Nonadiabatic donor–acceptor electron transfer mediated by a molecular bridge: A unified theoretical description of the superexchange and hopping mechanism

Multifunctional molecular carbon materials--from fullerenes to carbon nanotubes

This critical review covers the timely topic of carbon nanostructures-fullerenes and carbon nanotubes-in combination with metalloporphyrins as integrative components for electron-donor-acceptor ensembles. These ensembles are typically probed in condensed media and at semi-transparent electrode surfaces. In particular, we will present a comprehensive survey of a variety of covalent (i.e., nanoconjugates) and non-covalent linkages (i.e., nanohybrids) to demonstrate how to govern/fine-tune the electronic interactions in the resulting electron-donor-acceptor ensembles. In the context of covalent bridges, different spacers will be discussed, which range from pure "insulators" (i.e., amide bonds, etc.) to sophisticated "molecular wires" (i.e., p-phenylenevinylene units, etc.). Furthermore, we will elucidate the fundamental impact that these vastly different spacers may exert on the rate, efficiency, and mechanism of short- and long-range electron transfer reactions. Additionally, a series of non-covalent motifs will be described: hydrogen bonding, complementary electrostatics, pi-pi stacking and metal coordination-to name a few. These motifs have been successfully employed by us and our collaborators en route towards novel architectures (i.e., linear structures, tubular structures, rotaxanes, catenanes, etc.) that exhibit unique and remarkable charge transfer features.

A line-shape function in terms of changes in both molecular structure and force constants: a Gaussian approximation

We propose a new expression of a line-shape function (LSF) including the effects of changes in both force constants and the molecular structure within the harmonic-oscillator approximation. This expression enables us to calculate the LSF using only the data on molecular structures, force constants, and electronic energies in the initial and final electronic states without solving the eigenvalue equation for the normal vibration of a molecule. To derive the LSF expression, we consider one-photon emission from a polyatomic molecule thermalized in an electronic excited state, and derive the intensity distribution function for one-photon emission using not Lax and Kubo and Toyozawa's [J. Chem. Phys. 20, 1752 (1952); Prog. Theor. Phys. 13, 160 (1955)] generating function method but rather the density-matrix method. As a simple application, a Gaussian approximate LSF is applied to SO(2). As a result, it is found that the effect of change in force constants between the initial and final electronic states cannot be ignored, nor can the effect of change in the molecular structure between these two states. The LSF expression obtained is applicable to studies of not only radiative transition but also of electron-transfer and energy-transfer processes where both changes in molecular structure and force constants between the initial and final electronic states cannot be disregarded.

Theoretical studies of electron transfer through dendrimeric architecture

We have analyzed the steady-state electron transfer rate through a bridge of dendrimeric architecture. The difference between the linear chain and the dendrimeric architecture has also been demonstrated with steady-state rate as a main observable in the coherent and incoherent regimes of interactions. It is shown that generally the rate of electron transfer in dendrimeric architecture is faster than the rate associated with their linear chain counterpart with similar kind of bonding connectivities. The rate depends upon the size of the molecule, core branching, and the nature of the coupling among the different nodes on the dendrimer molecule. Depending upon the nature of the donor and acceptor, phenomenological dephasing coefficient due to environment and the geometry of the dendrimeric architecture, the modification of electron transfer rate has been studied. In the regime of fully coherent interactions where all quantum effects are considered the rate shows a multiple inversion due to the dendrimer architecture which is neither available in the regime of incoherent interaction nor in the linear chain case in similar condition. We have discussed about the applicability of our model in metal-molecule-metal junction, photoinduced electron transfer process, and molecular conductor.

Long distance electron transfer in cytochrome c oxidase immobilised on electrodes. A surface enhanced resonance Raman spectroscopic study

Cytochrome c oxidase was tethered to a functionalised Ag electrode via a histidine-tag on the C-terminus of subunit I or II and embedded in a phospholipid bilayer. The uniformly oriented membrane-bound proteins were studied by surface enhanced resonance Raman spectroscopy (SERRS) that reveals preservation of the native structures of the heme a and heme a(3) sites. On the basis of time-dependent SERRS measurements, the rate constant for the heterogeneous electron transfer to heme a was determined to be 0.002 s(-1) independent of the enzyme orientation and the overpotential. Taking into account that the electrode-to-heme a distance is larger than 50 A, these findings suggest an electron hopping mechanism in which the Cu(A) center is not involved. Electrochemical reduction is restricted to heme a whereas electron transfer from heme a to heme a(3), which in solution occurs on the nanosecond time scale, is drastically slowed down. It may be that the network of cooperativities that links intramolecular electron transfer and proton translocation is perturbed in the immobilised enzyme, possibly due to the effect of the interfacial electric field.

Effects of Dipole Moment, Linkers, and Chromophores at Side Chains on Long-Range Electron Transfer through Helical Peptides

Octadecapeptides carrying a ferrocene moiety at the molecular terminal were self-assembled on gold, and long-range electron transfer from the ferrocene moiety to gold was investigated by electrochemical methods. Effects on electron transfer of dipole moment of helical peptides, linkers connecting the peptide to gold, and chromophores introduced into the side chains were discussed. Cyclic voltammetry of the monolayers in an aqueous solution revealed that long-range electron transfer over 40 A occurred along the peptide molecule. Chronoamperometry showed that the long-range electron transfer should be ascribed to a hopping mechanism with use of amide groups as hopping sites. Electron transfer through the long peptide was not significantly accelerated by the dipole moment. However, the linker remarkably affected electron transfer depending on whether it was a methylene chain or a phenylene group, suggesting that local electron transfer between gold and the peptides should be the slowest step to determine the overall rate. Pyrenyl groups introduced into the side chains in the middle of the peptide molecule did not noticeably change electron transfer, probably because pyrenyl groups were too distant to allow direct electron transfer between them. Electrostatic potential profiles across the peptide monolayers were also calculated to explain reasonably the several interesting features in the present peptide systems.

Superexchange and sequential mechanisms in charge transfer with a mediating state between the donor and acceptor

The rate of intramolecular charge transfer from biphenyl to naphthalene was determined for the radical anions and radical cations of molecules with the general structure: (2-naphthyl)-(steroid spacer)-(4-biphenylyl). Varied degrees of unsaturation (one double bond, NSenB; two double bonds, NSen(2)B; and the b-ring completely aromatized, NSarB) were incorporated into the steroid spacer to examine the effect it would have on the charge transfer rate. The charge transfer rate, as inferred from the decay of the biphenyl radical ion absorption, increased in all cases relative to the completely saturated 3-(2-naphthyl)-16-(4-biphenylyl)-5alpha-androstane (NSB) reference molecule. For the anion charge transfer, the decay rates increased by factors of 1.4, 4.2, and 5.1, respectively, and for the cation, the decay rates increased by factors of 5, 276, and 470. To explain the results, the charge-transfer process was viewed as a combination of two independent mechanisms: a single-step, superexchange mechanism, and a two-step, sequential charge transfer. Using a low level of theory, simple models of the superexchange and two-step mechanisms were developed to elucidate the nature and differences between the two mechanisms. The critical variable for this analysis is the free energy of formation (DeltaG(I) degrees ) of the intermediate state: (2-naphthyl)-[spacer](1)+/--(4-biphenylyl). The conclusion from this treatment is that superexchange is the dominant mechanism when DeltaG(I) degrees is large, but at small DeltaG(I) degrees , the sequential mechanism will dominate. This is because the superexchange rate is shown to have a weak dependence on DeltaG(I) degrees , changing 10-fold for a change in DeltaG(I) degrees of 2 eV, compared to the sequential mechanism in which the rate can change over 10(3) for 0.5 V.

Protein dynamics and electron transfer: electronic decoherence and non-Condon effects

We compute the autocorrelation function of the donor-acceptor tunneling matrix element <T(DA)(t)T(DA)(0)> for six Ru-azurin derivatives. Comparison of this decay time to the decay time of the time-dependent Franck-Condon factor {computed by Rossky and coworkers [Lockwood, D. M., Cheng, Y.-K. & Rossky, P. J. (2001) Chem. Phys. Lett. 345, 159-165]} reveals the extent to which non-Condon effects influence the electron-transfer rate. <T(DA)(t)T(DA)(0)> is studied as a function of donor-acceptor distance, tunneling pathway structure, tunneling energy, and temperature to explore the structural and dynamical origins of non-Condon effects. For azurin, the correlation function is remarkably insensitive to tunneling pathway structure. The decay time is only slightly shorter than it is for solvent-mediated electron transfer in small organic molecules and originates, largely, from fluctuations of valence angles rather than bond lengths.

Understanding Electron Transfer across Negatively-Charged Aib Oligopeptides

The physicochemical effects modulating the conformational behavior and the rate of intramolecular dissociative electron transfer in phthalimide-Aibn-peroxide peptides (n = 0-3) have been studied by an integrated density functional/continuum solvent model. We found that three different orientations of the phthalimide ring are possible, labeled Phihel, PhiC7, and PhipII. In the condensed phase, they are very close in energy when the system is neutral and short. When the peptide chain length increases and the system is negatively charged, Phihel becomes instead the most stable conformer. Our calculations confirm that the 3(10)-helix is the most stable secondary structure for the peptide bridge. However, upon charge injection in the phthalimide end of the phthalimide-Aib3-peroxide, the peptide bridge can adopt an alpha-helix conformation as well. The study of the dependence of the frontier orbitals on the length and on the conformation of the peptide bridge (in agreement with experimental indications) suggests that for n = 3 the process could be influenced by a 3(10) --> alpha-helix conformational transition of the peptide chain.

Bridge mediated two-electron transfer reactions: On the influence of intersite Coulomb interactions

Donor-acceptor two-electron transfer (TET) mediated by a linear molecular bridge is described theoretically. The particular case is considered where the TET takes place in the presence of a strong electronic intersite coupling within the bridge and against the background of fast vibrational relaxation processes. For such a situation the coarse-grained description of bridge-assisted electron transfer in molecular systems can be utilized [Petrov et al., J. Phys. Chem. B 106, 3092 (2002)]. In the present case it leads to kinetic equations and rate expression for TET reactions. Our recent treatment of completely nonadiabtic TET reactions [Petrov et al., J. Chem. Phys. 120, 4441 (2004)] including a reduction to single-exponential kinetics (with overall transfer rate K(TET)) is generalized here to the case of strong intrabridge coupling and the presence of intersite Coulomb interactions. The dependence of K(TET) on the bridge length which is determined by a separate stepwise and concerted contribution is discussed in detail. It is found that the intersite Coulomb interaction favors the TET if the donor and the acceptor are uncharged in their completely reduced states (with two excess electrons present).

Thermally Activated Conduction in Molecular Junctions

We report temperature dependence measurements on the conductance of individual molecular wires. The results show for the first time in a molecular junction the theoretically predicted transition from coherent superexchange tunneling conductance to an activated hopping mechanism as temperature is increased.

Bridge mediated two-electron transfer reactions: Analysis of stepwise and concerted pathways

A theory of nonadiabatic donor (D)-acceptor (A) two-electron transfer (TET) mediated by a single regular bridge (B) is developed. The presence of different intermediate two-electron states connecting the reactant state D-(-)BA with the product state DBA-(-) results in complex multiexponential kinetics. The conditions are discussed at which a reduction to two-exponential as well as single-exponential kinetics becomes possible. For the latter case the rate KTET is calculated, which describes the bridge-mediated reaction as an effective two-electron D-A transfer. In the limit of small populations of the intermediate TET states D-B-A, DB-(-)A, D-BA-, and DB-A-, KTET is obtained as a sum of the rates KTET(step) and KTET(sup). The first rate describes stepwise TET originated by transitions of a single electron. It starts at D-(-)BA and reaches DBA-(-) via the intermediate state D-BA-. These transitions cover contributions from sequential as well as superexchange reactions all including reduced bridge states. In contrast, a specific two-electron superexchange mechanism from D-(-)BA to DBA-(-) defines KTET(sup). An analytic dependence of KTET(step) and KTET(sup) on the number of bridging units is presented and different regimes of D-A TET are studied.

Two-electron transfer reactions in proteins: bridge-mediated and proton-assisted processes

Nonadiabatic two-electron transfer (TET) reactions through donor-bridge-acceptor (DBA) systems is investigated within the approximation of fast vibrational relaxation. For TET reactions in which the population of bridging states remains small (less than 10(-2)) it is demonstrated that a multiexponential transition process reduces to three-state kinetics. The transfer starts at the state with two excess electrons at the D center (D(2-)BA), goes through the intermediate (transient) state with one electron at the D center and one at the A center (D-BA-), and ends up with the two electrons at the A center (DBA2-). Furthermore, if the population of the intermediate state becomes also small the two-exponential kinetics can be transformed with high accuracy to single-exponential D-A TET kinetics. The related overall transfer rate contains contributions from stepwise and from concerted TET. The latter process is determined by a specific two-electron superexchange coupling incorporating the bridging states (D-B-A and DB-A-) as well as the intermediate state (D-BA-). As an example, the reduction of micothione reductase by nicotinamide adenine dinucleotide phosphate is analyzed. Existing experimental data can be explained if one assumes that the proton-assisted reduction of the enzyme is realized by the concerted TET mechanism.

Long-range electron transfer over 4 nm governed by an inelastic hopping mechanism in self-assembled monolayers of helical peptides

Well-ordered self-assembled monolayers (SAMs) were prepared on gold from helical peptides carrying a ferrocene (Fc) moiety at the N- or C-terminal end, and long-range electron transfer (ET) from Fc to gold was investigated. Electrochemical studies revealed that an inelastic hopping mechanism dominated over the superexchange mechanism in the ET reactions in the present SAMs and the dipole moment of the helix accelerated the ET reactions probably due to the lowering of the barrier height between the gold surface and peptide layer.

Distance dependence of electron transfer across peptides with different secondary structures: the role of Peptide energetics and electronic coupling

The charge-transfer transition energies and the electronic-coupling matrix element, |H(DA)|, for electron transfer from aminopyridine (ap) to the 4-carbonyl-2,2'-bipyridine (cbpy) in cbpy-(gly)(n)-ap (gly = glycine, n = 0-6) molecules were calculated using the Zerner's INDO/S, together with the Cave and Newton methods. The oligopeptide linkages used were those of the idealized protein secondary structures, the alpha-helix, 3(10)-helix, beta-strand, and polyproline I- and II-helices. The charge-transfer transition energies are influenced by the magnitude and direction of the dipole generated by the peptide secondary structure. The electronic coupling |H(DA)| between (cbpy) and (ap) is also dependent on the nature of the secondary structure of the peptide. A plot of 2.ln|H(DA)| versus the charge-transfer distance (assumed to be the dipole moment change between the ground state and the charge-transfer states) showed that the polyproline II structure is a more efficient bridge for long-distance electron-transfer reactions (beta = 0.7 A(-1)) than the other secondary structures (beta approximately 1.3 A(-1)). Similar calculations on charged dipeptide derivatives, [CH(3)CONHCH(2)CONHCH(3)](+/)(-), showed that peptide-peptide interaction is more dependent on conformation in the cationic than in the anionic dipeptides. The alpha-helix and polyproline II-helix both have large peptide-peptide interactions (|H(DA)| > 800 cm(-1)) which arise from the angular dependence of their pi-orbitals. Such an interaction is much weaker than in the beta-strand peptides. These combined results were found to be consistent with electron-transfer rates experimentally observed across short peptide bridges in polyproline II (n = 1-3). These results can also account for directional electron transfer observed in an alpha-helical structure (different ET rates versus the direction of the molecular dipole).