Nonadiabatic Chemical Reaction

Further detailed discussions and applications to collinear chemical reactions are given in Reference [225]. Miller and co-workers have formulated the quantum instanton theory, starting from the expression of rate constant in terms of the flux operators, and applied it to various practical processes. Those who are interested in that should refer to References [226,227], The semiclassical instanton theory explained above can be conveniently generalized to the case of nonadiabatic chemical reaction. This is discussed in the next subsection. 

When the instanton trajectory X, jf(T)(T = 0 t = t ) is determined from this condition, the one-dimensional effective potential Veff Xi st) is obtained for the given temperature p. Thus the original multidimensional problem is reduced to the onedimensional transmission through this effective potential along the instanton trajectory and the rate constant k T) is given by 

X = xp, what we actually need is the half-period calculation from t = 0 to 

Assuming certain initial values for po, Q (/ = I,M) and x ia = i, f), we calculate approximate path, instanton energy, and avoided crossing point Xc. Then we can compute cr(x ) and / )[2f(T)], and determine the zeroth order approximate instanton path. 

Using the previous results of C , XqJ, we apply the following variational principle. Namely, for the variation 

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The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

A. Sergi and R. Kapral. Quantum-classical dynamics of nonadiabatic chemical reactions. J. Chem. Phys., 118 8566-8575, 2003. 

We generalize the path-branching formalism of nonadiabatic dynamics, which was developed in Chapter 6, so as to treat nonadiabatic chemical reactions in external laser fields. The extension to any other types of coupling is rather straightforward. 

The starting point is to consider the ET reactions as a special case of nonadiabatic chemical reactions on two electronic states. The potential for these states in the diabatic representation is written as ... 

Kramers theory to the two PESs. The Kramers theory can be considered as the special case of studying nonadiabatic chemical reactions where the non-adiabatic coupling is so strong that the reaction is controlled by the lower adiabatic PES. Similar to nonadiabatic TST, the influence of the upper PES to the ET rate is involved in nonadiabatic transition probability. 

Thebook reviews low-dimensional theories and clarifies their insufficiency conceptually and numerically. It also examines the phenomenon of nonadiabatic tunneling, which is common in molecular systems. The book describes applications to real polyatomic molecules, such as vinyl radicals and malonaldehyde, demonstrating the high efficiency and accuracy of the method. It discusses tunneling in chemical reactions, including theories for direct evaluation of reaction rate constants for both electronically adiabatic and nonadiabatic chemical reactions. In the final chapter, the authors touch on future perspectives. 

Comparison of (1.14), (2.47a) and (2.60a) reveals the universality of the golden rule in the description of both the nonadiabatic and adiabatic chemical reactions. However, the matrix elements entering into the golden-rule formula have quite a different nature. In the case of an adiabatic reaction it comes from tunneling along the reaction coordinate, while for a nonadiabatic... 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Since chemical reactions usually show significant nonadiabaticity, there are naturally quantitative errors in the predictions of the vibrationally adiabatic model. Furthermore, there are ambiguities about how to apply the theory such as the optimal choice of coordinate system. Nevertheless, this simple picture seems to capture the essence of the resonance trapping mechanism for many systems. We also point out that the recent work of Truhlar and co-workers24,34 has demonstrated that the reaction dynamics is largely controlled by the quantized bottleneck states at the barrier maxima in a much more quantitative manner than expected. 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

Typical chemical reactions are characterized by sharp reaction barriers, often arising in part from the existence of a reaction barrier in the gas phase. Thus, even though the magnitude ofthe reactive solute-solvent coupling is strong [large (t=0)], the intrinsic barrier is of such high frequency that the nonadiabatic solvation limit... 

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