Adiabatic and Nonadiabatic Reactions

A simple quadratic form of Eq. (34.10) is due to an identical parabolic form of the free-energy surfaces f/, and U. Since the dependence of the activation free energy on AF is nonhnear, the symmetry factor a may be introduced by a differential relationship, 

An equation of the type Eq. (34.9) (with instead of P) is valid for any shape of the free-energy surfaces as functions of the coordinates of any reactive modes provided that the motion along is classical. If the motion along some coordinates Q is quantum mechanical, these modes should be excluded from the free-energy surfaces. The transition along these modes has a tunnel character. 

It is determined by the overlap of the electron wave functions of the donor, qip, and acceptor, 9, and by the interaction V of the electron with the acceptor. 

The time during which nearly resonance configuration (within the energy interval on the order of V j) exists is 

the Landau-Zener parameter determining adiabatic or nonadiabatic character of transitions is [see Eqs. (34.15), (34.16), and (34.18)] 

The question arises above which interaction energy must a reaction be considered to be adiabatic This is difficult to answer, especially for electrode reactions, because it depends on the distance of the reacting species during the electron transfer. In the case of reactions in homogeneous solutions, Newton and Sutin [10] have estimated for typical transition-metal redox reactions that V p 0.025 eV is a reasonable limit above which a reaction must be considered to be adiabatic. This problem will be discussed again later in connection with some quantum mechanical models for electron transfer. 

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Figure 9.17. Adiabatic and nonadiabatic reaction profiles for the TICT process.

The concept of adiabacity in e.t. processes has gained importance in recent years, and the question does arise to what extent it may influence the observation of the M.I.R. In principle, the occurence of the M.I.R. is related only to the quadratic form of the activation energy, not to the form of the pre-exponential factor. The M.I.R. should therefore be observed for both adiabatic and nonadiabatic reactions. However, if the observable rate of an adiabatic process is controlled by the solvent relaxation time, the influence of the exponential factor may be negligible [18]. 

With proper assignment for transition probabilities, these parallel reaction models can be modified to describe simultaneous adiabatic and nonadiabatic reactions. For example, if one considers the reactant R and product B in the same electronic state and the product C in an excited electronic state, and if the spin-orbit interaction is negligible, then the reaction R B can be considered as adiabatic, whereas R -> C is a nonadiabatic crossing reaction. 

In the following sections we will consider some approximate methods for evaluation of the transmission coefficients and the tunneling corrections which will provide suitable criteria for electronically adiabatic and nonadiabatic reactions and will allow us a delimi-... 

Landau " and Zener independently gave the first mathematical formulation of the condition of adiabatic and nonadiabatic reactions in terms of probability factors. 

Elementary reactions can be classified by various properties. Some divisions have already been done earlier adiabatic and nonadiabatic reactions, exoergic and endoergic reactions. 

Unlike the simplest outer-sphere electron transfer reactions where the electrons are the only quantum subsystem and only two types of transitions are possible (adiabatic and nonadiabatic ones), the situation for proton transfer reactions is more complicated. Three types of transitions may be considered here5 ... 

For k(r) we shall assume at first, as in (19), that the reaction is adiabatic at the distance of closest approach, r = a, and that it is joined there to the nonadiabatic solution which varies as exp(-ar). The adiabatic and nonadiabatic solutions can be joined smoothly. For example, one could try to generalize to the present multi-dimensional potential energy surfaces, a Landau-Zener type treatment (41). For simplicity, however, we will join the adiabatic and nonadiabatic expressions at r = a. We subsequently consider another approximation in which the reaction is treated as being nonadiabatic even at r = a. 

The terms adiabatic and nonadiabatic are confusing. Thus, students who approach kinetics at an electrochemical interface via studies of chemistiy will be used to the term adiabatic. In thermodynamics, adiabatic indicates a process in which no heat enters or escapes from the system, e.g., from the vessel in which the reaction... 

At this stage we come back to the distinction between adiabatic and nonadiabatic e.t. reactions, alluded to in Sect. 1.4. The most important experimental clues about the role of the solvent are found in the temperature dependence of the rate constants of e.t., which broadly follow four distinct patterns ... 

One should add here that since parameter a may change with solvent, the analysis based on Eq. (46) may give an averaged value of that parameter. The problem of adiabatic and nonadiabatic charge-transfer reactions calls for further study. 

Kuznetsov, A.M. and Kharkats, Y.L (1975) Semiclassical theory of adiabatic and nonadiabatic electron-transfer bridging reactions. Soviet... 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

Kiefer, P. M., Hynes, J. T. (2006) Interpretation of primary kinetic isotope elfects for adiabatic and nonadiabatic proton transfer reactions in a polar environment, in Isotope Effects in Chemistry and Biology,... 

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. 

These two possibilities of reaction processes make much difference in calculating the rate of any chemical reaction. According to the transition state theory formalism, these two types of reactions (i.e., adiabatic and nonadiabatic) influence the value of the transmission coefficient, k, which is a preexponential term in the absolute rate expression. The value of k is considered unity for the adiabatic reaction and less than unity for a nonadiabatic reaction. 

Landau " and Zener made the first attempts to formulate the probability of transition of the reacting system from a lower electronic state to an upper one, and thus, to find the adiabaticity and nonadiabaticity of a reaction. 

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