Electron transfer reactions quantum transition-state theory

A. Dynamical Test of the Centroid Quantum Transition-State Theory for Electron Transfer Reactions... 

Applications are then presented in Section IV. These examples should served as a guide as to what kinds of problems can be studied with these techniques and the limitations and possibilities for these methods. We present three examples (1) a dynamical test of the centroid quantum transition-state theory for electron transfer (ET) reactions in the crossover regime between adiabatic and nonadiabatic electron transfer, (2) the primary electron transfer reaction in bacterial photosynthesis, and (3) the diffusion kinetics of a Brownian particle in a periodic potential. Finally, Section V offers an outlook and a perspective of the current status of the field from our vantage point. 

Figure 21. Adibatic solvent activation free-energy curves for the Fe /Fe electron transfer reaction with a platinum electrode at 300 K calculated obtained using the model of Ref. 50, path-integral quantum transition-state theory, and umbrella sampling. The solid line depicts the quantum adiabatic free-energy curve, while the dashed line depicts the curve in the classical limit. In both cases, the left-hand well corresponds to the Fe stable state, while the right-hand well is the Fe " stable state.

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

In 1969 Schmidt made the first attempt to calculate quantum mechanically the probability of the elementary act of the bridge-assisted electron transfer reaction. However, the transitions were only taken into account for which the energy is conserved in all the states including the intermediate one. In a number of subsequent papers, a more detailed quantum mechanical calculation for the reactions of this kind was performed. Recently the theory was extended to adiabatic bridge-assisted reactions. 

In the current understanding of PCET reactions, both electron and proton are treated quantum-mechanically, and therefore the tunnelling probability must be accounted for both particles. In fact, concerted processes can be described as double tunnelling (proton and electron), with a single transition state. " For a description of the reaction coordinate, four adiabatic states (reactants, products and intermediates) described by paraboloids, are usually considered. The expression for the semi-classical rate constant in this case incorporates elements derived from electron and proton transfer theories... 

Stefan Christov performed and also promoted profound scientific studies in the field of quantum electrochemistry, physical chemistry, theory of chemical reactions, and solid-state physics [36 3]. His contributions related to hydrogen evolution reactions, corrosion phenomena, and electron transfer reactions are well known to those who work in these particular scientific fields. The same is valid also for the so-called Christov s characteristic temperature related to the transition rate of over- and under-barrier tunneling reactions. 

Marcus[195] gave a quantitative interpretation of this idea and above all, the role of solvent rearrangement within the framework of the absolute rate theory. Later, he also extended these concepts to electrochemical processes[196]. Similar concepts were also developed by Hush[197,198]. An important result of this work was the establishment of the relation between the transfer coefficient for adiabatic reactions and the charge distribution in the transient state. Gerischer[93,199] proposed a very useful and lucid treatment of the process of electron transfer in reactions with metallic as well as semiconductor electrodes. While the works mentioned above were mainly based on transition state theory, a systematic quantum-mechanical analysis of the problem was started by Levich, Dogonadze, and Chizmadzhev[200-202] and continued in a series of investigations by the same group. They extensively used the results and methods of solid state physics, and above all the Landau-Pekar polaron theory[203]. 

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