The kinetics of nucleophilic substitution processes in the alkyl halides. Part A—theory

Activation energies for substitution reactions of the type AC + B → A + CB, occurring in polar media and characterized by an abrupt change of the term along two coordinates have been calculated within the framework of the quantum-mechanical theory of chemical reactions. In the case of nonadiabatic processes, the transmission coefficient and activation energy for these reactions are expressed in terms of characteristic parameters of the medium (reorganization energy, effective frequency of solvent fluctuation polarization) and the potential energy curves for intermolecular interactions between the reactants (AC and B) and between the products (A and BC).     

Nonadiabatic transitions in chemical reactions: A quantum mechanical study

This work presents an exact quantum mechanical treatment of a reactive three-atom collinear model system incorporating nonadiabatic couplings. It was assumed that nonadiabatic transitions are induced by the vibrational motion only. The main findings are: (i) The reaction process can create conditions in which weak nonadiabatic couplings terms ( for which the Massey parameter was round 10) may cause large probabilities (～0.5) for transitions from one electronic surface to the other. In other words, the reaction process is able in certain cases to create a near resonance situation which makes the non-adiabatic transition almost independent of the magnitude of the coupling term. For this to happen the two surfaces need not be proximate, nor need they “almost” cross along a certain line (ii) In cases where the main nonadiabatic transitions take place outside the interaction region one may, at least qualitatively, decouple the reaction process from the nonadiabatic one. Thus, under the conditions specified one may first treat the reactive system on the ground state surface without including the excited interacting surface and then treat the nonadiabatic process independently.     

Recent Developments in Kramers’ Theory of Reaction Rates

In this short review, we provide an update of recent developments in Kramers’ theory of reaction rates. After a brief introduction stressing the importance of this theory initially developed for chemical reactions, we briefly present the main theoretical formalism starting from the generalized Langevin equation and continue by showing the main points of the modern Pollak, Grabert and Hänggi theory. Kramers’ theory is then sketched for quantum and classical surface diffusion. As an illustration the surface diffusion of Na atoms on a Cu(110) surface is discussed showing escape rates, jump distributions and diffusion coefficients as a function of reduced friction. Finally, some very recent applications of turnover theory to different fields such as nanoparticle levitation, microcavity polariton dynamics and simulation of reaction in liquids are presented. We end with several open problems and future challenges faced up by Kramers turnover theory.     

A microscopic model of chemical reactions with reorganization. High-energy approximation

A dynamical model of a chemical reaction, accompanied by reorganization of the immediate environment of the isolated chemical subsystem, is proposed. The model enables studying the emergence of nonequilibrium distribution functions as a combined result of the interaction within the dynamical subsystem and the energy exchange with a subsystem of inactive degrees of freedom (thermal bath). The study is based on the quasiclassical high-energy approximation for nonadiabatic effects in the energy exchange within the dynamical subsystem, for strong and weak coupling of the oscillator mode with the thermal bath. Such an approximation allows for the important statement that nonequilibrium effects in thermal reactions are absent if the initial translational distribution along the reaction coordinate and the initial vibrational distribution in transversal degrees of freedom are Boltzmann-like with the same temperature. The results obtained in the absence of the initial equilibrium distribution have been used for interpreting the kinetics of endothermic plasmochemical reactions proceeding under nonequilibrium conditions.     

Exploring dynamical electron theory beyond the Born-Oppenheimer framework: from chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field

Chemical theory and its application to dynamical electrons in molecules under intense electromagnetic fields is explored, in which we take an explicit account of nuclear nonadiabatic (kinematic) interactions along with simultaneous coupling with intense optical interactions. All the electronic wavefunctions studied here are necessarily time-dependent, and thereby beyond stationary state quantum chemistry based on the Born-Oppenheimer framework. As a general and tractable alternative framework with which to track the electronic and nuclear simultaneous dynamics, we propose an on-the-fly method to calculate the electron and nuclear wavepackets coupled along the branching non-Born-Oppenheimer paths, through which their bifurcations, strong quantum entanglement between nuclear electronic motions, and coherence and decoherence among the phases associated with them are properly represented. Some illustrative numerical examples are also reported, which are aimed at our final goals; real time tracking of nonadiabatic electronic states, chemical dynamics in densely degenerate electronic states coupled with nuclear motions and manipulation and/or creation of new electronic states in terms of intense lasers, and so on. Other examples are also presented as to how the electron wavepacket dynamics can be used to analyze chemical reactions, shedding a new light on some typical and conventional chemical reactions such as proton transfer followed by tautomerization.     

Modeling of decoherence and dissipation in nonadiabatic photoreactions by an effective-scaling nonsecular Redfield algorithm

Nonadiabatic photoisomerization dynamics in a condensed-phase environment is studied within the framework of Redfield theory. Considering several measures of decoherence and dissipation, the relaxation behavior of various models of nonadiabatic cis–trans photoisomerization is investigated. Several levels of relaxation theory are compared: The full Redfield theory (including all terms of the relaxation tensor), the secular approximation to it (including only the resonant terms) and the popular Bloch model (which also neglects terms that are resonant by accidence). Although these approximations are shown to work well for the single-mode model, they are found to lead to significant deviations for various two-mode models. The latter behavior is expected to be generic for multidimensional systems, which typically exhibit numerous near-degeneracies of the level spacings. To correctly describe the relaxation dynamics, while still retaining the advantageous N² scaling of the Bloch model, a “nonsecular” algorithm is proposed that systematically includes the most important nonsecular terms. The algorithm correctly reproduces the results of the full Redfield theory, while the numerical effort is reduced by typically an order of magnitude for the examples considered.     

A new ab initio potential energy surface for studying vibrational relaxation in NO(v) + NO collisions

A new ab initio potential energy surface for the ground state of the NO-NO system has been calculated within a reduced dimensionality model. We find an unusually large vibrational dependence of the interaction potential which explains previous spectroscopic observations. The potential can be used to model vibrational energy transfer, and here we perform quantum scattering calculations of the vibrational relaxation of NO(v). We show that the vibrational relaxation for v = 1 is 4 orders of magnitude larger than that for the related O(2)(v) + O(2) system without having to invoke nonadiabatic mechanisms as had been suggested in the past. For highly vibrationally excited states, we predict a strong dependence of the rates on the vibrational quantum number as has been observed experimentally, although there remain important quantitative differences. The importance of a chemically bound isomer on the relaxation mechanism is analyzed, and we conclude it does not play a role for the values of v considered in the experiment. Finally, the intriguing negative temperature dependence of the vibrational relaxation rate constants observed in experiments was studied using an statistical model to include the presence of many asymptotically degenerate spin-orbit states.     

Dielectric study of side-group rotation of methyl methacrylate in copolymers

The dielectric absorption due to the side-group rotation of the methyl methacrylate (MMA) unit in MMA-styrene and MMA-p-chlorostyrene (pCS) copolymers was measured. The relaxation time for MMA-pCS copolymers with low MMA contents could be interpreted quantitatively in terms of the Kramers rate-constant theory at the low-friction limit. On the other hand, the frictional effect on the relaxation time could not be ignored for any copolymers except for the MMA-pCS copolymers with low MMA contents. The relaxation time for pure poly(methyl methacrylate) could be interpreted by the Kramers theory at the high-friction limit.     

Proton-coupled electron transfer in solution, proteins, and electrochemistry

Recent advances in the theoretical treatment of proton-coupled electron transfer (PCET) reactions are reviewed. These reactions play an important role in a wide range of biological processes, as well as in fuel cells, solar cells, chemical sensors, and electrochemical devices. A unified theoretical framework has been developed to describe both sequential and concerted PCET, as well as hydrogen atom transfer (HAT). A quantitative diagnostic has been proposed to differentiate between HAT and PCET in terms of the degree of electronic nonadiabaticity, where HAT corresponds to electronically adiabatic proton transfer and PCET corresponds to electronically nonadiabatic proton transfer. In both cases, the overall reaction is typically vibronically nonadiabatic. A series of rate constant expressions have been derived in various limits by describing the PCET reactions in terms of nonadiabatic transitions between electron-proton vibronic states. These expressions account for the solvent response to both electron and proton transfer and the effects of the proton donor-acceptor vibrational motion. The solvent and protein environment can be represented by a dielectric continuum or described with explicit molecular dynamics. These theoretical treatments have been applied to numerous PCET reactions in solution and proteins. Expressions for heterogeneous rate constants and current densities for electrochemical PCET have also been derived and applied to model systems.     

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.     

How CO binds to hexacoordinated heme in neuroglobin protein

In the present work, density functional theory (DFT) has been used to investigate CO binding to the hexacoordinated heme in neuroglobin (Ngb) protein. Structural relaxation of the selected model system in the protein environment has been fully included by the alternative quantum and molecular mechanical optimizations. The polarized continuum model (PCM) was used to simulate interaction between the model system and the protein environment. The CO binding could take place in a concerted way and a barrier of 17.9 kcal mol(-1) was predicted on the concerted singlet pathway, which is not favorable in energy. The adiabatically sequential pathway requires an energy of 14.5 kcal mol(-1) for formation of the singlet intermediate. There exist two nonadiabatic sequential pathways for the CO binding, which involves the triplet and quintet states of intermediate. Both the singlet/triplet and singlet/quintet intersections play an important role in nonadiabatic sequential processes, which enhance the probability that the processes occur. The nonadiabatic processes that involve the triplet and quintet states of intermediate are the most probable pathways for the CO binding to the hexacoordinated heme in Ngb to form the product complex.     

Combined nonadiabatic transition-state theory and ab initio molecular dynamics study on selectivity of the alpha and beta bond fissions in photodissociation of bromoacetyl chloride

The selectivity of the alpha C-Cl and beta C-Br bond fissions upon n-->pi(*) excitation of bromoacetyl chloride has been investigated with combined nonadiabatic Rice-Ramsperger-Kassel-Marcus theory and ab initio molecular dynamics calculations, which are based on the potential energy profiles calculated with the complete active space self-consistent field and multireference configuration interaction methods. The Zhu-Nakamura [J. Chem. Phys. 101, 10630 (1994); 102, 7448 (1995)] theory is chosen to calculate the nonadiabatic hopping probability. It is found that nonadiabatic effect plays an important role in determining selective dissociations of the C-Cl and C-Br bonds. The calculated rate constants are close to those from experimentally inferred values, but the branching ratio of the alpha C-Cl and beta C-Br bond fissions is different from the experimental findings. The direct molecular dynamics calculations predict that fission of the C-Cl bond occurs on a time scale of picoseconds and cleavage of the beta C-Br bond proceeds with less probability within the same period. This reveals that the initial relaxation dynamics is probably another important factor that influences the selectivity of the C-Cl and C-Br bond fissions in photodissociation of BrCH(2)COCl at 248 nm.     

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.     

Quantum chemistry studies of electronically excited nitrobenzene, TNA, and TNT

The electronic excitation energies and excited-state potential energy surfaces of nitrobenzene, 2,4,6-trinitroaniline (TNA), and 2,4,6-trinitrotoluene (TNT) are calculated using time-dependent density functional theory and multiconfigurational ab initio methods. We describe the geometrical and energetic character of excited-state minima, reaction coordinates, and nonadiabatic regions in these systems. In addition, the potential energy surfaces for the lowest two singlet (S(0) and S(1)) and lowest two triplet (T(1) and T(2)) electronic states are investigated, with particular emphasis on the S(1) relaxation pathway and the nonadiabatic region leading to radiationless decay of S(1) population. In nitrobenzene, relaxation on S(1) occurs by out-of-plane rotation and pyramidalization of the nitro group. Radiationless decay can take place through a nonadiabatic region, which, at the TD-DFT level, is characterized by near-degeneracy of three electronic states, namely, S(1), S(0), and T(2). Moreover, spin-orbit coupling constants for the S(0)/T(2) and S(1)/T(2) electronic state pairs were calculated to be as high as 60 cm(-1) in this region. Our results suggest that the S(1) population should quench primarily to the T(2) state. This finding is in support of recent experimental results and sheds light on the photochemistry of heavier nitroarenes. In TNT and TNA, the dominant pathway for relaxation on S(1) is through geometric distortions, similar to that found for nitrobenzene, of a single ortho-substituted NO(2). The two singlet and lowest two triplet electronic states are qualitatively similar to those of nitrobenzene along a minimal S(1) energy pathway.     

Nonadiabatic transitions in chemical reactions. A comparison between exact and approximate collinear calculations

The quantum mechanical study of nonadiabatic transitions during a reactive collision of an atom and a diatomic molecule is extended. The constant coupling term used in our previous work is replaced by a gaussian-type function to simulate the fact that the interaction becomes stronger as the two interacting surfaces become closer. the two surfaces are assumed to be one above the other but shifted along the vibrational coordinate. It is found that when the coupling is not too strong, the nuclear process taking place on the original adiabatic surface can be decoupled from the electronic process which is responsible for the transitions between the two surfaces. It is established that the strength of the coupling is defined in terms of the area under the coupling function and its width. Dependence on the latter is in general expected to be much stronger than on the former.     

The kinetics of overcoming the “entropy barrier”

The escape of a point particle from a cavity of an arbitrary configuration through a small orifice in a cavity wall is considered as a model of entropy barrier overcoming. The dynamics of the particle is determined by collisions with the wall and the dissipative action of the medium. As in reactions with overcoming energy barriers (the Kramers theory), three characteristic regimes can be identified depending on the intensity of friction: diffusion and intermediate regimes and the region of weak friction. For all of them, analytic dependences of rate constants on problem parameters were obtained. The procedure for sewing solutions together, similar to that employed in the Kramers theory, gives a unified equation for the transmission factor over the whole range of friction values. In the weak friction mode, overcoming the entropy barrier generally leads to a nonexponential kinetics.     

Modeling of surface exchange reactions and diffusion in composites including transport processes at grain and interphase boundaries

Surface exchange reactions and diffusion of oxygen in ceramic composites consisting of a dilute and random distribution of inclusions in a polycrystalline matrix (host phase) are modeled phenomenologically by employing the finite element method. The microstructure of the mixed conducting composite is described by means of a square grain model, including grain boundaries of the matrix and interphase boundaries between the inclusions and grains of the host phase. An instantaneous change of the oxygen partial pressure in the surrounding atmosphere may give rise to an oxygen exchange process, i.e., oxidation or reduction of the ceramic composite. Relaxation curves for the total amount of exchanged oxygen are calculated, emphasizing the role played by fast diffusion along the interfaces. The relaxation curves are interpreted in terms of effective medium diffusion, introducing appropriate equations for the effective diffusion coefficient and the effective surface exchange coefficient. When extremely fast diffusion along the grain and interphase boundaries is assumed, the re-equilibration process shows two different time constants. Analytical approximations for the relaxation process and relations for the separate relaxation times are provided for this limiting case as well as for blocking interphase boundaries. Furthermore, conductivity relaxation curves are calculated by coupling diffusion and dc conduction. In the case of effective medium diffusion, the conductivity relaxation curves do not deviate from those for the total amount of exchanged oxygen. On the contrary, the conductivity relaxation curves differ remarkably from the time dependence of the total amount of exchanged oxygen, when the different phases of the composite re-equilibrate with separate time constants.     

Adiabatic and nonadiabatic bond cleavages in Norrish type I reaction

One of the fundamental photoreactions for ketones is Norrish type I reaction, which has been extensively studied both experimentally and theoretically. Its α bond-cleavage mechanisms are usually explained in an adiabatic picture based on the involved excited-state potential energy surfaces, but scarcely investigated in terms of a nonadiabatic picture. In this work, the S(1) α bond-cleavage reactions of CH(3)OC(O)Cl have been investigated by using the CASSCF and MRCI-SD calculations, and the ab initio based time-dependent quantum wavepacket simulation. The numerical results indicate that the photoinduced dissociation dynamics of CH(3)OC(O)Cl could exhibit strong nonadiabatic bond-fission characteristics for the S(1) α C-Cl bond cleavage, while the dynamics of the S(1) α C-O bond cleavage is mainly of adiabatic characteristics. This nonadiabatic mechanism for Norrish type I reaction of CH(3)OC(O)Cl is uncovered for the first time. The quantum wavepacket dynamics, based on the reduced-dimensional coupled potential energy surfaces, to some extent illustrates the significance of the nonadiabatic effect in the transition-state region on the dynamics of Norrish type I reaction.     

Nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation method revisited: applications to Tully's three model systems

The nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method for nonadiabatic problems is reformulated. The method has the same spirit as Tully's surface hopping technique [J. Chem. Phys. 93, 1061 (1990)] and almost keeps the same structure as the original single-surface HK SC-IVR method except that trajectories can hop to other surfaces according to the hopping probabilities and phases, which can be easily integrated along the paths. The method is based on a rather general nonadiabatic semiclassical surface hopping theory developed by Herman [J. Chem. Phys. 103, 8081 (1995)], which has been shown to be accurate to the first order in h and through all the orders of the nonadiabatic coupling amplitude. Our simulation studies on the three model systems suggested by Tully demonstrate that this method is practical and capable of describing nonadiabatic quantum dynamics for various coupling situations in very good agreement with benchmark calculations.