D x 9 W c / 0 7 i z w J J u B K 5 + D 8 f / y J 5 3 S f t Y 4 z B p X B 5 n c S f Q w 4 r R N O 7 S H 2 z + i H F 1 Q A f X a U H 6 k J 3 r W H r Q X 7 V V 7 + 6 F q s S h n i / 4 M7 f 0 b s t a d 2 A = = < / l a t e x i t > Exciton < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 q M I u 9 6 + i K V P g e q T x L y N C / Y K N c Q = " > A A A C / X i c h V L L T s J A F L 3 U F + A L d e m m k Z i 4 I q 0 x 6 p I o G j c m m M g j A W L a M m B D a Z v p Q E R C / A u 3 u n F n 3 P o t / o R f 4 M I z Y z F R Q p h m e u + c e + 6 Z e 2 f G D j 0 3 E o b x k d D m 5 h c W l 5 K p 9 P L K 6 t p 6 Z m O z H A U 9 7 r C S E 3 g B r 9 p W x D z X Z y X h C o 9 V Q 8 6 s r u 2 x i t 0 5 l f F K n / H I D f x r M Q h Z o 2 u 1 f b f l O p Y A V K 8 L d i e G h c A P + O g m k z V y h h r 6 p G P G T p b i U Q w y n 1 S n J g X k U I + 6 x M g n A d 8 j i y J 8 N T L J o B B Y g 4 b A O D x X x R m N K I 3 c H l g M D A t o B / 8 2 V r U Y 9 b G W m p H K d r C L h 8 m R q d M u 5 r l S t M G W u z L 4 E e w X 5 r 3 C 2 l N 3 G C p l W e E A 1 o Z i S i l e A h d 0 C 8 a s z G 7 M H N c y O 1 N 2 J a h F x 6 o b F / W F C p F 9 O r 8 6 B U Q 4 s I 6 K 6 H S m m G 1 o 2 G r d x w n 4 s C V U I E 9 5 r K C r j p u w l r J M q f i x o g U 9 D i t P X 9 Y z v T s b P M l m 4 M r n Y P 6 / / E m n v J 8 z D 3 P m 1 U E 2 f x I / j C R t 0 w 7 t 4 f a P K E 8 X V E S 9 D v Z 6 p C d 6 1 h 6 0 F + 1 V e / u h a o k 4 Z 4 v + D O 3 9 G z R i n O w = < / l a t e x i t > Donor < l a t e x i t s h a 1 _ b a s e 6 4 = " q t 3 b Z U U n A R 0 2 h q g g W r g z z u o n Z x A = " > A A A D A n i c h V L L T s J A F L 3 U F + A D 1 K U b I j F x R V p j 1 C U + 4 8 Y E E 3 k k Q E h b B m w o b T M d i E j Y + R d u d e P O u P V H / A m / w I V n x m K i h D D N 9 N 4 5 9 9 w z 9 8 6 M F b h O K H T 9 I 6 b N z S 8 s L s U T y e W V 1 b V U e n 2 j F P o 9 b r O i 7 b s + r 1 h m y F z H Y 0 X h C J d V A s 7 M r u W y s t U 5 l f F y n / H Q 8 b 0 b M Q h Y v W u 2 P a f l 2 K Y A 1 E i n a o L d i e G x b b N A + H z U S G f 1 n K 5 G Z t I x I i d L 0 S j 4 6 U + q U Z N 8 s q l H X W L k k Y D v k k k h v i o Z p F M A r E 5 D Y B y e o + K M R p R E b g 8 s B o Y J t I N / G 6 t q h H p Y S 8 1 Q Z d v Y x c X k y M z Q D u a F U r T A l r s y + C H s F + a 9 w t p T d x g q Z V n h A N a C Y k I p X g E X d A v G r M x u x B z X M j t T d i W o R U e q G w f 1 B Q q R f d q / O m e I c G A d F c n Q u W K 2 o W G p d R 8 n 4 M E W U Y E 8 5 b F C R n X c h D W V Z U r F i x R N 6 H n 3 6 u p 6 / u / P I 0 2 x B r n 4 O z u / L 7 3 U O F 8 v O c t n Z X y q t r 2 U P I 4 d Z z G G B t 7 + C d e x g j / X 6 O M c d 7 v F g 3 V q P 1 p P 1 / E m 1 + r K c G f w Y 1 s s H 7 H 6 d i g = = < / l a t e x i t > loss < l a t e x i t s h a 1 _ b a s e 6 4 = " q H 2 U 9 k D y c r h T K C M b m a Z z p a m F 2 I l P e X i E l O 0 m D s m S 5 I R E R 3 x n 3 C 1 V 6 O a a 6 t Z u e y Y u 6 S c J T N 9 v O B 8 7 x Q F 2 X Z X S b + i / c 1 5 6 r D k z h 0 m T t l W e E I r q D j n F L e J G 3 w m 4 7 7 M r G Z e 1 3 J / p u 3 K 4 A i v X D e K 9 R U O s X 3 G N z p v G S m J j V z E x z v H T K g h 3 P q Y J 6 B p e 6 z A n v K 1 g u 8 6 H t J G z k q n o m v F i H o l r T 1 9 W 8 / d 3 Q n y L F u S a 5 9 D + P / l 3 3 b 6 L 7 v h e j f 8 u N b Z f F 0 / j C a e Y w W r v P 0 N b O I D d l h v j D N 8 w z m + e 1 + 9 H 9 5 P 7 + I v 1 W v U O c v 4 Z 3 i X f w D l k K E j < / l a t e x i t > Coulomb Binding < l a t e x i t s h a 1 _ b a s e 6 4 = " Z N 4 X u y e F 4 q B j p Z Z F I C 7 I r 6 w D g 8 X 8 U Z D W g O u V 2 w G B g 2 0 A 7 + H l b 1 D I 2 w l p q p y n a x S 4 D J k W n Q B u a x U n T A l r s y + C n s J + a d w r y R O / S V s q y w B + t A M a 8 U T 4 E L u g Z j X G a Y M X 9 q G Z 8 p u x L U p j 3 V j Y / 6 E o X I P t 1 f n S N E O L C O i h h U V k w P G o 5 a 3 + A E I t g q K p C n / K N g q I 5 b s L a y T K l E m a I N P Q 4 r T 1 / W M 7 o 7 B z z J Z u A O 8 B y s / 5 c / 7 N S 2 S t Z O y T r b L h 7 s Z w 8 j R 2 u 0 T p u 4 / V 0 6 o B O q o F 7 5 Q h 7 o k Z 6 0 e + 1 Z e 9 F e v 6 n a R J a z S n + G 9 v Y F g v i e E Q = = < / l a t e x i t > Energy V < l a t e x i t s h a 1 _ b a s e 6 4 = " I h l 2 0 q p 4 w a T y S L h o s J h l 0 g A U J 6 h t p h s r M z k + 5 e I m 6 4 8 w / 8 F 1 7 l 4 s 1 4 9 e y f 8 B d 4 4 H U z m A g h 9 K S n q l + 9 e l 3 V 3 U l d 5 M Z G 0 e + p Y H r m z t 1 7 s 3 P z 9 x 8 8 f P S 4 9 W R h 0 1 R j n U o / r Y p K b y f K S J G X 0 r e 5 L W S 7 1 q J G S S F b y b D r 4 l t H o k 1 e l T 1 7 X M v e S G V l P s h T Z Q n t t x Z 3 r X y y k + 6 h 0 p m E P a 1 K M x A d L n d 7 z 8 M N q 6 y Y k / 1 W O + p E f o T X n b h x 2 m j G e t X 6 g 1 0 c o E K K M U Y Q l L D 0 C y g Y f j u I E a E m t o c J M U 0 v 9 3 H B C e a Z O y Z L y F B E h / x n X O 0 0 a M m 1 0 z Q + O + U u B a d m Z o g l z v d e M S H b 7 S r 0 D e 1 f z s 8 e y 2 7 c Y e K V X Y X H t A k V 5 7 z i B + I W h 2 T c l j l q m J e 1 3 J 7 p u r I Y 4 I 3 v J m d 9 t U d c n FIG. 1. Coulomb potential and vibrational environments.(a) Schematic representation of a one-dimensional chain consisting of a donor and (N − 1) acceptors.The Coulomb-binding energy of electron and hole is modelled by Ω k = −V /k with V = 0.3 eV for k ≥ 1.The energy-gap between exciton and interfacial CT states is defined as driving force ∆ = Ω0 − Ω1.(b) Vibrational environments consist of low-frequency phonon baths with room temperature energy scales (kBT ≈ 200 cm −1) and high-frequency vibrational modes.In this work, the low-frequency phonons are modelled by an Ohmic spectral density J l (ω) with an exponential cutoff, while the high-frequency modes are described by Lorentzian spectral densities J h (ω) centered at vibrational frequency ω h = 1200 cm −1 (or 1500 cm −1 ).The vibrational damping rate of the high-frequency modes is taken to be γ = (50 fs) −1 or (500 fs) −1 , as shown in red and blue, respectively, to investigate the role of non-equilbrium vibrational motion in long-range charge separation.
t e x i t s h a 1 _ b a s e 6 4 = " S 0 j E o Y W / h 0 P 5 K 6 4 0 5 t W H r / B Z Q r w = " > A A A C 9 H i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I t Q N y U R U X F V 8 I E b o a J 9 Q C 0 y k 0 5 r a F 4 k a a G W f o J b 3 b g T t / 6 P P + E X u P D M m A p a p B M m 9 8 6 5 5 5 6 5 d 2 Z 4 4 N h R b B j v K W 1 m d m 5 + I Z 3 J L i 4 t r 6 z m 1 t a r k d 8 L L V G x f M c P 6 5 x F w r E 9 U Y n t 2 B H 1 I B T M 5 Y 6 o 8 e 6 x j N f 6 I o x s 3 7 u O B 4 F o u q z j 2 W 3 b Y j G g q 4 L Y u c 3 l j a K h h j 7 p m I m T p 2 S U / d w H 3 / l a t e x i t > (e) < l a t e x i t s h a 1 _ b a s e 6 4 = " l c G 3 s 71 Q k e 4 Y Q F T v c S C B 2 1 m 2 z x 8 = " > A A A C 9 H i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I t Q N y U R U X F V 8 I E b o a J 9 Q C 2 S p N M 6 N C8 m a a G W f o J b 3 b g T t / 6 P P + E X u P D M m A p a p B M m 9 8 6 5 5 5 6 5 d 2 b s 0 O V R b B j v K W 1 m d m 5 + I Z 3 J L i 4 t r 6 z m 1 t a r U d A T D q s 4 g R u I u m 1 F z O U + q 8 Q 8 d l k 9 F M z y b J f V 7 O 6 x j N f 6 T E Q 8 8 K / j Q c i a n t X x e Z s 7 V g z o q t D e u c 3 l j a K h h j 7 p m I m T p 2 S U g 9 w H 3 g n q a 1 p 6 + r e f + 7 m L y L F u S a 5 9 D e P v y 7 z p 7 K 5 1 w r R P u r r Y 3 N 5 q H M Y M X e I n X v P 1 1 b O I 9 d l h v g g t c 4 Q u + e p f e N + + 7 9 + M v 1 Z t o c p 7 j v + H 9 / A P x M K c z < / l a t e x i t > hxi t400 fs < l a t e x i t s h a 1 _ b a s e 6 4 = " b H E + 7 + m n b H m e 9 C W v e Z e D A + 2 L L S I = " > A A A D B H i c h V L L S s Q w F D 3 W 9 3 v U p Z v i I L g a W h F 1 O f j C j a D g q K A i b Y 1 j m E 5 a 2 o y v w a 1 / 4 V Y 3 7 s S t / + F P + A U u P I l V U B F T 0 n t z 7 r k n 9 y Y J 0 1 j m 2 v N e O p z O r u 6 e 3 r 7 + g c G h 4 Z H R 0 t j 4 T p 6 0 s k j U o i R O s r 0 w y E U s l a h p q W O x l 2 Y i a I a x 2 A 0 b y y a + e y a y X C s 1 z 8 H 9 e / m 9 n Z 7 b i z 1 f 8 r b l y d a l 4 G H 2 Y x B R m e P s L q G I d m 6 w 3 w j l u c Y d 7 5 8 Z 5 c B 6 d p w + q 0 1 H k T O D b c J 7 f A a d W n 0 Q = < / l a t e x i t > no exciton < l a t e x i t s h a 1 _ b a s e 6 4 = " + B j h e M p a T l 9 0 E z I D 9 6 a E 0 q O F J r I I O Y k p 6 b 8 4 9 9 + T e J F 4 W 8 V z a 9 m v J G B k d G 5 8 o T 1 a m p m d m 5 8 z 5 h V a e 9 o T P m n 4 a p e L E c 3 M a u 6 r / L 5 u z o b P M X m 4 K r n Y P 2 8 / N 9 O f a 1 k b Z a s k 4 1 i e S 9 5 G G l a o m V a x e 1 v U Z m O q I p 8 H f T 6 j u 7 p w b g 1 H o 0 n 4 / m T a q S S m D x 9 G 8 b L B 8 R J n u E = < / l a t e x i t > = 0.3 eV < l a t e x i t s h a 1 _ b a s e 6 4 = " t N 1 / F B 5 m r QB U e U r Q c M / s l L i q q J M = " > A A A D G n i c h V L L T h R B F D 0 0 y k u F A Z d u O k 6 I L s y k m x B g S S I Y N y a Q O E B C k 0 l 1 U 9 N 2 p v p B d Q 0 R J v M J / A F / w V Y 3 7 o x b N / 4 E X + D C U 2 V j I o R Q n e p 7 6 9 x z T 9 1 b V X G l s t o E w a 8 J b / L R 4 6 n p m d m 5 J 0 + f z S + 0 F p f 2 6 n K o E 9 l N S l X q g 1 j U U m W F 7 J r M K H l Q a Sn y W M n 9 e P D W x v d P p a 6 z s v h o z i p 5 l I u 0 y P p Z I g y h X u t V p E S R K u l / 9 i P t v N 7 I R E q e + K t B E L 0 h m I / 6 9 X j c a 7 W D T u C G f 9 c J G 6 e N Z u y U r W t E O E a J B E P k k C h g 6 C s I 1 P w O E S J A R e w I I 2 K a X u b i E m P M M X d I l i R D E B 3 w n 3 J 1 2 K A F 1 1 a z d t k J d 1 G c m p k + l j n f O c W Y b L u r p F / T / u Y 8 d 1 h 6 7 w 4 j p 2 w r P K O N q T j r F D 8 Q N / h E x k O Z e c O 8 q e X h T N u V Q R 8 b r p u M 9 V U O s X 0 m / 3 S 2 G N H E B i 7 i Y 9 s x U 2 r E b n 3 K E y h o u 6 z A n v K N g u 8 6 P q Y V z k q n U j S K g n q a 1 p 6 + r e f + 7 m L y L F u S a 5 9 D e P v y 7 z p 7 K 5 1 w r R P u r r Y 3 N 5 q H M Y M X e I n X v P 1 1 b O I 9 d l h v g g t c 4 Q u + e p f e N + + 7 9 + M v 1 Z t o c p 7 j v + H 9 / A P x M K c z < / l a t e x i t > hxi t400 fs < l a t e x i t s h a 1 _ b a s e 6 4 = " i 5 L u F h 8 L n X Q D y C p p i N s s b f U s r D M / v m b G p m E b t b o I D G 4 3 m n r R L J l q G H 8 d K 3 O K l I 2 z S H + n O r U o I p e 6 F B C n k F L 4 P j F K 8 N X I I p N i Y A 2 6 B S b g e S r O a U A F 5 H b B 4 m A w o B 3 8 2 1 jV M j T E W m o m K t v F L j 6 m Q K Z B 6 5 j H S t E B W + 7 K 4 S e w H 5 g 3 C m u P 3 O F W K c s K + 7 A O F P N K 8 R R 4 S t d g / J c Z Z M x h L f 9 n y q 5 S u q I 9 1 Y 2 H + m K F y D 7 d b 5 0 j R A S w j o o Y V F b M N j Q c t e 7 h B E L Y C i q Q p z x U M F T H L V i m L F c q Y a b I o C d g 5 e n L e k Z 3 5 4 A n 2 R z c A Z 6 D 9 f v y / z r 2 V s n a K V n n 2 8 W D / e x h 5 G i V 1 m g D t 7 9 L B 3 R C Z 6 j X h f o 9 P d C j d q c 9 a c / a y x d V G 8 t y l u n H 0 F 4 / A W F J n w k = < / l a t e x i t > E ↵ + V [eV]< l a t e x i t s h a 1 _ b a s e 6 4 = " H P Y m 0 H u H g 8 m C T 7 9 h F 3 k l I 0 I G 6 s p 7 Z 3 V n g S T Y H d 4 z n Y P 6 9 / G m n v l 8 y D 0 t m 9 a B Y P k k f R p a 2 a Y f 2 c P t H V K Z L q q B e G 5 o P 9 E h P m t C e t R f t 9 Z u q Z d K c L f o 1 t L c v e l + X x g = = < / l a t e x i t > k < l a t e x i t s h a 1 _ b a s e 6 4 = " t N 1 / F B 5 m r Q B U e U r Q c M / s l L i q q J M = "> A A A D G n i c h V L L T h R B F D 0 0 y k u F A Z d u O k 6 I L s y k m x B g S S I Y N y a Q O E B C k 0 l 1 U 9 N 2 p v p B d Q 0 R J v M J / A F / w V Y 3 7 o x b N / 4 E X + D C U 2 V j I o R Q n e p 7 6 9 x z T 9 1 b V X G l s t o E w a 8 J b / L R 4 6 n p m d m 5 J 0 + f z S + 0 F p f 2 6 n K o E 9 l N S l X q g 1 j U U m W F 7 J r M K H l Q a S ny W M n 9 e P D W x v d P p a 6 z s v h o z i p 5 l I u 0 y P p Z I g y h X u t V p E S R K u l / 9 i P t v N 7 I R E q e + K t B E L 0 h m I / 6 9 X j c a 7 W D T u C G f 9 c J G 6 e N Z u y U r W t E O E a J B E P k k C h g 6 C s I 1 P w O E S J A R e w I I 2 K a X u b i E m P M M X d I l i R D E B 3 w n 3 J 1 2 K A F 1 1 a z d t k J d 1 G c m p k + l j n f O c W Y b L u r p F / T / u Y 8 d 1 h 6 7 w 4 j p 2 w r P K O N q T j r F D 8 Q N / h E x k O Z e c O 8 q e X h T N u V Q R 8 b r p u M 9 V U O s X 0 m / 3 S 2 G N H E B i 7 i Y 9 s x U 2 r E b n 3 K E y h o u 6 z A n v K N g u 8 6 P q Y V z k q n U j S K g n q a 1 p 6 + r e f + 7 m L y L F u S a 5 9 D e P v y 7 z p 7 K 5 1 w r R P u r r Y 3 N 5 q H M Y M X e I n X v P 1 1 b O I 9 d l h v g g t c 4 Q u + e p f e N + + 7 9 + M v 1 Z t o c p 7 j v + H 9 / A P x M K c z < / l a t e x i t > hxi t400 fs < l a t e x i t s h a 1 _ b a s e 6 4 = " M y Q 5 Z h m 2 e M L o A W f f s c n L T 6 S I m 9 A = " > A A A D H n i c h V L L a t t A F D 1 W 2 s Z O X 0 6 6 7 E b U F B w K R g o l C Y G C I U 0 p h U I K d W K I g x n J E 1 l Y L 6 R x I D X + i P x B / 6 L b Z J N d y L b 9 i X 5 B F z 0 z l U s e B I 8 Y 3 T v n n n v m 3 p n x s i g s l O P 8 q l g L D x 4 + W q z W l h 4 / e f r s e X 1 5 Z a 9 I x 7 k v O 3 4 a p X n X E 4 W M w k R 2 V K g i 2 c 1 y K W I v k v v e a F v H 9 4 9 l X o R p 8 l W d Z P I w F k E S H o W + U I T 6 9 T e 9 W K i h L 6 L J p 2 m z l 8 Y y E K v 2 O / s a 2 h / O 8 H 6 9 4 b Q c M + y 7 j l s 6 D Z R j N 6 3 / R g 8 D p P A x R g y J B I p + B I G C 3 w F c O 7 w i C e j a z w b L 8 b r N 9 U Y y n O W 8 G c Y b 1 8 7 n Z n s < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B E l 5 U 1 O 7 Z Z L W u 9 c d P P

FIG. 2 .
FIG. 2. Charge separation under different vibrational environments and electronic eigenstate structures.(a) Time-averaged mean electron-hole distance x t≤400 fs is displayed as a function of driving force ∆ when vibrational environments are absent (J (ω) = 0) or only low-frequency phonon baths are present (J (ω) = J l (ω)), shown in yellow and gray, respectively.Here we consider a linear chain consisting of a donor and nine acceptors (N = 10 sites).(b-c) Time-averaged mean electron-hole distance when electronic states are coupled to (b) high-frequency vibrational modes only (J (ω) = J h (ω)) or to (c) the total vibrational environments (J (ω) = J l (ω) + J h (ω)).Here the vibrational damping rate of the high-frequency modes with frequency ω h = 1200 cm −1 is taken to be γ = (50 fs) −1 or (500 fs) −1 , shown in red and blue, respectively.(d) Electronic eigenstates in the absence of exciton-CT couplings where probability distributions for finding an electron at the k-th acceptor are vertically shifted depending on electronic energy-levels Eα. (e-f) Electronic eigenstates in the presence of exciton-CT couplings where driving force is taken to be (e) ∆e = 0.15 eV or (f) ∆v = 0.3 eV.In (e), hybrid exciton-CT states contributing to initial charge separation dynamics are colored in blue/green.In (f), the exciton and delocalised CT states, governing initial charge separation via a vibronic mixing, are highlighted in blue and green, respectively.In (d-f), the probabilities for finding an electron at donor/acceptor interface are shown in red.
t e x i t s h a 1 _ b a s e 6 4 = " B 5 6 G Y Y Q a N R c u W t Z W S l l j 1 m k G r i 8 = " > A A A C + n i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I v g q i Q i 6 r L 4 w o 1 Q w T 6 k L Z K k Y w 3 N i 8 m 0 U G u / w q 1 u 3 I l b f 8 a f 8 A t c e G Z M B S 2 l E y b 3 z r n n n r l 3 Z u z I c 2 N h G B 8 p b W Z 2 b n 4 h n c k u FIG. 3. Long-range charge separation dynamics.(a-b) Total probability for separating an electron-hole pair more than four molecular units, 9 k=5 P k (t), is shown as a function of time t and driving force ∆.Here a linear chain consisting of N = 10 sites is considered where electronic states are coupled to total vibrational environments (J (ω) = J l (ω) + J h (ω)).The damping rate of the high-frequency vibrational modes with ω h = 1200 cm −1 is taken to be (a) γ = (500 fs) −1 or (b) γ = (50 fs) −1 .
), we consider the eigenstates |E (CT) α of the electronic Hamiltonian where the exciton state |0 and its coupling J 0,1 to the CT states are ignored, namelyH CT = N −1 k=1 Ω k |k k| + N −2 k=1 J k,k+1 (|k k + 1| + h.c.).With a hole fixed at the donor site, the probability distributions | k|E (CT) α | 2 for finding an electron at the k-th acceptor site is displayed, which are vertically shifted by E (CT) α + V with E (CT) α representing the eigenvalues of H CT .The lowestenergy CT eigenstate is mainly localised at the interface due to the strong Coulomb-binding energy considered in simulations (Ω 2 − Ω 1 = V /2 = 0.15 eV > J 1,2 ≈ 0.06 eV).The other higher-energy CT eigenstates are significantly delocalised in the acceptor domain with smaller populations | 1|E (CT) α | 2 at hx(t)i t e x i t s h a 1 _ b a s e 6 4 = " 5 u 7 A 4 T v 4 l J 8 3 N U W D D N I 2 Y B Z X u U I = " > A A A C 9 3 i c h V L L S s N A F D 3 G V + u z 6 t J N s A i 6 K Y m I u p K C D 9 w I F W w r V J E k n d b Y N A n J t F D F f 3 C r G 3 f i 1 s / x J / w C F 5 4 Z o 6 A i T p j c O + e e e + b e m X H j w E + l Z b 0 9 X I 0 J B r p Z n q b I + 7 B J w J M 0 0 s c + 5 r R Z d s t a u g n 9 K + c V 5 p r P 3 n D t d a W V U 4 o H W p m N e K h 8 Q l L s j 4 L 7 O b M T 9 r + T 9 T d S X R w p b u x m d 9 s U Z U n 9 6 X z i 4 j C b G O j p j Y 0 8 w 2 N V y 9 7 v M E Q t o q K 1 C n / K l g 6 o 6 b t I 6 2 Q q u E m a J D v Y R W n b 6 q 5 + / u X P I U W 5 B 7 w + d g / 7 z 8 3 0 5 t r W R v l O y j 9 W J 5 O 3 s Y O S x i C S u 8 / U 2 U c Y A K 6 / V w i V v c 4 d 4 Y G A / G o / H 0 Q T W G s p w F f B v G 8 z t g G J k 2 < / l a t e x i t > P 1 (t) < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 2 Q L B s l v j x 1 3 6 3 6 P L G o 5 q E 5 w 4 4 I = " > A A A D B 3 i c h V L L S s N A F L 2 N r 7 a + 6 m P n p l i E S q E k 4 n M j B R + 4 E S r Y B 7 S l J O m 0 h q Z J S K a F W v o B / o V b 3 b g T t 3 6 G P + E X u P D M m A p a S i d M 7 p 1 z z z 1 z 7 8 w Y n m 0 F X F U / I s r M 7 N z 8 t > P 5 (t) + • • • + P 9 (t) OHMIC + HIGH-FREQUENCY MODE WITH = (50 fs) 1 OHMIC + HIGH-FREQUENCY MODE WITH = (500 fs) 1 P P P < l a t e x i t s h a 1 _ b a s e 6 4 = " w O0 R q F Q e A f i c l o 6 Z S o H I K o k u S h Y = " > A A A C / H i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I v g Q k o i o u K q 4 A M 3 Q g X7 g L Z I k k 5 r a F 5 M p o V a 6 l + 4 1 Y 0 7 c e u / + B N + g Q v P j K m g R T p h c u + c e + 6 Z e 2 f G j j w 3 F o b x n t J m Z u f m F 9 K Z 7 O L S 8 s p q b m 2 9 E o c 9 7 r C y E 3 o h r 9 l W z D w 3 Y G X h C o / V I s 4 s 3 / Z Y 1 e 6 e y H i 1 z 3 j s h s G 1 G E S s 6 V u d w G 2 7 j i U A 1 U V j t 9 7 g v h 7 F z Z t c 3 i g Y a u i T j p k 4 e U p G K c x 9 U I N a F J J D 7 g L Z I k k 5 r a F 5 M p o V a 6 l + 4 1 Y 0 7 c e u / + B N + g Q v P j K m g R T p h c u + c e + 6 Z e 2 f G j j w 3 F o b x n t J m Z u f m F 9 K Z 7 O L S 8 s p q b m 2 9 E o c 9 7 r C y E 3 o h r 9 l W z D w 3 Y G X h C o / V I s 4 s 3 / Z Y 1 e 6 e y H i 1 z 3 FIG. 4. Vibration-assisted exciton dissociation and charge localization towards donor/acceptor interfaces.(a) With a hole fixed at donor, the probability distribution for finding an electron at the donor (k = 0, corresponding to exciton) or at the k-th acceptor (k ≥ 1) is displayed as a function of time t, with mean electron-hole distance x(t) shown in red.With high driving force ∆v = 0.3 eV, here we consider a linear chain consisting of N = sites and total vibrational environments (J (ω) = J l (ω) + J h (ω)) including strongly damped high-frequency modes with ω h = 1200 cm −1 and γ = (50 fs) −1 .(b) Charge separation dynamics when the high-frequency vibrational modes are weakly damped with γ = (500 fs) −1 .(c) Population dynamics of interfacial CT state |1 .(d) Probability for separating an electron-hole pair more than four molecular units, 9 k=5 P k (t), shown as a function of time t.In both (c) and (d), the strongly (weakly) damped case with γ = (50 fs) −1 (γ = (500 fs) −1 is shown in red (blue).
energy Ω 0 is near-resonant with the CT energy E (CT) α and the CT state has sufficiently high population | 1|E (CT) α

ACKNOWLEDGMENTS
This work was supported by the ERC Synergy grant Hy-perQ (grant no.856432), the BMBF project PhoQuant (grant no.13N16110) under funding program quantum technologies -from basic research to market, and an IQST PhD fellowship.The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no.INST 40/575-1 FUGG (JUSTUS 2 cluster)

1 <
l a t e x i t s h a 1 _ b a s e 6 4 = " 6 6 X c 2 M 0 4 / Gx A L m A N Y z t H V / s / L d w = " > A A A C / H i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I v g Q k o i o i 6 L L 9 w I F e w D 2 i J J O q 2 h e T G Z F m q p f + F W N + 7 E r f / i T / g F L j w z p o I W 6 Y T J v X P u u W f u n R k 7 8 t x Y G M Z 7 S p u Z n Z t f S Ge y i 0 v L K 6 u 5 t f V K H P a 4 w 8 p O 6 I W 8 Z l s x 8 9 y A l Y U r P F a L O L N 8 2 2 N V u 3 s i 4 9 U + 4 7 E b B t d i E L G m b 3 U C t + 0 6 l g B U F 4 3 d e o P 7 e j t u 3 u T y R s F Q Q 5 9 0 z M T J U z J K Y e 6 D G t S i k B z q k U + M A h L w P b I o x l c n k w y K g D V p C I z D c 1 W c 0 Y i y y O 2 B x c C w g H b x 7 2 B V T 9 A A a 6 k Z q 2 w H u 3 i Y H J k 6 b W O e K 0 U b b L k r g x / D f m L e K a z z 7 w 5 D p S w r H M D a U M w o x U v g g m 7 B m J b

FIG. 7 .
FIG. 6. Charge separation dynamics in the presence of weakly coupled unstructured vibrational environments.The mean electronhole distances x(t) , computed by a non-perturbative method (HEOM), are shown in dashed lines, while the predictions of a perturbative method (Redfield equation) are shown in solid lines.Here we consider several values of bath relaxation times γ −1 ∈ {10, 20, 50, 100, 200} fs (see blue, red, yellow, purple, green lines, respectively), considered in Ref.[52].
(b) displaying the transient of [eV] < l a t e x i t s h a 1 _ b a s e 6 4 = " B 5 6 G Y Y Q a N R c u W t Z W S l l j 1 m k G r i 8 = " > A A A C + n i c h V L L S s N A F L 2 N r 7 a + q i 7 d B I v g q i Q i 6 r L 4 w o 1 Q w T 6 k L Z K k Y w 3 N i 8 m 0 U G u / w q 1 u 3 I l b f 8 a f 8 A t c e G Z M B S 2 l E y b 3 z r n n n r l 3 Z u z I c 2 N h G B 8 p b W Z 2 b n 4 h n c k u R J F C 3 o c V p 6 + r G d y d z Z 4 k s 3 A H e I 5 m P 8 v f 9 y p 7 B b M / Y J 5 u Z c v H i U P I 0 2 b t E U 7 u P 0 D K t I 5 l V C v g 4 o f 6 Y m e t Q f t R X v V 3 n 6 o W i r J 2 a A / Q 3 v / B m J O m x 0 = < / l a t e x i t > t[ps] < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 2 Q L B s l v j x 1 3 6 3 6 P L G o 5 q E 5 w 4 4 I = " > A A A D B 3 i c h V L L S s N A F L 2 N r 7 a + 6 m P n p l i E S q E k 4 n M j B R + 4 E S r Y B 7 S l J O m 0 h q Z J S K a F W v o B / o V b 3 b g T t 3 6 G P + E X u P D M m A p a S i d M 7 p 1 z z z 1 z 7 8 w Y n m 0 F X F U / I s r M 7 N z 8 Q j Q W X 1 x a X l l N r K 0 X A 7 f r m 6 x g u r b r l w 0 9 Y L b l s A K 3 u M 3 K n s / 0 j m G z k t E + E / F S j / m B 5 T q 3

9 k=5
P k (t), is shown as a function of time t and driving force ∆.

FIG. 9 .
FIG. 9. Vibronic eigenstate analysis.(a) Transient of mean electronhole distance x(t) of a linear chain consisting of a donor and nine acceptors where ∆v = 0.3 eV.Here we consider a reduced vibronic model where up to Nv vibrational excitations are considered in total.(b) Electron-hole distance xj of vibronic eigenstate |ψj is displayed as a function of the overlap | ψj|ψ initial | 2 with vibrationally cold initial exciton state |ψ initial .(c) Electronic eigenstate structure.(d) For vibronic eigenstates that have large overlaps with the initial state, denoted by |ψj=1,2,3 in (b), the populations Pj(Eα) in electronic eigenbasis are shown, which are dominated by exciton state |EXT = |E8 , low-lying CT states |ECT ∈ {|E3 , |E4 } and high-lying CT state |E CT = |E9 .

1 k=0FIG. 10 .
FIG.10.Charge separation dynamics in the presence of static disorder.(a-c) Ensemble-averaged mean electron-hole distance x(t) of one-dimensional chains consisting of (a) N = 10, (b) N = 20, (c) N = 30 sites.The degree of static disorder in the energy-levels Ω k of Coulomb potentials is quantified by the standard deviation σ of independent Gaussian distributions, which is taken to be σ = 50, 100, 200, 500, 1000 cm −1 , respectively, shown in red, orange, green, blue, black.For each ensemble-averaged transient, 1000 randomly generated sets of energy-levels Ω k were considered.
(f), where the eigenstate |E XT with the most strong excitonic character | 0|E XT | ≈ 1, marked in blue, has small amplitudes around the interface, | k|E XT | 1 for k ≥ 1.Here the energy-gaps between the exciton state |E XT , shown in blue, and lower-energy eigenstates |E CT with strong CT characters, shown in green, are nearresonant with the vibrational frequency of the high-frequency modes, E XT − E CT ≈ ω h .Therefore, the vibrationally cold exciton state |E XT , 0 v can resonantly interact with vibrationally hot CT states |E CT , 1 v where one of the highfrequency modes is singly excited.Here the CT states are delocalised in the acceptor domain, but have non-negligible amplitudes around the interface, leading to a moderate vibronic coupling to the exciton state, E XT | H e−v |E CT