118
4 The Treatment of Few-Body Reactions
use of NC coordinates, however, is so cumbersome and inefficient (especially in the
regions of branching between different arrangement channels) that they have never
been used systematically in reactive scattering investigations.
Obviously, in order to formulate and represent the interaction, one does not need
to retain the six dimensions of the two R τ and r τ vectors in the adopted functional
form of the PES. To this end one can write, in fact, the equations in terms of the
rotation angles (α, β and γ the three Euler angles) of the rigid three-body system
(i.e., molecular plane as it has been done for transforming from center-of-mass (CM)
to body-fixed (BF) formalism the two-body problem scattering equations) and formulate the PES only in terms of the other remaining three (internal) coordinates.
An alternative set of orthogonal coordinates more suitable by design to describe
reactive processes is the so-called hyperspherical coordinates. Hyperspherical coordinates change the perspective of looking at a chemical process by focusing on the
aggregated arrangement of the system and considering at its fragmentation into all
possible products (one of which in the traditional approach of the Jacobi coordinates
is considered the reactant arrangement). A particular set of hyperspherical coordinates, called Delves, at a fixed value of the Jacobi angle τ (for example collinear)
consists of a hyperradius ρ that is arrangement independent and is defined as
ρ
2
= S
2
τ + s
2
τ
(4.24)
for any value of τ and an arrangement (τ ) dependent angle θ τ defined as
θ τ = arctan
s τ
S τ
.
(4.25)
In the following, we give a short list of some popular atom–diatom quantum codes
made available for distribution by the authors.
ABC [46] is a time-independent atom–diatom quantum reactive scattering program using a coupled-channel hyperspherical coordinate method to solve the
Schrodinger equation for the motion of the three nuclei (A, B, and C) on a single
Born–Oppenheimer potential energy surface.
RWAVEPR [47] is a time-dependent atom–diatom quantum reactive scattering
program using Jacobi coordinates to integrate rigorously the three-dimensional timedependent Schrodinger equation by propagating wave packets.
DIFFREALWAVE [48] is a parallel real wavepacket code for the quantum
mechanical calculation of reactive state-to-state differential cross sections in atom–
diatom collisions using Jacobi coordinates and a real wavepacket.
A more general set of Adiabatically adjusting coordinates which follow the evolution of the Principal axis of inertia of the system Hyperspherical coordinates (APH)
[49] that will be discussed later has also been proposed and the related codes have
also been made available for circulation.
Another set of programs, of which some versions have been extended to more
than three atoms, is FLUSS-MCTDH [45, 51]. This is a pair of programs carrying
out a multiconfiguration time-dependent Hartree (MCTDH) calculation of thermally
4 The Treatment of Few-Body Reactions
use of NC coordinates, however, is so cumbersome and inefficient (especially in the
regions of branching between different arrangement channels) that they have never
been used systematically in reactive scattering investigations.
Obviously, in order to formulate and represent the interaction, one does not need
to retain the six dimensions of the two R τ and r τ vectors in the adopted functional
form of the PES. To this end one can write, in fact, the equations in terms of the
rotation angles (α, β and γ the three Euler angles) of the rigid three-body system
(i.e., molecular plane as it has been done for transforming from center-of-mass (CM)
to body-fixed (BF) formalism the two-body problem scattering equations) and formulate the PES only in terms of the other remaining three (internal) coordinates.
An alternative set of orthogonal coordinates more suitable by design to describe
reactive processes is the so-called hyperspherical coordinates. Hyperspherical coordinates change the perspective of looking at a chemical process by focusing on the
aggregated arrangement of the system and considering at its fragmentation into all
possible products (one of which in the traditional approach of the Jacobi coordinates
is considered the reactant arrangement). A particular set of hyperspherical coordinates, called Delves, at a fixed value of the Jacobi angle τ (for example collinear)
consists of a hyperradius ρ that is arrangement independent and is defined as
ρ
2
= S
2
τ + s
2
τ
(4.24)
for any value of τ and an arrangement (τ ) dependent angle θ τ defined as
θ τ = arctan
s τ
S τ
.
(4.25)
In the following, we give a short list of some popular atom–diatom quantum codes
made available for distribution by the authors.
ABC [46] is a time-independent atom–diatom quantum reactive scattering program using a coupled-channel hyperspherical coordinate method to solve the
Schrodinger equation for the motion of the three nuclei (A, B, and C) on a single
Born–Oppenheimer potential energy surface.
RWAVEPR [47] is a time-dependent atom–diatom quantum reactive scattering
program using Jacobi coordinates to integrate rigorously the three-dimensional timedependent Schrodinger equation by propagating wave packets.
DIFFREALWAVE [48] is a parallel real wavepacket code for the quantum
mechanical calculation of reactive state-to-state differential cross sections in atom–
diatom collisions using Jacobi coordinates and a real wavepacket.
A more general set of Adiabatically adjusting coordinates which follow the evolution of the Principal axis of inertia of the system Hyperspherical coordinates (APH)
[49] that will be discussed later has also been proposed and the related codes have
also been made available for circulation.
Another set of programs, of which some versions have been extended to more
than three atoms, is FLUSS-MCTDH [45, 51]. This is a pair of programs carrying
out a multiconfiguration time-dependent Hartree (MCTDH) calculation of thermally