Non adiabatic Reactions

Quantum mechanical approaches for describing electron transfer processes were first applied by Levich [4] and Dogonadze, and later also in conjunction with Kuznetsov [5]. They assumed the overlap of the electronic orbitals of the two reactants to be so weak that perturbation theory, briefly introduced in the previous section, could be used to calculate the transfer rate for reactions in homogeneous solutions or at electrodes. The polar solvent was here described by using the continuum theory. The most important step is the calculation of the Hamiltonians of the system. In general terms the latter are given for an electron transfer between two ions in solution by 

Since the ions are relatively heavy, they are considered to be stationary during electron transfer, i.e. the Hamiltonian for the electron He can be written in terms of the kinetic energy of the electron and its interaction with the ions as 

The Hamiltonian of the solvent Hie,n-f,a depends on two parts, one concerning the solvent itself, //joiv, and another term which specifies the role of the ion, Hjon, (see below), so that we have 

The Hamiltonian //sowhas been derived on the assumption that the solvent fluctuation occurs again as an harmonic motion. The harmonic of a single harmonic oscillator is given by 

The first term corresponds to the kinetic and the second to the potential energy. Since the frequency of the motion is given by coq = (kltn)V2 the Hamiltonian is given by 

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Thus, to calculate the transition probability for the non-adiabatic reaction it is sufficient to know the diabatic free energy... 

Furthermore, there are some effects related to the interaction of the reactants with the medium. We shall first consider the effects of the fluctuational preparation of the potential barrier in non-adiabatic reactions. 

The low pre-exponential factor associated with fc°° is an indication of a non-adiabatic reaction. Since there is good evidence122 against a significant extent of thermal dissociation of N20 to N+NO, the alternative dissociation, reaction (1), has been universally accepted as the initial step in the decomposition. The subsequent fate of the ground-state O atoms is unquestionably reaction with N20, viz. 

A special case of a non-adiabatic reaction is electron transfer. The dynamics of electron-transfer processes have been studied extensively, and the most robust model used to describe... 

In a thermal reaction R—>TS—>P, as shown in Figure 4.4, the transition state TS is reached through thermal activation, so that the general observation is that the rates of thermal reactions increase with temperature. The same is in fact true of many photochemical reactions when they are essentially adiabatic, for the primary photochemical process is then a thermally activated reaction of the excited reactant R. A non-adiabatic reaction such as R - (TS) —> P is in principle temperature independent and can be considered as a type of non-radiative transition from a state R to a state P of lower energy, for example in some reactions of isomerization (see section 4.4.2). 

Figure 2.1(a) above illustrates the potential energy surface for a diabatic electron transfer process. In a diabatic (or non-adiabatic) reaction, the electronic coupling between donor and acceptor is weak and, consequently, the probability of crossover between the product and reactant surfaces will be small, i.e. for diabatic electron transfer /cei, the electronic transmission factor, is transition state appears as a sharp cusp and the system must cross over the transition state onto a new potential energy surface in order for electron transfer to occur. Longdistance electron transfers tend to be diabatic because of the reduced coupling between donor and acceptor components this is discussed in more detail below in Section 2.2.2. 

These monolayers provide a significant opportunity to compare the extent of electronic communication across the p3p bridge when bound to a metal electrode as opposed to being coupled to a molecular species, e.g. within a dimeric metal complex. Electronic interaction of the redox orbitals and the metallic states causes splitting between the product and reactant hypersurfaces, which is quantified by HabL the matrix coupling element. The Landau-Zener treatment [15] of a non-adiabatic reaction yields the following equation ... 

A potential curve which may be realised more frequently is given as a full line in Fig. 5f/. Such a shape would occur always if two energy surfaces nearly cross (the other energy surface is dotted in Fig. 5 )-A non-adiabatic reaction is possible in such cases, but we shall disregard this. For such a potential curve one may expect some tunnelling near the top even for somewhat heavier atoms. 

On the other hand, for a non-adiabatic reaction, k i 1, /CeiVn = Vgi and the rate constant is given by Eq. 23 where Vei is the electron hopping frequency in the activated complex. The Landau-Zener treatment yields Eq. 24 for Vei [16, 17]. 

Fig. 6.5 Free energy profile along the reaction coordinate q for an adiabatic reaction (a) and a non-adiabatic reaction (b) AG = 0...

Accordingly, the effect of coupling is much more important if the two unperturbed levels have the same energy. This is exactly the situation in the energy surface diagram for non-adiabatic reactions (Fig. 6.1) where the two energy surfaces cross. The effect of perturbation is then first-order, as given by Eq. (6.54), while it is of second-order when A > V (Eq. 6.55). 

Eq. (6.123). As shown in Chapter 6, the pre-exponential factor in Eq. (6.123) depends primarily on the interaction between electrode and redox system (adiabatic or non-adiabatic reaction) and also on the characteristics of the specific model. Experimental values of k will be given in Section 7.3.4. 

We would like to complete this section by briefly describing some of the recent developments on electronically non-adiabatic reactions. From the standpoint of the coupled-channels method, there is in principle no added difficulty in treating more than one electronic state of the reactive system. This may be done, for example, by keeping electronic degrees of freedom in the Hamiltonian and expanding the total scattering wavefunction in the electronic states of reactants and products. In practice, however, some new difficulties may arise, such as non-orthogonality of vibrational states on different electronic potential surfaces. There is at present a lack of quantum mechanical results on this problem. 

This is the reverse process with respect to the dissociative attachment (2-66) and therefore it can also be illnstrated by Fig. 2-7. The associative detachment is a non-adiabatic process, which occnrs via intersection of electroiuc terms of a complex negative ion A -B and corresponding molecnle AB. Rate coefficients of the non-adiabatic reactions are qnite high, typically kd = 10 °-10 cm /s. The kinetic data and enthalpy of some associative detachment processes are presented in Table 2-7. 

Figure 6-6. Reaction path profile (potential energy curve) for the elementary process of NO synthesis O + N2 — NO + N, showing adiabatic and non-adiabatic reaction channels.

The Landau formula (159.11) for non-adiabatic reactions is also valid when the adiabatic potential curves and, actually cross, as can be the case when the electronic states a and b have different symmetry /36/. 

This expression differs by a factor of 2 from the corresponding semiclassical formula (159.11). This factor arises because in the quantum-mechanical treatment of the nuclear motion, there are two fluxes, corresponding to the incident and reflected waves, which are almost equal if the transition probability is small hence, the nuclear system crosses twice the coupling region in two opposite directions. Therefore, the formula (174.11) gives a "two-way transition probability for non-adiabatic reactions. 

We, therefore, conclude that the expression (178a.II) for non-adiabatic reactions (y<<1) applies well when the potential energy in both the initial and final state is described by nonlinear (parabolic) diabatic potentials. 

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