Variable-Range Hopping Electron Transfer through Disordered Bridge States: Application to DNA Thomas Renger* and R. A. Marcus Noyes Laboratory of Chemical Physics, Mail Code 127-72, Pasadena, California 91125 ReceiVed: August 19, 2002; In Final Form: July 25, 2003 A theory for electron transfer through a donor-bridge-acceptor system is described that involves tunneling and hopping-like transfers and an intermediate regime. The theory considers how a delocalization of electronic states and static and dynamic disorder in electronic energies influence the charge transfer rate and is used to study experiments on hole transfer through DNA. While an exponential distance dependence of the yield of hole trapping is observed experimentally for small bridges, the yield for long bridges is reported to be almost distance-independent. For long bridge lengths, for which thermally activating hopping dominates over tunneling, the model considers two competing channels, a hopping via localized states and a transfer through partly delocalized states. The variable-range hopping mechanism and the delocalized states aspect of the theory are used to interpret the flat rather than a slow decrease of yield with increasing distance reported in experiments with long bridges. I. Introduction In experiments on charge transfer through a DNA π-stack, a wide range of -values, values which characterize the exponent of the decay of the rate constant with distance, have been measured. In pioneering experiments of the Barton group, 1,2  values as small as 0.1 Å -1 were reported. Similarly small  values were reported by Schuster et al. 3,4 and were discussed in terms of a phonon-assisted polaron hopping model. Other experiments 5-11 yielded larger values up to  ) 1.4 Å -1 , for example, in Fukui and Tanaka 11 and indicated the superexchange transfer mechanism well-known from electron transfer in proteins. 12 It was understood from earlier theories 13,14 (see also the recent work in ref 15 and 16) that depending on the energetics of the system studied, either the superexchange mechanism or the hopping mechanism dominate the observed electron/hole trans- fer, leading to the strong or weak distance dependence of the rate constant, respectively. Numerical calculations applying a Redfield relaxation model were performed 13 on a model donor- bridge-acceptor system. The latter was coupled to one effective high-frequency mode, of which the potential energy minimum was shifted by the reaction. It was found that for large bridge lengths the rate constant becomes weakly dependent on distance and the transfer occurs in a hopping-like mechanism whereas for shorter bridges the tunneling dominates and a strong exponential distance dependence for the rate was obtained. The same result was obtained in theoretical studies performed using a Liouville pathway correlation-function approach. 14,17 The latter included the coupling of the electron to a manifold of vibrational modes and contained the reorganization of a whole set of nuclear coordinates involved in the electron transfer. 14 Experimental evidence on the critical role of the energetics of the system was provided by Meggers et al., 9 who demon- strated that the holes for short bridge lengths of adenines (A’s) will tunnel through the A’s and hop between guanines (G’s), an experiment that prompted extensive theoretical analyses. 18-21 The application of ultrafast time-resolved spectroscopy made it possible to resolve the charge-transfer dynamics in real time. In experiments on an ethidium system, 22 a distance-independent rate was found for hole transfer between a photoexcited ethidium and a deazaguanine for bridges of two, three, and four bases. There was also a decrease of the amplitude of the pump-probe signal for longer bridges, which was explained by static disorder. This finding shed some light on earlier frequency domain experiments 23 on the same system, which had yielded a  ≈ 0.1; the time-domain experiment suggested that this  value does not contain information on the intrinsic distance depen- dence of the rate constant per se but rather reflects the disorder. Under such conditions, the slow step is not the transmission along the chain of base pairs. In time-domain experiments on DNA hairpins, an exponential distance dependence of the rate constant of hole transfer between a stilbene and a guanine was found with a  ≈ 0.7. 7,8 A similar  value was obtained in recent time-resolved experiments on an aminopurine (Ap) system 10 for short bridge lengths. The rate constant between Ap and G in Ap(A) N G became distance- independent for N > 3, an indication of a change from superexchange to hopping-like transfer. Bixon and Jortner 24 explained an experiment 25 on hole transfer between guanine triplets in terms of thermally induced hopping. Recently, in another experiment 26 by the Giese group, experimental evidence for the presence of both transfer mech- anisms was obtained in measurements of the distance depen- dence of the yield of DNA cleavage triggered by hole transfer. As expected, for short bridges (N < 3) the superexchange mechanism dominates and the relative yield of DNA cleavage decreases exponentially ( ≈ 0.7) with distance. For bridge lengths N > 3, a transition occurs in which the relative yield becomes almost distance-independent. As shown below, the latter observation is not explained simply in terms of thermally activated nearest-neighbor hopping through the bridge. That result served as a stimulus for the present paper, although the * To whom correspondence should be addressed. Present address: Institut fu ¨r Chemie (Kristallographie), Freie Universita ¨t Berlin, Takustrasse 6, D-14195 Berlin, Germany. E-mail: rth@chemie.fu-berlin.de. 8404 J. Phys. Chem. A 2003, 107, 8404-8419 10.1021/jp026789c CCC: $25.00 © 2003 American Chemical Society Published on Web 09/19/2003