Formulation of Long-Range Transport Rates through Molecular

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Spectroscopy and Photochemistry; General Theory

Formulation of Long-Range Transport Rates through Molecular Bridges: From Unfurling to Hopping Ariel Levine, Michael Iv, and Uri Peskin J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01581 • Publication Date (Web): 01 Jul 2018 Downloaded from http://pubs.acs.org on July 3, 2018

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The Journal of Physical Chemistry Letters

Formulation of Long-Range Transport Rates through Molecular Bridges: from Unfurling to Hopping

Ariel D. Levine(1), Michael Iv(1) and Uri Peskin(1,2)

(1)

Schulich Faculty of Chemistry and (2)the Technion Grand Energy Program, Technion – Israel Institute of Technology Haifa 32000 Israel

Ariel D. Levine: [email protected] Michael Iv: [email protected] Uri Peskin: [email protected] *corresponding author

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Abstract Weak fluctuations about the rigid equilibrium structure of ordered molecular bridges drive charge transfer in donor-bridge-acceptor systems via quantum unfurling, which differs from both hopping and/or ballistic transfer. Yet, static disorder (low frequency motions) in the bridge is shown to induce a change of mechanism from unfurling to hopping when local fluctuations along the molecular bridge are uncorrelated. Remarkably, these two different transport mechanisms manifest in similar charge transfer rates, which are nearly independent on the molecular bridge length. We propose an experimental test for distinguishing unfurling from hopping in DNA models with different helix directionality. A unified formulation explains the apparent similarity in the length dependence of the transfer rate in spite of the difference in the underlying transport mechanisms. Table of Contents Graphics


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Charge transfer (CT) through molecules is an elementary step in many processes taking place in living cells1-4. Therefore, the mechanisms of CT through biomolecules have been intensively studied5-16 and yet, the complexity of molecular systems at the atomic level challenges attempts for a rigorous formulation of CT phenomena in biosystems. Particularly interesting in this context is CT in donor-bridge-acceptor (DBA) molecules17-26, where donor and acceptor moieties are connected via a molecular bridge. Typically, the electronic energy of the charged donor (or acceptor) state is well separated (off-resonant) from the energy of charged bridge states, such that the transfer is bridge mediated, and the rate of CT from the donor to the acceptor reveals the specific properties of the bridge and its coupling to the donor and/or the acceptor. When the DBA molecule is in solution or gas phase, one may observe an almost universal dependence of the CT on the molecular bridge length, where for short bridges the rate drops exponentially with bridge length, while for longer bridges the rate depends only weakly on the length3, 6, 12, 19, 20, 22, 27 . The exponential drop is attributed to a descent of a direct tunneling matrix element between donor and acceptor electronic wave functions28, whereas the weak length dependence is often associated with charge hopping29. Many different “hopping” based mechanisms such as: variable-range hopping30, partial delocalization6, flickering resonance31, and multistep electron tunneling8 were proposed over the years in order to better understand CT through molecular bridges. Other observations of weak length-dependence of CT rates are attributed to ballistic transport3, 6, 12, 19, 20, 22, 27. Indeed, when DBA molecules are attached to macroscopic leads in a molecular junction architecture, the overlap in energy between



the continuous band of lead states and the bridge states facilitates resonant electronic transport through the bridge. Recently, we proposed another transport mechanism, termed “quantum unfurling”, in order to explain length-independent CT rates in DBA molecules11. Weak fluctuations around the molecular equilibrium structure drive a localized charged donor state into delocalized bridge states in a single thermally activated (unfurling) step, whose rate is independent on the bridge length. In our earlier work, the assumption was that local fluctuations along the molecular bridge are correlated due to long-range forces. Therefore, transitions between different bridge units were effectively uncoupled from the environment, and the coupling was restricted to transitions into and out of the molecular bridge. In the realm of that assumption, the presence of disorder in the bridge, and the corresponding localization of bridge orbitals, renders the CT ineffective. Nevertheless, the assumption of full correlation between CT-enabling fluctuations at different locations along the molecular bridge may be too restrictive for many realistic DBA molecules in ambient environment. In the more common case, the uncorrelated motion of local modes can drive CT from one bridge unit to another. It is therefore important to explore the manifestation of quantum unfurling as well as the effect of structural disorder beyond the scope of fully correlated fluctuations. In this work we consider the contribution of uncorrelated local fluctuations along the bridge, which facilitates CT even in the presence of static disorder (owing to low frequency motions). Moreover, the rates of these CT processes are nearly independent of the molecular bridge length. Indeed, for ordered bridges the unfurling dominates CT, whereas in statically disordered bridges the local fluctuations can drive transitions


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between localized electronic bridge states, resulting in a hopping like CT mechanism. We show that both unfurling, and hopping can give rise to apparent length-independent CT rates, and we propose experimental tests that should pinpoint the dominance of unfurling vs hopping in a given system. As an example we consider hole-transport through DNA37,38 for a specific DBA like sequence, [5’-G(T)NGGG-3’]+ 27. Our approach is to model the electronic dynamics using a Hamiltonian, which assumes a rigid equilibrium structure, Hˆ rigid 11. Vibrations in the molecule or its surroundings are introduced by the coupling Hamiltonian, Hˆ nuc , such that the full Hamiltonian reads,

Hˆ = Hˆ rigid + Hˆ nuc

(1)

The coupling Hamiltonian assumes independent local environments (intra-molecular modes and/or solvent modes), for each local electronic subspace14,34, Nb

Hˆ nuc ≡ ∑ ( Hˆ Qn + µˆQn Pˆnb ) nb =1

b

b

(2)

Hˆ Qn represents a local “bath” of nuclear modes associated with the subspace, nb , where b

µˆQ is the corresponding vibrionic coupling operator, and the number of nuclear baths is nb

N b 41. The operator Pˆj projects onto a specific jth

molecular bridge electronic

subspace. It is convenient to follow the electronic dynamics in a reduced space of the electronic degree of freedom for the Hamiltonian in Eq. 1. Assuming that fluctuations 5 ACS Paragon Plus Environment


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around the equilibrium structure (deviations from the rigid structure) are sufficiently small, their effect on the electronic dynamics can be treated using perturbation theory. Following the Redfield approach40-45 (see SI) the reduced electronic dynamics in the basis of Hˆ rigid eigenstates ( Hˆ rigid n = En n ) follows quantum master equations for the populations kinetics, Nb d ρ n , n (t ) = ∑ [ ∑ k n( ,nmb ) ρ m , m (t ) − ∑ k m( n,bn) ρ n ,n (t )] . dt nb =1 m≠n m≠n

(3)

For boson baths model as implemented below (see SI) each population transfer rate corresponds to energy absorption ( En > Em ,

Bnb ( En − Em ) )

or

emission

( Em > En ,

kn(n,mb ) =

1 | m Pˆnb n |2 J n ( En − Em ) 2h

kn( n,mb ) =

1 | m Pˆnb n |2 J nb ( Em − En ) 2h

b

[ Bnb ( Em − En ) + 1] ) from/to the bath. The rates are proportional to the square of the

coupling matrix elements, | m Pˆ nb n |2 , where m and n represent the mth and nth eigenstates of the rigid electronic Hamiltonian. The nuclear environment is characterized by the boson occupation numbers, Bnb ( E ) = (e 11

for which we use an Ohmic model , i.e.,

E KTnb

− 1) −1 , and the bath spectral densities,

J nb ( E ) ≡

4πηnb hωnb

Ee

−E hωnb

for E > 0 , and

J nb ( E ) = 0 otherwise. The parameter ωnb is the characteristic cutoff-frequency, which was set to 0.1 eV , such that the spectral density covers the entire range of molecular and solvent vibrational frequencies, 0 < h ω 4) length-independent CT rates are apparent in both cases, consistent with the quantum unfurling mechanism, where a transition from the donor site into the bridge constitutes the rate limiting step (see the SI for the discussion of the negligible effect of transitions between bridge dominated eigenstates, Fig. S3). Notice that the bath correlation scheme has no effect on system eigenstates which are determined by the rigid structure; therefore, transitions into fully delocalized bridge orbitals (reflected in the length-independent CT rates) prevail also in the presence of uncorrelated fluctuations.

Scheme 1: Different schemes of coupling to the bath. Groups of nucleobases which are coupled to the same bath modes share the same blue area. Left: a ‘fully correlated’ systembath coupling scheme, Right: An “uncorrelated” system-bath coupling scheme.

Nevertheless, uncorrelated bath fluctuations lead to smaller unfurling rates. Considering the rate of transition between the donor-dominated eigenstate ( nD ) and any bridgedominated eigenstate ( nB ), a uniform system bath coupling operator results in higher rates ( k n , m ∝| ∑

nD Pˆ nb nB |2 ) in comparison to rates obtained from a set of local

nn

coupling operators ( knn,bm ∝| nD Pˆnb nB |2 ; nb = 1, 2,.., Nb ). In both cases the baths ({ nb })


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associated with bridge sites at the vicinity of the donor/acceptor dominate the rates, owing to the “tail” of the donor/acceptor wave functions at these sites. Since thermal transfer rates are dominated by lower energy bridge eigenstates ({ nB }), and since the sign of the corresponding wave functions does not alternate between neighboring bridge nD Pˆ nb nB |2 > ∑ | nD Pˆ nb nB |2 , which is

sites at bridge edges, it follows that, | ∑ nb

nb

consistent with higher rates for the correlated bath scheme.

Figure 1: Calculated effective donor-to-acceptor transfer rates for DNA model sequences of type [5’-G(T)NGGG-3’]+, plotted vs. the poly-A bridge length (N). Orange and red diamonds correspond to correlated and uncorrelated coupling schemes, respectively. Rates are in units of nsec-1. The nuclear baths were identical with the following o parameters: T = 298 [ K] , ηnb = 0.007 [eV ] , hωnb = 0.1 [eV ] .

Thus far, we have modeled fluctuations in the DNA about a single equilibrium structure. A more realistic account should consider a distribution of structures owing to the “floppiness” of long DNA molecules, attributed to low frequency modes. In order to account for such a distribution, static random noise was added to the local site energies11. The noise magnitude, ∆E , is regarded as the DNA “rigidity” parameter, meaning that low



∆E correlates to a more rigid DNA molecule while high ∆E correlates to a more floppy DNA molecule. The static (zero frequency) noise was introduced to the local ionization potentials of each Adenine (A) and Thymine (T) nucleobase at the bridge, as well as to the terminal donor Guanine (G) (the onsite energy at the terminal GGG was treated as a reference point common to all realizations). The changes in ionization potentials were randomly taken from a normal distribution, characterized by a zero mean value and a standard deviation,

σ . We chose the parameter, ∆E ≡ 3σ

as a measure for the magnitude of this

static noise. For each realization the rigid model Hamiltonian was diagonalized, and the orbitals were sorted according to their projection on the donor ([G]+), the poly-A bridge ([(T)N]+), or the acceptor ([GGG]+) sites (Fig. S1 in the SI). Focusing strictly on DBA like realizations of the DNA model, we excluded realizations for which the orbitals could not be sorted as above. In Fig. 2, we demonstrate the effect of the static noise on charge transfer, comparing between models of correlated and uncorrelated bath fluctuations. For fully correlated system-bath interactions, the CT rates decrease with bridge length, in contrast with the case, ∆E = 0 . However, uncorrelated system-bath interactions lead to a remarkable length-independence of the CT rates also in the presence of the static noise. The drop of the rate with increasing bridge length under static noise was already reported in Ref. 11, and was attributed to localization of the bridge orbitals, which prevents quantum unfurling from the localized donor orbital into orbitals delocalized over the entire bridge. Therefore, the length-independent CT rates observed for the uncorrelated system-bath interactions cannot be attributed to the unfurling mechanism. Instead, it


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points to a hopping like mechanism, driven by the local bath fluctuation, which facilitate CT between localized bridge orbitals.

Figure 2: Calculated averaged effective donor-to-acceptor transfer rates for DNA model sequences of type [5’-G(T)NGGG-3’]+, plotted vs. the poly-A bridge length (N). Blue and green circles correspond to correlated and uncorrelated coupling schemes, respectively. The rates were obtained by averaging over 1000 randomly set realizations (taken from a normal distribution with a standard deviation, ∆E /3 ). The error bars marked by vertical lines correspond to the actual standard deviation for each point. Rates are in units of nsec-1. o The following model parameters were used: ∆E = 0.1 [eV ] , T = 298 [ K] , ηnb = 0.007 [eV ]

, hωnb = 0.1 [eV ] .

Comparing the length-dependence between Fig. 1 (red) and Fig. 2 (green) in the case of uncorrelated system-bath interactions, it turns out that the mere observation of lengthindependent transport rates is not sufficient for determining the mechanism of fluctuation-driven transport. Indeed, both quantum unfurling (Fig. 1) and sequential hopping (Fig. 2) can explain a nearly length-independent phenotype, although the underlying physics is very different between these two mechanisms, as explained above.



Fig. 3 quantitatively demonstrates the similarity between the phenotypes of unfurling and hopping in the case of uncorrelated fluctuations. For each bridge length there is a range of hopping rates attributed to the different realization of the static noise. This range can be covered by a set of unfurling rates, attributed to zero noise but to different coupling strengths to the local bath modes (reorganization energy).

Figure 3: Calculated effective donor-to-acceptor transfer rates for DNA model sequences of type [5’-G(T)NGGG-3’]+, plotted vs. the poly-A bridge length (N) in the presence of coupling to uncorrelated local baths. Orange: Rates attributed to the hopping mechanism, where the circles and vertical lines correspond, respectively, to the average and standard deviation obtained from a distribution of structures with ∆E = 0.1 [eV ] , and a fixed reorganization energy as marked on the plot. Blue and green: Rates attributed to the unfurling mechanism for ∆E = 0 , and different reorganization energies (as marked on the plot). The shaded area demonstrates the overlap in the results of unfurling and hopping transfer rate for the same mode at different parameters. Rates are in units of nsec-1,

T = 298 [ oK] , hωnb = 0.1 [eV ] .

Since both unfurling and hopping can give rise to length independent CT rates it is challenging to distinguish between them experimentaly. In ref. 11 we demonstrated that unfurling rates in a correlated baths model depend on the directionality of the DNA double helix (5’to3’ vs. 3’to5’). Indeed, the rate limiting step for unfurling is the 12 ACS Paragon Plus Environment

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transition to the (de-localized) bridge-dominated orbital from the (localized) donordominated orbital. Changing the helix diretionality primarily affects the energy gap between these orbitals (see below ) and therefore directly influences the CT rate via unfurling. Things should be differnet for the hopping mechanism, where the rate limiting steps for the CT are transitions between (localized) bridge-dominated orbitals, and therefore the helix directionality should not be manifested in the CT rate.

Scheme 2: A scheme of the DNA model of [3’-G(T)NGGG-5’]+ and [5’-G(T)NGGG3’]+. The nucleobases marked in red have differnet on-site energy and coupling matrix elements to their nearest-neihgbors in the two helix directions..

The results in Fig. 4 demonstate the effect of helix directionality on the CT rates for DNA models with different static noise levels and uncorrelated bath fluctuations. Notice that the tight binding parameterization of the double helix46-48 implies that changing the helix directionality from [3’-G(T)NGGG-5’]+ to [5’-G(T)NGGG-3’]+ is reflected only in the onsite energy and inter-site coupling for specific sites at the donor/acceptor-bridge interface (see scheme 2, and Table S1 in the SI for the specific parameters). We present histograms of CT rates calculated independently for each model (5’to3’ and 3’to5’) in the presence of static noise (introduced as described above). For a low noise level (small ∆E , see Fig. 4C), the two distributions (corresponding to the two helix directions) are clearly distinguishable (narrow and well-separated) suggesting that the difference in the CT rate between the two molecules should be experimentally 13 ACS Paragon Plus Environment


detectible. In contrast, for increasing ∆E

values (see Fig. 4B), the two distributions

become broader and overlap each other, to the extent that the measured distributions will be apparently the same for the two molecules (see Fig. 4A). An experimental detection of a clear difference between the measured CT rates in the two molecules would suggest that the transport is dominated by the unfurling mechanism, whereas the lack of directionality effect would be consistent with a hopping mechanism.

Figure 4: Histograms of calculated transfer rates, and Gaussian fits to their mean and standard deviation. Blue and orange correspond respectively to the two different helix directions, [3’-G(T)NGGG-5’]+ (3’to5’) and [5’-G(T)NGGG-3’]+ (5’to3’) with N=16. The rates were obtained from 1000 random model realizations of the onsite energies, set independently for each helix direction. The model assumes an uncorrelated system bath interaction scheme. Rates are in units of nsec-1. The following parameters were used: o N = 16 , T = 298 [ K] , ηnb = 0.007 [eV ] , hωnb = 0.1 [eV ] . Plots (A), (B) and (C)

correspond, respectively, to ∆E = 0.175 ,0.075 ,0.005 [eV ] , demonstrating that the 14 ACS Paragon Plus Environment

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helix directionality becomes distinguishable at high molecular rigidity, where the unfurling mechanism becomes dominant.

We emphasize that G(T)NGGG serves here only as an example. In general, the unfurling should dominate in systems where the DNA sequence is more rigid, corresponding to lower ∆E values. Notice in Fig. 4 that for low ∆E values (where the helix directionality is apparent in the rate distributions), the CT rates increase with increasing ∆E (the centers of the Gaussian distributions are right-shifted). Indeed, for low ∆E values, the main effect of the static noise is to shift the energy gap between the donor and the bridge orbitals, whereas the bridge orbitals are still delocalized. This accelerates the unfurling rate. At yet higher ∆E values, the thermally activated transition into the bridge orbitals is no longer the rate limiting step (since the latter become localized), where the dominant CT mechanism is hopping, and the distributions become indistinguishable. In conclusion, the mere observation of CT rates that are weakly dependent on the bridge length is not sufficient for determining the mechanism of fluctuation-driven CT through DBA molecules. As we demonstrated for DNA models, both quantum unfurling, and sequential hopping give rise to a length-independent phenotype of the CT rate, although the underlying physics is very different. For floppy bridges, where low frequency modes lead to a broad distribution of geometrical structures, hopping is the dominant CT mechanism. However, for rigid ordered bridges delocalized orbitals give rise to quantum unfurling, driven by weak fluctuations about the equilibrium structure. An experimental detection of the underlying mechanism for long-range CT in DBA molecule cannot rely solely on the length-dependence of the CT rate. Comparing 15 ACS Paragon Plus Environment


molecules with similar bridges but different donor/acceptor-bridge energy gaps (e.g., comparing the two helix direction in the case DNA) should enable one to distinguish between hopping and unfurling. Notice that these conclusions are not restricted to the specific DNA models discussed above. In the SI, we show that they are typical for any DBA model. In particular, we demonstrate the manifestation of unfurling in correlated and uncorrelated bath models for different parameterizations of the rigid DNA double helix structure50-53, as well as for a generic DBA model.

Acknowledgments: This research was supported by the Israel Science Foundation Grant No. 1505/14 and

the Adelis Foundation. A.D.L. acknowledges scholarship by the Technion

Graduate School. We acknowledge Barbara A. Levine for reviewing the manuscript.

Supporting Information Available: The details of the Hamiltonian and the rate equations are provided in sections S1 and S2, respectively. In section S3 the role of intrabridge transitions is elucidated, and in section S4 the effect of a static noise on the bridge orbitals is demonstrated. In section S5 different DNA parameterizations from the literature are compared and in section S6 the transition from unfurling to hopping is demonstrated for a generic DBA model.

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Formulation of Long-Range Transport Rates through Molecular

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