Influence of Coherent Tunneling and Incoherent Hopping on the

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Influence of Coherent Tunneling and Incoherent Hopping on the Charge Transfer Mechanism in Linear Donor-Bridge-Acceptor Systems Guangqi Li, Niranjan Govind, Mark A. Ratner, Christopher J. Cramer, and Laura Gagliardi J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02154 • Publication Date (Web): 10 Nov 2015 Downloaded from http://pubs.acs.org on November 15, 2015

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The Journal of Physical Chemistry Letters

Influence of coherent tunneling and incoherent hopping on the charge transfer mechanism in linear donor-bridge-acceptor systems Guangqi Li,1 Niranjan Govind,2 Mark A. Ratner,3 Christopher J. Cramer,1 and Laura Gagliardi1 1

2

Department of Chemistry, University of Minnesota, Minneapolis, MN, 55455, USA

Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, WA, 99352, USA 3

Department of Chemistry, Northwestern University, Evanston IL, 60208, USA (Dated: November 2, 2015)

The mechanism of charge transfer has been observed to change from tunneling to hopping with increasing numbers of DNA base pairs in polynucleotides and with the length of molecular wires. The aim of this paper is to investigate this transition by examining the population dynamics using a tight-binding Hamiltonian with model parameters to describe a linear donor(D)-bridge(B)acceptor(A) system. The model includes a primary vibration and an electronvibration coupling at each site. A further coupling of the primary vibration with a secondary phonon bath allows the system to dissipate energy to the environment and reach a steady state. We apply the quantum master equation (QME) approach, based on second-order perturbation theory in a quantum dissipation system, to examine the dynamical processes involved in chargetransfer and follow the population transfer rate at the acceptor, ka , to shed light on the transition from tunneling to hopping. With a small tunneling parameter, V, the on-site population tends to localize and form polarons, and the hopping mechanism dominates the transfer process. With increasing V, the population tends to be delocalized and the tunneling mechanism dominates. The competition between incoherent hopping and coherent tunneling governs

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the mechanism of charge transfer. By varying V and the total number of sites, we also examine the onset of the transition from tunneling to hopping with increasing length.

I.

INTRODUCTION

Charge transfer has long been recognized as a key process in many research fields of chemistry1,2 , solid-state physics3,4 and biophysics5–7 , and its mechanism and dynamics are relevant to various applications such as molecular electronic devices8 , biosensing techniques9 , solar cells10–13 and various photoactivated processes. In DNA strands, the mechanism of charge transfer has been observed to change from tunneling to hopping with increasing numbers of base pairs14 and a similar phenomenon has also been observed in long donor-acceptor molecular wires8,15 with increasing monomer sites from which the wire is constructed. Understanding this mechanism is important to understand the charge transfer process in molecular systems. To capture the physics in realistic situations accurately, it is important to account for electron-vibronic interactions. Electron trapping and scattering are also other factors that are known to influence the charge transfer/transport characteristics in realistic systems.16–21 . Over the years Marcus theory22 has been utilized in several studies to describe charge transfer in artificial solar-energy conversion21,23,24 and molecular electronics17,18,25 . The success of this approach relies on the electron-vibration interaction, where the initial and final states of the electron transfer process are fully equilibrated polarons localized on different sites, and the transitions between them can be described as a succession of hopping steps.26 In this manuscript, the electron-phonon coupling and polaron theory are used to investigating the population localization and delocalization. Although the transfer rate can be obtained numerically, the analytic results can also be obtained via Marcus theory. The variable ranging hopping theory proposed by Yu and Song27 , and Renger


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The Journal of Physical Chemistry Letters 3

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and Marcus28,29 , can be taken as an motion in an extended system with the succession of hopping steps, each described as a Marcus process. In a simple pair of two level systems, the on-site population shows an oscillatory tunneling behavior depending on the strength of the inter-site tunneling parameter V. When the energy gap ∆ε (the absolute value of the energy difference between two nearest neighbor levels) is much larger than V, population tunneling between the two levels is inhibited. However, upon switching on the electron-vibration interaction, hopping is initiated, and the transfer rate can be expressed by Mar2

)22,30 . The reorganization energy λ includes two components: the cus theory as kET ∝ exp(− (λ−∆ε) 4λkB T electron-vibration interaction and the energy gained due to reorientation of the molecules in the surrounding environment, both of which can be obtained via detailed atomistic calculations31–33 . The reorganization energy effectively reduces the energy gap ∆ε, facilitating charge transfer. With strong electron-vibration interaction, the on-site population tends to form a polaron34–37 , inducing “self-trapping”at the local site26,38,39 . This trapping energy can also reduce the energy gap ∆ε and trigger charge transfer. The aim of this paper is to investigate the changing mechanism of charge transfer from tunneling to hopping, by studying the population dynamics using a tight-binding Hamiltonian with suitably chosen model parameters to describe a linear donor-bridge-acceptor (D-B-A) system. The transition from tunneling to hopping in DNA strands with increasing numbers of base pairs has been reported previously14 , and similar behavior can occur in long molecular wires8,15 . Jortner and co-workers have proposed super-exchange and multi-step hopping40,41 as possible mechanisms to explain charge transfer in these systems. This was further elaborated by Conwell and co-workers who proposed the polaron-assisted charge transfer mechanism5,42 and used that to explain the hole transport in DNA5,41 . The electron-transfer mechanism from tunneling to hopping when increasing the bridge length had also been proposed recently by skourtis and Beratan43 via the so-called flickering resonance



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model and applying to charge migration in DNA hairpins. Analytical results for the charge transfer rate have been reported by Segal and Nitzan17,18,44 via the steady-state flux method based on a multi-hopping model. Other recent simulation studies have also been reported to examine the charge transfer mechanism in DNA45–47 and in other biological systems43 . Here we have used a numerically exact and fully quantum approach based on the Bonca-Trugman method19,48 and a tight-binding model that includes a primary vibration and an electron-vibration coupling at each site. Further this vibration is coupled to the bath or environmental vibrations, by which the system is allowed to dissipate energy to the environment49,50 . Thermal relaxation is included by augmenting the Liouville equation for the oscillator density matrix with the corresponding kinetic terms that account for that relaxation26,39 . This model has been used to investigate the polaron formation process in solar cells26 , Raman scattering and heat conduction in molecular wires51–54 , and photoluminescence in semiconductors55–57 . With this model, the effect of the electron-vibration interaction on the population localization in the vibronic levels can be examined. This should be contrasted with a model where the electronic system is directly coupled to the bath vibrations.49,50,58–60 The quantum master equation (QME) approach is used to investigate the time-dependence of polaron formation, time-dependent population dynamics and transfer rate61–64 . This approach has been used to study the dynamics of a system coupled to a thermal bath18,65,66 , and is based on a second-order perturbation theory in the system-bath coupling. Other approaches include the nonequilibrium Green’s function approach (NEGF)67 , which has the advantage of being formally exact within the electron-vibration coupling. However, the dependence of Green’s functions on two time arguments makes it rather difficult to deal with a time-dependent Hamiltonian.53,54 . In the present study, the electron-phonon interaction (between the electronic system and the primary vibration) are accounted, and will be included in the primary Hamiltonian of the electric


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system. Such a system Hamiltonian can be applied to examine the influence from the electronphonon interaction to the phenomenons as localization and delocalization26 ; further the coupling between the primary vibration and the other rest vibrations will be taken as the bath coupling and deal with the second-order perturbation theory and the dissipation theory. In the real molecular system, there are many vibrational modes. Fully accounting all the modes will generate a huge density matrix, and is beyond the calculating ability with a quantum method. The recent works as Multi Configuration Time Dependent Hartree (MCTDH)68 and the variational approach (globallocal Ansatz)69 had been proposed to examine the electron-phonon interaction, in which all the vibrations are taken as bath environment. We start by formulating the basic Hamiltonian model in Section II. In section III, numerical results are presented and the mechanism of charge transfer from tunneling to hopping is discussed. Conclusions follow in section IV.

II.

D

1

THEORETICAL MODEL

2

3

4

N A

Vibration FIG. 1: Linear D (donor)-B (bridge)-A (acceptor) system is coupled to a primary vibration (the green color), with the electronvibration interactions permitted to occur on each site.



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The Hamiltonian of the D-B-A system (Fig. 1) is given by

H = HS + HB + HSB , HS = εD c†D cD +

N X

(1)

εl c†l cl + εA c†A cA + VD (c†D c1 + c†1 cD ) +

N−1 X

V(c†l cl+1 + c†l+1 cl ) + VA (c†A cN + c†N cA )

l=1

l=1

+ ~ω0 d†0 d0 + {αD c†D cD + αA c†A cA +

N X

αl c†l cl }(d†0 + d0 ) ,

(2)

l=1

HB =

∞ X

HSB =

∞ X

~ωs d†s ds ,

(3)

λs (d†0 ds + d†s d0 ) ,

(4)

s=1

s=1

where HS is the primary system Hamiltonian coupled by a single oscillator (primary vibration), HB is the secondary phonon bath and HSB is the coupling between vibrational modes. Operators c†D (cD ), c†l (cl ) and c†A (cA ) create (annihilate) an electron on sites D, l and A, with the on-site energy εD , εl and εA respectively; VD and VA represent the tunneling parameters between donor D and site 1, and site N and acceptor A, respectively; V is the tunneling parameter between any two nearest neighbor sites not including D or A. In this work, we will hereafter take VD = VA = V and report all values as simply V. The operators d†0 (d0 ) of the primary vibration create (annihilate) a vibronic state with energy ~ω0 ; the last term in the right side of Eq. 2 describes the interaction between the electronic states (D, l, A) and the single vibration, with parameters αA , αD and αl indicating the strength of the interaction. The secondary bath phonons are expressed in Eq. 3, and their couplings to the primary vibration are shown in Eq. 4 (using the rotating wave approximation with associated coupling strength λs ); The operators d†s (ds ) of the secondary phonon create (annihilate) one vibration with energy ~ωs . In the real molecular system, there are many vibrational modes. Fully accounting all the modes is beyond the calculating ability with a quantum method (for example using the quantum master


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equation). The semi-classical method (Ehrenfest approach) is the choice, but it fails in time scale compared to the quantum method26 . Thus, in our present model, only one primary vibration is accounted, and further it is coupled to the other phonon modes (environment), by which the system dissipates its energy and reaches its steady state49 . This primary vibration can be taken as the effective frequency, together with the effective Huang-Rhys factor70,71 in the real system with many modes. Such a model has been generally utilized to investigated the charge transfer17,72 , the exciton dissociation in solar cell38 , polaron formation of the charge transfer processes26 . The QME for calculating the density matrix of the primary system, ρS is50,64

i~

∂ρS (t) = [HS , ρS (t)] − iγ0 /2[d†0 d0 ρS (t) + ρS (t)d†0 d0 − 2d0 ρS (t)d†0 ] , ∂t

(5)

where the first term on the right side of Eq. 5 is relative to the system Hamiltonian HS (Eq. 2), which includes the contribution from the inter-site coupling V and the electron-vibration interaction. The second term comes from the phonon bath coupling. γ0 is the phonon relaxation rate induced by the active phonon bath coupling. We take the imaginary part of the active phonon self energy Σ as26,73

γ0 (ω)/2 =

1 1X |λs |2 δ(~ω − ~ωs ) , Im{Σphonon (ω)} = ~ ~ s

(6)

where δ is the Kronecker delta. This time-dependent equation can be numerically solved by the Runge-Kutta, which are methods for the numerical solution of the ordinary differential equation82 , In this paper, the iteration is done with the initial condition that there is one electron on the ground state of donor site. The time step are set as 0.01fs. In this paper, the electron-phonon interaction is accounted and included into the primary system Hamiltonian, which make it possible to investigate the population localization and delocalization.



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Two important parameter as the tunneling inter-site coupling V and the electron-vibration interaction α are also included into the primary Hamiltonian of the electronic system. Further the rest vibrations are taken as the phonon bath and its coupling to the primary vibrations is dealt with dissipation theory and second-order perturbation theory. Here the transfer mechanism is examined via the dynamical process of the populations. The mechanisms of the super-exchange and the hopping are also examined by Jang et. al.74 , in which all the vibrations are taken as bath, and dealt with second-order perturbation theory. The fully quantum method based on Bonca-Trugman method26,39,48 is used to investigate the polaron formation, which is a differential equations for the density matrix for the primary system. The dissipation theory49 and the second-order perturbation58 are used to deal with the coupling between the primary system and the bath environment. In this density matrix, the diagonal elements are normally relative to the site energies, which the off-diagonal elements are relative to the coupling between the different electronic states, and represents the quantum coherence in the system. The correlation function presents the influence of the bath environment to the primary system. Many difference forms such as the Ohmic spectral density75 and Drude form of the spectral density58 , had been proposed to investigate the comparison between the Markov and non-Markov process and the influence of the Matsubara terms58,76–78 . Here we use the simple approximation: wide band limit approximation, and we take the coupling as a constant. The population of each site at any time t is given by

PD =< c†D (t)cD (t) >; Pl (t) =< c†l (t)cl (t) >; PA =< c†A (t)cA (t) >;

(7)

The rate ka of the transfer processes to A can be calculated with the rising exponential


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function26,38,46

PA (t) = PA (∞)(1 − e−ka t ) ,

(8)

PA (t) and PA (∞) are the time-dependent population and the long-time stationary population of A respectively; both of them can be obtained numerically from Eq. 7 and ka can be obtained via fitting ρN (t) with the exponential function. When the tunneling parameter V is large, coherent tunneling dominates the transfer processes, and an oscillator behavior characterizes the population dynamics. In this case the population cannot be fitted to an exponential function. The transfer rate ka can be obtained by measuring the formation time τ with the relation τ = 1/ka . The formation time τ of the population is defined as the time point at which the population reaches a certain value. Experimentally the “formation time”is defined as the time at which the target population reaches ∼ (1 − e−1 ≈ 0.63) of its final value26,38 .

III.

NUMERICAL RESULTS

In the numerical simulation, the site energies εl , and the electron-vibration interaction αl (l = 1, 2, 3, 4, 5, 6) are kept fixed, as given in Table I. These parameters will be used for the following numerical calculations with a few exceptions, as noted below. The site energies are chosen so that the donor and acceptor sites are equal in energy, and each bridge site is higher in energy by 0.1 eV than the site before it along the chain. The model system based on the tight-binding model is set, together with the artificial parameters is used to examine and investigate the population dynamical process. Such uphill behaviors in site energies have been used to study the charge transfer in DNA46 , solar cells10 and photosynthesis79–81 . Addition, the coupling of each site to the primary polaronic vibration is assumed to increase along the chain until localization on the acceptor is achieved, and the increasing of the coupling will generate a large “self-trapping energy”to smoothen the increasing



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energy gaps. The energy of the primary vibration is set to ω0 = 0.02eV , and the coupling parameter to the bath environment is set to γ0 = 0.04eV . We also set VD = VA = V. We vary V and N (number of bridges) to investigate their influence on the transfer rate ka . The values of V have been chosen to vary between [0.002eV , 0.1eV ], with the minimum value 0.002 eV lower than kB T = 0.025eV (room temperature, and kB is the Boltzmann constant). The oscillatory behavior of the system depends on ω0 and V. When ω0 is smaller than V, the system exhibits classical behavior. When ω0 is larger than V, the system exhibits more quantum behavior26,39 . The value of γ0 will decide the speed of the relaxation process. The initial condition we have used is that there is one electron in the ground state of the donor, and charge transfer will occur from the left side (Donor), passing through the bridges (B) and finally arriving at the right side (Acceptor). We will calculate the transfer rate ka and examine the mechanism of charge transfer. In this present study, only one vibration mode is accounted, due to the ability of doing numerical calculation. The primary vibration energy shows influence to the charge localization, polaron formation, as have been discussed in out previous paper26,39 . Here we set the constant values 0.02eV, just in case that the very small value 0.002eV for the tunneling parameter V will be applied and discussed. In this case, V is 0.1 times of the primary vibration energy. Our model system is based on the tight-binding model and set up, together with the artificial parameters used to examine and investigate the population dynamical process. Although only one set of parameters shown in Table 1 are used, such a up-hill behavior of the site energies had been proposed in DNA46 , solar cells10 and photosynthesis79–81 .. TABLE I: Parameters used in simulations unless otherwise specified (unit in eV ) D εl 0

1

2

3

4

5

6

A

0.1 0.2 0.3 0.4 0.5 0.6 0

αl 0.01 0.07 0.12 0.16 0.2 0.24 0.28 0.01


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In the presence of a small V (or narrow band), and the electron-vibration interaction, the population tends to localize and form a polaron26,34,39 , and the hopping mechanism (in which, the line shapes of the charge transfer dynamics show exponential behavior, with the transfer rate decided by the Marcus theory) will dominate the transfer mechanism. With increasing V, the coherent tunneling increases and competes with the hopping mechanism. When V is larger than a certain threshold, the polaron does not form at all26,34,39 . The competition between the coherent tunneling (in which, the line shapes of the charge transfer dynamics show oscillator behavior, which strongly depend on the tunneling parameter V) and the incoherent hopping will govern and trigger the change of the charge transfer mechanism. In addition, the effect of coherent tunneling becomes weak with increasing site number N. 1 PD P1 P2 P3 P4 P5 PA

0.8

Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.6

0.4

0.2

0

0

200

400

600

Time [fs]

FIG. 2: Dynamical processes of populations (PD , Pl (l=1,2,3,4,5) and PA ) are shown. N=5; V=0.05eV ; αl = 0.0 for (l=D,1,2,3,4,5,A) is used only here in order to investigate the cases in absence of the electron-vibration interactions. Parameters of site energies used are shown in Table I.

In the absence of the electron-vibration interaction (i.e., setting all αl values to zero instead of using the values in Table I), the charges move in the energy band26,34,39 . Coherent tunneling occurs and dominates the transfer processes. As shown in Fig. 2, the populations on sites D, 1, 2, and to



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some extend 3, show strong oscillatory behavior. Because the site energies increase along the chain, the on-site populations become increasingly small with increasing N, and are negligible for N>3. We next consider turning on the electron-vibration interaction, and examine the degree to which incoherent hopping takes place in the transfer processes, as polaron formation generates trapping energyto smooth the energy gaps and to facilitate the charge transfer. as polaron formation generates “trapping energy”to smooth the energy gaps and to facilitate the charge transfer. Especially the polaron binding will depend on values of the site energies εl and the electron-vibration interaction coupling αl . In Fig. 3, results are shown for the same tunneling parameter as in Fig. 2 but with αl values as given in Table 1. Coherent tunneling (oscillator behavior) is visible and dominates the dynamical processes of populations (PD , P1 , P2 and P3 ) for a few hundred femtoseconds after the transfer processes start. Along the chain, the coherent effect becomes weaker and disappears due both to the increasing site energy parameters and the disorder in the system, The incoherent hopping process grows and dominates the transfer process for the sites in the right side of the chain. After few hundred femtoseconds, polaron formation occurs due to the electron-vibration interaction, and electrons move in the localized state via the incoherent hopping mechanism (see the insert figure of Fig. 3). Note that the long-time populations of sites at the right end of the chain exceed those at the left end of the chain owing to the increased strength of coupling to the polaronic vibration for those sites having larger N. For equivalent tunneling and coupling parameters, the charge transfer mechanism also changes from coherent tunneling to incoherent hopping with increasing the numbers of the total sites. In a system with a small number of sites, coherent tunneling dominates the charge transfer processes, not if the trapping energy is large or the tunneling is weak. Upon increasing the site numbers, the effect of the coherent tunneling becomes weaker and the incoherent hopping becomes stronger. Fig. 4 (left


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The Journal of Physical Chemistry Letters 13 N=5

V=0.05eV

0.4 Population

1 0.8 0.6 0.4 0.2 0 Population

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0

200

400 Time [fs]

600

PD P1 P2 P3 P4 P5 PA

0.2

0

0

100

200

300

Time [fs]

FIG. 3: Dynamical processes of populations Pl (l=D,1,2,3,4,5,A) are shown. Tunneling oscillator behaviors are visible with PD , P1 , P2 and P3 at the beginning. In P4 , P5 and PA , hopping mechanism (line shape of the population shows exponential behavior) dominates the population dynamics. The insert figure shows that in long time limit, polaron formation occurs and hopping mechanism dominates the population dynamics. N=5; V=0.05eV ; Other parameters shown in Table I.

panel) shows the dynamical processes of PA . When N is equal to 1 and 2, the oscillator behaviors of the coherent tunneling are visible for one hundred femtosecond. When N becomes larger than 2, coherent tunneling becomes weaker and incoherent hopping dominates the transfer processes when the transfer starts. The tunneling parameter V has a strong influence on this competition. When V has a value up to 0.1 eV (right panel of Fig. 4), the coherent tunneling becomes stronger and dominates the transfer process for a few hundred femtoseconds, even for a system with large N. After that, due to the electron-vibration interaction, the on-site population tends to localize to form the polaron and transfers through the localized state via the hopping mechanism. However when V is large enough, the polaron will not be formed, and coherent tunneling will dominate the whole transfer processes. The tunneling parameter V not only influences the dynamical process of PA , but also the dynamical processes of the other populations. As shown in Fig. 5 (left panel), when V is as small as 0.02eV , coherent tunneling is absent, and the incoherent hopping dominates the dynamical processes ACS Paragon Plus Environment


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V=0.04eV

V=0.1eV

0.75

0.75 N=1 N=2 N=3 N=4 N=5 N=6

0.6

N=1 N=2 N=3 N=4 N=5 N=6

0.6

0.3

0.3

0.15

0.15

PA

0.45

PA

0.45

0

0

200

400

0

100

Time [fs]

0 300

200 Time [fs]

FIG. 4: Dynamical processes of PA are shown with increasing N (number of bridges). Tunneling oscillator behaviors show up when N=1 and 2 when V=0.04eV . With increasing N, the oscillator behaviors disappear. When V=0.1eV , stronger tunneling oscillator behaviors show up with N=1,2,3,4. Other parameters shown in Table I.

when the transfer starts. With increasing V to 0.06eV (right panel of Fig. 5), coherent tunneling dominates the transfer processes, and oscillatory behavior appears in the dynamical processes of all the populations. N=2

V=0.02eV

V=0.06eV

1

1 PD P1 P2 PA

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

300 Time [fs]

0

40

80 Time [fs]

120

Population

PD P1 P2 PA

0.8

Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 160

FIG. 5: Dynamical processes of populations (PD , P1 , P2 and PA ) are compared between a small V (0.02eV ) and large V (0.06eV ). With large V, tunneling oscillator behaviors show up and tunneling mechanism dominates. With small V, hopping transfer dominates. N=2; Other parameters shown in Table I.


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The transition from tunneling charge transfer to hopping charge transfer may be judged by a sudden decrease in the exponential dependence of the rate constant on bridge length. Fig. 6 shows the change of the transfer rate log10 ks (from donor to acceptor) as a function of the total bridge site number N. At the positions (circles with cyan color), the line shape shows turning, which means the changing mechanism from tunneling to hopping takes places, with N equaling 2, 2, 3, and 4 from bottom to top with increasing V respectively. With larger V, the coherent tunneling mechanism becomes strong, and N shifts to large number, where the effect of the disorder energies in the longer system will cancel the effect of large V. In Fig. 6, one can also find that both the tunneling mechanism and the hopping mechanism give the exponential distance dependence based on this theoretical model with the artificial parameters. With increasing V, the transfer rate increases, reaching a higher value with 101 4s−1 when V=0.1eV. Under such a situation, the tunneling mechanism dominates the charge transfer processes. The charge will be delocalized and show oscillation behavior. Transfer rate as a function of the N (number of bridges) 14

-1

log10ka(s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

V=0.1eV V=0.04eV V=0.01eV V=0.002eV

12

10 1

2

3 4 N (number of bridges)

5

6

FIG. 6: Transfer rates log10 kα (kα is the transfer rate, with the unit second−1 ) are shown as a function of N (number of bridge sites) with different tunneling parameter V. The line shapes show turning at the positions (circles with cyan color), where the changing mechanism from tunneling to hopping takes place, with N equaling 2, 2, 3, and 4, from bottom to top with increasing V, respectively. Other parameters shown in Table I.



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IV.

CONCLUSIONS

In this paper, a tight-binding model with model parameters chosen to mimic various situations is used to examine population dynamical processes in a linear D-B-A system. The electronic states are coupled to a primary vibration. Further this vibration is coupled to the other bath or environmental vibrations, to dissipate the energy and allow the system to reach a steady state. The artificial parameters of the site energies εl and the electron-vibration interaction parameters αl are utilized in setting up a tight binding model with the uphill site energies. Such uphill behaviors in site energies have been used to study the charge transfer in DNA46 , solar cells10 and photosynthesis79–81 . This model allows us to examine the population localized on the vibronic levels and the polaron formation processes. QME approach is used to investigate the time-dependent charge transfer processes. We have examined the competition between coherent tunneling and incoherent hopping and their influence on the change of the charge transfer mechanism. For small V, the population tends to localize to form polarons, and the incoherent hopping dominates the transfer processes. For large V, the population tends to be delocalized for a few hundred femtoseconds and coherent tunneling dominates the transfer processes through the delocalized state. The effect of the coherent tunneling becomes weaker along a linear chain, because of the uphill site energies and localization. With increasing the site numbers, the change of the charge transfer mechanism from tunneling to hopping occurs, and this process is strongly dependent on the tunneling parameter V. Our analysis can also be applied to understand the charge separation in photovoltaic systems which can be useful to improve the efficiency of solar cells. Recent studies10,11 have suggested that exciton dissociation occurs within a few hundred femtoseconds when the delocalized states are first reached. After that, polaron formation can localize the electronic states; holes and electrons will move via incoherent hopping within these localized states. The transfer mechanism of this separation also changes from tunneling to hopping.


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As part of future work, we will focus on determining the model parameters used in this work from electronic structure calculations to set up a model to shed light on the rectifying behavior of long donor-acceptor molecular wires83,84 .

V.

ACKNOWLEDGEMENT

This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES), under SciDAC grant nos. DE-SC0008666 and KC030102062653 (NG).

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