Nonadiabatic ordered mechanism

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

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 physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

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. 

The structure of C given above, consists of two distinct components (i) classical propagation on mean surfaces accompanied by quantum mechanical phase oscillations with frequency Lvaa> = (Ea — Ea>)/h, and (ii) nonadiabatic transitions accompanied by changes in the momentum of the environment in order to conserve energy. The classical Liouville operator... 

As a reflection of these properties, direct information on Tad is not required in the semi-classical analytical theory, as demonstrated in the previous section. That information is replaced by the analytical continuation of the adiabatic potentials into the complex R-plane (see Eq. (24)). In order to carry out the quantum mechanical numerical calculations, however, we always stay on the real R-axis and we require explicit information on the nonadiabatic couplings. Even in the diabatic representation, which is often employed because of its convenience, nonadiabatic couplings are necessary to obtain the diabatic couplings. The quantum mechanical calculations are usually made by solving the coupled differential equations derived from an expansion of the total wave function in terms of the electronic wave functions. 

If the interaction between the reactants leading to the reaction is weak enough (nonadiabatic processes), the transition probability per unit time may be calculated using the formula of the first order in quantum mechanical... 

First we shall consider entirely nonadiabatic reactions for which formulas of the quantum mechanical perturbation theory in the first order in the interaction, leading to reaction may be used for the calculation of the... 

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