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Redfield Treatment of Multi-Pathway Electron Transfer in Artificial Photosynthetic Systems Daniel Douglas Powell, Michael R. Wasielewski, and Mark A. Ratner J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b02748 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on June 30, 2017
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Redfield Treatment of Multi-pathway Electron Transfer in Artificial Photosynthetic Systems Daniel D. Powell, Michael R. Wasielewski*, and Mark A. Ratner* Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center Northwestern University, Evanston, IL, 60208-3113 *Email:
[email protected];
[email protected] ABSTRACT Coherence effects on electron transfer in a series of symmetric and asymmetric two-, three-, four- and five-site molecular model systems for Photosystem I in cyanobacteria and green plants were studied. The total site energies of the electronic Hamiltonian were calculated using the DFT formalism and included the zero point vibrational energies of the electron donors and acceptors. Site energies and couplings were calculated using a polarizable continuum model to represent various solvent environments, and the site-to-site couplings were calculated using fragment charge difference methods at the DFT level of theory. The Redfield formalism was used to propagate the electron density from the donors to the acceptors, incorporating relaxation and dephasing effects to describe the electron transfer processes. Changing the relative energies of the donor, intermediate acceptor, and final acceptor molecules in these assemblies has profound effects on the electron transfer rates as well as on the amplitude of the quantum oscillations observed. Increasing the ratio of a particular energy gap to the electronic coupling for a given pair of states leads to weaker quantum oscillations between sites. Biasing the intermediate acceptor energies to slightly favor one pathway leads to a general decrease in electron transfer yield.
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INTRODUCTION The notion that electronic quantum coherences play a significant role in the charge transfer dynamics of biological systems is a relatively new idea, but extensive experimental work in the emerging field of quantum biology supports this concept, even when natural incoherent light is used as the excitation source.1-2 For example, Engel et al.3 demonstrated that long-lived excitonic coherences, lasting up to a picosecond, can occur at room temperature among the bacteriochlorophylls within the Fenna-Matthews-Olson (FMO) bacterial photosynthetic antenna complex. This finding spurred subsequent work demonstrating similarly long-lived excitonic coherences in several other bacterial antenna proteins,4-8 which are thought to be important for highly efficient energy transfer within these proteins.9-10 These long-lived coherences may ultimately prove useful for the creation of a biological quantum device that is sensitive to asymmetric perturbations. The membrane-bound Photosystem I (PSI) protein complex of cyanobacteria and green plants has a large array of chlorophylls (Chls),11 which serves as an antenna to increase the cross section for photon absorption and funnel the resulting excitons to the reaction center (RC) protein, wherein charge separation occurs.12,13 For example, PSI in Synechococcus elongatus contains 12 protein subunits and 96 Chls.14 The RC includes the PsaA and PsaB subunits, which along with their charge transfer cofactors, have pseudo-C2 symmetry between them.14 Following arrival of an exciton at the RC, the P700 Chl dimer primary electron donor transfers an electron to an adjacent monomeric Chl that is part of two nearly identical Chl → Chl → phylloquinone sequential electron transport pathways.15 Both pathways terminate in a common Fe4S4 cluster. More recently, the sequence of these events has been debated, largely because of spectral and redox congestion of the Chl cofactors.16-17 Electron transfer down either pathway can be selected
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using targeted chemical reduction procedures.18-19 The topological structure of the active electron transfer cofactors in PSI resembles a Mach-Zehnder interferometer in which electron propagation through the system may occur as an electronic superposition, and therefore may allow for quantum interference to take place between pathways.20-24 Developing molecular systems in which such pathway interference occurs would allow for the potential development of quantum devices, which would be very sensitive to small perturbations of the electronic structure or surrounding environment of a molecular interferometer.
D
The possibility of pathway interference in PSI is intriguing, but it is difficult to test in the PSI protein complex due to the large number of chromophores that make it difficult to isolate individual electronic processes. Our approach to understanding multi-pathway electron transfer involves developing a series of
e-
eA1’
A1 e-
eA2
dual pathway donor-acceptor systems that can function as molecular analogs of a Mach-Zehnder interferometer (Scheme 1).
Scheme 1. Design of a donoracceptor analog to a MachZehnder interferometer.
Before attempting to synthesize such systems, it is essential to develop appropriate design criteria by exploring the electronic structure and dynamics of dual electron transfer pathway systems for which A1 = A1’ and A1 ≠ A1’. In earlier work, we performed an analysis of the PSI RC cofactors utilizing a multi-site model, which has proven effective in describing electron transfer in such systems.22, 25 Here, we again use a multi-site tight binding model approximation with nearest-neighbor coupling to simulate the discrete charge carriers on which the electron resides. In addition, we use Bloch-Redfield relaxation theory to model the electron transfer dynamics in a series of D-A1/A1’-A2 molecules (Scheme 1), where the site energies are calculated by density functional theory (DFT) using the B3LYP functional
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and 6-31G* basis set. Redfield theory is used for these calculations because it has been proven to provide reliable results for open quantum system calculations and is readily derived from first principles of the system.26-29 The design of the small molecule D-A1/A1’-A2 systems studied here is based on our previous results with covalent donor-acceptor systems that demonstrate ultrafast electron transfer.30-33 We have modeled a series of increasingly complex D-A1/A1’-A2 molecules to capture the various characteristics that may exist in PSI. The donors used for these systems are perylene and tetracene, both of which have been shown to be effective electron donors in donoracceptor
systems.31,
34-38
The
well-characterized
and
versatile
electron
acceptors
naphthalenediimide (NDI) and pyromellitimide (PI) were employed because of their well-known electronic properties, ease of reduction, and potential for future device fabrication.30, 39-40 The intermediate acceptors (A1 and A1’) are naphthalene monoimide (NMI) derivatives, which were chosen because they provide a symmetric structure with convenient reduction potentials that can be adapted readily to our dual pathway interferometric design. m-Phenylene linkages are used to disrupt the electronic conjugation across the system π structure;41-43 thus, each site can be thought of as electronically unique, and a tight binding site model can be used to approximate the energy landscape.44 The energy levels described below were calculated using DFT with the energies of A1 and A1’ spanning the gap between D and A2. The phenyl group attached to the NMI nitrogen atoms in A1 and A1’ provides a site that can be readily modified to change the site energy and/or break the symmetry in the dual pathway systems without drastically changing the overall geometry of the system.
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RESULTS AND DISCUSSION Redfield Formalism Derivations and Energy Calculations. The total Hamiltonian of the system and the bath assuming a purely electronic basis is H = HS + HB + HI
(1)
where H S is the system Hamiltonian, H B represents the bath and HI is the interaction Hamiltonian between the system and the bath. After take the trace of the bath components to eliminate them from the standard Liouville von-Neumann equation and changing to the interaction picture, the full equation of motion of the system is represented by t
d ρ S ( t ) = − h −2 ∫ dτ TrB H I ( t ) , H I (τ ) , ρ (τ ) dt 0
(2)
In order to simplify the integral expression, the Born approximation is utilized. This implies that no entangled states exist between the system and the bath due to the assumed weak interaction between them, and allows the time dependency to be isolated to the system:
ρ ( t ) ≈ ρS ( t ) ⊗ ρ B .
Equation 2 still is non-Markovian and therefore difficult to solve
numerically due to the explicit time dependence of the bath. To address this, the Markov approximation is implemented by assuming a “memory-less” bath allowing ρS (τ ) to be replaced by ρS ( t ) which leads to the Redfield equation seen in eq 3: t
d ρ S ( t ) = − h −2 ∫ dτ TrB H I ( t ) , H I (τ ) , ρ S ( t ) ⊗ ρ B dt 0
(3)
This leads to a time local equation of motion, but not yet a true Markovian master equation due to the time dependence, t, in the integration limit. To remove this explicit dependence on initial time choice, we use a dynamical semigroup by substituting τ → t − τ and allow the upper boundary of integration to go infinity, removing this starting position dependence as seen in eq 4. 5 ACS Paragon Plus Environment
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This expression is the Markovian Redfield master equation starting point for the remainder of the work. ∞
d ρ S ( t ) = − h −2 ∫ dτ TrB H I ( t ) , H I ( t − τ ) , ρ S ( t ) ⊗ ρ B dt 0
(4)
From this starting expression, we then expand the expression by first assuming the interaction Hamiltonian between the system and the bath to be of the form
H I = ∑ Aα ⊗ Bα
(5)
α
in which the system operators are represented by Aα and the bath operators by Bα . The bath, ρ B , is assumed to be in a steady state and we have assumed the bath correlation functions to be of the form: ∞
Γαβ (τ ) ≡ ∫ dτ eiωτ TrB Bα ( t ) Bβ ( t − τ ) ρ B
(6)
0
This allows the full master equation to be written in eq 7 after operators and bath correlation functions are included. ∞
{
}
d ρ S ( t ) = − h −2 ∑ ∫ dτ Γαβ (τ ) Aα ( t ) Aβ ( t − τ ) ρ S ( t ) − Aα ( t − τ ) ρ S ( t ) Aβ ( t ) + h.c. dt αβ 0
(7)
Equation 7 is then converted into the matrix form of the Schrödinger picture from the interaction picture with the summation over the secular terms (sec) satisfying the relationship
ωab − ωcd
−1
