On the Role of Non-Diagonal System-Environment Interactions in Bridge-Mediated Electron Transfer

Bridge-mediated electron transfer (ET) between a donor and an acceptor is prototypical for the description of numerous most important ET scenarios. While multi-step ET and the interplay of sequential and direct superexchange transfer pathways in the donor-bridge-acceptor (D-B-A) model is increasingly understood, the influence off-diagonal system-bath interactions on the transfer dynamics is less explored. Off-diagonal interactions account for the dependence of the ET coupling elements on nuclear coordinates (non-Condon effects) and are typically neglected. Here we numerically investigate with quasi-adiabatic propagator path integral (QUAPI) simulations the impact of off-diagonal system-environment interactions on the transfer dynamics for a wide range of scenarios in the D-B-A model. We demonstrate that off-diagonal system-environment interactions can have profound impact on the bridge-mediated ET dynamics. In the considered scenarios the dynamics itself does not allow for a rigorous assignment of the underlying transfer mechanism. Further, we demonstrate how off-diagonal system-environment interaction mediates anomalous localization by preventing long-time depopulation of the bridge B and how coherent transfer dynamics between donor D and acceptor A can be facilitated. The arising non-exponential short-time dynamics and coherent oscillations are interpreted within an equivalent Hamiltonian representation of a primary reaction coordinate model that reveals how the complex vibronic interplay of vibrational and electronic degrees of freedom underlying the non-Condon effects can impose donor-to-acceptor coherence transfer on short timescales.


I. INTRODUCTION
Photosynthetic solar energy conversion relying on molecular charge carriers starts with the light-induced generation of Frenkel-type excitons, followed by the irreversible fixation of excitonic energy in multi-step electron transfer (ET) reactions. In reaction centres (RC), of purple bacteria or plants, the irreversible fixation proceeds across a phospholipid membrane in a sequence of directional and highly efficient ET reactions, mediated by spatially well organized (bacterio-) chlorophyll and pheophytin molecules. Pioneering electrostatic considerations 1 , supported by multi-objective evolutionary algorithm optimizations 2 suggest that efficient and irreversible charge separation requires a sequence of at least three molecular charge carriers. The emerging prototypical setup of bridge B mediated multi-step or superexchange ET between a donor D and an acceptor A (see Fig. 1), thus serves as minimal model for the description of charge separation in RC and has shown relevance for numerous bridge-mediated ET processes [3][4][5][6][7][8] .
While nonadiabatic ET theory provides a valuable starting point for the descriptions of multi-step ET reactions, e.g. in the ET kinetics of the bacterial RC 9-12 , limitations arise from the inherent assumptions of instantaneous medium relaxation, the perturbative nature of Diagonal and off-diagonal system-bath interactions lead to fluctuations of state energies i and electron transfer coupling elements V ij that mediate the dynamics. coherent oscillations observed in pump-probe and, more recently, two-dimensional electronic spectroscopy experiments of bacterial RC suggest a more complex picture where details of the ET reaction are potentially affected by the interplay of coherent electronic and nuclear motion. [13][14][15][16] Particular theoretical challenges arise for a strong interaction of vibrational modes and electronic degrees of freedom that can impose oscillatory dynamics in bridge mediated ET. 15 Tanimura investigated multistate ET for a system coupled to a heat bath with a non-Ohmic spectral density with the numerically exact reduced hierarchy equations of motion (HEOM) approach 17,18 that is numerically efficient for Debye spectral densities. Nevertheless, the strong coupling to the environment as typically realized in ET reactions is a persistent challenge for convergence 19,20 . Pioneering quasi-adiabatic propagator path integral (QUAPI) simulations by Makri and coworkers provided valuable insight into details of ET in bacterial RC. 21,22 The QUAPI method formally does not assume a particular form of the environment spectral density or the interaction strength but the long system bath memory times of a sluggish environment pose persistent challenges to QUAPI methods. [23][24][25] Recently, we investigated bridge-mediated ET in a model of the bacterial RC, where charge separation is initiated from a non-equilibrium excitonic superposition. 26 The path integral simulations with particularly longtime system-environment correlations allowed to explore the influence of discrete vibrational modes on the transfer dynamics. While excitonic energy transfer was found to be strongly affected, the kinetics of ET dynamics appeared exceptionally robust to the details of the spectral density function, suggesting a picture where intramolecular vibrations assure the robustness of optimal, non-activated ET reactions.
Hence, multi-step ET in the D-B-A model is increasingly understood beyond perturbative approaches and for various (non-equilibrium) conditions of the environment. Nevertheless, the impact of off-diagonal system-bath interactions on the transfer dynamics is less explored.
Such off-diagonal system-environment interactions are associated with non-Condon effects, i.e., the dependence of the coupling matrix element mediating the transfer dynamics on the nuclear coordinates. The nuclear coordinate dependence of electronic coupling was explored in the context of nonadiabatic transitions, indicating that promoting modes can dominate the vibrational effects. 27 Milischuk and Matyushov explored non-Condon effects for nonadiabatic electron transfer reactions in donor-bridge-acceptor systems. 28 The importance of non-Condon effects was further highlighted for ET at oligothiophene-fullerene interfaces via multi-layer MCTDH simulations 29 and the role of off-diagonal couplings was emphasized for the formation of charge-transfer states in polymeric solar cells. 30 Condensed phase ET beyond the Condon approximation was recently explored by Mavros and Van Voorhis. 31 Off-diagonal environment fluctuations were further identified to induce unexpectedly fast Förster resonant energy transfer between orthogonal oriented photoexcited molecules. 32 A review highlighting limitation of the Condon approximation in biological and bioinspired ET reactions is given in Ref. 33.
Here, we show via non-perturbative QUAPI simulations that non-diagonal systemenvironment interactions can have profound impact on bridge-mediated ET dynamics.
We investigate the dynamics in different regimes of the prototypical Donor-Bridge-Acceptor (D-B-A) model and demonstrate, that induced by non-Condon effects, the dynamics itself precludes a rigorous assignment of the underlying transfer mechanism. Further, we demonstrate how anomalous localization, mediated by off-diagonal system-environment interaction prevents the depopulation of the bridge B and how coherent transfer between donor D and acceptor A can be facilitated in presence of off-diagonal system-environment interactions.

A. Multi-Step Electron Transfer with Non-Diagonal System-Environment
Interaction.
We consider a three-state D-B-A model (see schematic in Fig. 1) that interacts bi-linearly via diagonal and off-diagonal interactions with the environment: In eq. 1, i denote site energies of donor D, bridge B and acceptor A, and V ij are the respective electron transfer coupling elements. The bath is composed of harmonic oscillators with momenta p j , position x j , mass m j and frequency ω j . The linear coupling constant c j mediates the interaction between system and the environment where d i specify the diagonal system-bath interaction inducing fluctuation of energy levels i and C ij represent off-diagonal system bath interactions that induce fluctuations of the transfer coupling elements V ij . We restrict the Hamiltonian (eq. 1) to the case with V DA = 0, i.e., for diagonal system-environment interaction only the sequential pathway contributes to the dynamics.
We consider the coupling to a single bath affecting site energies i and transfer couplings V ij . For linear coupling all information about the environment is contained in the spectral density.
For Ohmic dissipation α characterizes the system-bath interaction strength and ω c specifies the cut-off frequency that is related to the inverse Drude memory time τ D = 1/ω c . A key quantity in the description of ET is the reorganization energy λ R , defined as λ R = 1 π J(ω)

B. Primary Reaction Coordinate Model.
The Hamiltonian with bi-linear diagonal and off-diagonal system-environment interactions (eq. 1) can be equivalently mapped onto a model where the electronic system interacts with a primary reaction coordinate Q, which couples and dissipates into a bath 20,34,35 the form of the primary reaction coordinate model (eq. 4) directly reveals the coordinate dependence of transfer coupling elements V ij , i.e., non-Condon effects.
The information about the interaction of the primary mode with the environment is contained in the modified spectral density When the relaxation dynamics of the bath is fast, the low frequency part of J(ω) becomes relevant which yields an Ohmic spectral density where γ is the damping coefficient of the primary mode.
The primary mode can be expressed as a linear combination of the bath normal modes Using this canonical transformation (eq. 7), the Hamiltonian of the primary reaction coordinate model (eq. 4) can be recast in the form of eq. 1 where Let us denote the eigenvalues of the interaction matrix (second term in eq. 4) as {e 1 , e 2 , e 3 }. In the overdamped limit, γ Ω, the parameters of the spectral density J(ω) (eq. 3) are determined by where D 1 denotes the lowest eigenvalue of the original system-bath interaction matrix (see below, eq. 14). On the other hand, a diagonal shift

System-Environment Interaction
We are interested in the time evolution of the reduced density matrix with H given by eq. 1, which determines observables. For diagonal interaction with the environment (C ij = 0) numerical exact simulations are facilitated with the QUAPI method. [37][38][39] For the general interaction matrix which satisfy we define the unitary matrix U which diagonalizes the system-bath interaction M : 40 Upon transformation of the total Hamiltonian (eq. 1), we obtain The assumption of a single spectral density affecting diagonal and off-diagonal elements of the Hamiltonian (eq. 1) facilitates accurate QUAPI simulations for diagonal and off- Employing factorized initial conditions ρ(0) = |D D| and assuming the bath in thermal equilibrium at temperature T , ρ(t) was evaluated numerically by transforming the initial conditions ρ (t = 0) = U † ρ(t = 0)U. (17) followed by solving for ρ (t) in transformed basis using QUAPI methods, and followed by reverse transform to obtain ρ(t). In particular, numerical propagation was performed with the recently introduced mask assisted coarse graining of influence coefficients (MACGIC)-  Table S.3), respectively, where in the case of direct donor-acceptor coherence transfer the memory time has to cover the entire non-Markovian oscillatory dynamics.

III. RESULTS AND DISCUSSION
In the following we investigate the impact of non-diagonal system-bath interactions on the prototypic dynamics of multi-step electron transfer in the bridge-mediated three level system (see Fig. 1). We start by investigating the regime of sequential, bridge-mediated transfer dynamics (Sec. III A) and further examine how off-diagonal system-bath interactions can activate the direct superexchange transfer pathway |D → |A (Sec. III B). Section III C presents scenarios how off-diagonal system bath interactions can induce anomalous popula- tion localization in the bridge state B and Section III D demonstrates off-diagonal mediated coherence transfer between donor and acceptor states.

A. Sequential Donor-Bridge-Acceptor (D-B-A) Model
Sequential ET via a low-energy bridge ( D > B ). We start by briefly summarizing the well-known dynamics in the sequential D-B-A model of multi-step electron transfer (Fig. 2,  Fig. 2   Considering off-diagonal coupling to the environment (C 12 = 0, C 23 = 0, Fig. 3 (b)), substantially accelerated dynamics can be realized compared to the diagonal interaction case and the dynamics can closely resemble the sequential ET dynamics via a low-energy bridge ( D > B , cf. Fig 2). In particular, the characteristic time constant for depopulation    In presence of off-diagonal system-bath interactions, significant differences to the diagonal interaction case are identified. The system exhibits overdamped decay dynamics towards the thermal equilibrium state if α|D 1 − D 2 | > α c and α|D 2 − D 3 | > α c but α|D 1 − D 3 | < α c (solid lines in Fig. 7 (a), with D i being the eigenvalues of the system-environment interaction matrix, eq. 14). Localization in the bridge B via a freezing of dynamics due to a renormalization of tunneling amplitudes (V ij → 0) is only observed if α|D α − D β | > α c for all α = β (α, β = 1, 2, 3) ( Fig. 7 (b)). In analogy to the diagonal interaction case, the tunneling processes is suppressed due to tunneling amplitude renormalization but initial short time dynamics persists that determines the degree of localization. 54 Figure 7 (c) presents the dynamics for α|D 1 − D 2 | > α c and α|D 1 − D 3 | > α c but α|D 2 − D 3 | < α c . In this scenario, population transfer between the donor D and bridge B is suppressed after the initial short time dynamics but subsequent population transfer among bridge B and the acceptor state A is facilitated.
In presence of off-diagonal system-bath interactions, we thus identify distinct mechanistic differences for the anomalous population localization via the renormalization of tunneling amplitudes V ij → 0: (i) due to the basis transformation of initial conditions (eq. 17), coherences are imposed in the initial conditions that induce short time dynamics and consequently a population of all basis states |D , |B and |A ; (ii) stricter requirements are to be satisfied by the system-bath interactions in order to realize anomalous bridge localization due to a freezing of the dynamics, arising from the strong interaction in the BKT localized phase.
For example, the condition for the diagonal interaction case, α|D 1 − D 2 | > α c with arbitrary D 3 , is not sufficient because this only renormalizes V DB to zero but not V DA whose magnitude can become significant due to basis rotation. Thus, in presence of non-diagonal system-bath interactions, the conditions for bridge localization are stricter: α|D 1 −D 2 | > α c , α|D 2 − D 3 | > α c and α|D 1 − D 3 | > α c need to be simultaneously satisfied to sufficiently renormalize V BA and V DA . Alternatively, when α|D 1 − D 2 | > α c and α|D 1 − D 3 | > α c but (Fig. 7 (c)), the donor population freezes subsequent to the initial shorttime dynamics as both V DB and V DA are renormalized to zero, which suppresses tunneling processes involving the donor state D but still facilitates equilibration between the bridge B and acceptor A.

D. Donor-Acceptor Coherence Transfer
The prototypical dynamics in the sequential D-B-A model (Fig. 2) proceeds overdamped and is reasonably accounted for by pseudo-activationless, non-adiabatic ET theory. In the following, we explore possibilities of coherent transfer dynamics between donor D and acceptor A and the impact of non-diagonal system-environment interactions. Figure 8 presents population dynamics of the sequential D-B-A model shows coherent oscillatory modulations involving donor D, bridge B and acceptor A with diagonal system-environment interaction ( Fig. 8 (a), C 12 = C 23 = 0). Oscillatory dynamics occurs on the few-hundred fs timescale and is followed by exponential equilibration dynamics occurring on the ≈ 1 ps timescale.
Oscillations appear particularly pronounced in the donor and bridge population while the acceptor population follows a step-function increase during the first ≈ 300 fs.
Off-diagonal system-environment interactions allow to modulate the relative amplitudes of the oscillatory donor, bridge and acceptor population dynamics ( Fig. 8 (b), C 12 = 0.2, C 23 = 0.9). In the particular realization, the oscillatory decay of the donor state D is largely preserved while the oscillation amplitude of the bridge state B is decreased and the oscillation amplitude of the acceptor state A is increased. Figure 9 presents the frequency domain representation of the dynamics for varying system-environment interaction strength α. We find that for increasing α the non-zero frequency components (centered around . For weak-to-moderate interaction with the environment (α = 0.8 − 1.6) the oscillation frequency appears nearly unperturbed by the interaction with bath while for strong system-bath interaction deviations from the system Hamiltonian eigenvalues become larger (up to 60 cm −1 ) due to a bath induced renormalization of system frequencies. Thus, upon increase of the system-environment interaction strength α (Fig. 9) the coherence frequency monotonically shifts away from the system resonance frequency to higher frequencies and the amplitude of coherences is reduced due to increased interaction strength with the environement.   Fig. 8). We find that the oscillation frequency and the amplitude of oscillations increase with growing values of the coupling strength α, the oscillator displacement d 0 and the magnitude of off-diagonal system-environment interactions C ij (Fig. 10 (a-c)).
The oscillatory dynamics depicted in Fig. 10  tor Ω and its damping coefficient γ due to subsequent dissipation into the bath: ω c = Ω 2 /γ.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.