Topological Effects in Electron Transfer: Applications to Dendrimers

Quantum Chemistry Research Group, Albuquerque High-Performance Computing Center,. Albuquerque, New Mexico 87131, and The Santa Fe Institute, Santa ...
0 downloads 0 Views 118KB Size
Download PDF
370

J. Phys. Chem. B 2001, 105, 370-378

Topological Effects in Electron Transfer: Applications to Dendrimers and Branched Molecules T. Sean Elicker,†,‡,| Jean-Sebastien Binette,† and Deborah G. Evans*,†,§ Department of Chemistry, UniVersity of New Mexico, Albuquerque, New Mexico 87131-1096, Quantum Chemistry Research Group, Albuquerque High-Performance Computing Center, Albuquerque, New Mexico 87131, and The Santa Fe Institute, Santa Fe, New Mexico 87501 ReceiVed: July 5, 2000; In Final Form: October 6, 2000

Dendrimers belong to a class of complex hyperbranched structures through which electron transfer has been observed; in this work, we study the effect of the bridge topology on long-range transfer. Electron-transfer rates for bridge-mediated electron transfer are calculated by solving the steady-state Liouville equation with phenomenological corrections for solvent interaction. While the behavior of linear bridge groups is wellknown, branched structures display interesting electron transport properties. Addition of a small side group with nodes equivalent to the bridge nodes increases the steady-state rate, and this effect is amplified by increasing the site-site couplings in the side group. The effect for a single side group is small, but increases with each additional side group producing a noticeable increase in electron transfer rates for structures with several side groups such as dendrimers. Addition of an electron withdrawing side group at the acceptor end of the bridge creates a structure that can manifest asymmetric tunneling in the weak solvent limit. We demonstrate that this potential for rectification as well as the electron transfer rates can readily be manipulated by varying the on-site energies of the side group.

I. Introduction As the quest for smaller and faster electronic devices continues, interest in linear-chain molecules as “wires” increases.1,2 The electronic structure of these molecules is very different from that of a regular metallic wire, and consequently electron-transfer characteristics are not well-explained by typical models of conduction.3,4 A number of recent studies have examined the properties of long-range electron-transfer systems.5-8 Electron transfer in proteins is a well-studied problem both experimentally and theoretically.9,10 Recent work on electron transfer in DNA has resulted in many theoretical and experimental studies to determine whether DNA can in fact function as a “molecular wire”.11,12 Investigations of molecular wires have not been limited to these biological molecules. A number of studies of electron transfer through molecules on surfaces13 have been used to demonstrate electron transmission through single molecules. Systems have also been studied beyond the molecular level; for example, the recent conduction measurements of carbon nanotubules.14 Theoretical studies of conduction through molecular systems have highlighted key differences at the molecular scale. These include the effects of solvent interactions and molecular vibrational motion on the conductance. The rich and varied behavior of these molecular systems is complicated, but they provide the potential for new forms of control and switching that exploit the particular properites of these molecular systems. Previous theoretical treatment of electron transfer in bridged donor-acceptor molecular systems where the coupling to the solvent is explicitly taken into account15 have demonstrated well-known behavior: McConnell* Corresponding author. Fax: (505)-277-2609. E-mail: [email protected]. † University of New Mexico. ‡ The Santa Fe Institute. § Albuquerque High-Performance Computing Center. | Deceased.

like scaling16 with molecular length in the weak solvent regime, and a turnover region of the electron tranfer rates as higher couplings are approached. These studies have focused exclusively on models for systems like thiols and carbon nanotubules where the bridge is treated as a linear chain of sites. Recently, we investigated the possibility of using dendrimers17san interesting class of molecules18 that have great potential in many areas of molecular-scale engineeringsas molecular conductors. Electron transfer through dendrimeric species has been measured for a number of different structures, including photoinduced electron-transfer studies of structures with encapsulated electroactive cores.19 These observations demonstrate that dendrimers can act as suitable bridging media for electron transfer between redox centers, and indicate the feasibility of controlling photoinduced electron transfer in dendrimers with a suitable choice of electron/donor functionality at the core and on the periphery. Our previous time-dependent simulations17 have shown that electron transfer in dendrimers depends on both the specific connectivity of the dendrimer structure and the nature of the solvent fluctuations. These calculations focused on electron-transfer dynamics following photoexcitation in the typical photoinduced experiments mentioned above.19 While pertinent to such experiments, the timedependent method is not applicable to the study of dendrimers as molecular wires, where for example dendrimers are adsorbed onto surfaces,20 or attached to contacts by two termini. This study focuses on electron transfer through dendrimers in such systems. To understand electron transfer in these dendrimeric structures, we have carried out a systematic study of electron transfer through molecular bridges with a generic side group topology. This study is therefore relevant to more general questions concerning long-range electron transfer especially given the recent experimental studies on switching in branched topologies.18

10.1021/jp002398r CCC: $20.00 © 2001 American Chemical Society Published on Web 12/21/2000

Topological Effects in Electron Transfer

Figure 1. Comparison of linear and dendrimeric electron transfer between a donor [D] and acceptor [A]; the thick lines show the D f A paths of equivalent lengths.

In section II, we outline a tight-binding approach for studying steady-state electron transfer in general chain structures. Using a model analogous to that developed by Ratner et al. for linear chains,15 we consider a system where two terminal nodes in a dendrimeric system are connected to a circuit. The two termini are connected to separate branches of the dendrimer, and the resulting system has a unique nonrecurrent path of length equal to the diameter of the dendrimer. In this way, the dendrimer topology is simply an extension of the typical linear-chain polymer with side groups [see Figure 1]. By setting up steadystate boundary conditions, electron-transfer rates for these complex dendrimeric systems are extracted from our tightbinding models in many different solvent coupling regimes. To understand the electron-transfer rates in dendrimers, in section III this methodology is applied to address a more general question: namely, what are the effects of bridge topology on steady-state electron transfer in donor-bridge-acceptor systems? This study is timely, given the intense interest in molecular electronics and studies on molecular wires and switches.2,21,22 It is also relevant to the recent femtosecond studies of electron transfer systems where the effects of the bridge structure are explicitly used to generate switching behavior.23 While motivated by the experimental work on the electrochemical behavior of dendrimers,19 this study explicitly examines the role of bridge topology on electron transfer rates. It demonstrates that interesting physical manipulations are possible with complex bridge geometries. The behavior found for the general systems studied here is therefore not only applicable to dendrimers but can be extended to other systems with more general topologies, including branched thiol wires and branched carbon nanotubule structures.14 II. Methods The steady-state model we have used to calculate electrontransfer rates is based on a conventional N-site tight-binding Hamiltonian, where the on-site energy, E, of the two nodes connected to the external contacts is -1500 cm-1 (relative to the bridge site energies, which for convenience are taken as zero) and all nearest-neighbor (n-n) couplings, W, are 300 cm-1, unless otherwise explicitly stated. N n-n only

H ˆ ) E(|1〉〈1| + |N〉〈N|) + W

∑ ∑ i)1 j< j| + |j >< i|]

From this Hamiltonian we generate the Liouville superoperator L ˆ , which is an N2 × N2 matrix that operates on the density

J. Phys. Chem. B, Vol. 105, No. 2, 2001 371 vector b F ) (F11, F12, ..., F1N, F21, ..., FNN) in the Liouville equation ˘F ) -iL ˆb F, where p ) 1. Following the approach of Ratner et B ˆ ) al.,15 the standard Liouville superoperator is replaced by A L ˆ - iL ˆ D, where L ˆ D is a diagonal matrix containing phenomological parameters to account for the dissipation effects of the solvent (hereafter referred to by the parameter γ) and the connection to complete the circuit (κ). Throughout this work, we use κ ) -400 cm-1, to allow direct comparison with the previous studies of linear bridges.15 γ is taken as a variable parameter over the same range as the previous studies on linear chains.15 Within this range, asymptotic behavior can be examined and these values span from a description of low γ [tunneling], to intermediate γ [hopping] to strong γ [the incoherent regime]. To determine the steady-state electron˘ transfer rate, we set B F equal to zero to reduce the system to a set of linear equations; A ˆb F ) -c b, where b c ) (c, 0, ..., 0) accounts for the influx of electron density from the exterior circuit. The electron-transfer rate can be calculated as the ratio between the steady-state flux through the system and the population of the -1 donor level: rss ) c/Fss 11. This rate is calculated in units of ps throughout, unless otherwise stated. A comparison of the above steady-state rates for dendrimers and linear bridged systems (as shown in Figure 1) is given in Figure 2. For the sites of the dendrimer sidebranches, the onsite energies are equivalent to the other bridge nodes, and only nearest neighbor site-site couplings are considered. As shown in the figure, at low to intermediate dissipation (low to intermediate values of γ), the dendrimer shows enhanced electron-transfer rates relative to its linear counterpart, and this enhancement is accentuated as the value of the electronic coupling is increased. This is an exciting observation in view of the potential use of branched polymers as molecular wires. It is also a reasonable observation based on previous work by Risser et al.24 on electron coupling in dendrimer materials which shows that, for energy ranges outside the band, starburst materials are expected to have larger coupling and faster electron-transfer rates than their linear counterparts. While these results refer to materials of infinite extent in the absence of solvent coupling, Risser et al. have shown that these results carry over surprisingly well to dendrimers of finite extent. It is clear that our model, involving steady-state electron flow from one terminal to another, is not exactly the same. However, our work does show an increase in transfer rates for the systems shown in Figure 2, even when the solvent coupling is nonzero. Note also that as the high coupling limit is approached and the sequential hopping mechanism predominates, the rates for the branched chains and their linear counterparts approach the same limit. This seems reasonable given previous results for linear polymers that show the rate essentially independent of the length and size of the bridge in the incoherent limit.15 At some value of the dephasing the rates are no longer affected by the size and molecular details of the bridge. These observations concerning dendrimers are interesting, but difficult to predict given the structural complexity of these macromolecules. To investigate more fully the possible effects of bridge topology on electron transfer, we expand our study to “generalized chains,” a larger class of structures that includes both linear chains and hyperbranched dendrimers. We define a generalized chain as a linear chain of length N, where nodes 1 and N (which also assume the roles of the donor and acceptor) are connected to the external circuit. A side group is then attached to each node k, 2 e k e N - 1, and the null side group is permitted. Note that this definition includes dendrimers as a particular type of generalized chain. For example in Figure

372 J. Phys. Chem. B, Vol. 105, No. 2, 2001

Elicker et al.

Figure 2. Comparison of steady-state transfer rates for the linear chains with their dendrimer counterparts as shown in Figure 1. The rate is enhanced in the dendrimers, and the enhancement increases as the nearest-neighbor coupling [W, cm-1] is increased.

at node k to node N - k + 1 for each k. If such a structure has ∆ > 1, its mirror image will have ∆ < 1 , and vice versa. Moreover, a generalized chain that is symmetric under this transformation (this includes all dendrimers with chemically equivalent nodes) will not show asymmetric transfer and will have ∆ ) 1. III. Results and Discussion

Figure 3. Three-generation dendrimer can be “unfolded” into a generalized chain with N ) 5 and three side groups.

3, a three-generation dendrimer can be “unfolded” and viewed as a generalized chain with N ) 5, with single-node side groups attached at k ) 2 and k ) 4, and a three-node branched side group attached at k ) 3. With these models, we have studied not only the effect of topology on electron transfer rates in generalized chains, but we have also identified possible rectifying structures. These structures show asymmetric electron transfer in that the rate in a forward direction through the structure is not necessarily the same as the electron transfer rate in the reverse direction, even when the donor and acceptor on-site energies are equivalent. For an isolated molecule, this clearly will not be the case, but for our particular model, where there is relaxation from the acceptor to the solvent with a rate κ, unless the bridge is symmetrical, the observed rates will depend on the proximity and position of particular side groups relative to this acceptor site. To quantify the inherent rectification possibilities for a particular structure, we compute the steady-state rate rss for two systems: the “forward” (1 f N) rate rss F and the “backward” (1 . The asymmetry in the forward-backward r N) rate rss B electron-transfer rates is then denoted as ∆, which is the ratio of these two quantities. Therefore, ∆ > 1 indicates a faster rate in the forward direction (an in a circuit this would correspond to a forward rectifier) and ∆ < 1 shows a slower rate in the backward direction and as part of circuit would be reverse rectification. Note that when E1 ) EN, the forward and backward rates refer to structures that differ only by moving the side group

Although the quantity of interest is the steady-state rate rss, in some instances the differences between systems being compared are extremely small. In these cases we will refer instead to the “gain” gss ) rss(gc)/rss(linear), which is simply the ratio of the rate in the generalized chain (gc) to the rate in a linear chain of the same length. We only compare forward and backward currents in generalized chains of the same length, and the denominators in the two gain expressions will be the same for any given structure. Therefore, ∆ may be computed ss ss ss as either rss F /rB or gF /gB . (i) Generalized Chains: Uniform On-Site Energies. The simplest nonlinear generalized chain is one with a single-node side group (SG), where that single node has an on-site energy, ESG ) Ebridge. (We assume here that all the bridge nodes in the underlying chain are identical.) In Figure 4, the gain as a function of the dephasing parameter, γ, is plotted for this structure with a single-node side group at position k ) 2 along the bridge, for both forward and backward flow. The maximum gain in both directions is less than 1.1 for all three chain lengths, with N ) 5, 6, and 7. There is slight asymmetry in forwardbackward rates for γ values near 1.0, but this is unlikely to be useful to device applications due its relatively small value. When the site-site coupling between the side group node and the bridge is increased to twice the normal value [the coupling is increased from 300 cm-1 to 600 cm-1], the gain at lower γ values increases to roughly 1.4. The reverse current is slighly larger than the forward current in the weakly solvated regime, while the forward and reverse rates behave quantitatively differently in the intermediate solvent dephasing regime. In the regime of 10-1 e γ e 104 the forward current gain increases while the backward current gain falls off to 1.0. However, the maximum value of ∆ in these systems is still below 1.65, which is of academic interest but not sufficient for workable device applications. As seen in Figure 5, the behavior of a generalized chain with a single three-node side group is very similar to the behavior

Topological Effects in Electron Transfer

J. Phys. Chem. B, Vol. 105, No. 2, 2001 373

Figure 4. Comparison of gains for various systems having one single-node side group with Ebridge.

Figure 5. Comparison of gains for various systems with one three-node branched side group in which all nodes have ESG ) Ebridge.

of the single-node system. This three-node side group is a branched side group with one of the nodes connected directly to the bridge. It is placed at the position k ) 2 along the chain and the on-site energies ESG ) Ebridge ) 0 for all side group nodes. There is some difference when we increase all the side group nearest-neighbor couplings (Wij, where i and/or j is a node in the side group) by a factor of 2 from 300 cm-1 (regular coupling) to 600 cm-1 (increased coupling). As before, the asymmetry ∆ is slightly less than unity for small γ, but is greater than one for γ between 10-2 and 102. In this case, however, the maximum asymmetry, ∆ is approximately 2.25, which is nonnegligible. The steady-state solution of the Liouville equation can also provide the steady-state electron densities on each site of the bridge. For the three-site side group, we have found that the increased gain that is observed is accompanied by an increase in the steady-state density present on the side group nodes. An increase in electron density on the side group is correlated with an increase in the observed electron-transfer rate from the donor to the acceptor through the bridge in comparison with systems that do not have this side group. As the coupling to the side

group is increased, the electron density on this side group increases at steady state, and the steady-state electron-transfer rate is observed to be higher. In light of these observations, we can now reexamine the behavior of the dendrimer presented in the previous section. For generalized chain structures, we have found the presence of each of the three side groups contributes an increase in gain. It would therefore be expected that addition of side groups would have an cumulative effect. This hypothesis is confirmed by the data in Figure 6, which compares the gain of the dendrimer studied in the previous section with several intermediate structures that have one or more of the requisite side groups. The gain in the low dissipation regime increases from 1.0 to the dendrimer value as the structure of the generalized chain approaches that of the dendrimer. The figure shows the results for several branched structures of a chain of length N ) 5. These structures are denoted by five integers (0, n2, n3, n4, 0), where the integers nj signify the number of nodes nj for a side group placed at position j along the linear chain. These results show a smooth increase in the gain from the linear chain with no side groups (00000) to the dendrimeric structure (01310) through

374 J. Phys. Chem. B, Vol. 105, No. 2, 2001

Figure 6. Gain plots for the three-generation dendrimer and a number of quasi-dendrimeric generalized chains. These chains are denoted by (0, n2, n3, n4, 0), where the integers nj signify the number of nodes nj for a side group placed at position j along the 5-node linear chain.

structures with an increasing number of side groups: for example (01000) and (01300), respectively. (ii) Generalized Chains: Inequivalent On-Site Energies. The above studies have examined systems where all the bridge sites and the side group nodes have equal on-site energies. The addition of a single-node side group trap [which could be an electron-withdrawing side group], with on-site energy equal to E1 and/or EN, can show much more interesting effects. It is important to note that even though these side groups may have on-site energies equal to those of the sites 1 and N, we have not examined systems with multiple contact points. As for all the systems studied above, sites 1 and N are the only points that contact the external circuit. If this single-node trap is attached at k ) 2, the behavior for large γ values is the same as the previously discussed systems. In Figure 7, we show that rss F decreases sharply with γ in the weakly solvated regime. As we move toward the coherent (very small values of γ) limit, gss F becomes less than unity and approaches zero. This same behavior does not occur for gss B . Instead, the gain for the N f 1 current is higher than the rate for a regular chain. Thus, this structure (a single-node trap at k ) 2) could in a circuit behave as a reverse rectifier, and by symmetry the structure with a single-node trap at k ) N - 1 will be (for small

Elicker et al. γ) a rectifier, with ∆ . 1. Although Figure 7 resembles Figures 4 and 5 qualitatively, there is a large quantitative difference. The gain for a single-node trap is an order of magnitude larger than the gain for a bridgelike side group. Since the absolute rates in all cases are still on the order of 10-1 or less, we note that the device applications may be limited despite this large value for the gain. Nevertheless, this asymmetry should be observable, and may be useful in molecular-scale experiments. As shown in Figure 7, increasing the site-site coupling to the side group from 300 cm-1 (regular coupling) to 600 cm-1 has a significant effect. We see that the gain now depends inversely on the side group coupling, which is contrary to the behavior of the systems studied in Figures 4 and 5, where the side groups have on-site energies equivalent to the other bridge sites. This observation is consistent with the correlation noted before between the steady-state density on the side groups, and the observed electron-transfer rate. When the side group has a lower on-site energy, an increased site-site coupling results in less density being located on the side group node, and a lower rate is observed. In the previous section we noted that when the side group has an on-site energy equivalent to the rest of the bridge sites, increased coupling increases the steady-state density on the side group and an increase in the rate from the donor to the acceptor is observed. Figure 8 shows the gain as a function of the dephasing parameter γ, for a three-node branched side group where only the terminal nodes are electron traps [i.e. their on-site energies are equal to E1]. Although the behavior is qualitatively similar to the single-node data in Figure 7, we see that the overall magnitude of the gain is much smaller. This is largely a consequence of the absolute rates being much smaller for the larger side group trap. Unlike the single-node case, this threenode side group also does not exhibit the sharp decrease of rss F with γ in the weakly solvated regime. A natural extension of the above studies with electronwithdrawing side group traps is a dendrimeric structure in which all the terminal nodes have energies equal to E1 and/or EN. This type of structure is in fact a more realistic representation of dendrimers that have been synthesized in the laboratory than the dendrimers already discussed. In a dendrimer created via normal convergent synthesis,25 all nodes in a given generation are chemically equivalent. Therefore, a dendrimer with all the

Figure 7. Comparison of gains for various systems with a single-node electron trap (ESG ) E1) as a side group.

Topological Effects in Electron Transfer

J. Phys. Chem. B, Vol. 105, No. 2, 2001 375

Figure 8. Comparison of gains for various systems with a three-node branched side group in which the two terminal nodes have ESG ) E1.

Figure 9. Comparison of rates between a ten-node dendrimer where E ) -1500 cm-1 for only the two terminal nodes connected to the circuit and a dendrimer in which E ) -1500 cm-1 for all six terminal nodes.

nodes in the terminal generation being equivalent is typical for structures that have been synthesized. In Figure 9, we have compared such a dendrimer with all terminal nodes having equivalent energies EN ) -1500 cm-1 to the previously studied dendrimer shown in Figures 2 and 6, where all terminal nodes except the Nth are equivalent to the other bridge sites with E ) 0. The observed results are consistent with the behavior when single side group traps are added. At low values of the dephasing parameter γ, the observed rate is lower, which is the same behavior that is observed for a single-node trap at k ) 2. Note, however, that for moderate values of γ the rate for this dendrimer becomes larger than that of both the linear chain and the idealized dendrimer. This suggests that enhanced electrontransfer rates for solvated dendrimers can be observed. (iii) On-Site Energy Changes on the Branches. In all of the above systems we used only two on-site energies for the side group nodes; ESG ) E1()EN) and ESG ) Ebridge. The behavior of systems with single-node side groups shows interesting variations when we scan over a wide range of side group-node energies. The electron-transfer rates in these systems exhibit a resonance peak that is related to the energies of those nodes where electron density enters or exits the molecule. This behavior is to be expected for isolated molecular systems given that the side-group energies affect the eigenstates of the entire

bridge and in the absence of solvent, these are known to give rise to enhanced transmission rates.8 Here we show that these properties can be harnessed in specific molecular geometries to manipulate current directionality. Figure 10 shows the behavior for a weakly solvated (γ ) 10-4) system with a single 1-node side group. The peak in rss F (rss B ) occurs at or near E1 (EN). (Of course, in the reversecurrent system the roles of node 1 and node N are reversed.) The large changes in the electron transfer rates allow for the possibility of creating a molecular switch. For example, a side group with a large dipole moment could have its orbital energies manipulated by an external electric field. By tuning the energy of the node to one of the two peaks, one could effectively shut off electron transfer in a particular direction or reverse the direction of the flow. As shown in Figure 11, modulating the energy of the side group, E, in a symmetric system (E1 ) EN ) -1500 cm-1) can change rss F by several orders of magnitude, effectively turning electron transfer “on” and “off.” In the asymmetric system (E1 ) -1500 cm-1, EN ) 2250 cm-1), moving between two side group-node energies changes the sign of ∆, with the magnitude of ∆ close to 10.0 in both cases. While the values of the absolute rates are small in both cases, the potential for molecular switching is apparent. Experimentally, for a polar side group, these required energy changes of 1000 cm-1 can be induced in a side group branch with a dipole moment of 5-10 D by an external electric field of 0.1 - 1 MV/cm. This is quite reasonable, given that fields of this magnitude have been used in many studies of field effects on electron-transfer rates in the photosynthetic reaction center.26 IV. Conclusions Transfer of electrons from one localized state to another through a molecular bridge is a very common model system for a number of chemical processes. With the increased interest in molecular-scale wires and fundamental questions concerning electron transfer in proteins and DNA, a number of theoretical and experimental studies have provided insight into bridgemediated electron transfer. During the past few years, interest in electron transfer through a class of hyperbranched macromolecules called dendrimers18 has been fueled by their synthetic richness and their interesting structure. This complex structure

376 J. Phys. Chem. B, Vol. 105, No. 2, 2001

Elicker et al.

Figure 10. Dependence of the rate on side group on-site energy (in cm-1) for four different (E1, EN) settings. The solid line shows the forward rate ss rss F , and the dotted line, the backward rate, rB . Note the location of the resonance peaks in relation to the donor and acceptor energies.

Figure 11. Two applications to molecular switching: the symmetric system yields an on/off switch, while the asymmetric system produces a switch capable of reversing the direction of electron transfer.

lends itself to the possibility of manipulating electron transport in these molecules and makes their behavior as molecular wires extremely interesting. In this study we have used a simple model to study the differences in the behavior of these molecules and their linear counterparts. Understanding their electron transport is directly related to how the topology of the bridge affects their behavior. We have therefore conducted a systematic study of the effect of side-branches from a bridge on the electron-transfer rate between donor and acceptor sites of a molecule. Given very recent studies on switching in a set of branched oligomers,23 a study of systems with branched topology is relevant. Unlike previous studies of electron transfer in dendrimers,24 we have adopted a model where a steady-state current is set up between a donor and an acceptor as part of a circuit. This model has been very successful in elucidating electron transfer in linear bridge-mediated electron transfer in all solvent coupling regimes by incorporating the solvent influences phenomenologically. This approach therefore provides a way of studying topological effects in the presence of solvent, and allows a direct comparison of branched systems with linear bridges. It allows an examina-

tion of the behavior in many solvent regimes, and has allowed us to examine the effects of side-groups for molecular systems, including dendrimers, incorporated into molecular circuits. The effects of branches on electron-transfer pathways in proteins have also been studied recently.27 Here, in the absence of solvent, it is shown by Lo¨wdin decomposition techniques that the changes induced by the branches can be described in terms of renormalized matrix elements along the main pathway. By including solvent dissipation in our model, a simple Lo¨wdin decomposition is no longer possible. However, the physical origin for the changes in the observed behavior for these two systems is obviously the same: the branches affect the bridge eigenstate energies and consequently the electron-transfer rate. Our studies have shown that for electron transfer in molecular systems with typical donor-bridge-acceptor energies and couplings, the rate in dendrimeric structures is enhanced relative to the linear chain counterparts. This agrees with previous work on dendrimers in the absence of solvent dissipation.24 We have shown enhanced rates for dendrimers in all but the highly dissipative regime, where the branched topology of the bridge has little effect on the rates. This is consistent with observations

Topological Effects in Electron Transfer for linear chains where, at high solvent coupling, the rate becomes only weakly dependent on the distance between the donor and acceptor.15 We have also shown that each side-group increases the electron transfer rate, so that for dendrimers [which have side-groups on all bridge nodes] the electron transfer rate increase is significant. When considering realistic models of synthetic dendrimers where all the terminal nodes are equivalent, we have also found that even larger deviations from linear chain behavior are found. In intermediate solvent dephasing regimes, these realistic dendrimer structures show an even greater enhancement of electron transfer rates. We have shown that the influence of side groups on the electron transfer rates is cumulative, and the enhanced electron transfer in dendrimers is a simple extension of the behavior found for simpler branched structures with only one side group. From our studies of these more general branched bridge topologies, we have also found that these side groups have an effect on the rate which depends on their position along the major chain backbone. Consequently, for a given branched structure with a side group, the electrontransfer rate depends on whether the current is passed in a forward or backward direction. This behavior can be enhanced by varying the nearest-neighbor coupling within the side groups and thereby used to create a molecular rectifier. This asymmetry in the electron-transfer rates occurs despite the donor and acceptor having the same onsite energies. It is a consequence of the relaxation to the solvent from the acceptor site. In view of this, the mechanism is not the same as that proposed for molecular rectifiers by Aviram and Ratner,28 but nevertheless, in solvent, it should be possible to detect this asymmetry using experimental techniques similar to those used in studies of Aviram-Ratner molecular rectifiers.29 Interesting methods of manipulating the electron-transfer rates are found in our study of systems where the on-site energies of the side group nodes are not equivalent to the other bridge sites. For example, we have found that these systems show more pronounced asymmetry in the forward-backward electrontransfer rates. We have found that electron-transfer rates in these structures have resonances when the on-site energy of the side group node is near the energy of the donor or acceptor node, and that these resonances may be exploited to create two types of molecular switch (on/off and forward/backward) in different systems. Manipulating the on-site energies of terminal groups on the dendrimer structure would allow for the possibility of many interesting changes in electron-transfer rate. Although originally motivated by the study of electron transfer in dendrimeric macromolecules, this work constitutes a more general study of the effects of branching in bridge-mediated electron transfer. Since the model is a tight-binding representation of a system, the effects that we find due to topology or changes of the on-site side group energies are not restricted to a particular chemical system, and may be applicable to a variety of molecular wire systems and mesoscale systems like carbon nanotubes.30 This model has focused on the essential differences between linear chain bridge systems and branched chain or dendrimeric systems. All our calculations have used only nearest-neighbor coupling. Given that very interesting properties have been observed for linear chain models where secondneighbor and higher order interactions are included,31 a future study will focus on this issue, particularily on the effect of longer-range couplings on luminescence decay times. Other studies will include the application of a similar theoretical approach32,33 (which uses the Redfield equations of motion) to study branched systems, and dendrimers in particular. An advantage of this new approach will be the ability to directly

J. Phys. Chem. B, Vol. 105, No. 2, 2001 377 compare the time-dependent and steady-state electron-transfer behavior within the same model,33 and thereby simulate both photoinduced and steady-state electron-transfer experiments. Acknowledgment. We thank the National Science Foundation and the Petroleum Research Fund administered by donors of the ACS for partial financial support of this research. D.G.E. is a Cottrell Scholar of the Research Corporation and a Camille Dreyfus Teacher-Scholar. We would also like to thank the Santa Fe Institute and the Albuquerque High-Performance Computing Center for use of their computational resources. Sean, you were our friend and co-worker. We fondly remember and admire your love of science and your creativity. References and Notes (1) A small sampling of some of the more recent theoretical developments include: (a) Tour, J. M.; Kozaki, M.; Seminario, J. M. J. Am. Chem. Soc. 1998, 120, 8486-8493. (b) Larsson, S.; Klimkans, A. Theochem J. Mol. Struct. 1999, 464, 59-65. (c) Ratner, M. A.; Davis, B.; Kemp, M.; Mujica, V.; Roitberg, A.; Yaliraki, S. Ann. New York Acad. Sci. 1998, 852, 22-37. (d) Magoga, M.; Joachim, C. Phys. ReV. B 1999, 59, 16011-16021. (2) A few examples of the experimental literature include these and the references therein: (a) Reed, M. A.; Tour, J. M. Sci. Am. 2000, 282, 86-93. (b) Schumm, J. S.; Pearson, D. L.; Jones, L.; Hara, R.; Tour, J. M. Nanotechnology 1996, 7, 430-433. (c) Bumm, L. A.; Arnold, J. J.; Cygan, M. T.; Dunbar, T. D.; Burgin, T. P.; Jones, L.; Allara, D. L.; Tour, J. M.; Weiss, P. S. Science 1996, 271, 1705-1707 . (3) Imry, Y.; Landauer, R. ReV. Mod. Phys. 1999, 71, S306-S312. (4) Tian, W. D.; Datta, S.; Hong, S. H.; Reifenberger, R.; Henderson, J. I.; Kubiak, C. P. J. Chem. Phys. 1998, 109, 2874-2882. (5) (a) Bumm, L. A.; Arnold, J. J.; Dunbar, T. D.; Allara, D. L.; Weiss, P. S. J. Phys. Chem. B 1999, 103, 8122-8127. (b) Smalley, J. F.; Feldberg, S. W.; Chidsey, C. E. D.; Linford, M. R.; Newton, M. D.; Liu, Y. P. J. Phys. Chem. 1995, 99, 13141-13149. (6) (a) Skourtis, S. S.; Beratan, D. N.; Onuchic, J. N. Chem. Phys. 1993, 176, 501-520. (b) Electron-transfer tubes. Regan, J. J.; Onuchic, J. N. AdV. Chem. Phys. 1999, 107, 497-553. (7) (a) Felts, A. K.; Pollard, W. T.; Friesner, R. A. J. Phys. Chem. 1995, 99, 2929-2940. (b) Pollard, W. T.; Friesner, R. A. J. Chem. Phys. 1994, 100, 5054-5065. (c) Skourtis, S. S.; Mukamel, S. Chem. Phys. 1995, 197, 367-388. (d) Kuznetsov, A. M.; Ulstrup, J. J. Inclusion Phenom. Macrocycl. Chem. 1999, 35, 45-54. (8) Evenson, J. W.; Karplus, M. J. Chem. Phys. 1992, 96, 5272. (9) See for example: (a) deAndrade, P. C. P.; Onuchic, J. J. Chem. Phys. 1998, 108, 4292-4298. (b) Balabin, I. A.; Onuchic, J. N. J. Phys. Chem. B 1998, 102, 7497-7505. (c) Regan, J. J.; Beratan, D. N.; Onuchic, J. N. Biophys. J. 1993, 64, A129-A129. (d) Onuchic, J. N.; Beratan, D. N.; Winkler, J. R.; Gray, H. B. Annu. ReV. Biophys. Biomol. Struct. 1992, 21, 349-377. (10) (a) Medvedev, E. S.; Stuchebrukhov, A. A. Pure Appl. Chem. 1998, 70, 2201. (b) Lee, E.; Medvedev, E. S.; Stuchebrukhov, A. A. J. Chem. Phys. 2000, 112, 9015. (11) (a) Grinstaff, M. W. Angew. Chem., Int. Ed. Engl. 1999, 38, 36293635. (b) Risser, S. M.; Beratan, D. N.; Meade, T. J. J. Am. Chem. Soc. 1993, 115, 2508-2510. (c) Beratan, D. N.; Priyadarshy, S.; Risser, S. M. Chem. Biol. 1997, 4, 3-8. (d) Berlin, Y. A.; Burin, A. L.; Ratner, M. A. J. Phys. Chem. A 2000, 104, 443-445. (e) Priyadarshy, S.; Risser, S. M.; Beratan, D. N. J. Phys. Chem. 1996, 100, 17678-17682. (12) Lewis, F. D.; Letsinger, R. L. J. Bio. Inorg. Chem. 1998, 3, 215221. (13) (a) Chen, J.; Calvet, L. C.; Reed, M. A.; Carr, D. W.; Grubisha, D. S.; Bennett, D. W. Chem. Phys. Lett. 1999, 313, 741-748. (b) Chen, J.; Reed, M. A.; Asplund, C. L.; Cassell, A. M.; Myrick, M. L.; Rawlett, A. M.; Tour, J. M.; VanPatten, P. G. Appl. Phys. Lett. 1999, 75, 624-626. (14) (a) Mirsky, S. Sci. Am. (June 2000), 282, 40-42. (b) Dekker, C. Phys. Today 1999, 52, 22-28. (c) Tans, S. J.; Dekker, C. Nature 2000, 404, 834-835. (d) Yao, Z.; Postma, H. W. C.; Balents, L.; Dekker, C. Nature 1999, 402, 273-276. (15) Davis, W. B.; Wasielewski, M. R.; Ratner, M. A.; Mujica, V.; Nitzan, A. J. Phys. Chem. A 1997, 101, 6158-6164. (16) McConnell, J. Chem. Phys. 1961, 35, 508-515. (17) Elicker, T. S.; Evans, D. G. J. Phys. Chem. A 1999, 103, 94239431. (18) See for example: (a) Tomalia, D. A.; Naylor, A. M.; Goddard, W. A.; Angew. Chem., Int. Ed. Engl. 1990, 29, 138-175. (b) Smith, D. K.; Diederich, F. Chem. A Eur. J. 1998, 4, 1353-1361 and references therein. (19) (a) Gorman, C. B.; Parkhurst, B. L.; Su, W. Y.; Chen, K. Y. J. Am. Chem. Soc. 1997, 119, 1141-1142. (b) Sadamoto, R.; Tomioka, N.; Aida, T. J. Am. Chem. Soc. 1996, 118, 3978-3979.

378 J. Phys. Chem. B, Vol. 105, No. 2, 2001 (20) (a) Takada, K.; Diaz, D. J.; Abruna, H. D.; Cuadrado, I.; Casado, C.; Alonso, B.; Moran, M.; Losada, J. J. Am. Chem. Soc. 1997, 119, 1076310773. (b) Tokuhisa, H.; Zhao, M.; Baker, L. A.; Phan, V. T.; Dermody, D. L.; Garcia, M. E.; Peez, R. F.; Crooks, R. M.; Mayer, T. M. J. Am. Chem. Soc. 1998, 120, 4492-4501. (21) Davis, W. B.; Svec, W. A.; Ratner, M. A.; Wasielewski, M. R. Nature 1998, 396, 60-63. (22) Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M. Science 1999, 286, 1550-1552. (23) Lukas, A. S.; Miller, S. E.; Wasielewski, M. R. J. Phys. Chem. B 2000, 104, 931-940. (24) Risser, S. M.; Beratan, D. N.; Onuchic, J. N. J. Phys. Chem. 1993, 97, 4523-4527. (25) Xu, Z. F.; Kahr, M.; Walker, K. L.; Wilkins, C. L.; Moore, J. S. J. Am. Chem. Soc. 1994, 116, 4537-4550. (26) See for example: (a) Franzen, S.; Martin, J. L. Annu. ReV. Phys. Chem. 1995, 46, 453. (b) Bublitz, G. U.; Boxer, Steven G. Annu. ReV. Phys. Chem. 1997, 48, 213-242. (c) Moser, C. C.; Sension, R. J.; Szarka, A. Z.; Repinec, S. T.; Hochstrasser, R. M.; Dutton, P. L. Chem. Phys. 1995, 197,

Elicker et al. 343. (d) Ogrodnik, A.; Langenbacher, T.; Bieser, G.; Siegl, J.; Eberl, U.; Volk, M.; Michel-Beyerle, M. E. Chem. Phys. Lett. 1992, 198, 653. (27) (a) Coutinho-Neto, M. D. da Gama, A. A. S. Chem. Phys. 1996, 203, 43. (b) Skourtis, S. S.; Beratan, D. N. AdV. Chem. Phys. 1999, 106, 377. (28) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277. (29) Metzger, R. M.; Chen, B.; Hopfner, U.; Lakshmikantham, M. V.; Vuillaume, D.; Kawai, T.; Wu, X. L.; Tachibana, H.; Hughes, T. V.; Sakurai, H.; Baldwin, J. W. J. Am. Chem. Soc. 1997, 119, 10455. (30) Nardelli, M. B. Phys. ReV. B 1999, 60, 7828-7833. (31) (a) Lopez-Castillo, J. M.; Filali-Mouhim, A.; Jay-Gerin, J. P. J. Phys. Chem. 1993, 97, 9266. (b) Kemp, M.; Roitberg, A.; Mujica, V.; Wanta, T.; Ratner, M. A. 1996, 100, 8349. (c) Balabin, I. A.; Onuchic, J. N. J. Phys. Chem. 1996, 100, 11573. (d) Jordan, K. D.; Paddon-Row, M. N. Chem. ReV. 1992, 92, 395. (32) Segal, D.; Nitzan, A.; Ratner, M. A.; Davis, W. B. J. Phys. Chem. B 2000, 104, 2790-2793. (33) Segal, D.; Nitzan, A.; Davis, W. B.; Wasielewski, M. R.; Ratner, M. A. J. Phys. Chem. B 2000, 104, 3817.