Breakdown of the Born–Oppenheimer–Condon–Marcus approximation in long distance electron transfer

Singlet Oxygen in Electrochemical Cells: A Critical Review of Literature and Theory

Rechargeable metal/O₂ batteries have long been considered a promising future battery technology in automobile and stationary applications. However, they suffer from poor cyclability and rapid degradation. A recent hypothesis is the formation of singlet oxygen (¹O₂) as the root cause of these issues. Validation, evaluation, and understanding of the formation of ¹O₂ are therefore essential for improving metal/O₂ batteries. We review literature and use Marcus theory to discuss the possibility of singlet oxygen formation in metal/O₂ batteries as a product from (electro)chemical reactions. We conclude that experimental evidence is yet not fully conclusive, and side reactions can play a major role in verifying the existence of singlet oxygen. Following an in-depth analysis based on Marcus theory, we conclude that ¹O₂ can only originate from a chemical step. A direct electrochemical generation, as proposed by others, can be excluded on the basis of theoretical arguments.

Tunneling Time and the Breakdown of Born-Oppenheimer Approximation

In electron-transfer reactions in proteins and other molecular systems involving long-distance electron tunneling, the tunneling time, i.e., the time that an electron spends in the barrier region between redox centers, can be comparable to vibrational periods of the nuclei. One consequence of this is the breakdown of the Born-Oppenheimer (BO) approximation at the far tails of the tunneling electronic wave functions. These tails define the coupling of redox centers exchanging electrons and hence the rates of electron transfer. We discuss the transition in the distance dependence of the rate of electron transfer that separates the BO and non-BO behavior of the tunneling reaction and show how the crossover is related to tunneling time.

Long-range electron tunneling

Electrons have so little mass that in less than a second they can tunnel through potential energy barriers that are several electron-volts high and several nanometers wide. Electron tunneling is a critical functional element in a broad spectrum of applications, ranging from semiconductor diodes to the photosynthetic and respiratory charge transport chains. Prior to the 1970s, chemists generally believed that reactants had to collide in order to effect a transformation. Experimental demonstrations that electrons can transfer between reactants separated by several nanometers led to a revision of the chemical reaction paradigm. Experimental investigations of electron exchange between redox partners separated by molecular bridges have elucidated many fundamental properties of these reactions, particularly the variation of rate constants with distance. Theoretical work has provided critical insights into the superexchange mechanism of electronic coupling between distant redox centers. Kinetics measurements have shown that electrons can tunnel about 2.5 nm through proteins on biologically relevant time scales. Longer-distance biological charge flow requires multiple electron tunneling steps through chains of redox cofactors. The range of phenomena that depends on long-range electron tunneling continues to expand, providing new challenges for both theory and experiment.

Interference, fluctuation, and alternation of electron tunneling in protein media. 2. Non-condon theory for the energy gap dependence of electron transfer rate

Developing the quantum transition rate theory of Prezhdo and Rossky (J. Chem. Phys. 1997, 107, 5863), we produced a new non-Condon theory of the rate of electron transfer (ET) which happens through a protein medium with conformational fluctuation. The new theory is expressed by a convolution form of the power spectrum for the autocorrelation function of the electronic tunneling matrix element T(DA)(t) with quantum correction and the ordinary Franck-Condon factor. The new theory satisfies the detailed balance condition for the forward and backward ET rates. The ET rate formula is divided into two terms of elastic and inelastic tunneling mechanisms on the mathematical basis. The present theory is applied to the ET from Bph(-) to Q(A) in the reaction center of Rhodobacter sphaeroides. Numerical calculations of T(DA)(t) were made by a combined method of molecular dynamics simulations and quantum chemistry calculations. We showed that the normalized autocorrelation function of T(DA)(t) is almost expressed by exponential forms. The calculated energy gap law of the ET rate is nearly Marcus' parabola in most of the normal region and around the maximum region, but it does not decay substantially in the inverted region, which is called the anomalous inverted region. We also showed that the energy gap law at the high uphill energy gap in the normal region is elevated considerably from the Marcus' parabola, which is called the anomalous normal region. Those anomalous energy gap laws are due to the inelastic tunneling mechanism which works actively at the energy gap far from zero. We presented an empirical formula for easily calculating the non-Condon ET rate, which is usable by many researchers. We provided experimental evidence for the anomalous inverted region which was basically reproduced by the present theory. The present theory was extensively compared with the previous non-Condon theories.

Conformationally averaged score functions for electronic propagation in proteins

We explore the influence of conformational dynamics on protein-mediated electron donor-acceptor interactions. We introduce a thermally averaged score function to characterize electronic propagation from redox cofactors into the protein and solvent. The score function is explored for myoglobin at the extended-Hückel level, and the results are compared with those of simpler models. The conformationally averaged quantum results are consistent with the empirical analysis of the Pathways model. Notably, subtle effects of quantum interference among multiple coupling pathways that arise in static structures are largely averaged out when protein thermal motion is included. Propagation through bulk water near the single-protein interface decays rapidly with distance.

Interfacial bridge-mediated electron transfer: mechanistic analysis based on electrochemical kinetics and theoretical modelling

Understanding the physical and chemical factors that control the kinetics of interfacial electron-transfer (ET) reactions is important for a large number of technological applications. The present article describes electrochemical kinetic studies of these factors, in which standard interfacial ET rate constants (k(0)(l)) have been measured for ET between substrate Au electrodes and various redox couples attached to the electrode surfaces by variable lengths (l) of oligomethylene (OM), oligophenylenevinylene (OPV) and oligophenyleneethynylene (OPE) bridges, which were constituents of mixed self-assembled monolayers (SAMs). The k(0)(l) measurements employed the indirect laser-induced temperature jump (ILIT) technique, which permits the measurement of interfacial ET rates that are orders of magnitude faster than those measurable by conventional techniques using the macroelectrodes that are the most convenient substrates for the mixed SAMs. The robustness of the measured rate constants (k(0)(l)), together with the Arrhenius activation energies (E(a)(l)) and preexponential factors (A(l)), is demonstrated by their invariance with respect to several experimental system parameters (including the chemical nature and length of the diluent component of the mixed SAM). Analysis of the kinetic results demonstrates that all of the observed interfacial ET processes proceed through a common type of transition state (predominantly associated with solvent reorganization around the redox moiety) and that the actual ET step involves direct electronic tunnelling between the Au electrode and the redox moiety. However, for the full range of l investigated, a global exponential decay of A(l) is not found for any of the three types of bridges. Possible reasons for this behavior, including the role of rate determining steps associated with adiabatic mechanisms within or beyond the transition state theory framework, are discussed, and comparisons with related conductance measurements are presented.

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.

Electron tunneling dynamics in anharmonic bath

Tunneling transition probability for a particle interacting with an anharmonic bath is found in a time-dependent Hartree approximation. The general expression is presented in terms of medium Keldysh functions that are assumed to be known. Furthermore, the transition probability is calculated in the noninteracting-blip approximation where the rate constant does not exhibit an activation dependence at high temperatures. The reorganization energy E(r) and the renormalized reaction heat epsilon are expressed in terms of the correlation matrix for a solvent and internal modes in both quantum and classical regimes. It is shown that E(r) and epsilon are temperature dependent.

Theory of torsional non-Condon electron transfer: A generalized spin-boson Hamiltonian and its nonadiabatic limit solution

The paper develops a theory of electron transfer with torsionally induced non-Condon (NC) effects. The starting point of the theory is a generalized spin-boson Hamiltonian, where an additional torsional oscillator bilinearly coupled to other bath modes causes a sinusoidal NC modulation. We derive closed form time dependent nonadiabatic rate expressions for both sudden and relaxed initial conditions, which are applicable for general spectral densities and energetic condition. Under the assumption that the torsional motion is not correlated with the polaronic shift of the bath, simple stationary limit rate expression is obtained. Model calculations of this rate expression illustrate the effects of torsional quantization and gating on the driving force and temperature dependences of the electron transfer rate. The classical limit of the rate expression consists of three Marcus-type terms, and is shown to agree very well with the exact numerical result.