Intra-molecular electron transfer and electric conductance via sequential hopping: Unified theoretical description
Introduction
Electron transfer (ET) is one of the most basic chemical processes, particularly important for radiation physics and chemistry. Mechanisms of this process have been under active theoretical and experimental study for over half of a century, for review see e.g. Barbara et al. (1996); Bixon and Jortner (1999); Kuznetsov (1995); Kuznetsov and Ulstrup (1999); Lamb (1995); Miller et al. (1995); Schmickler (1996). At the early stage of investigations main efforts of theoreticians and experimentalists were concentrated on ET between donor and acceptor species existing in the solution as free solutes or generated in the “pure” liquid under irradiation. As follows from theoretical analysis of Marcus (1956) and Hush (1958) confirmed by a series of elegant experiments (Asahi and Mataga, 1989; Gould et al., 1990; McCleskey et al., 1992; Gould and Farid, 1996; Mataga et al., 1986; Miller et al., 1984a, Miller et al., 1984b; Turró et al., 1996; Guldi and Asmus, 1997), the rate of inter-molecular ET from donor to acceptor is mainly controlled by such factors as the donor–acceptor distance, the free energy change of the reaction, molecular reorganization of reactants and their environment, and the temperature. In many cases, especially for certain types of radiation-induced and photo-induced ET reactions (Barzykin et al., 2002), transport of negative charge towards the acceptor can be important as well (Berlin, 1975; Burshtein and Krissinel, 1996; Nishikawa et al., 1991; Pilling and Rice, 1975; Tachiya and Murata, 1992). This is particularly true for the attachment of excess electrons to the so-called scavengers in non-polar solvents, where according to the two-state model (Schiller, 1977) electron can be transferred to the acceptor as a quasi-free delocalized particle or as a low mobile localized charge carrier. Schiller and Nyikos (1977) have proposed a phenomenological description for kinetics of these reactions proceeding, for instance, in irradiated solutions of NO2, SF6, C2HCl3, and biphenyl in hydrocarbons (Allen and Holroyd, 1974; Allen et al., 1975; Beck and Thomas, 1972, Beck and Thomas, 1974; Yakovlev, 1979). The key element of the Schiller–Nyikos description is the postulate that the apparent rate for the transfer of an excess electron to the acceptor depends on the particular electron state involved in the process. More recent calculations (Plotnikov and Ovchinnikov, 1985) based on the theory of radiationless transitions (Lax, 1952; Kubo and Toyozawa, 1955; Bixon and Jortner, 1968) show that quasi-free and localized states of excess electrons indeed differ in reactivity, thus providing a theoretical justification of the conjecture used in the phenomenological description of electron scavenging in non-polar liquids.
Although investigations of inter-molecular ET in solutions still contribute significantly to the understanding of the reaction mechanism (Barzykin et al., 2002), the advent of molecular electronics and photonics as well as the increasing interest in charge migration in complex biological molecules (e.g. proteins, DNA, and photosynthetic systems) have shifted the focus of research towards intra-molecular ET. In the latter process electron is transferred from donor to acceptor species that exist in solution as spatially separated sites of a bigger molecule rather than as free solutes involved in inter-molecular ET.
A convenient model system often used to probe intra-molecular ET reactions by different experimental techniques comprises of a donor (D) and an acceptor (A) of electrons connected by molecular bridge (B) (for details see e.g. Jordan and Paddon-Row, 1992; Wasielewski, 1992; Miller et al., 1984a, Miller et al., 1984b; Pullen et al., 1999; Warman et al., 1999; Wegewijs and Verhoeven, 1999; Sikes et al., 2001; Walters et al., 2003; Hviid et al., 2004; Weiss et al., 2004). The number and variety of such donor-bridge-acceptor (DBA) systems have grown explosively in recent years. They can be divided roughly into those with π-bond conjugated bridges and those with σ-bonded bridges, although hybrid combinations are also known (Kilsa et al., 2001).
Extensive investigations of intra-molecular ET between donor and acceptor sites through bridges of different chemical structures reveal two extremes for the mechanism of the process. One extreme is the superexchange mediated tunneling (Bixon and Jortner, 1999; Kuznetsov and Ulstrup, 1999), which occurs if is much higher in energy as compared with and . For this coherent mechanism, the electron transfer rate, , decreases with the length of the bridge, R, as where is the pre-exponential factor and is the falloff parameter. The value of the parameter in Eq. (1) is known to be most sensitive to the structure of the bridging media. Highly conjugated organic bridges have the smallest falloff parameters ranging from 0.2 up to 0.6 Å−1 (Helms et al., 1992; Davis et al, 1998; Wasielewski et al., 1989; Grosshenny et al., 1996; Sachs et al., 1997), while free space is characterized by the largest value of ∼2 Å−1 (Newton, 1991). Lying between these two limits are many motifs, both synthetic and natural, including cytochromes, docked proteins, DNA with adenine/thymine base pairs sequences connecting D and A sites, and saturated organic molecules. Each displays its own characteristic range of values and hence its own time scale and distance dependencies of ET.
It is worth mentioning that in the case of superexchange mediated tunneling the bridge remains physically unpopulated during the entire ET process. Therefore for the simplest kinetic scheme of charge separation, DBA→D+B−A→D+BA−, the intermediate state D+B−A should be considered as virtual and ET from D to A can be treated as a single-step process.
Another situation arises if B is comparable in energy with D. Now the electron can be injected from the donor to the bridge, and the state D+B−A becomes physically occupied, thus representing a genuine intermediate. Once the bridge has been populated, sequential incoherent hopping between bridge sites allows the electron to reach the remote A unit, where this charge carrier will react with the acceptor and charge transport will be terminated. Thus, unlike the single-step coherent superexchange, the regime of incoherent hopping involves several steps including electron injection to the bridge, migration of the injected charge carrier along the bridge from the site of its generation to the distant acceptor site, and the subsequent reaction with A leading to the formation of the anion A−. As a consequence, the distance dependence of the ET rate for incoherent multi-step hopping turns out to be distinct from the exponential law (1) typical for coherent single-step superexchange. Calculations performed for long bridges with the large number of energetically identical hopping sites separated by the distance show that in this particular case the dependence vs. can be approximated by with the numerical parameter ranging from 1 (irreversible hopping) to 2 (unbiased hopping) (see e. g. Berlin et al., 2000; Segal et al., 2000; Berlin et al., 2001; Wang and Nau, 2001; Bicout and Kats, 2002; Petrov et al., 2003). Experimental evidences for the existence of this algebraic distance dependence has been obtained for several DBA systems with . A short list of examples includes series of molecules based on organic donors and acceptors linked by p-phenylenevinylene oligomers of increasing length (Davis et al., 1998), assemblies of donor and acceptor units joined together by peptide bridge (Malak et al., 2004.), and periodic nucleotide base stacks of alternating adenine/thymine and guanine/cytosine pairs connecting donor and acceptor sites in DNA molecules (Giese et al., 1999).
There is no dichotomy between coherent and incoherent mechanisms of electron migration in DBA systems. On the contrary, each can contribute to the mechanism of the intra-molecular ET process. The contribution depends on what is measured and particularly on the specific relative energies of D, A and B units. If the “bridging states” are very high in energy compared with D and A, the coherent mechanism will dominate. Otherwise, charge migration will mainly proceed by incoherent hopping. Note, however, that in a number of DBA systems the resonance coupling between D and certain bridging states provide conditions wherein both limiting extremes are operative: The superexchange mediated tunneling controls the rate of the elementary jump between proximate sites with appropriate energetics, while sequential hopping is responsible for the long-range electron migration along the bridge. This situation is typical for ET processes in DNA (see e.g. Jortner et al., 1998; Berlin et al., 2000, Berlin et al., 2001; Bixon and Jortner, 2002; Berlin et al., 2004) and probably occurs in other biological molecules (Yanagisawa et al., 2004).
Superexchange and sequential hopping together provide not only an adequate background for the description of electron-transfer properties of DBA systems, but also a general theoretical framework for estimating the ability of the bridge molecule to conduct electric charge between two electrodes (see e.g. Ratner et al., 1998; Nitzan and Ratner, 2003). Usually these two properties are discussed in terms of two different quantities determined in two different experiments. Intra-molecular ET is typically monitored in the time domain, yielding the electron transfer rate as the main kinetic characteristic of the process. By contrast, conductive properties of the molecular bridge connecting two macroscopic electrodes are probed in a steady-state situation by applying the voltage across the bridge molecule and measuring a resulting steady-state current . In the limit of the Ohmic regime, which often exists near zero bias because of either tunneling or thermal activation, this gives the molecular conductance as the prime quantity characterizing the ability of the molecule to transmit electrons between two contacts in nanojunctions.
If, however, electrodes replace donor and acceptor species in the DBA system, so that the same bridge molecule B is studied in two experiments mentioned above, one can expect that g and will be closely related, since both are controlled by the common process of electron motion from one end of the bridge to another. Indeed, Segal et al. (2000), Nitzan (2001) and Nitzan and Ratner (2003) have derived the relationship between the ET rate and the zero-bias molecular conduction for coherent superexchange regime of electron migration along the given bridge. Their analysis has been extended to the particular case of hopping with the equal transition rates for all successive elementary steps involved in electron transport (Nitzan, 2002). This restricts applications of theory to the bridges with the energy landscape formed by energetically identical hopping sites separated by equal distances. Therefore theoretical results obtained so far are relevant only to the spatial type of electron transport, which can be viewed as a series of unbiased hopping steps, each proceeding with the same transition probability per unit time.
In the present paper we consider the hopping regime of intra-molecular ET and molecular conductance beyond these limitations using the discrete steady-state flux method for analytical study of the problem. This method allows us to express both g and in terms of the rates of all elementary hopping steps involved in the process of electron migration along the bridge that either links D and A units of the DBA system or connects two metal electrodes in the molecular nanojunction. Based on the results obtained, the relationship between the intra-molecular ET rate and the molecular conductance is established for the general form of energy landscape governing hopping motion. Our theoretical findings provide practical expressions for estimating zero-bias conductance of molecular bridges in the metal–molecule–metal junction based on the data of the ET rate along the same bridge in the DBA system.
Section snippets
Model
To calculate the rate, , of intra-molecular ET and the zero-bias molecular conductance g in the case, where the DBA molecule and the nanojunction contain the same molecular bridge, either of the two systems under investigation is considered as a chain of sites with the position of each site within the chain being defined by the index (see Fig. 1). Then the donor unit of the DBA system and/or the left electrode of the nanojunction will correspond to the site , while the
Discussion
In this paper we developed the unified description of inter-molecular ET and molecular conductance in the case of sequential electron hopping along the bridge, which connects either donor and acceptor units of the DBA system or two metal electrodes. Our description is based on the discrete variant of the steady flux method. This enables us to use the same approach to derive the expressions for two quantities, i.e. for the intra-molecular ET rate and for the zero-bias molecular conductance
Acknowledgments
The authors are grateful to the Chemistry Division of the ONR, MOLETRONICS program at DARPA, to the DoD/MURI and DURINT programs for support of the research. We thank many colleagues, particularly J. Jortner, A. Nitzan and M. Galperin, for useful discussions.
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