Chapter 4
The Treatment of Few-Body Reactions
This chapter focuses on the problem of determining the reactive dynamics of the
simplest prototypes of elementary chemical reactions starting from a general nonBorn–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.
4.1 The Combined Dynamics of Electrons and Nuclei
4.1.1 The N-Body Dynamical Equations
In the previous chapter, we have discussed the numerical integration of the polyelectronic multidimensional Schrödinger equation. During a chemical process, the
faster electronic motion rapidly adjusts to the sluggish movement of the heavy nuclei
and the two motions interplay within a common game. Accordingly, in general, for a
system made of N nuclei (each of mass M i and charge Z i q e ), K electrons (with mass
m e and charge −q e ), localizable in space (with respect to an arbitrary axis systems)
using the nuclei position vector W
‡ , and the electron position vector w
‡ (the nuclei
position vector is of length N W
‡
= (W
‡
1 , . . . , W
‡
N ) and the electron position vector is of length K w
‡
= (w
‡
1 , . . . , w
‡
K )), the equation of Schrödinger in its general
time-dependent form is
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6_4
111
The Treatment of Few-Body Reactions
This chapter focuses on the problem of determining the reactive dynamics of the
simplest prototypes of elementary chemical reactions starting from a general nonBorn–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.
4.1 The Combined Dynamics of Electrons and Nuclei
4.1.1 The N-Body Dynamical Equations
In the previous chapter, we have discussed the numerical integration of the polyelectronic multidimensional Schrödinger equation. During a chemical process, the
faster electronic motion rapidly adjusts to the sluggish movement of the heavy nuclei
and the two motions interplay within a common game. Accordingly, in general, for a
system made of N nuclei (each of mass M i and charge Z i q e ), K electrons (with mass
m e and charge −q e ), localizable in space (with respect to an arbitrary axis systems)
using the nuclei position vector W
‡ , and the electron position vector w
‡ (the nuclei
position vector is of length N W
‡
= (W
‡
1 , . . . , W
‡
N ) and the electron position vector is of length K w
‡
= (w
‡
1 , . . . , w
‡
K )), the equation of Schrödinger in its general
time-dependent form is
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6_4
111