Theoretical study of electron, proton, and proton-coupled electron transfer in iron bi-imidazoline complexes.

A comparative theoretical investigation of single electron transfer (ET), single proton transfer (PT), and proton-coupled electron transfer (PCET) reactions in iron bi-imidazoline complexes is presented. These calculations are motivated by experimental studies showing that the rates of ET and PCET are similar and are both slower than the rate of PT for these systems (Roth, J. P.; Lovel, S.; Mayer, J. M. J. Am. Chem. Soc. 2000, 122, 5486). The theoretical calculations are based on a multistate continuum theory, in which the solute is described by a multistate valence bond model, the transferring hydrogen nucleus is treated quantum mechanically, and the solvent is represented as a dielectric continuum. For electronically nonadiabatic electron transfer, the rate expressions for ET and PCET depend on the inner-sphere (solute) and outer-sphere (solvent) reorganization energies and on the electronic coupling, which is averaged over the reactant and product proton vibrational wave functions for PCET. The small overlap of the proton vibrational wave functions localized on opposite sides of the proton transfer interface decreases the coupling for PCET relative to ET. The theory accurately reproduces the experimentally measured rates and deuterium kinetic isotope effects for ET and PCET. The calculations indicate that the similarity of the rates for ET and PCET is due mainly to the compensation of the smaller outer-sphere solvent reorganization energy for PCET by the larger coupling for ET. The moderate kinetic isotope effect for PCET arises from the relatively short proton transfer distance. The PT reaction is found to be dominated by solute reorganization (with very small solvent reorganization energy) and to be electronically adiabatic, leading to a fundamentally different mechanism that accounts for the faster rate.     

Nonadiabatic proton-coupled electron transfer reactions: impact of donor-acceptor vibrations, reorganization energies, and couplings on dynamics and rates

Fundamental aspects of proton-coupled electron transfer (PCET) reactions in solution are analyzed with molecular dynamics simulations for a series of model systems. The analysis addresses the impact of the solvent reorganization energy, the proton donor-acceptor mode vibrational frequency, and the distance dependence of the nonadiabatic coupling on the dynamics of the reaction and the magnitude of the rate. The rate for nonadiabatic PCET is expressed in terms of a time-dependent probability flux correlation function. The time dependence of the probability flux correlation function is determined mainly by the solvent reorganization energy and is not significantly influenced by the proton donor-acceptor frequency or the distance dependence of the nonadiabatic coupling. The magnitude of the PCET rate becomes greater as the solvent reorganization energy decreases, the proton donor-acceptor frequency decreases, and the distance dependence of the nonadiabatic coupling increases. The approximations underlying a previously derived analytical PCET rate expression are also investigated. The short-time approximation for the solvent is valid for these types of systems. In addition, solvent damping effects on the proton donor-acceptor motion are not significant on the time scale of the probability flux. The rates calculated from the molecular dynamics simulations agree well with those calculated from the analytical rate expression.     

Dynamics of proton transfer and vibrational relaxation in dilute hydrofluoric acid

The molecular mechanisms in both vibrational relaxation and proton transfer (PT) associated with infrared (IR)-induced PT in a dilute hydrofluoric acid solution at ambient temperature are studied by molecular dynamics (MD) simulations with the multistate empirical valence bond model. To investigate the solvation dynamics, a collective solvent coordinate and its perpendicular bath modes are defined from the diabatic energy gap and their motions are examined by the generalized Langevin equation (GLE) formalism. The GLE analysis using the equilibrium MD simulation shows that the major solvent reorganizations in the PT are represented by the libration and hindered translation. In particular, the libration gives the stronger coupling to the solvent reorganization and the faster relaxation. The nonequilibrium MD simulation demonstrated that both the HF stretching vibration and the solvent reorganization relax on a similar time scale and thus compete in the PT. It also supported the "presolvation mechanism" for the PT in this system.     

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.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.     

Proton-coupled electron transfer in soybean lipoxygenase: dynamical behavior and temperature dependence of kinetic isotope effects

The dynamical behavior and the temperature dependence of the kinetic isotope effects (KIEs) are examined for the proton-coupled electron transfer reaction catalyzed by the enzyme soybean lipoxygenase. The calculations are based on a vibronically nonadiabatic formulation that includes the quantum mechanical effects of the active electrons and the transferring proton, as well as the motions of all atoms in the complete solvated enzyme system. The rate constant is represented by the time integral of a probability flux correlation function that depends on the vibronic coupling and on time correlation functions of the energy gap and the proton donor-acceptor mode, which can be calculated from classical molecular dynamics simulations of the entire system. The dynamical behavior of the probability flux correlation function is dominated by the equilibrium protein and solvent motions and is not significantly influenced by the proton donor-acceptor motion. The magnitude of the overall rate is strongly influenced by the proton donor-acceptor frequency, the vibronic coupling, and the protein/solvent reorganization energy. The calculations reproduce the experimentally observed magnitude and temperature dependence of the KIE for the soybean lipoxygenase reaction without fitting any parameters directly to the experimental kinetic data. The temperature dependence of the KIE is determined predominantly by the proton donor-acceptor frequency and the distance dependence of the vibronic couplings for hydrogen and deuterium. The ratio of the overlaps of the hydrogen and deuterium vibrational wavefunctions strongly impacts the magnitude of the KIE but does not significantly influence its temperature dependence. For this enzyme reaction, the large magnitude of the KIE arises mainly from the dominance of tunneling between the ground vibronic states and the relatively large ratio of the overlaps between the corresponding hydrogen and deuterium vibrational wavefunctions. The weak temperature dependence of the KIE is due in part to the dominance of the local component of the proton donor-acceptor motion.     

Solvent effects on electron-driven proton-transfer processes: adenine-thymine base pairs

Time-Dependent Density Functional Theory (TD-DFT) computations, with M05-2X and PBE0 functionals, have been employed for a detailed study of the Electron-Driven Proton-Transfer (PT) processes in an Adenine-Thymine Watson-Crick Base Pair in the gas phase and in solution, with the bulk solvent described by the polarizable continuum model. In the gas phase, TD-DFT computations predict that the Adenine → Thymine Charge Transfer (CT) excited state undergoes a barrierless PT reaction, in agreement with CC2 computations (S. Perun, A. Sobolewski, W. Domcke, J. Phys. Chem. A, 2006, 110, 9031.). The good agreement between the TD-DFT approach and CC2 results validates the former for the studies of excited state properties, excited state proton transfer reaction, and deactivation mechanisms in the DNA base pairs. Next, it is shown that inclusion of solvent effects significantly influences the possibility of both barrier-less excited state proton transfer and radiation-less deactivation through conical intersection with the ground state, affecting the energy of the CT excited state in the Franck-Condon region, the energy barrier associated to the PT process and the energy gap with the ground electronic state. These findings clearly indicate that environmental effects, with a special attention to proper treatment of dynamical solvation effects, have to be included for reliable computational analysis of photophysical and photochemical processes occurring in condensed phases.     

Theoretical analysis of proton relays in electrochemical proton-coupled electron transfer

The coupling of long-range electron transfer to proton transport over multiple sites plays a vital role in many biological and chemical processes. Recently the concerted proton-coupled electron transfer (PCET) reaction in a molecule with a hydrogen-bond relay inserted between the proton donor and acceptor sites was studied electrochemically. The standard rate constants and kinetic isotope effects (KIEs) were measured experimentally for this double proton transfer system and a related single proton transfer system. In the present paper, these systems are studied theoretically using vibronically nonadiabatic rate constant expressions for electrochemical PCET. Application of this approach to proton relays requires the calculation of multidimensional proton vibrational wave functions and the incorporation of multiple proton donor-acceptor motions. The decrease in proton donor-acceptor distances due to thermal fluctuations and the contributions from excited electron-proton vibronic states play important roles in these systems. The calculated KIEs and the ratio of the standard rate constants for the single and double proton transfer systems are in agreement with the experimental data. The calculations indicate that the standard PCET rate constant is lower for the double proton transfer system because of the smaller overlap integral between the ground state reduced and oxidized proton vibrational wave functions, resulting in greater contributions from excited electron-proton vibronic states with higher free energy barriers. The theory predicts that this rate constant may be increased by modifying the molecule in a manner that decreases the equilibrium proton donor-acceptor distances or alters the molecular thermal motions to facilitate the concurrent decrease of these distances. These insights may guide the design of more efficient catalysts for energy conversion devices.     

Proton transfer in a polar solvent from ring polymer reaction rate theory

We have used the ring polymer molecular dynamics method to study the Azzouz-Borgis model for proton transfer between phenol (AH) and trimethylamine (B) in liquid methyl chloride. When the A-H distance is used as the reaction coordinate, the ring polymer trajectories are found to exhibit multiple recrossings of the transition state dividing surface and to give a rate coefficient that is smaller than the quantum transition state theory value by an order of magnitude. This is to be expected on kinematic grounds for a heavy-light-heavy reaction when the light atom transfer coordinate is used as the reaction coordinate, and it clearly precludes the use of transition state theory with this reaction coordinate. As has been shown previously for this problem, a solvent polarization coordinate defined in terms of the expectation value of the proton transfer distance in the ground adiabatic quantum state provides a better reaction coordinate with less recrossing. These results are discussed in light of the wide body of earlier theoretical work on the Azzouz-Borgis model and the considerable range of previously reported values for its proton and deuteron transfer rate coefficients.     

Solvation and proton transfer in polar molecule nanoclusters

Proton transfer in a phenol-amine complex dissolved in polar molecule nanoclusters is investigated. The proton transfer rates and mechanisms, as well as the solvation of the complex in the cluster, are studied using both adiabatic and nonadiabatic dynamics. The phenol-amine complex exists in ionic and covalent forms and as the size of the cluster increases the ionic form gains stability at the expense of the covalent form. Both the adiabatic and nonadiabatic transfer reaction rates increase with cluster size. Given a fixed cluster size, the stability of the covalent state increases with increasing temperature. The proton transfer rates do not change monotonously with an increase in temperature. A strong correlation between the solvent polarization reaction coordinate and the location of the phenol-amine complex in the cluster is found. The ionic form of the complex strongly prefers the interior of the cluster while the covalent form prefers to lie on the cluster surface.     

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.     

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.     

Activation energy for a model ferrous-ferric half reaction from transition path sampling

Activation parameters for the model oxidation half reaction of the classical aqueous ferrous ion are compared for different molecular simulation techniques. In particular, activation free energies are obtained from umbrella integration and Marcus theory based thermodynamic integration, which rely on the diabatic gap as the reaction coordinate. The latter method also assumes linear response, and both methods obtain the activation entropy and the activation energy from the temperature dependence of the activation free energy. In contrast, transition path sampling does not require knowledge of the reaction coordinate and directly yields the activation energy [C. Dellago and P. G. Bolhuis, Mol. Simul. 30, 795 (2004)]. Benchmark activation energies from transition path sampling agree within statistical uncertainty with activation energies obtained from standard techniques requiring knowledge of the reaction coordinate. In addition, it is found that the activation energy for this model system is significantly smaller than the activation free energy for the Marcus model, approximately half the value, implying an equally large entropy contribution.     

Quantized time correlation function approach to nonadiabatic decay rates in condensed phase: application to solvated electrons in water and methanol

A new, alternative form of the golden rule formula defining the nonadiabatic transition rate between two quantum states in condensed phase is presented. The formula involves the quantum time correlation function of the energy gap, of the nonadiabatic coupling, and their cross terms. Those quantities can be inferred from their classical counterparts, determined via molecular dynamics simulations. The formalism is applied to the problem of the nonadiabatic p-->s relaxation of an equilibrated p-electron in water and methanol. We find that, in both solvents, the relaxation is induced by the coupling to the vibrational modes and the quantum effects modify the rate by a factor of 2-10 depending on the quantization procedure applied. The resulting p-state lifetime for a hypothetical equilibrium excited state appears extremely short, in the sub-100 fs regime. Although this result is in contrast with all previous theoretical predictions, we also illustrate that the lifetimes computed here are very sensitive to the simulated electronic quantum gap and to the strongly correlated nonadiabatic coupling.     

Theoretical Study of the Intramolecular Proton Transfer in the Tautomers of Cytosine Assisted by Water

Ab initio MP2 and DFT studies on the tautomers of cytosine and the related hydrated tautomers have been carried out. The ground‐state structures of four tautomers of cytosine and related transition states were fully optimized. The vibrational frequency analysis was performed on all the optimized structures. Detailed intrinsic reaction coordinate (IRC) calculations were carried out to guarantee the optimized transition‐state structures being connected to the related tautomers. We obtained the relative stability order for the tautomers of cytosine and the related hydrated tautomers. In the isolated and hydrated condition, the bond types of C(2) O(7) and C(4) N(8) greatly affect the stability of the cytosine tautomers. Moreover, we have explored the influence of the water molecules on the intramolecular proton transfer between the keto and enol forms of the cytosine tautomers. The first water molecule obviously decreases the isomerization activation energy for the monohydrated cytosine tautomers. It is shown that the isomerization energy barrier changes only a little when the second and third water molecules are added in the reaction loop. The solvent effects have an obvious influence on the proton‐transfer barrier of the isolated cytosine. However, the solvent effects seem to be insignificant for the isomerization energy barriers of the monohydrated, dihydrated and trihydrated cytosine. The water molecule in these complexes can be looked on as the explicit water. Therefore, the explicit water model may be more credible to explore the intramolecular proton transfer, in comparison with the PCM which is the implicit water model.     

Vibrational energy relaxation of the ND-stretching vibration of NH(2)D in liquid NH(3)

The vibrational energy relaxation from the first excited ND-stretching mode of NH(2)D dissolved in liquid NH(3) is studied using molecular dynamics simulations. The rate constants for inter- and intramolecular energy transfer are calculated in the framework of the quantum-classical Landau-Teller theory. At 273 K and an ammonia density of 0.642 g cm(-3) the calculated ND-stretch lifetime of τ = 9.1 ps is in good agreement with the experimental value of 8.6 ps. The main relaxation channel accounting for 52% of the energy transfer involves an intramolecular transition to the first excited state of the umbrella mode. The energy difference between both states is taken up by the near-resonant bending vibrations of the solvent. Less important for the ND-stretch lifetime are both the direct transition to the ground state and intramolecular relaxation via the NH(2)D bending modes contributing 23% each. Our calculations imply that the experimentally observed weak density dependence of τ is caused by detuning the resonance between the ND-stretch-umbrella energy gap and the solvent accepting modes which counteracts the expected linear increase of the relaxation rate with density.     

Reorganization of intramolecular high frequency vibrational modes and dynamic solvent effect in electron transfer reactions

The possibility of the multichannel stochastic model to adequately describe all principal regularities observed in thermal electron transfer kinetics has been demonstrated. The most important are as follows: (i) the model predicts the solvent controlled regime in the Marcus normal region and its almost full suppression in the Marcus inverted region as well as a continuous transition between them in the vicinity of the activationless region; (ii) the suppression of dynamic solvent effect (DSE) is principally caused by the reorganization of high frequency vibrational modes; (iii) an additional factor of the DSE suppression stems from fast solvent relaxation component; (iv) in the inverted region, the multichannel stochastic model predicts the apparent activation energy to be much less than that calculated with Marcus equation. The exploration of the multichannel stochastic model has allowed one to conclude that the reorganization of high frequency vibrational modes can (i) raise the maximum rate constant above the solvent controlled limit by 2 and more orders of magnitude, (ii) shift the rate constant maximum to larger values of the free energy gap, and (iii) approach the electron transfer kinetics to the nonadiabatic regime.     

Exploring excited-state hydrogen atom transfer along an ammonia wire cluster: competitive reaction paths and vibrational mode selectivity

The excited-state hydrogen-atom transfer (ESHAT) reaction of the 7-hydroxyquinoline(NH(3))(3) cluster involves a crossing from the initially excited (1)pipi(*) to a (1)pisigma(*) state. The nonadiabatic coupling between these states induces homolytic dissociation of the O-H bond and H-atom transfer to the closest NH(3) molecule, forming a biradical structure denoted HT1, followed by two more Grotthus-type translocation steps along the ammonia wire. We investigate this reaction at the configuration interaction singles level, using a basis set with diffuse orbitals. Intrinsic reaction coordinate calculations of the enol-->HT1 step predict that the H-atom transfer is preceded and followed by extensive twisting and bending of the ammonia wire, as well as large O-H...NH(3) hydrogen bond contraction and expansion. The calculations also predict an excited-state proton transfer path involving synchronous proton motions; however, it lies 20-25 kcal/mol above the ESHAT path. Higher singlet and triplet potential curves are calculated along the ESHAT reaction coordinate: Two singlet-triplet curve crossings occur within the HT1 product well and intersystem crossing to these T(n) states branches the reaction back to the enol reactant side, decreasing the ESHAT yield. In fact, a product yield of approximately 40% 7-ketoquinoline.(NH(3))(3) is experimentally observed. The vibrational mode selectivity of the enol-->HT1 reaction step [C. Manca, C. Tanner, S. Coussan, A. Bach, and S. Leutwyler, J. Chem. Phys. 121, 2578 (2004)] is shown to be due to the large sensitivity of the diffuse pisigma(*) state to vibrational displacements along the intermolecular coordinates.     

Electronic to vibrational energy transfer assisted by interacting transition dipole moments: a quantum model for the nonadiabatic I2(E) + CF4 collisions

We report a theoretical study of nonadiabatic transitions within the first-tier ion-pair states of molecular iodine induced by collisions with CF(4). We propose a model that treats the partner as a spherical particle with internal vibrational structure. Potential energy surfaces and nonadiabatic matrix elements for the I(2)-CF(4) system are evaluated using the diatomics-in-molecule perturbation theory. A special form of the intermolecular perturbation theory for quasi-degenerate electronic states is implemented to evaluate the corrections to the long-range interaction of transition dipole moments of colliding molecules. The collision dynamics is studied by using an approximate quantum scattering approach that takes into account the coupling of electronic and vibrational degrees of freedom. Comparison with available experimental data on the rate constants and product state distributions demonstrates a good performance of the model. The interaction of the transition dipole moments is shown to induce very efficient excitation of the dipole-allowed upsilon(3) and upsilon(4) modes of the CF(4) partner. These transitions proceed predominantly through the near-resonant E-V energy transfer. The resonant character of the partner's excitation and the large mismatch in vibrational frequencies allow one to deduce the partner's vibrational product state distributions from the distributions measured for the molecule. The perspectives of the proposed theoretical model for treating a broad range of molecular collisions involving the spherical top partners are discussed.