Charge Transfer Through Molecules with Multiple Pathways: Quantum

Apr 12, 2010 - It is found that despite the clear effect of dephasing on the charge ..... The European Union FP6 Marie Curie Research Training Network...
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J. Phys. Chem. C 2010, 114, 7973–7979

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Charge Transfer Through Molecules with Multiple Pathways: Quantum Interference and Dephasing Aleksey A. Kocherzhenko, Ferdinand C. Grozema,* and Laurens D. A. Siebbeles Opto-Electronic Materials, Department of Chemical Engineering, Delft UniVersity of Technology, Julianalaan 126, 2628BL Delft, The Netherlands ReceiVed: December 11, 2009; ReVised Manuscript ReceiVed: March 20, 2010

Quantum interference effects occurring in molecules through which a charge can travel via multiple pathways can be the basis for new unconventional design principles in molecular scale electronics. However, these quantum interference effects can be reduced by interaction between the charge and molecular vibrations. In this work dephasing (decoherence) effects have been studied using a model that combines a (classical) molecular mechanics description of molecular vibrations with a quantum mechanical propagation of the charge. It is found that despite the clear effect of dephasing on the charge propagation, interference effects are largely retained at room temperature if vibrations are accounted for. Additionally, it is shown that taking electronic interactions between non-nearest neighbor atoms into account also diminishes interference effects but not sufficiently to destroy them completely. It is concluded that interference effects are strong enough to use them in a functional manner in molecular electronics. This opens up new ways to design molecular electronic components that exploit quantum interference. 1. Introduction Molecular electronics provides an attractive alternative to conventional silicon-based electronics, as it offers cheap and efficient (chemical) production technologies. Additionally, this allows us to go beyond the size limitations of traditional techniques that will be approached in the near future if the integration density in microchips is to continue increasing at the same rate as has been seen over the past few decades.1 Therefore, it is of both fundamental and practical interest to design and study the properties of single-molecule electronic devices. A variety of such devices has been proposed over the years, most notably, molecular rectifiers,2 switches,3 field effect transistors,4 electromechanical single-molecule transistors5 and logic gates.6 Most molecular electronic components are currently designed with the layout of their conventional microelectronic counterparts in mind. In such designs, quantum mechanical effects that are operative on the molecular scale, such as wave function interference, are not always accounted for. This is unfortunate, since it has been shown that taking advantage of these effects can give molecular electronic devices new functionality, which goes beyond that on a macroscopic level.3,7-10 Quantum interference effects are often encountered in organic molecules, although they are not always regarded as such. The most common example is that of coupling of substituents to benzene. The electronic coupling between, for instance, an electron donor and an acceptor is known to be effective in the para and ortho positions but very bad when they are coupled to the benzene in the meta configuration. This effect has been demonstrated in single molecule conductance experiments where the molecule was coupled to electrodes by phenyl moieties either in the para or meta configuration.11 Coupling through the para position resulted in a conductance that was 2 orders of magnitude higher than for the meta configuration. Similar effects have been shown for intramolecular charge transfer.12 In the * To whom correspondence should be addressed. E-mail: F.C.Grozema@ tudelft.nl.

case of the meta configuration the donor and acceptor are said to be connected through a cross-conjugated pathway, which is much less efficient than a linearly conjugated pathway, as found for the para or ortho configuration.13,14 This difference between cross- and linear conjugation can be described in a quantitative way by considering the wave function of the charge that propagates from the donor in both directions along the benzene ring (see Figure 1A).15 The components of the wave function propagating along the two pathways are coherent, resulting in interference when the two waves meet. In the steady state this leads to the formation of a standing wave pattern with antinodes at the ortho and para positions (constructive interference) and nodes at the meta positions (destructive interference). This effect can be expected to occur not only in benzene but also in larger cyclic compounds, such as davidene (Figure 1B) and in noncyclic compounds with the same symmetry, such as hexabenzocoronene (Figure 1C). Fully destructive interference leads to a zero wave function amplitude at the meta position (and, consequently, to the charge density at that position being zero at all times), making charge transfer to an acceptor at the meta position impossible.16 However, this only occurs for perfectly symmetric and completely rigid systems with electronic coupling present only between nearest neighbors. In real systems the coherence of the wave function components traveling along different pathways will always be partially lost due to coupling of the charge to its environment through interactions with vibrations (or phonons) in the molecule and the surrounding medium.17 In a first approximation, molecular vibrations can be considered as time-dependent perturbations,18 displacing nuclei from their equilibrium positions, and thus preventing the formation of a standing wave. As a result, the charge transfer rate to an acceptor at a position where a node would be expected in the absence of vibrations, becomes nonzero. This makes dephasing effects an important aspect to consider when designing molecular components that exploit quantum interference.

10.1021/jp9117216  2010 American Chemical Society Published on Web 04/12/2010

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Figure 1. The chemical structures of the molecules studied: (A) benzene, (B) davidene, and (C) hexabenzocoronene.

The exploitation of quantum interference effects in molecular scale electronics and in donor-bridge-acceptor systems was first considered already in 198819 and is presently attracting growing attention from theoreticians.16,20-27 Originally interference effects in molecules were studied using electron scattering quantum chemistry.3,19 Currently Green’s function methods are predominantly used for exploring single molecule conduction.24 These methods provide a good description of charge transfer by elastic tunneling in rigid molecules. However, the effects of dephasing due to molecular vibrations are largely ignored.16,24 Several Green’s function methods have been developed that include a description of inelastic tunneling. However, most of these methods are only valid in the weak coupling limit where vibrations introduce only a small perturbation to the charge transfer.28 One approach in which dephasing is explicitly considered is the density matrix propagation method.20 However in this case the effect of dephasing cannot be related to the molecular structure and dynamics, since it enters as an empirical parameter. In this paper a fast computational method is proposed, which offers an intuitive understanding of interference and dephasing in molecular systems. The importance of dephasing effects in donor-bridge-acceptor systems with multiple charge transfer pathways is explicitly established using a model that combines a realistic (classical) molecular mechanics description of the molecular vibrations with a quantum mechanical propagation of the charge.

It is important to note that the representation of the charge carrier wave function in terms of pz orbitals on individual carbon atoms, given by eq 1, is identical to a representation as a linear combination of all (occupied as well as unoccupied) delocalized π orbitals. Both descriptions can be converted into one another by a basis set transformation. Using atomic orbitals as a basis set is convenient when describing molecular vibrations since these orbitals are independent of the molecular geometry. In typical experiments on donor-bridge-acceptor systems the charge is created on the donor by, e.g., photoexcitation.31 In most simulations presented in this paper, the charge is assumed to be initially localized on a single carbon atom; thus, the initial conditions are taken in the form

cD(0) ) 1;

Ψ(t) ) cD(t)φD +

∑ cn(t)φn + cA(t)φA

ip

n

with time-dependent expansion coefficients.

(1)

cA(0) ) 0

(2)

Depending on the actual donor-bridge-acceptor system, the charge can also be taken to be delocalized over several atoms that make up the donor moiety. The effect of the actual structure of the donor group on charge transport will be considered below. The transfer of the charge from the donor to the acceptor through a molecular bridge is simulated by propagating the charge wave function, eq 1, according to the time-dependent Schro¨dinger equation

2. Computational Methodology To describe charge transport through a molecular bridge it is assumed that charge transfer can only occur between sp2 carbon atoms. Other atoms influence the geometry and the dynamics of the molecule, however, contributions of the atomic orbitals on these atoms to the molecular orbitals involved in charge transfer is assumed to be negligible. Each sp2 carbon atom is described as a charge localization site, which is coupled to other charge localization sites. To study charge transfer through a molecular bridge a tightbinding model is used.29,30 For the sake of specificity, in further discussion the charge is considered to be a hole (considerations for an electron are similar). The wave function of the hole, Ψ(t), is taken to be a linear combination of (nonhybridized) pz orbitals, φn, on all sp2 carbon atoms, as well as the highest occupied molecular orbitals of the donor, φD, and acceptor, φA, sites

cn(0) ) 0, ∀n;

∂Ψ(t) ) H(t)Ψ(t) ∂t

(3)

with a time-dependent Hamiltonian H(t) and initial conditions given by eq 2. The two important parameters that characterize the rate of charge transfer between neighboring localization sites are the energy εn ) 〈φn|H|φn〉 of a charge localized on the nth site, and the electronic coupling for charge transfer Jmn ) 〈φm|H|φn〉 (charge transfer integral) between sites m and n. In terms of these parameters, the Hamiltonian of the donor-bridge-acceptor system can be written as

(

εD JD1 JD2 ... JDN J1D ε1 J12 ... J1N J2D J21 ε2 ... J2N H) l l l ··· l JND JN1 JN2 ... εN

JDA J1A J2A l JNA

JAD JA1 JA2 ... JAN εA - ip τ

)

(4)

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A complex part is added to the diagonal element of the Hamiltonian matrix, eq 4, on the acceptor site (A) in order to ensure that the charge is irreversibly trapped when it arrives at the acceptor. A decay time τ of 1.3 fs was used in the simulations described below (small enough that the charge disappears from the acceptor site instantaneously, yet not small enough to cause reflections of the wave function on the acceptor site). Variation of τ within reasonable limits has no effect on the simulation results.30 The irreversible decay of the charge at the acceptor site leads to a decay of the total charge density on the donor-bridgeacceptor system. The survival probability, P(t), is defined as the probability that the charge remains in the system at time t, which can be expressed as

P(t) ) |cD(t)| 2 +

∑ |cn(t)|2 + |cA(t)|2

(5)

n

The rate at which the survival probability decays is equivalent to the charge transfer rate through a molecular bridge (the rate of arrival of a charge initially localized on the donor site to the acceptor site). Note that charge transfer is not always a first order process, characterized by a single rate constant, kP. Since charge transfer from the donor to the acceptor does not necessarily occur in a single step, kP does not have to be the same as the rate at which the charge leaves the donor.32 The Hamiltonian given by eq 4 is time-dependent, as both the site energies εn and the charge transfer integrals Jmn are sensitive to the molecular geometry, which changes with time due to thermal vibrations. However, in what follows it is assumed that the site energy is independent of the environment of a specific carbon atom, i.e., εn(t) ) ε ) const, ∀n. This approximation is reasonable, since the variations of site energies are several times smaller than the typical charge transfer integral values.33 The interaction between orbitals on different charge localization sites is assumed to be pairwise and to depend only on the distance between the sites. Thus, the charge transfer integral, Jmn, dependence on the distance, lmn, between two sp2 carbon atoms can be calculated for an ethylene molecule. The structure of this molecule was optimized with DFT, using the program Spartan ’02 for Macintosh34 (BLYP density functional and 6-31G** basis set).35 The distance between the two carbon atoms in the ethylene molecule was then varied between 1.0 and 2.0 Å. The charge transfer integrals between these atoms were calculated as a function of the interatomic distance, lmn, by density functional theory (DFT), using the Amsterdam density functional program (ADF)36 with the asymptotically correct statistical average of orbital potentials (SAOP) model functional and DZP basis set.37 The obtained charge transfer integral values were corrected to account for nonzero spatial overlap of orbitals on neighboring carbon atoms:33,38,39 J′mn ) Jmn - 1/2Smn(εm + εn), where J′mn is the effective charge transfer integral (the prime will be omitted in further discussion), and S ) (Smn) is the orbital spatial overlap matrix. It was found that the effective charge transfer integral dependence on the interatomic distance can be fit to an exponential

Jmn(lmn) ) J0 exp(-lmn /l0)

(6)

where J0 ) 16.69 eV and l0 ) 0.71 × 10-10 m. At equilibrium bond lengths for CdC bonds (alkene or aromatic), eq 6 yields charge transfer integral values of about 2.5 and 2.3 eV,

respectively. These values are in good agreement with those used in the Su-Schrieffer-Heeger Hamiltonian.40 Assuming εn(t) ) ε ) const, ∀n and using eq 6 to calculate the charge transfer integrals between any pair of sp2 carbon atoms m and n, one can now construct the Hamiltonian of the donor-bridge-acceptor system, given by eq 4, for any molecular geometry. In our simulations the wave function of the charge Ψ is propagated in time quantum-mechanically using the time-dependent Schro¨dinger equation, eq 3, with the Hamiltonian given by eq 4, and initial conditions given by eq 2. The time step used for the integration of the timedependent Schro¨dinger equation is one atomic time unit (2.419 × 10-17 s). The geometry of the system is propagated in time classically, using Newton’s formalism. The equations of motion are solved with the Adams-Bashforth-Moulton predictor-corrector method (with the initial points calculated using the Euler method), with a time step of 1-100 atomic time units.41 The forces acting on the atoms are calculated using the Tinker molecular modeling package,42 with the MM2 force field,43 in every step of the geometry propagation (via an interface with our charge propagation program). An additional force is introduced to account for the influence of charge presence in the system on the nuclear motion, it is expressed as40,44

Fq ) -

dJmn(lmn) (cmc*n + c*c m n) dlmn

(7)

where cm and cn are the charge wave function expansion coefficients; see eq 1. At the start of the simulation, the molecules are restricted to a planar conformation, which ensures the absence of out-of-plane forces in all simulation steps. The charge Hamiltonian is updated after every geometry propagation step. 3. Results and Discussion Many previous theoretical studies of charge transfer through molecular bridges are restricted to rigid molecules in their equilibrium conformation, often with only interactions between covalently bound atoms (nearest neighbors) taken into account.16,24 In this case the Hamiltonian of the donor-bridge-acceptor system is time-independent. Results of charge transport simulations for this case, performed according to the methodology described in Section 2, are shown in Figure 2, panels A (benzene), B (davidene), and C (hexabenzocoronene), blue lines. It can be seen (Figure 2A) that in the case of benzene with an acceptor attached at the ortho or para position (Figure 1A) the decay of the survival probability, eq 5, of the charge on the donor-bridge-acceptor system is expected to be very fast (decay rate kP ) 3.0 × 1013 s-1 for an acceptor at the ortho and kP ) 5.4 × 1013 s-1 for an acceptor at the para position). The decay dynamics of the survival probability for the case of the acceptor attached at the meta position can not be characterized by a single rate constant. After an initial fast decay (with kP ) 2.0 × 1013 s-1) by approximately 40% within the first 0.5 ps, the survival probability remains constant (kP ) 0). This effect can be understood if the amplitude of the wave function coefficients, |cn(t)|2, at the meta carbon atoms in the benzene ring are considered as a function of time. After some initial fluctuations when a charge is injected onto the benzene ring from the donor, |cn(t)|2 becomes zero. This means that the wave function of the charge on the benzene ring forms a standing wave with nodes at the meta positions. This is a result of destructive interference

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Figure 2. Decay of the survival probability for bridges consisting of benzene, davidene, and hexabenzocoronene to which a donor and an acceptor are coupled in ortho (dashed), meta (full), and para (dotted) configurations. The curves are plotted for fully rigid molecules (blue) and molecules including vibrations (300 K, red).

of waves traveling from the donor to the acceptor along different pathways (see Figure 1A). This standing wave pattern does not form until components of the wave function traveling along both pathways (of different length) reach the meta carbon atom. In terms of charge transfer, standing wave formation means, that if a charge instantaneously injected into the bridge is not transferred through it within a short initial time interval, it should remain on the bridge infinitely long. The probability of such an event is then given by the equilibrium survival probability value. Since |cn(t)|2 gives the probability of finding a charge at site n, it also becomes evident why charge transfer to an acceptor at the meta position is much slower than to an acceptor at the para and ortho positions.16 In Figure 2B it can seen that the survival probabilities for davidene with acceptors at the ortho, meta, and para positions (Figure 1B) are virtually the same. The initial survival probability decay rate is slower than for benzene (kP ) 6.0 × 1012 to 9.6 × 1012 s-1), due to the larger distance between the donor and the acceptor. The survival probability at longer times is very similar to that for benzene with an acceptor at the meta position. This can be understood, since the individual phenyl rings in davidene are coupled via the meta carbon atoms. It should also be noted that the time required for a standing wave to form in davidene is longer than for benzene, due to its larger size. Eventually (after about 2.5 ps) a standing wave is formed, see Figure 3, green dashed line.

Figure 3. Time-dependence of the survival probability of charge in donor-bridge-acceptor complexes with the acceptor at the meta position as a function of time: benzene (red), davidene (green), and hexabenzocoronene (blue). Electronic couplings between nearest-neighbor atoms only (dashed lines) and between all atoms (solid lines).

In hexabenzocoronene (Figure 1C) the number of pathways along which a charge can travel from the donor to the acceptor is significantly higher than for benzene or davidene. However, adding up the charge transfer along all these pathways leads to destructive interference for an acceptor at the meta position. This is in fact equivalent to saying that all pathways are crossconjugated. In Figure 2C one can see that this again leads to a much slower decay of the survival probability for the acceptor

CT Through Molecules with Multiple Pathways at the meta position than at the ortho or para positions. The charge wave function also forms a standing wave in this case (after approximately 4 ps). The results of the calculations discussed above agree well with previous calculations for benzene and related molecules, carried out using Greens function and density matrix methods, in which only nearest-neighbor interactions were considered.19,20,24 However, since the charge transfer integral values, Jmn, are significant (of the order of several tenths of an electronvolt) for some non-nearest neighbor sp2 carbon atoms, it is critical to take into consideration the full Hamiltonian given by eq 4. The results of simulations at T ) 0 K, taking into account all Jmn, are shown in Figure 2, panels D (benzene), E (davidene), and F (hexabenzocoronene), blue lines. It can be seen (Figure 2D) that for benzene with an acceptor at the ortho or para position, the decay rate of the survival probability increases only slightly upon inclusion of non-nearest neighbor interactions (kP ) 3.3 × 1013 and 5.5 × 1013 s-1, for ortho and para respectively, as opposed to kP ) 3.0 × 1013 and 5.4 × 1013 s-1 in the case when only nearest-neighbor interactions were considered). However, there is a significant increase in the survival probability decay rate for the case of an acceptor at the meta position at longer times (from kP ) 0 to 9.2 × 1012 s-1). This is because for an acceptor at the ortho or para position the charge transfer pathways via nearestneighbors are more favorable, due to the fact that charge transfer integrals Jmn for nearest neighbors are an order of magnitude higher than for non-nearest neighbors. On the other hand, for an acceptor at the meta position charge transfer pathways via nearest-neighbors are forbidden (see Figure 2A). Thus, all charge transfer for a rigid molecule in its equilibrium conformation occurs via non-nearest neighbor interactions. It should be noted, however, that the quantum interference effects are not lost completely, as charge transfer from the donor to an acceptor at the meta position is still expected to be 3 to 6 times slower than to an acceptor at the ortho or para position. For davidene (Figure 2E), the survival probability decay rates for an acceptor at meta or para positions remain relatively small (below 1010 s-1) upon inclusion of non-nearest neighbor interactions. For an acceptor at the ortho position kP ) 6.8 × 1012 s-1 when non-nearest neighbor interactions are taken into account, whereas no decay of the survival probability is found after about 2 ps in the nearest neighbor model, see Figure 3. This is caused by the fact that even in the case where all couplings are included, charge transfer through a meta-coupled phenyl is relatively slow (see Figure 2E). Thus, charge transfer from the donor to an acceptor at the para position, through four meta-coupled phenyls, is much slower than to an acceptor at the ortho position where the charge only passes two metacoupled phenyls. Charge transfer to an acceptor at the para position may be slowed down even further due to slower charge injection into phenyl rings further away from the donor (the effects of a distributed charge wave function on the rate of charge transport are discussed in more detail below). Charge transport to an acceptor at the meta position is slow due to destructive interference of the charge wave function on the scale of the davidene ring as a whole, as well as on the level of the individual phenyl rings. In the case of hexabenzocoronene (Figure 2F), inclusion of non-nearest neighbor interactions has the largest effect on the survival probability when the acceptor is at the meta position (for which charge transport is forbidden at longer time scales when only nearest-neighbor interactions are taken into account). The reason for this is the same as for benzene, discussed above.

J. Phys. Chem. C, Vol. 114, No. 17, 2010 7977 Rates of charge transfer from the donor to an acceptor at the ortho or para positions are still much faster than to an acceptor at the meta position when non-nearest neighbor interactions are included. It is interesting to note that charge transfer to an acceptor at the ortho position becomes slower compared to the case when only nearest neighbor interactions are accounted for. There is no obvious explanation for this fact, it is most probably related to a loss of constructive interference at the ortho site due to the presence of additional pathways for charge transport. In the discussion above only charge transport for completely rigid molecules in their equilibrium conformations was considered. To describe the affects of thermal vibrations of the nuclei on charge transfer through benzene, davidene and hexabenzocoronene molecular bridges, simulations of charge transfer at T ) 300 K were performed (see Figure 2, red curves). In the case of benzene, the effect of vibrations is relatively weak. For an acceptor at the ortho or para position, the decay rate of the survival probability is very high, and changes little upon the introduction of vibrations, both in the case when only nearest neighbor interactions are taken into account (Figure 2A) and in the case when non-nearest neighbor interactions are included (Figure 2D). There is a slightly larger effect of vibrations on the survival probability decay rate for an acceptor at the meta position, better visible for the case when only nearest neighbor interactions are taken into account (Figure 2A). In the more realistic case, when non-nearest neighbor interactions are included (Figure 2D), the effect of vibrations on charge transfer through the bridge is negligible. Such a weak effect of nuclear vibrations is explained by the high rigidity and relatively small size of benzene. As a consequence of this, vibrations do not change significantly the geometry of the molecule, and thus the interference patterns of the wave function are preserved. In the case of davidene, the effect of vibrations is much more pronounced than for benzene, particularly for the case when non-nearest neighbor interactions are included in the simulation (Figure 2E). It is interesting to note that the introduction of vibrations decreases the rate of the (fast) survival probability decay when the acceptor is at the ortho position, and increases the relatively slow survival probability decay rate for an acceptor at the meta or para position. A similar effect is also seen for hexabenzocoronene (Figure 2C,F). This means that the partial loss of coherence of the charge wave function leads to a loss of both constructive and destructive interference, thus diminishing the difference between charge transport pathways through the molecular bridge. Hexabenzocoronene exhibits the strongest effect of vibrations on charge transport (Figure 2C,F). This is due to the large size of this molecule, and a large number of charge transfer pathways through it, all of which experience geometric change when vibrations are introduced. Consequently, the coherence of the waves traveling along different pathways, and thus the interference pattern, is lost to a greater extent than for smaller molecules. It can be seen from Figure 2 that in the realistic case when all interatomic interactions are taken into consideration charge transfer through molecular bridges is relatively fast (the survival probability of the charge on the donor-bridge-acceptor system in most cases decays completely on the time scale of one to several picoseconds). The effect of non-nearest neighbor interactions is significant, and generally exceeds the effect of thermal vibrations at room temperature (at least for small molecules). The effect of vibrations increases with the size of the molecule, this is particularly evident if longer time scales are considered (see Figure 3).

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Figure 4. Survival probability of a charge on a davidene molecular bridge as a function of time: planar conformation (red line) and nonplanar conformation (green line).

The effect of the presence of a charge on the molecular bridge on the molecular bridge geometry has been included in the simulations by means of eq 7. However, for the molecular bridges considered this effect was found to be negligibly small (within the error of the calculation). This is most likely due to the fast delocalization of the charge over the whole molecular bridge, which makes the additional forces given by eq 7 2-3 orders of magnitude smaller than the classical interatomic forces. In all simulations the molecular geometry was restricted to a planar conformation. This is a good approximation for benzene (Figure 1A) and hexabenzocoronene (Figure 1C); however, the equilibrium conformation of davidene (Figure 1B) is nonplanar. Moreover, torsional degrees of freedom are known to account for a significant fraction of the total off-diagonal (charge transfer integral) disorder in molecules where such degrees of freedom are present.45,46 To account for the effect of relative rotations of phenyl rings in the davidene molecule dihedral angles θn,n+1 between neighboring phenyl rings n and (n + 1) in the davidene molecule were introduced into the simulations. These degrees of freedom were assumed to be independent, and free random diffusive rotation (no potential barrier for rotation) was assumed46,47

Figure 5. Survival probability of a charge on a benzene molecular bridge as a function of time: “point” donor (red line) and “extended” donor (green line).

molecule is constrained, the dihedral angles θn,n+1 are not independent, and the equilibrium value of θn,n+1 is close to 40°. Charge transfer rates through a molecular bridge strongly depend on the properties of the donor (and acceptor). In all of the above simulations the charge was assumed to be initially localized on a “point” donor site, see eq 2. The energy of a charge localized on the donor site was assumed to be the same as that of a charge localized on an sp2 carbon atom. The initial distribution of the charge can significantly change the charge transfer rate through a molecular bridge. Figure 5 shows a comparison of the charge transfer rates through a benzene molecule with an acceptor attached at the para position for a charge that is initially localized on a single carbon atom and for a charge that is initially equally distributed over a phenyl ring attached at the para position. Thus, charge transfer rates through molecular bridges strongly depend on the initial distribution of the charge. A similar effect can be observed upon introduction of an energy barrier for injection of a charge onto a molecular bridge. This means that, when charge transfer in real donor-bridge-acceptor complexes is studied, the nature of the specific donor (and acceptor) should be taken into account. 4. Conclusions

diff 〈(∆θn,n+1 )2〉 ) ∆t/τrot

(8)

where the time step ∆t was taken to be the same as the time step in the in-plane molecular dynamics simulation (see section 2) and the diffusion rotation time τrot was taken to be 200 ps.46 The charge transfer integrals between carbon atoms belonging to adjacent phenyl rings where then scaled with the dihedral angle θn,n+1 as follows:33,46

Jn,n+1 ) Jn,n+1(θn,n+1 ) 0) cos θn,n+1

(9)

The results of this simulation for davidene with an acceptor at the para position are shown in Figure 4. It can be seen that the charge transfer rate decreases as a result of the dihedral angle variation given by eq 8 and consequent scaling of the charge transfer integral given by eq 9. This means that even though the rotational disorder leads to additional loss of coherence between waves traveling along different pathways for charge transfer, which can be expected to eliminate destructive interference and increase the survival probability decay rate, the dominant effect of rotational disorder is still simply a reduction in the average charge transfer integral values. In reality, this effect can be expected to be even more pronounced, as the relative rotation of neighboring phenyl rings in a davidene

In this paper the feasibility of exploiting quantum interference effects in molecular scale electronics was examined. Simulations of charge transfer in donor-bridge-acceptor systems with multiple pathways, along which a charge could travel from the donor to the acceptor, were performed. The interaction between the charge and molecular vibrations was explicitly taken into account in a model, which combines a classical molecular mechanics description of the vibrations with a quantum mechanical propagation of the charge. Simulations confirm again that quantum interference effects have a determining effect on charge transfer in symmetric molecules such as benzene, davidene, and hexabenzocoronene. Destructive interference is observed when the donor and the acceptor are coupled in the meta configuration, whereas constructive interference occurs when they are coupled in the ortho or para configuration. At higher temperatures, molecular vibrations leads to a time-dependent perturbation of the Hamiltonian, partially destroying the coherence between charge transfer pathways. However, at room temperature the loss of interference (or decoherence) occurs on a relatively long time scale (at least several picoseconds). Thus control of charge transfer by interference effects is possible in molecular devices (such as the molecular transistor proposed in ref 18). It is shown that taking into consideration non-nearest neighbor electronic couplings is crucial for an adequate description of

CT Through Molecules with Multiple Pathways charge transfer through molecular bridges. It is also important to note that charge transfer in donor-bridge-acceptor systems strongly depends on the properties of the donor (and the acceptor), namely, their energy levels and the initial charge distribution. Acknowledgment. We thank Prof. David Beljonne and Michael Wykes of Materia Nova (Mons, Belgium) for useful discussions. The European Union FP6 Marie Curie Research Training Network “THREADMILL” (Contract Number MRTNCT-2006-036040) is acknowledged for financial support. Supporting Information Available: Molecular coordinates for benzene, davidene, and hexabenzocoronene (optimized geometries). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Moore, G. E. Electronics 1965, 38, 114. (2) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277. (3) Sautet, P.; Joachim, C. Chem. Phys. 1989, 135, 99. (4) Aviram, A. J. Am. Chem. Soc. 1988, 110, 5687. (5) Joachim, C.; Gimzewski, J. K.; Tang, H. Phys. ReV. B 1998, 58, 16407. (6) Carter, F. L. Physica D 1984, 10, 175. (7) Duchemin, I.; Joachim, C. Chem. Phys. Lett. 2005, 406, 167. (8) Remacle, F.; Levine, R. D. Phys. ReV. A 2006, 73, 033820. (9) Andrews, D. Q.; Solomon, G. C.; Van Duyne, R. P.; Ratner, M. A. J. Am. Chem. Soc. 2008, 130, 17309. (10) Solomon, G. C.; Andrews, D. Q.; Van Duyne, R. P.; Ratner, M. A. J. Am. Chem. Soc. 2008, 130, 7788. (11) Mayor, M.; Weber, H. B.; Reichert, J.; Elbing, M.; Von Hanisch, C.; Beckmann, D.; Fischer, M. Angew. Chem., Int. Ed. 2003, 42, 5834. (12) Patoux, C.; Coudret, C.; Launay, J.-P.; Joachim, C.; Gourdon, A. Inorg. Chem. 1997, 36, 5037. (13) Gholami, M.; Tykwinski, R. R. Chem. ReV. 2006, 106, 4997. (14) van der Veen, M. H.; Rispens, M. T.; Jonkman, H. T.; Hummelen, J. C. AdV. Funct. Mater. 2004, 14, 215. (15) Ke, S.-H.; Yang, W. Nano Lett. 2008, 8, 3257. (16) Cardamone, D. M.; Stafford, C. A.; Mazumdar, S. Nano Lett. 2006, 6, 2422. (17) Zurek, W. H. ReV. Mod. Phys. 2003, 75, 715. (18) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics: Non-RelatiVistic Theory, 3rd ed.; Butterworth-Heinemann: Oxford, 1981; Vol. 3. (19) Sautet, P.; Joachim, C. Chem. Phys. Lett. 1988, 153, 511. (20) Goldsmith, R. H.; Wasielewski, M. R.; Ratner, M. A. J. Phys. Chem. B 2006, 110, 20258. (21) Yoshizawa, K.; Tada, T.; Staykov, A. J. Am. Chem. Soc. 2008, 130, 9406. (22) Weiss, E. A.; Katz, G.; Goldsmith, R. H.; Wasielewski, M. R.; Ratner, M. A. J. Chem. Phys. 2006, 124, 074501. (23) Walter, D.; Neuhauser, D.; Baer, R. Chem. Phys. 2004, 299, 139. (24) Solomon, G. C.; Andrews, D. Q.; Hansen, T.; Goldsmith, R. H.; Wasielewski, M. R.; Van Duyne, R. P.; Ratner, M. A. J. Chem. Phys. 2008, 129, 054701. (25) Skourtis, S. S.; Waldeck, D. H.; Beratan, D. N. J. Phys. Chem. B 2004, 108, 15511.

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