10
1 From the Phenomenology of Chemical Reactions …
interactions.
2 In order to illustrate most of the concepts developed by Molecular
Science in its approach to “understanding” the nature of chemical processes, we
shall focus in the following on the two particles (two bodies) before undertaking the
study of more complex systems. The greater simplicity of this study will allow us to
approach in a smoother way the basic concepts of chemical transformations and of
the related computational approaches.
1.2 Classical Mechanics of Two-Particle Collisions
1.2.1 Reference Frame and Elementary Interactions
At present, we assume that the two bodies are represented as two points or spherical
objects with masses in the three-dimensional physical space (after all a large part of
the collision process takes place at distances at which the structure of the colliding
bodies has hardly any effect) and that the behavior of such systems can be described
by the laws of classical mechanics (which is a reasonable starting point for most of
the dynamical computational chemistry applications).
In an arbitrary laboratory fixed reference system (X, Y, Z) lab (thick arrows of
Fig. 1.4), according to classical mechanics, a system formed by two colliding particles
A and B is uniquely defined in physical space by the two position vectors r A and r B
and the two momentum vectors p A and p B defined, respectively, as m A v A and m B v B
with m i being the mass and v i the velocity of either particle A or B).
These vectors are usually referred to as laboratory (lab) axis frame (X, Y, Z) lab , a
frame of Cartesian orthogonal X, Y, and Z axes having a fixed origin and orientation
like those of the physical laboratory (in the following, we shall omit the specification
lab when not strictly necessary). The position vectors r A and r B of the particles A
and B, respectively, can be represented either in terms of their projections X A , Y A ,
Z A , and X B , Y B , Z B over the X, Y, and Z axes (not shown in Fig. 1.4 for the sake
of simplicity) or in terms of the corresponding spherical polar coordinates (i.e., the
moduli r A and r B of the two vectors r A and r B and the respective angles A , A and
B , B .
3 Similar representations can be adopted for the vectors p A and p B .
A representation of the two-body system isomorphous with the above-described
one can be obtained using r C M , the position vector of the center-of-mass (CM) of
the system, and r AB (or r for short), the position vector of particle B with respect to
particle A.
2 Ionic is the interaction between charged particles (ions) in which the number of positive components
(e.g., protons) differs from that of negatively charged particles (e.g., electrons). Covalent is the
interaction associated with evenly shared particles (e.g., two atoms equally sharing the electrons).
Permanent is a stable feature of the particles (e.g., the dipole moment). Induced is a temporary
feature associated with the presence of another particle. Short and long ranges refer to the distance
between the particles.
3 The spherical polar coordinates of the particle i make use of r i (the module of the position vector
r i ) and of its orientation angles i and i .
1 From the Phenomenology of Chemical Reactions …
interactions.
2 In order to illustrate most of the concepts developed by Molecular
Science in its approach to “understanding” the nature of chemical processes, we
shall focus in the following on the two particles (two bodies) before undertaking the
study of more complex systems. The greater simplicity of this study will allow us to
approach in a smoother way the basic concepts of chemical transformations and of
the related computational approaches.
1.2 Classical Mechanics of Two-Particle Collisions
1.2.1 Reference Frame and Elementary Interactions
At present, we assume that the two bodies are represented as two points or spherical
objects with masses in the three-dimensional physical space (after all a large part of
the collision process takes place at distances at which the structure of the colliding
bodies has hardly any effect) and that the behavior of such systems can be described
by the laws of classical mechanics (which is a reasonable starting point for most of
the dynamical computational chemistry applications).
In an arbitrary laboratory fixed reference system (X, Y, Z) lab (thick arrows of
Fig. 1.4), according to classical mechanics, a system formed by two colliding particles
A and B is uniquely defined in physical space by the two position vectors r A and r B
and the two momentum vectors p A and p B defined, respectively, as m A v A and m B v B
with m i being the mass and v i the velocity of either particle A or B).
These vectors are usually referred to as laboratory (lab) axis frame (X, Y, Z) lab , a
frame of Cartesian orthogonal X, Y, and Z axes having a fixed origin and orientation
like those of the physical laboratory (in the following, we shall omit the specification
lab when not strictly necessary). The position vectors r A and r B of the particles A
and B, respectively, can be represented either in terms of their projections X A , Y A ,
Z A , and X B , Y B , Z B over the X, Y, and Z axes (not shown in Fig. 1.4 for the sake
of simplicity) or in terms of the corresponding spherical polar coordinates (i.e., the
moduli r A and r B of the two vectors r A and r B and the respective angles A , A and
B , B .
3 Similar representations can be adopted for the vectors p A and p B .
A representation of the two-body system isomorphous with the above-described
one can be obtained using r C M , the position vector of the center-of-mass (CM) of
the system, and r AB (or r for short), the position vector of particle B with respect to
particle A.
2 Ionic is the interaction between charged particles (ions) in which the number of positive components
(e.g., protons) differs from that of negatively charged particles (e.g., electrons). Covalent is the
interaction associated with evenly shared particles (e.g., two atoms equally sharing the electrons).
Permanent is a stable feature of the particles (e.g., the dipole moment). Induced is a temporary
feature associated with the presence of another particle. Short and long ranges refer to the distance
between the particles.
3 The spherical polar coordinates of the particle i make use of r i (the module of the position vector
r i ) and of its orientation angles i and i .