116
4 The Treatment of Few-Body Reactions
(W, t) = e
−i Et/
(W),
(4.19)
where E is the total energy of the system. Substituting this into the time-dependent
Schrödinger equation and dividing by the functional dependence of time, one obtains
the Schrödinger equation for stationary states of the system
H n (W) =
ˆ
T n + V (W)
(W) = E(W).
(4.20)
4.2 Three-Atom Systems
4.2.1 Three-Body Orthogonal Coordinates
The most important pros and cons of using internuclear coordinates for an atom–
diatom system are easy to specify: we have already mentioned that they are particularly suited to formulate the interaction (see Fig. 4.1 for the N + N 2 reaction channel)
though, as already pointed out, they do not cover homogeneously the space of molecular geometries due to the triangular rule (in other words they cannot freely vary
individually) and they are not orthogonal (and are therefore less suited for quantum dynamical calculations because leading to matrices full of nonzero off-diagonal
elements).
On the contrary, orthogonal coordinates have by definition the advantage of providing diagonal representations (no crossed terms) of dynamical problems. The most
popular sets of orthogonal coordinates are the Jacobi ones sketched in Fig. 4.2 for
the three different arrangements τ = 1 (A,BC), τ = 2 (B,CA), τ = 3 (C,BA). More
in general, the definition of the Jacobi coordinates in terms of the CM position vectors W is R τ = R τ ,(τ +1)(τ +2) = W τ − (m τ +1 W τ +1 + m τ +2 W τ +2 )/(m τ +1 + m τ +2 )
and r τ = r (τ +1)(τ +2) = W τ +1 − W τ +2 (see Fig. 4.2) with τ labeling also the isolated atoms in a modulus 3 sequence. Accordingly, the angle τ is defined as
1
2
arctan(R τ r τ )/|R τ − r τ |.
A key feature of the Jacobi coordinates is that they are arrangement dependent
(arrangements have been labeled above with different values of τ ) and are therefore
unsuitable for a full description of reactive processes involving a breaking of an
existing bond and the forming of a new one (see Fig. 4.2).
It is often useful to scale Jacobi coordinates R τ and r τ by the mass coefficient
appearing in the formulation of the kinetic operator (see some examples later in this
chapter) S τ and s τ defined as S τ = d τ R τ and s τ = d
−1
τ r τ in which the dimensionless
scaling factor d τ is chosen so as to stretch/compress the coordinates to the end of
making more democratic their weight in the Hamiltonian (very useful for an intuitive
description of isotopic effects in the graphical representation of the scattering).
The different sets of Jacobi coordinates are related by the so-called kinematic rotations (which are not physical rotations but matrix transformations relating different
arrangements) like the following ones:
4 The Treatment of Few-Body Reactions
(W, t) = e
−i Et/
(W),
(4.19)
where E is the total energy of the system. Substituting this into the time-dependent
Schrödinger equation and dividing by the functional dependence of time, one obtains
the Schrödinger equation for stationary states of the system
H n (W) =
ˆ
T n + V (W)
(W) = E(W).
(4.20)
4.2 Three-Atom Systems
4.2.1 Three-Body Orthogonal Coordinates
The most important pros and cons of using internuclear coordinates for an atom–
diatom system are easy to specify: we have already mentioned that they are particularly suited to formulate the interaction (see Fig. 4.1 for the N + N 2 reaction channel)
though, as already pointed out, they do not cover homogeneously the space of molecular geometries due to the triangular rule (in other words they cannot freely vary
individually) and they are not orthogonal (and are therefore less suited for quantum dynamical calculations because leading to matrices full of nonzero off-diagonal
elements).
On the contrary, orthogonal coordinates have by definition the advantage of providing diagonal representations (no crossed terms) of dynamical problems. The most
popular sets of orthogonal coordinates are the Jacobi ones sketched in Fig. 4.2 for
the three different arrangements τ = 1 (A,BC), τ = 2 (B,CA), τ = 3 (C,BA). More
in general, the definition of the Jacobi coordinates in terms of the CM position vectors W is R τ = R τ ,(τ +1)(τ +2) = W τ − (m τ +1 W τ +1 + m τ +2 W τ +2 )/(m τ +1 + m τ +2 )
and r τ = r (τ +1)(τ +2) = W τ +1 − W τ +2 (see Fig. 4.2) with τ labeling also the isolated atoms in a modulus 3 sequence. Accordingly, the angle τ is defined as
1
2
arctan(R τ r τ )/|R τ − r τ |.
A key feature of the Jacobi coordinates is that they are arrangement dependent
(arrangements have been labeled above with different values of τ ) and are therefore
unsuitable for a full description of reactive processes involving a breaking of an
existing bond and the forming of a new one (see Fig. 4.2).
It is often useful to scale Jacobi coordinates R τ and r τ by the mass coefficient
appearing in the formulation of the kinetic operator (see some examples later in this
chapter) S τ and s τ defined as S τ = d τ R τ and s τ = d
−1
τ r τ in which the dimensionless
scaling factor d τ is chosen so as to stretch/compress the coordinates to the end of
making more democratic their weight in the Hamiltonian (very useful for an intuitive
description of isotopic effects in the graphical representation of the scattering).
The different sets of Jacobi coordinates are related by the so-called kinematic rotations (which are not physical rotations but matrix transformations relating different
arrangements) like the following ones: