Abstract
This chapter focuses on the problem of determining the reactive dynamics of the simplest prototypes of elementary chemical reactions starting from a general non-Born–Oppenheimer (mixed electron–nuclei) approach first and then formulating the problem using a separating from that of the nuclei. To this end, the problem of adopting coordinate sets suited for describing both the interaction and the dynamics of the simplest reactive systems is discussed. Typical features of the atomistic phenomenology of atom–diatom systems such as the effect of a different allocation of energy to the various degrees of freedom in promoting reactivity, the importance of providing an accurate representation of the potential energy, the merits and demerits of reduced dimensionality calculations, and the importance of periodic orbits are analyzed.
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Notes
- 1.
Note that for the generic core \( \overline{ i} \), the following relation holds
$$\begin{aligned} \mathbf{W} _ { \overline{i} } = \frac{ 1}{ M_ {\overline{i} } } \sum _ { i = 1 \ne \overline{i} } ^ IM_i \mathbf{W} _i \end{aligned}$$and then \(I-1\) vectors \(\mathbf{W} _i \) are sufficient to define the system.
- 2.
Which rotate the coordinates (passive rotations) to body-fixed coordinates where the \(z_{Q}\) axis points along the smallest principle moment of inertia and the \(y_{Q}\) is perpendicular to the plane formed by the 3-particle system. For simplicity, we will hereafter drop the subscript Q.
- 3.
The logarithmic derivative is
$$\begin{aligned} y=\frac{d \psi }{d\rho }{\psi }^{-1}. \end{aligned}$$(4.42) - 4.
The semiclassical connection between the deflection angle \(\theta \) and the JWKB approximation to the phase shift \(\delta _l \) can be obtained from the semiclassical formulation of the wavefunction
that gives
where \(k_l(r)\) is the Langer-corrected wavenumber function (resulting from the transformation of r into \(e^x\) and to the scaling of the wavefunction by \(e^{x/2}\) to the end of taking into account the singularity occurring at \(\theta =0\)) that reads
from which a comparison with the quantum solution gives
- 5.
PODS stands for periodic orbits dividing the surface (for surface here is meant an isoenergetic cut of the PES) a family of bound periodic trajectories (whose number and location depend on the energy at which the analysis is carried out) separating the phase space of reactants and products.
- 6.
For a detailed discussion on the use of PODS for defining the converging sequences of the number of odd crossing (forth) and even crossing (back) trajectories to improve the accuracy of the estimated reactive and non-probabilities see Ref. [3].
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Laganà, A., Parker, G.A. (2018). The Treatment of Few-Body Reactions. In: Chemical Reactions. Theoretical Chemistry and Computational Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-62356-6_4
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DOI: https://doi.org/10.1007/978-3-319-62356-6_4
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