4.2 Three-Atom Systems
127
matrix. Currently, there exist implementations of this method to treat the reactions
of quantum systems of either three and four atoms.
The corresponding code for triatomic systems, APH3D [49] is available for distribution.
4.2.4 The Atom–Diatom Time-Dependent APH Method
Time-independent methodologies tend to deal with characteristic properties of the
system under study like ground state equilibrium properties, energy level structure,
and statistical thermodynamical aspects which are invariant to initial conditions set
by the experiment. On the contrary, one may want to study the properties of a system
prepared in specific initial conditions. In this case, time-dependent calculations are
typically the way to proceed.
The formalism of the APH coordinates can also be used for a time-dependent
approach. In this case, we start with the Hamiltonian of Eq. (4.36) and instead of
calculating surface functions that parametrically depended on ρ one discretizes the
hyperradius, and the two internal angles θ and χ. To this end, one can use an equally
spaced grid in ρ, a DVR grid in θ, a periodic DAF grid in χ and the analytic Wigner
rotation functions for the three Euler angles. In this way, one ends up with the very
large set of coupled time-dependent Schrödinger equations
− i
∂ψ
J
(ρ, θ, χ, t)
∂t
= Hψ
J
(ρ, θ, χ, t)
(4.43)
of dimension N ρ N DV R N χ (J + 1). For time-independent Hamitonians, we use the
time evolution operator
ψ
J
(ρ, θ, χ, t) = U(t, t 0 )ψ
J
(ρ, θ, χ, t) = e
−i
Ht
ψ
J
(ρ, θ, χ, t 0 ),
(4.44)
where ψ
J
(ρ, θ, χ, t 0 ) is any desired initial wavepacket (that is often taken as a linear
combination of initial quantum states with an initial Gaussian energy distribution and
momentum) and the exponential of the H matrix is evaluated using the Taylor series
expansion of the exponential
exp [x] = 1 + x +
x
2
2!
+
x
3
3!
+
x
4
4!
+
x
5
5!
+ · · ·
(4.45)
where in our case
x = −i
H
t
(4.46)
In actuality, one can use a Chebychev economization of the Taylor series expansion to decrease the number of terms necessary for convergence. As the wavepacket
127
matrix. Currently, there exist implementations of this method to treat the reactions
of quantum systems of either three and four atoms.
The corresponding code for triatomic systems, APH3D [49] is available for distribution.
4.2.4 The Atom–Diatom Time-Dependent APH Method
Time-independent methodologies tend to deal with characteristic properties of the
system under study like ground state equilibrium properties, energy level structure,
and statistical thermodynamical aspects which are invariant to initial conditions set
by the experiment. On the contrary, one may want to study the properties of a system
prepared in specific initial conditions. In this case, time-dependent calculations are
typically the way to proceed.
The formalism of the APH coordinates can also be used for a time-dependent
approach. In this case, we start with the Hamiltonian of Eq. (4.36) and instead of
calculating surface functions that parametrically depended on ρ one discretizes the
hyperradius, and the two internal angles θ and χ. To this end, one can use an equally
spaced grid in ρ, a DVR grid in θ, a periodic DAF grid in χ and the analytic Wigner
rotation functions for the three Euler angles. In this way, one ends up with the very
large set of coupled time-dependent Schrödinger equations
− i
∂ψ
J
(ρ, θ, χ, t)
∂t
= Hψ
J
(ρ, θ, χ, t)
(4.43)
of dimension N ρ N DV R N χ (J + 1). For time-independent Hamitonians, we use the
time evolution operator
ψ
J
(ρ, θ, χ, t) = U(t, t 0 )ψ
J
(ρ, θ, χ, t) = e
−i
Ht
ψ
J
(ρ, θ, χ, t 0 ),
(4.44)
where ψ
J
(ρ, θ, χ, t 0 ) is any desired initial wavepacket (that is often taken as a linear
combination of initial quantum states with an initial Gaussian energy distribution and
momentum) and the exponential of the H matrix is evaluated using the Taylor series
expansion of the exponential
exp [x] = 1 + x +
x
2
2!
+
x
3
3!
+
x
4
4!
+
x
5
5!
+ · · ·
(4.45)
where in our case
x = −i
H
t
(4.46)
In actuality, one can use a Chebychev economization of the Taylor series expansion to decrease the number of terms necessary for convergence. As the wavepacket