Redfield Treatment of Multipathway Electron Transfer in Artificial

Article pubs.acs.org/JPCB

Redfield Treatment of Multipathway Electron Transfer in Artificial Photosynthetic Systems Daniel D. Powell, Michael R. Wasielewski,* and Mark A. Ratner*

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Department of Chemistry and Argonne−Northwestern Solar Energy Research (ANSER) Center Northwestern University, Evanston, Illinois 60208-3113, United States 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 density functional theory (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 (Chl’s),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 Chl’s.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 © 2017 American Chemical Society

debated, largely because of spectral and redox congestion of the Chl cofactors.16,17 Electron transfer down either pathway can be selected 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. 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 multipathway electron transfer involves developing a series of dual pathway donor−acceptor systems that can function as molecular analogues of a Mach−Zehnder interferometer (Scheme 1). 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 multisite model, which has proven effective in describing electron transfer in such systems.22,25 Here, we again use a multisite tight binding model approximation with nearestReceived: March 22, 2017 Revised: June 28, 2017 Published: June 29, 2017 7190

DOI: 10.1021/acs.jpcb.7b02748 J. Phys. Chem. B 2017, 121, 7190−7203

Article

The Journal of Physical Chemistry B

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) ⊗ ρB. 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:

Scheme 1. Design of a Donor−Acceptor Analogue to a Mach−Zehnder Interferometer

d ρ (t ) = −ℏ−2 dt S

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 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 donor−acceptor 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.

d ρ (t ) = −ℏ−2 dt S

∫0

∞

dτ

TrB[HI(t ), [HI(t − τ ), ρS (t ) ⊗ ρB ]]

(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 HI =

∑ Aα ⊗ Bα (5)

α

in which the system operators are represented by Aα and the bath operators by Bα. The bath, ρB, is assumed to be in a steady state and we have assumed the bath correlation functions to be of the form: Γαβ(τ ) ≡

∫0

∞

dτ eiωτ TrB[Bα (t ) Bβ (t − τ )ρB ]

(6)

This allows the full master equation to be written in eq 7 after operators and bath correlation functions are included. d ρ (t ) = −ℏ−2 ∑ dt S αβ

∫0

∞

dτ {Γαβ(τ )[Aα (t ) Aβ (t − τ )

ρS (t ) − Aα (t − τ ) ρS (t ) Aβ (t )] + h.c}

(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 ≪ τdecay. sec

d ρ (t ) = −iωabρab (t ) − ℏ−2 ∑ ∑ dt ab α ,β c ,d

where HS is the system Hamiltonian, HB represents the bath, and HI is the interaction Hamiltonian between the system and the bath. After taking 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

∫0

∞

dτ

α β iωcnτ {Γαβ(τ )[δbd ∑ Aan A nc e − Aacα Adbβ eiωcnτ ] + h.c}ρcd (t ) n

(8)

It is convenient to decompose the bath correlation function in eq 8 into a noise-power spectrum of the environment. The one-sided Fourier transform can be separated into two components,

t

dτ TrB[HI(t ), [HI(τ ), ρ(τ )]]

dτ TrB[HI(t ), [HI(τ ), ρS (t ) ⊗ ρB ]]

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. This expression is the Markovian Redfield master equation starting point for the remainder of the work.

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)

∫0

t

(3)

■

d ρ (t ) = −ℏ−2 dt S

∫0

(2) 7191


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Figure 1. Site populations as a function of time for (A) a perylene donor with a Ph-NMI acceptor, (B) a tetracene donor with a Ph-NMI acceptor, and their respective Hamiltonians (C) and (D). Energies are given in electronvolts. The solvent is CH2Cl2.

Γαβ(ω) =

1 Sαβ(ω) + iλαβ (ω) 2

point vibrational energy (ZPVE), as calculated by DFT. The ZPVE is the correction applied to the DFT calculations accounting for the intrinsic vibrational energy of the calculated molecules at 0 K. Similarly, EAX is the electronic affinity of the acceptor and was obtained by subtracting the total energy of the anion from that of the neutral state with the ZPVE contribution included. Eℏν is the lowest singlet excited state energy of the donor molecule calculated by TDDFT, and EC represents the Coulombic energy of the charge separated species on the donor and acceptor. EC is proportional to 1/εr with ε being the solvent dielectric constant and r the ion pair distance obtained from the DFT-minimized structures. The total energy EX is given as a negative value that represents the energy difference between the donor and the local site of interest in the system. The electronic coupling between the subunits was obtained using the fragment charge difference (FCD) formalism through DFT calculations.46 Different phonon couplings for the donor D (excited state) and acceptor A (charge-transfer state) result in different displacements of their potentials along nuclear coordinates. Exciton mixing occurs only near the crossing point of D and A potentials, whereas relaxation to the bottom of the A potentials produces dynamic localization. This feature is not included in Redfield theory, where the mixing of the D and A states is assumed to independent of nuclear coordinates. This frequently overestimates both the D-to-A transfer rate and amplitude of the coherences. More accurate results can be obtained using Redfield theory in an exciton−vibrational basis, or exact methods, such as exact hierarchical equation of motion (HEOM).47 The molecular systems that were studied use perylene or tetracene donors and NDI or PI acceptors, while the NMI intermediate acceptor molecules are kept the same in all cases with variations made to the phenyl attached to their nitrogen atoms to slightly modify the site energies and/or intentionally break the symmetry in the dual pathway systems. Numerical calculations were performed using dynamic open quantum system code developed for the Quantum Toolbox in Python (QuTiP).48,49 This work is intended to lay down a theoretical framework to guide synthetic design of molecules for the purpose of demonstrating simultaneous multipathway electron transfer.

(9)

with iλαβ(ω) being a Lamb energy shift that is small for the given system and can be ignored. The spectral density S(ω) is taken to be a smooth (ohmic) function of ω, which is taken to be 0.2 eV (∼1600 cm−1) and is typical of C−C stretching frequencies in aromatic donors and acceptors. The spectral density is assumed to be the same for the donors and acceptors because both species are benzenoid derivatives. This approximation leads to a form of the bath correlation which can be readily computed numerically and provides information on the direct nature of the system−bath interactions. Having made these assumptions, the following working form of the Bloch− Redfield equation emerges with “sec” above the summation representing the secular terms satisfying |ωab − ωcd|−1 ≪ τdecay: d ρ (t ) = −iωabρab (t ) + dt ab

sec

∑ Rabcdρcd (t ) c ,d

(10)

The Redfield tensor Rabcd is used to solve for the electronic densities of the system and is represented by Rabcd = −

ℏ−2 2

+ δac ∑

∑ {δbd ∑ Aanα A ncα Sα(ωcn) − Aacα Adbα Sα(ωca) α

n

α α Adn A nbSα(ωdn)

α − Aacα Adb Sα(ωdb)}

n

(11)

To calculate the system Hamiltonian for the various model complexes being studied, a tight-binding, nearest-neighbor coupling model was used. The initial energy of the photoexcited electron on the donor was calculated by timedependent DFT (TDDFT) using the B3LYP functional and double-ζ basis set. The initial energy of the excited state of the donor was set to zero for this work with the subsequent site energies being energetically downhill relative to this starting point and calculated as follows:45 E X = IPD − Eℏν − EAX + EC(D+ /X−)

(12)

In eq 12 IPD is the ionization potential of the system donor, which was calculated by subtracting the total energy of the neutral molecule from that of the cation, including the zero 7192


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Figure 2. Site populations as a function of time for the symmetric three-site systems shown above. (A) Population at the perylene donor, (B) population at the acceptors, and Hamiltonians for (C) A1 = A1′ = Ph-NMI, (D) A1 = A1′ = Me3Ph-NMI, and (E) A1 = A1′ = (CO2H)3Ph-NMI. All energies are given in electronvolts. The solvent is CH2Cl2.

Figure 3. Site populations as a function of time for the symmetric three-site systems shown above. (A) Population at the tetracene donor, (B) population at the acceptors, and Hamiltonians for (C) A1 = A1′ = Ph-NMI, (D) A1 = A1′ = Me3Ph-NMI, and (E) A1 = A1′ = (CO2H)3Ph-NMI. All energies are given in electronvolts. The solvent is CH2Cl2.

Two-Site Calculations on Single Donor−Acceptors. The results from a pair of two-site calculations are displayed in Figure 1, demonstrating electron transfer from the photoexcited donor state to Ph-NMI. These calculations involve a simple

donor−acceptor transfer process and hence there is no chance of interferometric behavior because only one pathway is available. The oscillations present in the plots arise from the coherent nature of the electron transfer, as there is sufficiently 7193


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Figure 4. Site populations as a function of time for three-site electron transfer with a perylene donor in CH2Cl2 with asymmetrically substituted acceptors. (A) Me3Ph on A1′, (B) (CO2H)3Ph on A1′, (C) comparison of all substituted A1′ acceptors, and (D) Hamiltonians for the indicated systems. All energies are given in electronvolts.

strong donor−acceptor electronic coupling relative to the energy gap between them. The coupling between the tetracene and Ph-NMI is much smaller and is likely due to poor overlap resulting from the near-perpendicular geometry of the tetracene relative to the phenyl linker resulting from steric hindrance.31 The larger coupling/energy gap ratio for the perylene system leads to both a faster rate of electron transfer and higher frequency coherent oscillations. Three-Site Calculations on Symmetric Dual Pathway Systems. With a view toward building a system with two pathways, we performed electron transfer calculations for a series of symmetric branched acceptors. The molecules have either a perylene or tetracene donor connected to two NMI groups via a m-phenylene linker. In this series of calculations, the geometry of the system is such that the electron from the photoexcited donor has the chance to propagate down both branches equally and will lead to electron population on both branches of the system. The site energies of both acceptors are the same for these calculations, so the electron populates each acceptor equally. It should also be noted that there is an intrinsic asymmetry present based upon the fact that the donor is not symmetrically attached to the m-phenylene linker. DFT calculations indicate that there is no appreciable difference in electronic coupling between the donor and each of the acceptors, so the value is taken to be the same. The calculations are performed in simulated CH2Cl2 because it has a moderate dielectric constant, and would readily solubilize these systems. Methyl and carboxylic acid substituents are attached to the NMI phenyl group to vary the acceptor lowest unoccupied molecular orbital (LUMO) energy, and thus the energy gap for the electron transfer reaction. Later, we will describe how these substituents are used to intentionally break the electronic symmetry of the two electron transfer pathways. Molecular orbital calculations indicate that there is minimum throughspace interaction directly between the donor and the NMI

acceptor, so the methyl and carboxylic acid groups attached to the NMI are not expected to change the energy of the donor. Figure 2 demonstrates the electron propagation in the branched system with a perylene donor and the series of three NMI acceptors. Within the series of substituted phenyls, the oscillation amplitude is slightly larger for (CO2H)3Ph-NMI than for Me3Ph-NMI or Ph-NMI. The electron transfer rate is being driven mostly by the large energy gap between the excited perylene and the acceptors. The small changes in acceptor energy due to the substituents on the Ph-NMI do not have a large effect on this rate. This is in contrast to tetracene as the donor (Figure 3). In this case, there are both longer-lived coherent oscillations and a higher sensitivity to the LUMO energies of the acceptor. The oscillations are preserved up to 100 fs for the tetracene systems, whereas they are effectively damped out by 40 fs in the perylene case. The smaller energy gap between tetracene and the acceptors results in the longer duration oscillations. Moreover, the electron transfer rate from the tetracene to the acceptors slows (Figure 3B) relative to that of perylene because the coupling between the tetracene and NMI acceptors is roughly half of that in the perylene series. Three-Site Calculations on Asymmetric Dual Pathway Systems. Using different substituents on the phenyl group attached to one of the NMI electron acceptors breaks the electronic symmetry between them. In these molecules, the site energies of both acceptors are no longer equal and the site populations between them are no longer in phase. Figure 4 illustrates the results for the three-site system using the perylene donor. Interesting dynamics begin to emerge in Figure 4A, where even the decrease in energy of acceptor A1′ by 0.05 eV begins to show a profound change in the site population between the A1 and A1′ intermediate acceptors. As expected, the decrease in energy of A1′ does increase the driving force and more population resides on A1′ than when the acceptors are both Ph-NMI. However, there appears to be a large periodic 7194


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Figure 5. Site populations as a function of time for three-site electron transfer with a tetracene donor in CH2Cl2 with asymmetrically substituted acceptors. (A) Me3Ph on A1′, (B) (CO2H)3Ph on A1′, (C) comparison of all substituted A1′ acceptors, and (D) Hamiltonians for the indicated systems. All energies are given in electronvolts.

fluctuation where the two populations begin to even out around 30 fs. As expected, further decreasing the site energy of A1′ biases the transfer to that site (Figure 4B). The large fluctuation in density also appears here although at a slightly earlier time than is the case for Figure 4A. Figure 5 shows the electron transfer dynamics in the threesite asymmetric system with tetracene as the donor. This change in donor decreases the energy gap between the donor and both acceptors, as well as the donor−acceptor electronic coupling. Electronic structure calculations indicate that the tetracene donor has poorer orbital overlap with the acceptors, and this is why the coupling is significantly lower. Similar to the data for perylene, the tetracene systems indicate that a periodic shift in the site populations occurs. This is especially pronounced in Figure 5B, where at 30−40 fs there is a large difference in population between A1 and A1′, which diminishes at later times. This oscillatory pattern is conserved at much longer time scales, and is likely due to the nature of the Redfield approach and how it handles relaxed system energies. Figure 5C also provides insight into the nature of the electron density residing on A1′, which for the case of (CO2H)3Ph-NMI gains more population with respect to the sites that have higher energy LUMOs. The periodicity of the quantum coherences observed is longer, as expected from the larger energy gap. Four-Site Calculations on Symmetric Dual Pathway Systems. Time dependent electron transfer dynamics in a series of four-site symmetric molecular systems were calculated next (Figure 6). The geometry of these systems now uses electron acceptors A1 and A1′ shown in Figures 4 and 5 as intermediary electron acceptors, which are both covalently bound to a terminal acceptor A2. Thus, this structure closely resembles the structure of the PSI electron transfer center, where the electron may move from the photoexcited donor down two (nearly) equivalent pathways through two intermediate acceptors to reach a common terminal acceptor.

This type of system also resembles a Mach−Zehnder interferometer, where interference patterns may occur depending on the sign of the coupling between the sites of the system. Our calculations determined that the signs of the coupling down both pathways are the same, indicating that there is no deconstructive interference. The terminal acceptors A2 are either NDI or PI; once again all of the molecules are assumed to be dissolved in CH2Cl2. Figure 6 shows the results for electron transfer from the perylene donor to the NDI terminal acceptor using three substituted NMIs as intermediate electron acceptors, where A1 = A1′. The electronic behavior of the donor is similar between the three-site and four-site perylene systems for the Ph-NMI and Me3Ph-NMI acceptors. The intensity of the oscillations is slightly lower in the four-site system and the coherent oscillations damp out more quickly, but neither shows major changes. The (CO2H)3Ph-NMI intermediate acceptor does lead to some interesting behavior for this system. The coherent oscillations are preserved for a longer time with this system, in contrast to those of Ph-NMI or Me3Ph-NMI. With respect to the electron density on the NDI acceptor, the (CO2H)3Ph-NMI intermediate acceptor leads to more density on the final state as well as a faster decay of the population on NMI. This is likely due to the lower energy of the (CO2H)3Ph-NMI acceptor, which leads to stronger relative coupling between this intermediate acceptor and the terminal NDI acceptor. Changing the terminal acceptor to PI effectively raises the energy of the final charge separated state to be closer to that of the intermediate charge separated state (Figure 7). Decreasing the energy gap between the LUMO of the reduced intermediate acceptors and that of the terminal acceptor by 0.25 eV produces distinctive coherent oscillations present for longer times between these states. This results in faster electron transfer to the PI terminal acceptor because now the energy gaps between the various charge separated states are smaller, while the coupling between them remains essentially the same. 7195


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Figure 6. Site populations as a function of time for the symmetric foursite systems shown above. (A) Electron density at the three intermediate acceptors and (B) electron density on the NDI acceptor. The energy of the NDI acceptor is −1.27 eV for each case. All energies are given in electronvolts. The solvent is CH2Cl2.

Figure 7. Site populations as a function of time for the symmetric foursite systems shown above. (A) Electron density on the three intermediate acceptors and (B) electron density on the PI acceptor. The energy of each PI acceptor is −1.01 eV. All energies are given in electronvolts. The solvent is CH2Cl2.

There are weaker oscillations between the donor and the intermediate acceptors. This is a consequence in part of the coupling/energy gap ratio between the intermediate and terminal electron acceptors being large enough that the electronic density can make its way to the acceptor in larger quantity, compared to the lower energy NDI acceptor. For these molecules there does not appear to be a large dependence on the intermediate electron acceptor substituents and the system dynamics seem to be primarily driven by the energy gap between the intermediate and final acceptors. Results for the corresponding systems using the tetracene donor are shown in Figure 8. In this case the donor energy is lower, which affects the overall dynamics of the system in a manner similar to changing the terminal acceptor energies. Much stronger coherent oscillations between the donor and the intermediate acceptors are observed compared to the perylene donor systems. Also there appears to be a slower, lower yield electron transfer from the intermediate acceptors to the

terminal acceptor. The intensity of the oscillations decreases compared to the perylene cases, and the electron density changes take longer to plateau. It appears that the highest yield electron transfer occurs for the Ph-NMI intermediate acceptors. The Ph-NMI system demonstrates the fastest rise in density on the NDI acceptor, and has the strongest and longest-lasting oscillations in the electron density on the donor and the intermediate electron acceptors. The final symmetric four-site case uses tetracene as the donor and PI as the terminal acceptor (Figure 9). This molecule has the smallest energy gaps between the various states. As a result, the coherent oscillations between the tetracene donor and the intermediate acceptors are quite strong. These oscillations are the strongest for both the PhNMI and Me3Ph-NMI acceptors. The interaction between the 7196


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Figure 8. Site populations as a function of time for the symmetric foursite systems shown above. (A) Electron density of the three intermediate acceptors and (B) electron density at the NDI acceptor. The site energy of the NDI acceptor is −1.10 eV. All energies are given in electronvolts. The solvent is CH2Cl2.

Figure 9. Site populations as a function of time for the symmetric foursite systems shown above. (A) Electron density on the three intermediate acceptors and (B) electron density on the PI acceptor. Site energy of the PI acceptor is −0.88 eV All energies are given in electronvolts. The solvent is CH2Cl2.

tetracene donor and the (CO2H)3Ph-NMI acceptor is significantly weaker. Again, as the (CO2H)3Ph-NMI acceptor is the lowest in energy it leads to weaker quantum oscillations, and hence to faster electron transfer to the terminal PI acceptor. For all these systems, there seems to be a correlation between the amplitude of the quantum oscillations and size of the coupling/energy gap ratio between the donor−acceptor pairs. Thus, in order to observe quantum interference in a synthetic molecule, this ratio should be maximized, while ensuring that there is sufficient driving force to allow electron transfer to occur. Four-Site Calculations on Asymmetric Dual Pathway Systems. The same set of asymmetric substituent effects that were probed with the three-site model were also calculated for the four-site model (Figure 10). Some interesting differences in site populations arise between A1′ = Me3Ph-NMI (Figure 10A) and A1′ = (CO2H)3Ph-NMI (Figure 10B). The perylene donor populations decay somewhat faster when A1′ = Me3Ph-NMI rather than in the symmetric case A1 = A1′ = Ph-NMI (Figure 6), but it seems that the decrease in energy of intermediate

acceptor A1′ by 0.05 eV does not have a profound impact on either the rate or yield of electronic population that reaches both A1 and A1′ or the NDI acceptor. This changes drastically when A1′ = (CO2H)3Ph-NMI because the energy of A1′ is 0.14 eV lower than that of A1 = Ph-NMI. This change leads to an increase in population that traverses A1. Given that the electronic couplings between the donor and A1 and A1′ are the same, this is most likely due to the fact that the large decrease in energy of A1′ drives the electron population to A1 which is closer in energy to the donor state. If the energy of the bath is increased by raising the temperature, transfer to A1′ should begin to be more favored as it is the pathway with greater driving force for electron transfer. Figure 10C also shows more fine structure in the time dependent population that arrives at the NDI acceptor. This most likely results from interferences between the competing A1 and A1′ pathways. Tetracene is introduced as the donor in Figure 11, and similar to the data in Figure 9, the energy gap between sites is narrowed, but so is the coupling between the subunits. As the 7197


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Figure 10. Site populations as a function of time for four-site electron transfer with a perylene donor with asymmetrically intermediate acceptors and a terminal NDI acceptor. (A) Me3Ph on A1′, (B) (CO2H)3Ph on A1′, (C) comparison of all substituted A1′ acceptors, and (D) Hamiltonians for the indicated systems. All energies are given in electronvolts. The solvent is CH2Cl2.

Figure 11. Site populations as a function of time for four-site transfer with tetracene as the donor with asymmetrically substituted intermediate acceptors and a terminal NDI acceptor. (A) Me3Ph on A1′, (B) (CO2H)3Ph on A1′, and (C) site population on NDI acceptor for the three asymmetric systems. (D) Hamiltonians for the indicated systems. All energy values are given in electronvolts. The solvent is CH2Cl2.

energy gap between tetracene and A1′ increases with the perturbations to that site, so does the population decay at the donor site accompanied by an increase in population at the A1 and A1′ sites. Similar to Figure 10, the decrease in energy that occurs when A1′ = (CO2H)3Ph-NMI (Figure 11B) leads to a faster population transfer to A1, which is closer in energy to the tetracene donor excited state energy level. Once again, more fine structure is seen in the time dependent population that

arrives at the NDI acceptor that may also derive from the reason given above. Solvent Effects on Symmetric Four-Site Systems. All of the calculations presented so far used CH2Cl2 as the solvent. The solvent effects are incorporated into the various DFT calculations using a polarizable continuum model (PCM). For this series of calculations tetrahydrofuran (THF; ε = 7.58), CH2Cl2 (ε = 8.93), and CH3CN (ε = 37.5) were chosen to 7198


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Figure 12. Solvent effects on four-site electron transfer in the indicated perylene−NDI system in CH3CN, CH2Cl2, and THF showing (A) site population on perylene as a function of time and (B) population on NDI as a function of time.

Figure 13. Solvent effects on four-site electron transfer in the indicated tetracene−PI system in CH3CN, CH2Cl2, and THF showing (A) site population on tetracene as a function of time and (B) population on PI as a function of time.

Figure 14. Five-site electron transfer in the indicated system, where the site Y energy is varied, while the coupling to A1′−site Y coupling is constant. (A) Site populations on the NDI acceptor as a function of time varying the site Y energy. (B) Hamiltonian for the indicated system. The energy of Y is given in (A). The solvent is CH2Cl2.

span a reasonable range of dielectric constants. The molecular systems computed in these various solvents have the widest variation in energy gaps. The perylene−NDI system has the largest gaps, while the tetracene−PI system has the smallest gaps, where for both systems A1 = A1′ = Ph-NMI. The data for the perylene−NDI system in three solvents shows a strong correlation between the dielectric constant of the solvent and the transfer rate and intensity of quantum oscillations (Figure 12). In CH3CN the greater stabilization of the charge separated state leads to a faster transfer rate and more pronounced

quantum oscillations at early times. On the other hand, in tetrahydrofuran, longer-lasting oscillations are observed on the terminal acceptor. Figure 13 shows the electron transfer for tetracene−PI in the various solvents. With the small energy gap between the donor and acceptors subunits, the amplitude of the oscillations increases. Figure 13A demonstrates that faster coherent oscillations occur in the higher polarity solvents. Interestingly, there is a lower yield in the moderate dielectric CH2Cl2 versus THF and CH3CN seen in Figure 13B. This suggests that there may be multiple electron transport 7199


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Figure 15. Five-site electron transfer within the indicated system in CH2Cl2 varying the coupling V between acceptor A1′ and site Y, while site energies remain constant. (A) Site population of the NDI acceptor as a function of time and A1′−site Y electronic coupling V. (B) Site population as a function of time on site Y as V varies. C) Hamiltonian of the indicated system. All energy values are given in electronvolts.

Figure 16. Five-site electron transfer of the indicated system in CH2Cl2. The A1′−site Y coupling V is constant, while the site Y energies are varied. (A) Site population of the NDI acceptor as a function of time and site Y energy. (B) Site population of site Y as a function of time and its energy. (C) Hamiltonian for the indicated system. All energy values are given in electronvolts.

dephasing effects on the preservation of quantum coherences in a variety of solvents is an interesting subject for future work. Five-Site Calculations on Asymmetric Dual Pathway Systems. The calculations presented below for the five-site transfer are more phenomenological in their nature than those presented above, but are useful to probe the various ways that asymmetries can affect electron transfer in these classes of molecules. Two cases were examined in which perylene or tetracene is used as the donor, A1 = Ph-NMI, A2 = NDI, and A1′ is bound to site Y. The effects of the site Y energy relative to A1′ and the site Y−A1′ electronic coupling on electron

mechanisms to the acceptor at higher and lower dielectric environments and preferred charge recombination back to the donor in the moderate dielectric CH2Cl2 as seen in Figure 13A. A similar trend is seen in Figure 12B with the total yield of CH2Cl2 being lowest after 60 fs, although the magnitude is less pronounced. This suggests that there may be multiple molecule-specific solvent environments that promote high yield electron transport. Dephasing should be significant especially in high polarity solvents. Some of this is captured by the effective energetic shift that the solvent environment has on the site energies, but a more rigorous examination of the 7200


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Figure 17. Five-site electron transfer within the indicated system in CH2Cl2 varying the coupling between acceptor A1′ and site Y, while site energies remain constant. (A) Site population of the NDI acceptor as a function time and A1′−site Y electronic coupling V. (B) Site population as a function of time on site Y as V varies. C) Hamiltonian of the indicated system. All energy values are given in electronvolts.

transfer from perylene or tetracene to NDI were examined. This calculated fifth site can have electronic density reside on it uniquely, which leads to a complex interaction with the A1′ site to which it is attached. One could envision an A1′ molecule that could bind a small molecule that would affect the electronic coherence of electron transfer through the A1 and A1′ pathways and could serve as a quantum sensor. In the Hamiltonian shown in Figure 14, the Y in row 5, column 5 is the local site energy of the site Y perturbation and varies from −0.5 to −0.9 eV in increments of 0.1 eV, while the coupling between site Y and A1′ remains 0.25 eV. This value was chosen to be on the same order as the nearest-neighbor couplings throughout the rest of the system. As the energy of this site changes so does the population that resides on this site, although, interestingly, having site Y isoenergetic with A1′ does not equate with the greatest transfer rate or yield. The greatest yield occurs when the energy of site Y is much lower than that of A1′, making site Y a trap site for electron density. When the site Y energy is −0.8 and −0.9 eV, the electron transfer yield to NDI more than doubles. The data presented in Figure 15 keeps the same molecular structure as seen in Figure 14, but now all the site energies are held constant while the coupling (V) between A1′ and site Y is changed. The values of V in the Hamiltonian shown in Figure 15 are varied from 0.15 to 0.35 eV. One could imagine these coupling changes might result from steric changes at site Y modifying the interaction with A1′. Figure 15A shows that at times

Redfield Treatment of Multipathway Electron Transfer in Artificial

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