Quantum Interferences and Electron Transfer in Photosystem I - The

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Quantum Interferences and Electron Transfer in Photosystem I Nicolas Renaud, Daniel Douglas Powell, Mahdi Zarea, Bijan Movaghar, Michael R. Wasielewski, and Mark A. Ratner J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 08 Nov 2012 Downloaded from http://pubs.acs.org on November 11, 2012

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The Journal of Physical Chemistry

Quantum Interferences and Electron Transfer in Photosystem I Nicolas Renaud,∗ Daniel Powell, Mahdi Zarea, Bijan Movaghar, Michael R. Wasielewski, and Mark A. Ratner Northwestern University, Department of Chemistry 2145 Sheridan Road, Evanston, IL, 60208-3113 E-mail: [email protected]

∗ To

whom correspondence should be addressed

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Abstract We have studied the electron transfer occurring in the Photosystem I (PSI) reaction center from the special pair to the first iron-sulfur cluster. Electronic structure calculations performed at the DFT level were employed to determine the on-site energies of the fragments composing PSI, as well as the charge transfer integrals between neighboring pairs. This electronic Hamiltonian was then used to compute the charge transfer dynamics, using the Stochastic Surrogate Hamiltonian approach to account for the coherent propagation of the electronic density but also for its energy relaxation and decoherence. These simulations give reasonable transfer time ranging from sub-picoseconds to nanoseconds, and predict coherent oscillations for several picoseconds. Due to these long-lasting coherences, the propagation of the electronic density can be enhanced or inhibited by quantum interferences. The impact of random fluctuations and asymmetries on these interferences is then discussed. Random fluctuations lead to a classical transport where both constructive and destructive quantum interferences are suppressed. Finally it is shown that an energy difference of 0.15 eV between the on-site energies of the phylloquinones leads to a highly-efficient electron transfer even in presence of strong random fluctuations.

Keywords: charge separation, quantum beating, decoherence, biomolecule

Introduction Quantum effects in biological processes have been a prolific source of exciting new ideas. 1 Attention has been focused on the potential role that quantum coherence may play during exciton transfer, considerably modifying our understanding of this biological process. 2–4 Recent experiments 2,3 have demonstrated that quantum coherence may be preserved for up to a picosecond during exciton transfer across several large bio-molecules, raising questions concerning the utilization of quantum pathways in natural systems. Very different ideas have been proposed to understand the efficiency of energy transfer in these complexes, emphasizing the importance of quantum co-


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herence 5 or the role of the bath spectral density. 6 Despite their conceptual differences, all of these approaches agree on one point: the weak interactions between the core of the complex, generally composed of several interacting pigments, and its protein environment. Due to these weak interactions, the propagation of the initial excited state may remain coherent for a long time, allowing observation of quantum effects in biomolecules.

The excitonic transfer is the first step of the transformation of light energy in chemical energy but electron transfer (ET) plays also an important role in this energy transduction. Photosystem I (PSI) is one of the dedicated complex in charge of creating a radical pair from the initial exciton. The main part of the complex is composed of ∼ 100 chlorophylls that form the antenna, the locus for the absorption of the incident photons and excitonic propagation. 7 At the center of this antenna lies the electron transfer chain (ETC) where the charge separation occur. This ETC, represented in Fig. 1, contains 6 chlorophylls (Chl), 2 phylloquinones (PhQ) and three Fe4 S4 clusters arranged in two similar branches related by a pseudo-C2 symmetry axis. The Chl’s are separated into three groups: |P1 ⟩ and |P2 ⟩ form the special pair, |C3 ⟩ and |C4 ⟩ are usually labeled Chl-A, and |A5 ⟩ and |A6 ⟩ Chl-A0 . The two PhQ’s are labeled |Q7 ⟩ and |Q8 ⟩. Only one iron-sulfur cluster, labeled |Fx⟩, is considered in this article since electron hopping from cluster to cluster occurs on a much longer time scale than the one explored here. 8 Extensive work has been done for more than 50 years 9 to reveal the route the electron takes starting from the special pair to reach the cluster. 8 Many crucial points have been intensely debated: the propagation of the electron over one single branch or both of them, 10–14 the utilization of the PhQ’s and the Fx cluster as intermediary electron acceptors, 15–20 the location of the principal electron donor 21 and the importance of exciton propagation over the Chl’s. 22–24 The current consensus is that starting from the special pair, the electron propagates over both branches and reaches the Chl-A0 ’s after 0.5 to 20 ps (with a complex unexplored subpicosecond dynamics). The electron is then transferred to the PhQ’s after 20 to 200 ps 21 and finally to the Fx cluster after 20 to 200 ns. 25,26 Asymmetries between the propagation over the two branches are most often observed, leading to a biphasic rate, 21,27 probably due to slight differences



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in the environment of the phylloquinones. 28,29 However this biphasic rate is significantly reduced for given species or by fully reducing the clusters Fa and Fb . 20

Other important open questions remain, such as the capability of the PSI to shut down whenever subjected to intense illumination. 30 The aim of this article is not to shed new light on these fundamental questions, nor to make any claim on the in-vivo mechanisms used by these large biomolecules, but rather to investigate the possibility to observe quantum interferences in PSI due to its unique topology. Quantum interference is a well-known phenomenon encountered in many different fields from quantum optics 31,32 to molecular junctions. 33–35 When different pathways are available for a particle, for example an electron, to travel between two points of a quantum system, the components of its wave function propagating on the different pathways may cancel each other when reaching the final electron acceptor. The electronic density is completely reflected by the electron acceptor and the electron is unable to cross the system despite the presence of several pathways. This quantum mechanical behavior is impossible to describe using a classical kinetic rate approach 36 and only a quantum description of ET can simulate such counterintuitive effect. In PSI, the charge transfer from the principal donor to the iron-sulfur cluster occurs on the nanosecond timescale. Simulating the propagation of the electronic density for such a long evolution time using a quantum description is challenging since an appropriate description of energy relaxation and decoherence is required. Several theories are available to account for such irreversible processes during the quantum propagation. 37 The Redfield theory is usually used in physical chemistry but is only valid in the limit of slow relaxation processes and can lead to unphysical results such as negative populations 38 or to erroneous conclusions when applied to energy transfer. 39 The trace preserving Lindblad equation is a general type of Markovian master equation and important efforts have been made to determine the precise definition of its operators from microscopic arguments in biosystems. 38,40 Recently, several non-Markovian analytical and numerical approaches 41,42 have been proposed to solve efficiently the dynamics of an open quantum system. Due to its low numerical cost and long accessible evolution time, the Stochastic Surrogate Hamiltonian (SSH) is


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an attractive method to simulate electron transfer along the ETC of the PSI system, where both coherent and incoherent propagation play an important roles. Initially proposed in the framework of grid-based wave function propagation, 43,44 and recently adapted to tight-binding Hamiltonians to tackle larger systems, 6 the SSH treats explicitly a few well-resolved bath modes and introduces stochastic quantum jumps in the bath manifold to increase the accessible evolution time at a reasonable computational cost. Without these quantum jumps, thousands of bath modes would be required to simulate the evolution over a few nanoseconds and the computational cost would be excessive. 45,46

The article is organized as follows: the charge separation steps and charge transfer integrals governing ET in PSI are calculated in section II. The SSH approach and its application to the ETC of the PSI are presented in section III. Section IV presents the charge separation dynamics obtained using the SSH approach for different initial conditions of the electronic density. Quantum interferences are observed for specific initial states but require C2 symmetry to be clearly observed. Hence we will first focus on a symmetric model of the PSI ETC and then explore the impact of deviations of this perfect symmetry on the magnitude of the interference pattern.

Electronic propagation on the open system In the SSH framework the dynamics of the open quantum system is obtained via the solution of the full Liouville equation for the density matrix ρ : d ρ (t) 1 = [H , ρ (t)] dt i¯h

(1)

To model the electronic propagation along the ETC of the PSI, each fragment composing this large complex is represented by a single quantum state, labeled |sn ⟩, assuming that the electron propagates only on the lowest accessible electronic state of each site. To simulate decoherence and relaxation, this electronic Hamiltonian interacts with a bath composed of a set of Q harmonic 5 ACS Paragon Plus Environment


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oscillators, or quantum modes. The total Hamiltonian of the system and the bath reads: 6

H

N

= +

N

∑ En|sn⟩⟨sn| + ∑

N

∑ (tnm|sn⟩⟨sm| + tmn|sm⟩⟨sn|)

n=1 Q

n=1 m>n N N Q

q=1

n=1 m=1 q=1

q |sn ⟩⟨sm |(σ †q + σ q ) ∑ ωqσ †qσ q + ∑ ∑ ∑ Knm

(2)

where En is the on-site energy of the |sn ⟩ site of the ETC; tnm the charge transfer integral between sites |sn ⟩ and |sm ⟩; ωq is the frequency of the q-th bath mode with σ †q (σ q ) its creation (annihilaq

tion) operator; and Knm defines the interactions between the propagating electron and the vibration q

modes of the bath. The diagonal terms, Knn , represent intramolecular modes that acts on a sinq

gle site, |sn ⟩ of the system. At the contrary, the off-diagonal terms, Knm , represent intermolecular modes, sometimes referred as spatially correlated or non-local modes. These terms allow the electronic density to transfer from the site to site by emitting or absorbing a phonon in the bath. 47,48 Recent experiments have demonstrated the presence of such correlated modes that may enable highly efficient energy transfer in large biomolecules. 49 Due to the straightforward treatment of the system-bath interactions the SSH approach allows to account easily for both intramolecular and intermolecular modes.

However the SSH is a statistical approach to the dynamics of an open quantum system. To reduce the computational cost, only a few number of modes are explicitly incorporated in the Hamiltonian (2). The frequencies of these modes are randomly chosen, and quantum jumps are performed in the bath manifold during the propagation of the density matrix. Each of these quantum jumps reset the state of a given bath mode to its thermal state, and simulate the energy transfer from the explicit bath to a larger unresolved environment. 6 As described in the Supplementary Material, the random sampling of the bath modes frequencies is performed using a Gaussian distributed random generator whose parameters have been tuned to fit as well as possible the bath spectral density defined below in eq. 4. Since it involves a random sampling of the bath spectral 6 ACS Paragon Plus Environment

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density and stochastic jumps, the SSH approach requires to solve N times the Liouville equation. The evolution of the system is then obtained by averaging the total dynamics over these N realizations and by tracing out the bath modes. 5 bath modes were explicitly incorporprated in the Hamiltonian to achieve a compromise between computational cost and accuracy. 500 realizations were computed to obtain the final dynamics. In order to observe quantum effects during the electron transfer along PSI, the temperature of the bath was set to 77 K. These quantum effects would also be observable at room temperature but would not last as long as at cryogenic temperature. 6 A more detailed description of the SSH algorithm is presented in the supplementary information.

Charge separation steps A quantitative determination of the on-site energies, En , is required to simulate the ET accurately. Kinetic approaches have provided models for the on site energies for photosystem II. 50,51 These models lead to small energy differences among the Chl’s, and steeper downhill steps to reach the quinones. These values differ quite significantly from the traditional model of PSI where the site energies vary substantively over the Chl’s. 27,28 The electronic structure of the iron-sulfur cluster, and its interaction with the organic part of the ETC, are hard to obtain from first principle 52–56 but generally show a large energy difference between the acceptor site of these clusters and the electronic states of the organic fragments. 57 A method already used in the determination of the energy of the charge separation steps in PSII 51 is employed here. The electronic structure calculations underlying the determination of the on-site energies have been performed at the DFT level using the B3LYP functional and a double-ζ basis set in a solvent of dielectric constant ε = 4. This value of the dielectric constant is usually used for photosynthetic complexes as it is between that of water (ε = 78.8) and a protein (ε ≃ 3) environment. The crystal structure of the PSI complex determined by X-ray diffraction 7 has been used to perform these calculations, optimizing the structure of the pigments when required. For this geometrical optimization, the carotene tails of the fragments have been removed to facilitate relaxing the structure. The on-site energy of the two Chl’s of the special pair was set as the zero-point energy reference and the energy of the fragment X was computed 7 ACS Paragon Plus Environment


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following: 51

EX = IPP700 − Ehν − EAX + Ec (P700+ /X − )

(3)

In eq. (3), IPP700 the ionization potential (IP) of the special pair, Ehν is the energy of the incident photon,EAX the electron affinity (EA) of the fragment X and Ec (P700+ /X − ) the Coulombic attraction between the hole created on the special pair and the propagating electron located on the fragment X. The Coulombic potential is model here by a simple point charge approximation: Ec (P700+ /X − ) = Q1 Q2 /ε R12 where R12 is the center-to-center distance between the two fragments. The energy of the incident photon is known experimentally: Ehν = 1.77 eV but the other quantities have to be computed from first principles. To clearly observe the interferences, the two branches were supposed identical during these calculations. Deviations from this ideal case will be considered in section V.

The ionization potential of the special pair has been found to be IPP700 = 4.38 eV, i.e. 0.35 eV lower than the IP of the special pair present in Photo-System II. 51 The EA of the Chl-A0 has been found to be: EAChl-A0 = 2.46 eV which is in good agreement with previous calculations. 51 According to the crystal structure of PSI, the Chl-A0 are located 0.71 nm away from the special pair. This leads to an electrostatic interaction of -0.16 eV. Following eq. (3), the on-site energy of the Chl-A0 is: EChl-A0 = −11 meV. According to the crystal structure of the complex, the auxiliary Chl-A’s are coordinated with water molecules. As already reported elsewhere, 58 this coordination decreases their EA: EAChl-A = 2.25 eV. The Chl-A’s are in close proximity (0.71 nm) of the special pair leading to a large electrostatic term of -0.35 eV. The on-site energy of the Chl-A reads: EChl-A = +5 meV. Small energy differences are consequently obtained among the Chl’s with a positive on-site energy for the Chl-A. These results are rather different from the traditional model of PSI where the onsite energies of the Chl-A0 are supposed to be 0.25 eV lower that P700, 27 however they are in agreement with early measurements performed on PSI 59 and electronic structure calculations carried on PSI 28 and PSII. 51,60 The EA of the PhQ has been found to be EAPhQ = 2.97 8 ACS Paragon Plus Environment

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eV. Located at 2.31 nm away from the special pair, the electrostatic term of the PhQ is only of 0.1 eV. The on-site energy of the PhQ is then: EPhQ = −460 meV. It is experimentally known that the driving force of the PhQ’s is surprisingly large in PSI ranging from -0.6 to -0.8 eV. 27 Our approach leads to a smaller on-site energy and much more advanced electronic structure calculations would be required to obtain values in better agreement with the experiment. 28 Depending on the methods employed, the on-site energy of the Fx cluster can range from -0.585 eV to -0.700 eV. 27,28 According to these values the charge separation step going from the PhQ’s to the Fx can either be slightly downhill, isoergic or even uphill. 27,28 The uncertainties on this value are reinforced by the difficulty to compute appropriately the electronic structure of the iron-sulfur cluster. 57 Consequently the experimental value of -0.7 eV, generally well accepted in the community, is adopted in our model. The on-site energies of the different fragments are reported in Fig. 1.

Charge transfer integrals The coupling strengths between the different pigments composing the PSI cannot be directly measured experimentally and are rather hard to evaluate theoretically. 60–64 An estimation of these couplings has been obtained using the split orbital method. 29 Contrary to early models, 60,64 the split orbital method leads to weak couplings between the Chl’s, and even weaker interactions between the Chl’s and PhQ’s and between the PhQ’s and the iron sulfur cluster. 29 We have computed these charge transfer integrals using fragment approach implemented in the Amsterdam Density functional (ADF) code. 65 The charge transfer integrals are directly computed via the off diagonal matrix elements of the Fock operator and consequently accounts for the non-zero spatial overlap between the molecular orbitals of the fragments. Contrary to the split orbital methods this approach can be used to compute the coupling strength between molecular orbitals presenting different energies. In evaluating the charge transfer integrals directly from the molecular orbital overlaps, the sign of the coupling strength can be either positive or negative. These signs are extremely important when dealing with quantum interferences since they are responsible for the phase inversion 9 ACS Paragon Plus Environment


between the propagation in the two branches. These charge transfer integrals have been computed using a double-ζ basis set and the B3LYP functional with a solvent of dielectric constant ε = 4. A perfect C2 symmetry was assumed, so we obtain the final values of the charge transfer integral by averaging our results over the two branches. The values of the charge transfer integrals are reported in Fig. 1. A relatively strong coupling is obtained between the two Chl’s of the special pair and between the Chl-A and Chl-A0 of each branch. The interactions between two Chl’s of the special pair and the Chl-A show positive and negative signs. This difference in the coupling signs leads eventually to the interference pattern, as we will demonstrate in the section IV. However the coupling between the Chl-A0 ’s and the PhQ’s are one order of magnitude lower and the coupling between the PhQ’s and the final cluster is close to 0. Using different methods to determine the parameters of the Hamiltonian would lead to slightly different values of the on-site energies and coupling strength between pairs. These slight differences would affect the frequencies of the

energies (meV)

coherent oscillations and their amplitude but not the general charge separation mechanism.

coupling (meV)

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|P1 ⟩ = 0 |P2 ⟩ = 0 |C3 ⟩ = 5.1 |C4 ⟩ = 5.1 |A5 ⟩ = −11 |A6 ⟩ = −11 |Q7 ⟩ = −460 |Q8 ⟩ = −460 |Fx⟩ = −700 |P1 ⟩ ↔ |P2 ⟩ = 6.9 |P1 ⟩ ↔ |C3 ⟩ = −4.0 |P1 ⟩ ↔ |C4 ⟩ = 2.5 |C3 ⟩ ↔ |A5 ⟩ = 6.3 |A5 ⟩ ↔ |Q7 ⟩ = −0.1 |Q7 ⟩ ↔ |Fx⟩ = 10−2

|P2 ⟩ ↔ |C4 ⟩ = −4.0 |P2 ⟩ ↔ |C3 ⟩ = 2.5 |C4 ⟩ ↔ |A6 ⟩ = 6.3 |A6 ⟩ ↔ |Q8 ⟩ = −0.1 |Q8 ⟩ ↔ |Fx⟩ = 10−2

Figure 1: Graphical representation of the Photosystem I. Each fragment is here represented by a single quantum state and all the states interact with each other due to the overlap of their respective molecular orbitals.


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Bath spectral density and electron-phonon interactions In the SSH approach, only a few modes, whose frequencies are randomly chosen to span efficiently the bath spectral density, are explicitly treated. The dynamics of the system are then computed a few hundred times, each time for different values of the ωq , to completely cover the bath spectral density. Different bath spectral densities, J (ω ), have been used to simulate exciton or electron dynamics in photosynthetic systems like PSI 66,67 and an accurate determination of these quantities, as well as the interaction strength between the moving charges and their environment, is still a challenging problem. During its propagation along PSI, we suppose that the electronic density principally interacts with the vibration modes of the Chl’s 68,69 and the PhQ’s 70–72 whose energies are between ∼ 0.1 and 0.5 eV (see SI). Intermolecular vibrational modes also assist the electron in transferring between neighboring fragments.These modes generally present a low vibration energy and interact only weakly with the propagating electron. To model this complex bath spectral density, a super-ohmic density of states is used:

J (ω ) = λ (ω/ωc )2 e−(

ω/ωc )2

(4)

with: ωc = 0.15 eV and λ = 50 meV to fit the vibration modes of the Chl and the PhQ (see SI for details). Super-Ohmic spectral densities are widely used for quantum Brownian motion 73 and describe accurately a phonon bath. 74 In the SSH approach, the electron-phonon interaction of eq. q

(2), Knm , reads: 44 rnm q Knm = √ J (ωq ) Q

(5)

where rnm is a dimensionless parameter and J (ωq ) the spectral density of the bath, evaluated at the bath-mode frequency ωq . Q is the number of modes explicitly treated in the simulation and this q

renormalization term is introduced for the simulation to scale with the size of the explicit bath. Knm represents the probability for the electron to transit from the state |n⟩ to state |m⟩ by emitting or q

absorbing a phonon in the bath. Due to its dependency on J (ω ), weak values of Knm are obtained 11 ACS Paragon Plus Environment


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for low frequency modes, that mainly correspond to intermolecular and protein modes, 75,76 and stronger interaction strengths are obtained with the intramolecular modes that are centered around 0.15 eV. 77,78

Each rnm is defined via the overlap of two spherical Gaussian centered on the fragments n and m: rnm = e− 4 β 3

2d2 nm

, where dnm is the center to center distance between the two fragments and β the

spatial dimension of the Gaussian. This definition of the rnm matrix elements is similar to the exponentially decaying functions generally used to account for spatially correlated fluctuations. 79–81 Such definitions ensure that the correlated modes affect close packed fragments but are unable to transfer energy between infinitely separated sites. The width of the Gaussian is set to β = 1.7 nm−1 , a value for which direct relaxation between pigments separated by more than 2 nm is almost completely prohibited. Following this definition the diagonal terms, rnn , are all equal to unity and the the off diagonal terms vary from 0.42 to 0.04. All these values are reported in the SI.

Initial state and quantum interferences The initial excitation at 700 nm occurs principally on the special pair, making the latter the principal electron donor. The spacial distribution of the excited state over the two Chl’s of the special pair is still under investigation, some studies leading to a strong asymmetry in the initial state 82,83 where others suggest a more even balance of the initial state over the special pair. 84 We here investigate the impact of the initial superposition on the ET rate in the model presented above. Both Chl’s of the special pair, |P1 ⟩ and |P2 ⟩, can be excited independently or as a superposition. In a density matrix formalism a general expression for the initial state is :

ρS (0) = α |P1 ⟩⟨P1 | + (1 − α )|P2 ⟩⟨P2 | ( ) √ iϕ −iϕ + κ α (1 − α ) e |P1 ⟩⟨P2 | + e |P2 ⟩⟨P1 |


(6)

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where α controls the bias of the initial superposition over the two Chl’s: for α = 1 the electron is initially located only on |P1 ⟩, and is completely localized on |P2 ⟩ for α = 0. The parameter ϕ controls the phase between the two components of the superposition and κ the degree of coherence: ρS (0) is a pure state for κ = 1 and a mixed state for κ ̸= 1. Let’s first illustrate our approach with examples obtained for a coherent symmetric distribution of the initial excitation over the two Chl’s, i.e. κ = 1 and α = 1/2.

If ϕ = 0, α = 0.5 and κ = 1, the initial state can be written simply as: ρS (0) = |Ψ+ (0)⟩⟨Ψ+ (0)| with |Ψ+ (0)⟩ =

√1 (|P1 ⟩ + |P2 ⟩). 2

The propagation of the electronic density obtained from this ini-

tial superposition is represented in Fig. 2 with four snapshots of the electronic density at different times in the evolution. In these snapshots, only the electronic density is represented and the hole density localized on the special pair is not shown. Since the two branches are assumed rigorously equivalent, the population of |P1 ⟩ and |P2 ⟩ are exactly the same and only their sum is represented. For the same reason the sites: |C3 ⟩, |C4 ⟩ are perfectly equivalent as are the sites |A5 ⟩, |A6 ⟩ and |Q7 ⟩, |Q8 ⟩. Hence, only the sums of the populations of these equivalent sites are represented.

Large coherent oscillations between the special pair and the Chl-A’s, are observed for about 2 ps. Such coherent oscillations have been experimentally recorded during exciton transfer in light harvesting complexes 2,3 but have not yet been observed during electron transfer. These oscillations are due to the strong interactions and small energy differences between the pigments and are only weakly damped by the weak system-bath interactions. The initial state can propagate coherently among the Chl’s without a dramatic loss of energy, leading to the large beats represented in Fig. 2. The important coherent transfer between the special pair and the Chl-A’s is due to the near resonance between the anti-bonding initial state considered here, whose energy is +6.9 meV and the Chl-A’s whose energies are +5.1 meV, leading to Rabi-like oscillations. Energy relaxation and decoherence induced by the bath drive the electronic density on the Chl-A0 . The electronic population on these Chl’s increases slowly and reaches 0.21 after 5 ps. This transfer time agrees with



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experimental results that detect the presence of the electron on the Chl-A0 after few ps. 27

On a very different timescale, the energy relaxation slowly drives the electronic density to the PhQ’s and finally to the iron-sulfur cluster. Due to the very weak coupling between the Chl-A0 ’s and the PhQ’s, the coherent propagation of the electronic density alone would not be able to transfer the electronic density onto the PhQ’s. This charge transfer, as well as the transfer from the PhQ’s to the Fx, is principally mediated by the correlated modes i.e. the off-diagonal elements q

Knm . The time-scales for these two irreversible charge transfer steps are respectively ∼ 400 ps and ∼ 10 ns which roughly correspond to the experimental values. 27 We can therefore consider that in the framework of this perfectly symmetric model, our calculations give satisfying results, simulating the charge separation dynamics over few ns and accounting correctly for the subpicosecond dynamics. For this initial state, the quantum yield, i.e. the percentage of the electronic density finally localized on the iron-sulfur cluster is rather high at about 0.6.

Since the propagation of the electron remains coherent over a few picoseconds, quantum effects may arise during the charge separation dynamics. Let’s consider the case ϕ = π α = 0.5 and κ = 1 where the initial state can be written as: ρS (0) = |Ψ− (0)⟩⟨Ψ− (0)| with |Ψ− (0)⟩ = √12 (|P1 ⟩ − |P2 ⟩). The initial populations on the two Chl’s of the special pair remain unchanged compared to the previous situation but their relative phase has changed sign. Such a simple change could be thought of as negligible but leads in fact to significant modifications of the dynamics, as represented in Fig. 3. First, the sub-picosecond propagation is modified compared to the previous case. The energy of the initial state is now -6.9 meV and is in near resonance with the Chl-A0 ’s whose energies are -11 meV. This resonance is responsible for the large indirect coherent transfer from the special pair to the Chl-A0 ’s where the Chl-A’s are used as virtual states to transfer. The principal difference between the populations shown in Fig. 2 and Fig. 3 is the very weak quantum yield: quantum interference between the two branches of the PSI ETC prevent the electronic density from recombining on the cluster. To understand the origin of these interferences, one can compute the coupling 14 ACS Paragon Plus Environment

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Figure 2: a) Evolution of the electronic density on each fragment obtained following the SSH approach (details in the text). The initial state of the evolution is here: |Ψ+ (0)⟩ = √12 (|P1 ⟩ + |P2 ⟩). After the fast coherent oscillations on the Chl’s (few ps), energy relaxation drives the electron density toward the PhQ (200 ps) and finally to the iron-sulfur cluster (10 ns). b) Snapshots of the electronic density at different time illustrating the different charge transfer steps across the ETC.



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strengths between the initial state and the two branches by block diagonalizing the Hamiltonian over |P1 ⟩ and |P2 ⟩ (see SI for more details). The initial state interacts with a positive coupling of +4.2 meV with |C3 ⟩ and with a negative coupling of -4.2 meV with |C4 ⟩. Due to their positive and negative coupling strengths, the electronic density propagates across the two branches with a π phase shift difference. Due to their phase difference, the two components of the electronic density on the two branches exactly cancel when recombining on the iron-sulfur cluster whose population therefore remains null. In the previous case, where no interferences were observed, the two components of the electronic density evolved in phase allowing for constructive interference on the iron-sulfur cluster. However the possibility to observe such completely destructive quantum interferences relies on two conditions: a nearly perfect symmetry in the system and a non classical charge transfer mechanism. The first condition is satisfied, as the two branches are identical in our model: the coherent propagation and the energy relaxation are strictly equivalent on both branches. In other words, two equivalent sites in the system are populated and decohere simultaneously. However, the quantum transport eventually ends when all the coherences of the system vanish. This is what is observed in Fig. 3 where after 1 ns where the electronic density on the iron-sulfur cluster rises slowly. This marks the transition between the quantum the classical transport regime where quantum effects, such as destructive interference, are absent.

The magnitude of the quantum interferences presented above can be monitored by varying continuously the population bias, α , of the initial preparation and the degree of coherence, κ , of its two components. The variations of the quantum yield with α and κ are reported in Fig. 4a for both cases ϕ = π and ϕ = 0. In the case ϕ = π quantum interferences are obtained for α = 0.5 and

κ = 1 and the quantum yield remains almost null. Keeping α = 0.5 and decreasing κ to 0 leads to a quantum yield of 0.3: quantum interferences disappear for a mixed initial state. Similarly, introducing a bias in the initial state superposition, i.e. increasing or decreasing α , leads to a quantum yield of 0.3 by breaking the C2 symmetry. However, the quantum yield does not reach 0.6 16 ACS Paragon Plus Environment

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Figure 3: Evolution of the electronic density in the fragment under the initial condition: |Ψ− (0)⟩ = √1 (|P1 ⟩ − |P2 ⟩) which leads to quantum interferences. Due to these interferences, the electronic 2 density accumulates on the PhQ’s. Only when all coherence is lost, i.e. after ∼ 10 ns, the quantum interferences cease to exist and the electronic density start recombining on the Fx. b) Snapshots of the electronic density at different time illustrating the different charge transfer steps across the ETC.



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as was the case in Fig. 2 where the initial state was |Ψ+ (0)⟩. This partial transfer of the electronic density comes from the fact that the state |P1 ⟩ can be expressed as: |P1 ⟩ = √12 (|Ψ+ (0)⟩ + |Ψ− (0)⟩). Hence, when the initial electronic density is located on |P1 ⟩ half of the electronic density interferes destructively and half is transfered on the cluster. In the case ϕ = 0 constructive interference are obtained and a maximum quantum yield of 0.6 is recovered for α = 0.5 and κ = 1. Decreasing

κ lowers the transfer amplitude to 0.3 for κ = 0. The large quantum yield observed in Fig. 1a is hence due to coherent constructive interferences. Snapshots of the electronic density obtained when |Ψ(0)⟩ = |P1 ⟩ are represented in Fig. 4b. Even when starting exclusively on the left branch, the electronic density can delocalize over the two sides of the PSI ETC due to the strong coupling in the special pair. Note that the electronic density remaining on the PhQ’s is larger here than in Fig. 2b due to the mix between constructive and destructive interference.

a)

b)

Figure 4: a) Map of the quantum yield depending on the initial superposition bias (α ) and degree of coherence (κ ) for ϕ = π (up) and ϕ = 0 (down). b) Electronic density snapshots obtained for α = 1 and κ = 0.


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Noise and asymmetry To clearly observe the destructive quantum interferences responsible for the enhancement or suppression of the ET to the iron-sulfur cluster, the ETC of the PSI has been assumed to be perfectly C2 symmetric. To study the impact of asymmetries on the quantum interferences, two phenomena are considered: small random Gaussian fluctuations on the on-site energies and coupling strengths; and a large detuning on the on-site energies of the PhQ’s.

The random fluctuations are induced by the geometrical deformations of the complex that modify the coupling between the fragments and their on-site energies. In our model, the fluctuations in a given coupling strength, tnm , are proportional to its unperturbed value reported in Fig. 1 and follow a Gaussian distribution of width: η × tnm . This ensures that large fluctuations on weak coupling are unlikely to happen since they would require large deformations of the complex. On the contrary, the fluctuations on the on-site energies are independent of their original values and follow a Gaussian distribution of width: η × E0 with E0 = 0.1 eV. The introduction of a detuning in the on-site energy of the PhQ is motivated by recent experimental studies that tend to show that the on-site energy of one the PhQ could lie as much as few hundred meV below the other, probably due to slight asymmetric modifications on the two PhQ’s local environments. 16,29,85 This detuning, noted ∆E, is introduced in our simulation by decreasing the energy of one of the PhQ from -460 meV to -860 meV. The equivalent cases, where either the energy of |Q7 ⟩ or |Q8 ⟩ is modified, are considered and correspond respectively to a negative or a positive ∆E.

Fig. 5a shows the variation of the quantum yield with η and ∆E, and where the initial state was |Ψ(0)− ⟩. The raw data obtained from our simulations is represented as a transparent surface and has been smoothed by a simple function represented as a wireframe. The plane located at 0.6



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marks the highest value of the quantum yield obtained for constructive interference as shown in Fig. 2. As explained above, destructive interference leads to an almost null quantum yield when the initial state is |Ψ(0)− ⟩. Therefore, the quantum yield is almost null at the center of the map in Fig. 5a where η = ∆E = 0. Increasing the random fluctuations breaks the C2 symmetry and leads to a small population on the iron sulfur cluster. For η = ± 10% the quantum yield of 0.3 which correspond to the value of the quantum yield obtain when quantum effects do not enhance nor prohibit the ET.

If the random fluctuations have a small effect on the quantum yield, the effect of the detuning is much more important. Let’s consider the case ∆E > 0, where the on-site energy of |Q8 ⟩ decreases. Upon such a detuning, the quantum yield reaches a maximum value of 0.6 for ∆E = 0.15 eV. In this case, the effect of the destructive interferences completely disappear and the quantum yield reaches a value comparable to the one obtained with the initial condition |Ψ(0)+ ⟩. This high value of the quantum yield is maintained even in the presence of strong random fluctuations and remains around 0.6 even for η = ± 10% as long as ∆E = 0.15. For this value of the detuning, the maximum population on |Q8 ⟩ only reaches 0.1 whereas the population of the |Q7 ⟩ reaches 0.4. This weak accumulation of the electronic density on the perturbed PhQ is simply due to a decrease of the injection rate from |A6 ⟩ to |Q8 ⟩ and an increase of the transfer rate from |Q8 ⟩ to the iron-sulfur cluster. This two effects combined lead naturally to a weaker maximum population on |Q8 ⟩. This results confirms that the bi-phasic arrival rate observed on Fx 8 may be due to a small energetic modification on one of the PhQ’s. 17,85 However decreasing further the on-site energy of |Q8 ⟩ is thermodynamically unfavorable: the quantum yield reaches 0.1 for ∆E = 0.4 eV. For such a large value of the detuning a significant percentage of the electronic density is trapped on |Q8 ⟩ and never reaches the iron sulfur cluster. A similar behavior has been observed for different initial states with a significant increase of the quantum yield for a detuning of 0.15 eV (see SI): decreasing the on-site energy of one of the PhQ is a way to obtain highly efficient electron transfer along the ETC PSI from any initial condition. Hence, this detuning may be very important in natural condition


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where the initial excitation of the special pair is poorly controlled.

a)

b)

Figure 5: a) Variations of the quantum yield with the magnitude of the random fluctuations η and the detuning ∆E. b) Snapshots of the electronic density in the case where the on-site energy of the PhQ is 0.15 eV lower on the right branch than on the left.

Conclusion We have computed the electron transfer dynamics of a PSI reaction center model, using the SSH approach that accounts both for the coherent propagation and energy relaxation and decoherence due to the interactions between the system and the bath. This scheme allows to simulate both the complex subpicosecond coherent dynamics and the incoherent propagation that dominates the charge transfer at the nanosecond timescale. Satisfying transfer times have been obtained: around 5 ps to reach the Chl-A0 ’s, then 200 ps to reach the PhQ’s and 10 ns to finally arrive at the ironsulfur cluster. These calculations rely on the determination of the electronic structure of the system 21 ACS Paragon Plus Environment


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that has been calculated at the DFT level using the B3LYP functional and double-ζ basis set in a solvent with dielectric constant of 4. These calculations lead to small on-site energies for the Chl’s and steeper steps to PhQ’s and to the FeX cluster. The charge transfer integrals between neighboring fragments have also been calculated at the DFT level using the same functional and a double-ζ basis set. Interactions of ∼ ± 7 meV have been found between the two Chl’s composing the special pair and also between the neighboring Chl-A and Chl-A0 . The impact of the initial state preparation on the charge transfer dynamics have been investigated. Starting from the initial state |Ψ− (0)⟩ =

√1 (|P1 ⟩ − |P2 ⟩) 2

leads to quantum destructive interferences that prevent the elec-

tronic density to recombine on the iron-sulfur cluster. The population of the cluster remains then completely null for about a nanosecond. For longer evolution time quantum coherence essentially vanishes, and the quantum transfer is replaced by an incoherent classical transport mechanism. However the destructive interference disappears for any significantly biased or incoherent initial state. At the contrary constructive interference can be obtained when the initial electronic density reads: |Ψ+ (0)⟩ =

√1 (|P1 ⟩ + |P2 ⟩) 2

leading to a quantum yield of 0.6. The efficiency of the elec-

tronic transport decreases for biased or incoherent initial state. The impact that random noise and static energy detuning on the PhQ’s have on the ET quantum yield have been explored. The random fluctuations break the C2 symmetry and perturbs significantly the ET. Hence, even for weak fluctuations the destructive and constructive interferences disappear leading to a quantum yield of 0.3 indifferently of the of the initial state. However, a detuning of 0.15 eV on the on-site energies of the PhQ’s lead to a strong biphasic regime and gives a large quantum yield, even in the presence of strong random fluctuations. The difference in the on-site energies of the PhQ’s, observed in most of the PSI, could be a way to obtain highly-efficient electron transfer indifferently of the initial state of the electronic propagation.


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Acknowledgement This work was supported by: the Non-equilibrium Energy Research Center (NERC) an Energy Frontier Research Center funded by the U.S. Department of Energy, the Office of Science and Office of Basic Energy Sciences under Award Number DE-SC0000989; DARPA under Award Number N66001-10-1-4066 for the QuBE project; NSF grant Number: CHE-1112258 (MRW). NR built the model did the calculations and drafted the paper, MW and MR introduced the topic and suggested the methods. DP, MZ and BM made major suggestions and interpretations.

Supporting Information Available Implementation of the Stochastic Surrogate Hamiltonian approach. Details on the computation of the electronic structure of the fragments composing PSI, the charge transfer integrals between pairs and the relaxation matrix. Parametrization of the bath spectral density. Block diagonalization of the model Hamiltonian. Destructive interference in a 4-level system. Effect of the energy detunning between the PhQ’s. This material is available free of charge via the Internet at http: //pubs.acs.org/.



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Figure 6: TOC


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Quantum Interferences and Electron Transfer in Photosystem I - The

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