Coherence in electron transfer pathways

Central to the view of electron-transfer reactions is the idea that nuclear motion generates a transition state geometry at which the electron/hole amplitude propagates coherently from the electron donor to the electron acceptor. In the weakly coupled or nonadiabatic regime, the electron amplitude tunnels through an electronic barrier between the donor and acceptor. The structure of the barrier is determined by the covalent and noncovalent interactions of the bridge. Because the tunneling barrier depends on the nuclear coordinates of the reactants (and on the surrounding medium), the tunneling barrier is highly anisotropic, and it is useful to identify particular routes, or pathways, along which the transmission amplitude propagates. Moreover, when more than one such pathway exists, and the paths give rise to comparable transmission amplitude magnitudes, one may expect to observe quantum interferences among pathways if the propagation remains coherent. Given that the effective tunneling barrier height and width are affected by the nuclear positions, the modulation of the nuclear coordinates will lead to a modulation of the tunneling barrier and hence of the electron flow. For long distance electron transfer in biological and biomimetic systems, nuclear fluctuations, arising from flexible protein moieties and mobile water bridges, can become quite significant. We discuss experimental and theoretical results that explore the quantum interferences among coupling pathways in electron-transfer kinetics; we emphasize recent data and theories associated with the signatures of chirality and inelastic processes, which are manifested in the tunneling pathway coherence (or absence of coherence).     

Extension of solution‐reaction‐surface description to examination of nonequilibrium solvation effect for microsolvated reaction

A microscopic method to examine a nonequilibrium solvation effect is reported. The solution reaction is simplified as a barrier‐crossing reaction within a solution reaction surface that corresponds to a two‐dimensional space determined by solute and solvent reactive coordinates. For this simplification, the motions within the space spanned by nonreactive coordinates are frozen. We derive three rate constant expressions: (1) in the nonadiabatic solvation limit, (2) in the equilibrium solvation limit, and (3) of the transition‐state theory. This method was applied to the examination of the contact‐ion‐pair formation of t‐BuCl in four waters. We found that the nonadiabatic solvation picture overestimates the nonequilibrium solvation effect. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 791–796, 2000     

Rate processes with non-Markovian dynamical disorder

Rate processes with dynamical disorder are investigated within a simple framework provided by unidirectional electron transfer (ET) with fluctuating transfer rate. The rate fluctuations are assumed to be described by a non-Markovian stochastic jump process which reflects conformational dynamics of an electron transferring donor-acceptor molecular complex. A tractable analytical expression is obtained for the relaxation of the donor population (in the Laplace-transformed time domain) averaged over the stationary conformational fluctuations. The corresponding mean transfer time is also obtained in an analytical form. The case of two-state fluctuations is studied in detail for a model incorporating substate diffusion within one of the conformations. It is shown that an increase of the conformational diffusion time results in a gradual transition from the regime of fast modulation characterized by the averaged ET rate to the regime of quasistatic disorder. This transition occurs at the conformational mean residence time intervals fixed much less than the inverse of the corresponding ET rates. An explanation of this paradoxical effect is provided. Moreover, its presence is also manifested for the simplest, exactly solvable non-Markovian model with a biexponential distribution of the residence times in one of the conformations. The nontrivial conditions for this phenomenon to occur are found.     

Dynamical arrest of electron transfer in liquid crystalline solvents

We argue that electron transfer reactions in slowly relaxing solvents proceed in the nonergodic regime, making the reaction activation barrier strongly dependent on the solvent dynamics. For typical dielectric relaxation times of polar nematics, electron transfer reactions in the subnanosecond time scale fall into nonergodic regime in which nuclear solvation energies entering the activation barrier are significantly lower than their thermodynamic values. The transition from isotropic to nematic phase results in weak discontinuities of the solvation energies at the transition point and the appearance of solvation anisotropy weakening with increasing solute size. The theory is applied to analyze experimental kinetic data for the electron transfer kinetics in the isotropic phase of 5CB liquid crystalline solvent. We predict that the energy gap law of electron transfer reactions in slowly relaxing solvents is characterized by regions of fast change of the rate at points where the reaction switches between the ergodic and nonergodic regimes. The dependence of the rate on the donor-acceptor separation may also be affected in a way of producing low values for the exponential falloff parameter.     

Semiclassical treatment of thermally activated electron transfer in the inverted region under the fast dielectric relaxation

The previously formulated semiclassical theory (Zhao, Liang, and Nakamura, J. Phys. Chem. A 2006, 110, 8204) is used to study electron transfer in the Marcus inverted case by considering multidimensional potential energy surfaces of donor and acceptor. The Zhu-Nakamura formulas of nonadiabatic transition in the case of Landau-Zener type are incorporated into the approach. The theory properly takes into account the nonadiabatic transition coupled with the nuclear tunneling and can cover the whole range from weak to strong coupling regime uniformly under the assumption of fast solvent relaxation. The numerical calculations are performed for the 12-dimensional model of shifted harmonic oscillators and demonstrate that the reaction rate with respect to the electronic coupling shows a maximum, confirming the adiabatic suppression in the strong coupling limit. The adiabatic suppression is dramatically reduced by the effect of nuclear tunneling compared to the case that the Landau-Zener formula is used. The possible extension and applications to the case of the slow solvent dynamics are discussed.     

Dynamics of nonadiabatic chemical reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.     

Unified semiclassical theory for the two-state system: an analytical solution for general nonadiabatic tunneling

Unified semiclasical solution for general nonadiabatic tunneling between two adiabatic potential energy surfaces is established by employing unified semiclassical solution for pure nonadiabatic transition [C. Zhu, J. Chem. Phys. 105, 4159 (1996)] with the certain symmetry transformation. This symmetry comes from a detailed analysis of the reduced scattering matrix for Landau-Zener type of crossing as a special case of nonadiabatic transition and nonadiabatic tunneling. Traditional classification of crossing and noncrossing types of nonadiabatic transition can be quantitatively defined by the rotation angle of adiabatic-to-diabatic transformation, and this rotational angle enters the analytical solution for general nonadiabatic tunneling. The certain two-state exponential potential models are employed for numerical tests, and the calculations from the present general nonadiabatic tunneling formula are demonstrated in very good agreement with the results from exact quantum mechanical calculations. The present general nonadiabatic tunneling formula can be incorporated with various mixed quantum-classical methods for modeling electronically nonadiabatic processes in photochemistry.     

Electron tunneling in lithium-ammonia solutions probed by frequency-dependent electron spin relaxation studies

Electron transfer or quantum tunneling dynamics for excess or solvated electrons in dilute lithium-ammonia solutions have been studied by pulse electron paramagnetic resonance (EPR) spectroscopy at both X- (9.7 GHz) and W-band (94 GHz) frequencies. The electron spin-lattice (T(1)) and spin-spin (T(2)) relaxation data indicate an extremely fast transfer or quantum tunneling rate of the solvated electron in these solutions which serves to modulate the hyperfine (Fermi-contact) interaction with nitrogen nuclei in the solvation shells of ammonia molecules surrounding the localized, solvated electron. The donor and acceptor states of the solvated electron in these solutions are the initial and final electron solvation sites found before, and after, the transfer or tunneling process. To interpret and model our electron spin relaxation data from the two observation EPR frequencies requires a consideration of a multiexponential correlation function. The electron transfer or tunneling process that we monitor through the correlation time of the nitrogen Fermi-contact interaction has a time scale of (1-10) × 10(-12) s over a temperature range 230-290 K in our most dilute solution of lithium in ammonia. Two types of electron-solvent interaction mechanisms are proposed to account for our experimental findings. The dominant electron spin relaxation mechanism results from an electron tunneling process characterized by a variable donor-acceptor distance or range (consistent with such a rapidly fluctuating liquid structure) in which the solvent shell that ultimately accepts the transferring electron is formed from random, thermal fluctuations of the liquid structure in, and around, a natural hole or Bjerrum-like defect vacancy in the liquid. Following transfer and capture of the tunneling electron, further solvent-cage relaxation with a time scale of ～10(-13) s results in a minor contribution to the electron spin relaxation times. This investigation illustrates the great potential of multifrequency EPR measurements to interrogate the microscopic nature and dynamics of ultrafast electron transfer or quantum-tunneling processes in liquids. Our results also impact on the universal issue of the role of a host solvent (or host matrix, e.g. a semiconductor) in mediating long-range electron transfer processes and we discuss the implications of our results with a range of other materials and systems exhibiting the phenomenon of electron transfer.     

Low temperature electron transfer in strongly condensed phase

Electron transfer coupled to a collective vibronic degree of freedom is studied in strongly condensed phase and at lower temperatures where quantum fluctuations are essential. Based on an exact representation of the reduced density matrix of the electronic + reaction coordinate compound in terms of path integrals, recent findings on the overdamped limit in quantum dissipative systems are employed. This allows us to give a consistent generalization of the well-known Zusman equations to the quantum domain. Detailed conditions for the range of validity are specified. Using the Wigner transform these results are also extended to the quantum dynamics in full phase space. As an important application electronic transfer rates are derived that comprise adiabatic and nonadiabatic processes in the low temperature regime including nuclear tunneling. Accurate agreement with precise quantum Monte Carlo data is observed.     

Semiclassical treatment of thermally activated electron transfer in the intermediate to strong electronic coupling regime under the fast dielectric relaxation

The generalized nonadiabatic transition-state theory (NA-TST) (Zhao, Y.; et al. J. Chem. Phys. 2004, 121, 8854) is used to study electron transfer with use of the Zhu-Nakamura (ZN) formulas of nonadiabatic transition in the case of fast dielectric relaxation. The rate constant is expressed as a product of the well-known Marcus formula and a coefficient which represents the correction due to the strong electronic coupling. In the case of general multidimensional systems, the Monte Carlo approach is utilized to evaluate the rate by taking into account the multidimensionality of the crossing seam surface. Numerical demonstration is made by using a model system of a collection of harmonic oscillators in the Marcus normal region. The results are naturally coincident with the perturbation theory in the weak electronic coupling limit; while in the intermediate to strong electronic coupling regime where the perturbation theory breaks down the present results are in good agreement with those from the quantum mechanical flux-flux correlation function within the model of effective one-dimensional mode.     

Semiclassical treatments of electron transfer rate from weak to strong electronic coupling regime

Electron transfer (ET) rate is a fundamental parameter to characterize ET processes in physical, chemical, material and biologic sciences. It is affected by a number of quantum phenomena, such as nuclear tunneling, curve crossing, quantum interference, and the coupling to the environment. It is thus a challenge to accurately evaluate the ET rate since one has to incorporate both quantum effects and dissipation. In this review article, we present several semiclassical theories proposed in our group to cover the regime from weak to strong electronic coupling. Their applications to some concrete systems are also shown.     

Dynamical theory of proton tunneling transfer rates in solution: general formulation

A general dynamical theory is presented for the rate constant of weak coupling, nonadiabatic proton-tunneling reactions in solution. The theory incorporates the critical role of the solvent and the vibration of the separation of the heavy particles between which the proton transfers, including their dynamics. The formulation which allows the computation of the quantum rate constant k via classical molecular dynamics simulation techniques is presented, as are a number of approximate analytic results for k in a variety of different important regimes. The frequent appearance of (nearly) classical Arrhenius behavior for k — even though the intrinsic reactive event is quantum proton tunneling — is discussed, together with the solvent and vibrational contributions to the apparent activation energy. In certain weak solvation limits, however, non-Arrhenius behavior for k is found and is related to vibrational Franck-Condon features in the reaction.     

Tunneling and classical paths for proton transfer in an enzyme reaction dominated by tunneling: oxidation of tryptamine by aromatic amine dehydrogenase

Proton tunneling dominates the oxidative deamination of tryptamine catalyzed by the enzyme aromatic amine dehydrogenase. For reaction with the fast substrate tryptamine, a H/D kinetic isotope effect (KIE) of 55 +/- 6 has been reported-one of the largest observed in an enzyme reaction. We present here a computational analysis of this proton-transfer reaction, applying combined quantum mechanics/molecular mechanics (QM/MM) methods (PM3-SRP//PM3/CHARMM22). In particular, we extend our previous computational study (Masgrau et al. Science 2006, 312, 237) by using improved energy corrections, high-level QM/MM methods, and an ensemble of paths to estimate the tunneling contributions. We have carried out QM/MM molecular dynamics simulations and variational transition state theory calculations with small-curvature tunneling corrections. The results provide detailed insight into the processes involved in the reaction. Transfer to the O2 oxygen of the catalytic base, Asp128beta, is found to be the favored reaction both thermodynamically and kinetically, even though O1 is closer in the reactant complex. Comparison of quantum and classical models of proton transfer allows estimation of the contribution of hydrogen tunneling in lowering the barrier to reaction in the enzyme. A reduction of the activation free energy due to tunneling of 3.1 kcal mol-1 is found, which represents a rate enhancement due to tunneling by 2 orders of magnitude. The calculated KIE of 30 is significantly elevated over the semiclassical limit, in agreement with the experimental observations; a semiclassical value of 6 is obtained when tunneling is omitted. A polarization of the C-H bond to be broken is observed due to the close proximity of the catalytic aspartate and the (formally) positively charged imine nitrogen. A comparison is also made with the related quinoprotein methylamine dehydrogenase (MADH)-the much lower KIE of 11 that we obtain for the MADH/methylamine system is found to arise from a more endothermic potential energy surface for the MADH reaction.     

A quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution based on the framework of mixed quantum-classical molecular dynamics

We present a quantum equation of motion for chemical reaction systems on an adiabatic double-well potential surface in solution in the framework of mixed quantum-classical molecular dynamics, where the reactant and product states are explicitly defined by dividing the double-well potential into the reactant and product wells. The equation can describe quantum reaction processes such as tunneling and thermal excitation and relaxation assisted by the solvent. Fluctuations of the zero-point energy level, the height of the barrier, and the curvature of the well are all included in the equation. Here, the equation was combined with the surface hopping technique in order to describe the motion of the classical solvent. Applying the present method to model systems, we show two numerical examples in order to demonstrate the potential power of the present method. The first example is a proton transfer by tunneling where the high-energy product state was stabilized very rapidly by solvation. The second example shows a thermal activation mechanism, i.e., the initial vibrational excitation in the reactant well followed by the reacting transition above the barrier and the final vibrational relaxation in the product well.     

Approach to nonadiabatic transitions by density matrix evolution and molecular dynamics simulations

Many biological processes are characterized by an essentially quantum dynamical event, such as electron or proton transfer, in a complex classical environment. To treat such processes properly by computer simulation, allowing nonadiabatic transitions involving excited states, we recently developed a density matrix evolution (DME ) method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem, 97 , 13464 (1993)] which simulates the dynamics of quantum systems embedded in a classical environment. The formalism of the method is presented and an overview of the applications ranging from collisions of a quantum harmonic oscillator with noble gas atoms to proton tunneling in a double-well hydrogen bond is given. The methodology for treatment of proton-transfer processes with inclusion of excited states is presented. Future application of the method on biologically interesting processes, such as proton transfer in enzymatic reactions, is discussed. © 1996 John Wiley & Sons, Inc.     

Calculation of vibronic couplings for phenoxyl/phenol and benzyl/toluene self-exchange reactions: implications for proton-coupled electron transfer mechanisms

The vibronic couplings for the phenoxyl/phenol and the benzyl/toluene self-exchange reactions are calculated with a semiclassical approach, in which all electrons and the transferring hydrogen nucleus are treated quantum mechanically. In this formulation, the vibronic coupling is the Hamiltonian matrix element between the reactant and product mixed electronic-proton vibrational wavefunctions. The magnitude of the vibronic coupling and its dependence on the proton donor-acceptor distance can significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of proton-coupled electron transfer reactions. Both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. For the phenoxyl/phenol system, the electrons are unable to rearrange fast enough to follow the proton motion on the electronically adiabatic ground state, and the excited electronic state is involved in the reaction. For the benzyl/toluene system, the electrons can respond virtually instantaneously to the proton motion, and the proton moves on the electronically adiabatic ground state. For both systems, the vibronic coupling decreases exponentially with the proton donor-acceptor distance for the range of distances studied. When the transferring hydrogen is replaced with deuterium, the magnitude of the vibronic coupling decreases and the exponential decay with distance becomes faster. Previous studies designated the phenoxyl/phenol reaction as proton-coupled electron transfer and the benzyl/toluene reaction as hydrogen atom transfer. In addition to providing insights into the fundamental physical differences between these two types of reactions, the present analysis provides a new diagnostic for differentiating between the conventionally defined hydrogen atom transfer and proton-coupled electron transfer reactions.     

NO2^-/NO2体系电子转移的外电场效应和温度效应

用半经典模型研究NO2^-/NO2体系自交换电子转移的反应机理和速率常数, 探讨了外电场对反应过程势能面, 反应能垒及电子转移速率常数的影响, 确定了能垒崩溃的阈值; 讨论了低能垒条件下电子转移反应的速率常数随温度变化的特征。研究表明, 一定方向的外电场能显著降低电子转移反应的活化能垒并提高反应速率常数k, 而对非绝热电子转移反应, 当温度T和活能能垒Ec满足T=2Ec/R时, k取得极大值。