126
4 Chaotic Scattering
where 2
R =
μ 1 a 2
0 D e
¯
h 2
= 1930.43 and 2
θ =
m d
μ 1
r 2
o 2
R = 543.44. Thus, the HOCl
system is reduced to a system with two degrees of freedom.
A contour plot of the potential energy, V (R, θ), for the case when the HO bond
is in the ground state, is shown in Fig. 4.18. The potential energy is symmetric about
θ = 0, has a high wall at θ = π , and a saddle point and potential hill along θ = 0.
These potential barriers at θ = π and θ = 0 rise high above the HO+Cl dissociation
energy and play a significant role in the scattering dynamics of the molecule. The
potential energy minimum occurs at (R m = 3.232 d.u., θ m = 1.347 rad).
The classical dynamics of the 2D HOCl molecule is dominated by KAM tori for
energies below E = 14,000 cm −1 . However, for energies above E = 14,000 cm −1
and for energies approaching the dissociation energy E = 20,312.3 cm −1 , the
molecular dynamics becomes increasingly chaotic. In Fig. 4.19, we show stretch
and bend surfaces of section of the 2D model of the molecule for energies E =
14,000 cm −1 , E = 17,020 cm −1 , and E = 20,150 cm −1 . In the stretch surfaces
of section (Fig. 4.19a, c, e), values of (P R , R) are plotted each time the bend angle
(p θ > 0, θ = θ m ). In the bend surfaces of section, values of (p θ , θ) are plotted each
time (p R > 0, R = R m ). Below E = 14,000 cm −1 , the dynamics is dominated by
nonlinear resonances and by the bifurcation of periodic orbits as energy increases
(Barr et al. 2009). Above E = 14,000 cm −1 , resonances start to form overlapping
self-similar structures and chaos begins to occupy ever larger regions of the phase
space with increasing energy. In Fig. 4.19, some of the dominant remaining resonant
orbits are indicated by black dots.
4.5.1 Homoclinic Tangles
For energies above the dissociation energy E = 20,312.3 cm −1 , the dynamics of
the HO–Cl complex is governed by the stable and unstable manifolds associated
with an unstable periodic orbit at R = ∞. The stable and unstable manifolds of the
unstable periodic orbit at R = ∞ form a homoclinic tangle in phase space.
Figures 4.20a, b show the stable (mapped backward in time) and unstable
(mapped forward in time) manifolds, respectively, which form the homoclinic
tangle, in the stretch SOS at E = 21,000 cm −1 . Using the Hamiltonian in Eq. (4.25),
the zeroth order stable (unstable) tendril (thick black line) is integrated backward
(forward) in time to find each trajectory’s next intersection with the SOS to produce
the first order tendril t s
1 (t u
1 ) (dashed line). Continuing in this way, we obtain the
second order tendrils t s
2 and t u
2 (grey lines). Already at second order, the tangle
displays a great deal of structure. The first and second order tendrils stretch into
the asymptotic region with both tendrils exhibiting multiple folds. As the stable and
unstable manifolds evolve towards smaller R values, they are repeatedly deflected
by collisions with potential barriers at θ = π and at θ = 0. These deflections
dominate the structure of the homoclinic tangle.
4 Chaotic Scattering
where 2
R =
μ 1 a 2
0 D e
¯
h 2
= 1930.43 and 2
θ =
m d
μ 1
r 2
o 2
R = 543.44. Thus, the HOCl
system is reduced to a system with two degrees of freedom.
A contour plot of the potential energy, V (R, θ), for the case when the HO bond
is in the ground state, is shown in Fig. 4.18. The potential energy is symmetric about
θ = 0, has a high wall at θ = π , and a saddle point and potential hill along θ = 0.
These potential barriers at θ = π and θ = 0 rise high above the HO+Cl dissociation
energy and play a significant role in the scattering dynamics of the molecule. The
potential energy minimum occurs at (R m = 3.232 d.u., θ m = 1.347 rad).
The classical dynamics of the 2D HOCl molecule is dominated by KAM tori for
energies below E = 14,000 cm −1 . However, for energies above E = 14,000 cm −1
and for energies approaching the dissociation energy E = 20,312.3 cm −1 , the
molecular dynamics becomes increasingly chaotic. In Fig. 4.19, we show stretch
and bend surfaces of section of the 2D model of the molecule for energies E =
14,000 cm −1 , E = 17,020 cm −1 , and E = 20,150 cm −1 . In the stretch surfaces
of section (Fig. 4.19a, c, e), values of (P R , R) are plotted each time the bend angle
(p θ > 0, θ = θ m ). In the bend surfaces of section, values of (p θ , θ) are plotted each
time (p R > 0, R = R m ). Below E = 14,000 cm −1 , the dynamics is dominated by
nonlinear resonances and by the bifurcation of periodic orbits as energy increases
(Barr et al. 2009). Above E = 14,000 cm −1 , resonances start to form overlapping
self-similar structures and chaos begins to occupy ever larger regions of the phase
space with increasing energy. In Fig. 4.19, some of the dominant remaining resonant
orbits are indicated by black dots.
4.5.1 Homoclinic Tangles
For energies above the dissociation energy E = 20,312.3 cm −1 , the dynamics of
the HO–Cl complex is governed by the stable and unstable manifolds associated
with an unstable periodic orbit at R = ∞. The stable and unstable manifolds of the
unstable periodic orbit at R = ∞ form a homoclinic tangle in phase space.
Figures 4.20a, b show the stable (mapped backward in time) and unstable
(mapped forward in time) manifolds, respectively, which form the homoclinic
tangle, in the stretch SOS at E = 21,000 cm −1 . Using the Hamiltonian in Eq. (4.25),
the zeroth order stable (unstable) tendril (thick black line) is integrated backward
(forward) in time to find each trajectory’s next intersection with the SOS to produce
the first order tendril t s
1 (t u
1 ) (dashed line). Continuing in this way, we obtain the
second order tendrils t s
2 and t u
2 (grey lines). Already at second order, the tangle
displays a great deal of structure. The first and second order tendrils stretch into
the asymptotic region with both tendrils exhibiting multiple folds. As the stable and
unstable manifolds evolve towards smaller R values, they are repeatedly deflected
by collisions with potential barriers at θ = π and at θ = 0. These deflections
dominate the structure of the homoclinic tangle.