Adiabatic reactions, theory

Baer M 1985 The theory of electronic non-adiabatic transitions in chemical reactions Theory of Chemical Reaction Dynamics vol II, ed M Baer (Boca Raton, FL CRC Press) p 281... 

The exchange reactions (6.20) and (6.21) have been among the basic objects of chemical-reaction theory for half a century. Clearly further investigation is needed, incorporating real crystal dynamics. It is worth noting that the adiabatic model, upon which the cited results are based, can prove to be insufficient because of the low frequency of the promoting vibrations. 

We have established an important principle in electron transfer theory that is not present in conventional one-dimensional models. The reaction coordinate is always localizing and corresponds to coordinate Aj. The coordinate X2 corresponds to the direction in which the matrix element between ground and excited states is switched on. If this coordinate has zero length then the branching space becomes one dimensional and an adiabatic reaction path does not exist. We now consider two examples. 

In calculating the transition probability for the nonadiabatic reactions, it is sufficient to use the lowest order of quantum mechanical perturbation theory in the operator V d. For the adiabatic reactions, we must perform the summation of the whole series of the perturbation theory.5 (It is insufficient to retain only the first term of the series that appeared in the quantum mechanical perturbation theory.) Correct calculations in both adiabatic and diabatic approaches lead to the same results, which is evidence of the equivalence of the two approaches. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

The theory of adiabatic reaction developed in the previous article is here generalized to the case when heat transfer is present. Consideration of the heat transfer leads to the appearance of new features in the consumptiontime kinetic curves, specifically, the possibility of extinction as the residence time is increased and of self-ignition when the reaction time is decreased (in the previous article, in the adiabatic case, extinction occurred only for a decrease in the reaction time, and self-ignition only for an increase). 

For a small initial concentration a reaction which is close to isothermic is always realized, i.e., a curve of type 2. Increasing the initial concentration, with small heat transfer, we move into the region of curves of type 3, analyzed earlier in the theory of adiabatic reaction. 

The transmission coefficient ktI — 1 in TST approach, but ktr < 1 in reality. It can be obtained in the more exact approach of Kramers equation [27]. The detailed account of the theory of adiabatic reactions of electron transfer in polar media may be found in the already mentioned Kuznetsov s monograph [13]. 

The form of the reaction-rate function is germane to equation (18) the nonadiabatic form for a thermal theory possesses a greater degree of arbitrariness than does that for a complete theory with one-step, Arrhenius kinetics. To achieve agreement with the nonadiabatic version of the theory whose adiabatic reaction-rate function is given by equation (5-66), we may put... 

As already mentioned above, the derivation of the Butler-Volmer equation, especially the introduction of the transfer factor a, is mostly based on an empirical approach. On the other hand, the model of a transition state (Figs. 7.1 and 7.2) looks similar to the free energy profile derived for adiabatic reactions, i.e. for processes where a strong interaction between electrode and redox species exists (compare with Section 6.3.3). However, it should also be possible to apply the basic Marcus theory (Section 6.1) or the quantum mechanical theory for weak interactions (see Section 6.3.2) to the derivation of a current-potential. According to these models the activation energy is given by (see Eq. 6.10)... 

Adiabatic, or as it has been termed vibrationally adiabatic,15 transition state theory has its origin in a paragraph in an article by Hirschfelder and Wigner.16 The treatment was developed further by a number of authors.17 In this type of transition state theory the eigenvalues of the system at each R, which are the vibrationally adiabatic eigenvalues, are plotted versus R. The N j in Eq. (2.1) then becomes the number of such states whose maximum energy on this plot does not exceed E, that is, N%j now denotes the sum of all open adiabatic reaction channels. 

Since the solvation time correlation function is known both from experiments and from computer simulations, we can easily carry out the above exercise. When this is done, the theory predicts a lack of, or weak, dependence of the electron transfer rate on solvent dynamics, for weakly adiabatic reactions the reason being the dominance of the ultrafast component in SD of water, so the solvent moves too fast to offer any retardation ... 

In this way, a complete treatment of the one-frequency oscillator system was made by CHRISTOV /67/ using the semi-classical Lan-dau-Zener theory however, the nuclear tunneling has been considered for the whole energy range only in the limiting case of adiabatic reactions 1). A recent extension of these results to include the... 

Adiabatic Statistical Theory of Reaction Rates 5.1. Exact Formulations of the Adiabatic Rate Theory... 

The expressions (106.Ill) and (124.Ill) represent two equivalent formulations of an exact adiabatic statistical theory of reaction rates based on two different definitions of the activated complex . Therefore, there exists the relation... 

The advantage of the statistical theory appears for fully (electronically and vibration-rotationally) adiabatic reactions, involving activation energy, at sufficiently high temperatures at which a solution of the dynamical problem may be avoided, since the correction factor to any of the statistical formulations comes close to unity. In this situation the less restricted and most useful of these formulations is certanly the Eyring rate equation, which follows from the exact expression (67.Ill) if the condition (82.Ill) is valid only from reactants to transition region of configuration space. Since... 

---