The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

The absorption of a photon by a hydroxy-aromatic photoacid triggers a cascade of events contributing to the overall phenomenon of intermolecular excited-state proton transfer. The fundamental steps involved were studied over the last 20 years using a combination of theoretical and experimental techniques. They are surveyed in this sequel in sequential order, from fast to slow. The excitation triggers an intramolecular charge transfer to the ring system, which is more prominent for the anionic base than the acid. The charge redistribution, in turn, triggers changes in hydrogen-bond strengths that set the stage for the proton-transfer step itself. This step is strongly influenced by the solvent, resulting in unusual dependence of the dissociation rate coefficient on water content, temperature, and isotopic substitution. The photolyzed proton can diffuse in the aqueous solution in a mechanism that involves collective changes in hydrogen-bonding. On longer times, it may recombine adiabatically with the excited base or quench it. The theory for these diffusion-influenced geminate reactions has been developed, showing nice agreement with experiment. Finally, the effect of inert salts, bases, and acids on these reactions is analyzed.

Elementary steps in excited-state proton transfer.

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Biochemistry and theory of proton-coupled electron transfer.

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.

Proton-Coupled Electron Transfer in Solution, Proteins, and Electrochemistry

6.2. Reactive Flux Method 3156 6.3. Model Theories 3156 6.4. Survey 3156 7. Applications 3157 7.1. Yeast Enolase 3157 7.2. Triosephosphate Isomerase 3157 7.3. Methylamine Dehydrogenase 3158 7.4. Alcohol Dehydrogenase 3158 7.5. Thermophilic Alcohol Dehydrogenase 3158 7.6. Haloalkane Dehalogenase 3159 7.7. Dihydrofolate Reductase from E. Coli 3159 7.8. Hyperthermophilic DHFR from Thermotoga Maritima 3160

https://chem.ccu.edu.tw/~hu/Web_Lib/literature/quantum/Enzyme_Tunneling_Truhlar_Chem_Rev_2006.pdf

Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions.

Theory of tunnel transitions of atoms in solids

Effect of pressure and temperature on the H-atom tunneling in solid phase chemical reactions. The acridine/fluorene system

Tunneling of a hydrogen atom in low temperature processes

The pressure and temperature dependence of the reaction rate of the photochemical intermolecular H-transfer reaction in acridine-doped fluorene single crystals exhibit all typical features of solid state H-tunneling. Independent Raman data including the pressure dependence have identified three low energy inter- and intramolecular vibrational fluctuations as reaction promoting modes. The mode identification allows a more subtle analysis of the H-transfer reaction rate behavior on the basis of recent theoretical concepts. Surprisingly, the complete data set k(P. T) can be satisfactorily reproduced by an average one frequency mode model with parameters consistent with an average of the parameters calculated for the individual reaction promoting modes.

V.L. Klochikhin

Papers

Theory of tunnel transitions of atoms in solids

Effect of pressure and temperature on the H-atom tunneling in solid phase chemical reactions. The acridine/fluorene system

Tunneling of a hydrogen atom in low temperature processes

Vibration-assisted intermolecular hydrogen tunneling in photoreactive doped molecular crystals : Effect of temperature and pressure

Diffusion-controlled kinetics of systems containing a finite number of reacting particles