On the Theory of Non-adiabatic Chemical Reactions

The static theory of atomic forces, which has been considered almost exclusively up to now with the methods of quantum mechanics, needs the addition of a dynamics of a chemical reaction. The collision methods, used by several groups, do not appear to be a useful method for the treatment of chemical processes (Section 1). In what follows, a dynamical theory for the simplest cases of bimolecular reactions is developed (Sections 2 and 3), in which the problem of chemistry is expressed immediately, and clearly discussed with the help of elementary examples (Section 4). Sections 5 and 6 contain a perturbation approach, whic converges for small and large collision velocities and allows for a relatively simple sqiproximation method for the reaction velocity of non-adiabatic processes. On the other hand the theory contains a quantitative description of the connection to the adiabatic reaction process in the limit of low velocities or separated characteristics of the potential. In this way it yields a conditional justification for the application of potential theoretical representations to chemical processes and at the same time a fixation of the limits of such idealisations. 

In the theory of chemical kinetics, interest has recently been limited especially to processes that can be conceived of potential theoretically or, as is sand sometimes as well, adiabatically . This means that in these processes the dynamical behaviour can essentially be characterised by a single functionj dependent only on the positions of the atomic centres of mass. This function is interpreted as a potential in the sense of classical mechanics, or, more precisely and more accurately, in the sense of wave mechanics as the determining factor for the index of refraction of the de Broglie waves of the atomic centres of mass.  

We have to be grateful that quantum mechanics allows us to base chemical processes on a potential theoretical picture to a relatively large extent. The atoms can be viewed as solid units, and particularly the act of valence bonding and even some processes of catalysis can be described bjy conservative forces without fundamentally having to consider quantum jumps or similar processes in the internal coordinates. 

But on the other hand there are important reactions for which it is impossible to make such idealisations even approximately. These reactions are the subject of the present note. 

Energy transfers Hg -I- Na = Hg + Na (activation, stimulated fluorescence, depolarisations, etc.)  

---

The Landau-Zener model illustrates the important variables influencing the probability of non-adiabatic transitions, but as a ID model it is only applicable to bimolecular reaction of two atoms. For most reactions of interest it is too simple to provide accurate results. For reactions involving more than two atoms the PESs are multidimensional, as we have seen above, and the avoided crossing region on a multidimensional surface is described as a conical intersection [61]. The best method for handling this complex multidimensional reactive scattering problem is trajectory calculations. Fernandez-Ramos et al. [52] has discussed approaches to this problem as part of a recent review of bimolecular reaction rate theory. It is fortunate that the vast majority of chemical reactions occur adiabatically. It will only be necessary to delve into the theory of non-adiabatic reactions when a non-adiabatic reaction is present in a reaction model, experimental data are not available, and the reaction rate influences the overall rate appreciably. 

The general treatment of the theory of chemical reactions presented in this book is based on the usual adiabatic separation of nuclear and electronic motions which permits a definition of the potential energy as a function of internuclear distances This approach proves to be very useful for the study of electronically adiabatic reactions, provided a separation of the rotation of the reacting system, treated as a supermolecule, is possible. In general, such a separation seems to be a bad approximation /10/. A consideration of the coupling of the overall rotation with the internal motions of the system means taking into account the possibility of non-adiabatic transitions from one to another potential energy surface. This is still an unsolved problem of theoretical chemistry which is open for discussion. 

The theory of electron transfer in chemical and biological systems has been discussed by Marcus and many other workers 74 84). Recently, Larson 8l) has discussed the theory of electron transfer in protein and polymer-metal complex structures on the basis of a model first proposed by Marcus. In biological systems, electrons are mediated between redox centers over large distances (1.5 to 3.0 nm). Under non-adiabatic conditions, as the two energy surfaces have little interaction (Fig. 5), the electron transfer reaction does not occur. If there is weak interaction between the two surfaces, a, and a2, the system tends to split into two continuous energy surfaces, A3 and A2, with a small gap A which corresponds to the electronic coupling matrix element. Under such conditions, electron transfer from reductant to oxidant may occur, with the probability (x) given by Eq. (10),... 

The modem theory of chemical reaction is based on the concept of the potential energy surface, which assumes that the Born-Oppenheimer adiabatic approximation [16] is obeyed. However, in systems subjected to the Jahn-Teller effect, adiabatic potentials have the physical meaning of the potential energy of nuclei only under the condition that non-adiabatic corrections are small [28]. In the vicinity of the locally symmetric intermediate, these corrections will be very large. The complete description of nuclear motion, i.e. of the mechanism of the chemical reaction, can be obtained only from Schroedinger s equation without applying the Born-Oppenheimer approximation in the vicinity of the locally... 

Quantum theory of an elementary electron transfer act confirms this suggestion. In the early 1970s, using Marcus idea on the fluctuations of solvent energy as a driving force for electron transfer [1], Vorotyntsev and Kuznetsov [2] showed theoretically that, for non-adiabatic reactions, the elementary two-electron step is highly improbable, while Dogonadze and Kuznetsov proved that the steps with more than two transferred electrons are practically impossible [3]. It is consistent with the rules of chemical kinetics mentioned above two-electron elementary step can formally be presented as almost improbable reaction of third order, and three or more electron steps as the impossible reactions of more than third order. 

---