4.5 Chaos in the HOCl Molecular System
123
(2011), it is determined that, for the line of initial conditions shown in Fig. 4.16, of
those crossing the tendril, 44.24% are transmitted and 55.76% are reflected.
The PSS used here to analyze the scattering dynamics (we plot p and x each
time φ = π/2 + n2π , with n = 0, 1, 2, . . .) corresponds to a particular phase of the
radiation field at time t = 0. If the phase of the field is different at time t = 0, then
the asymmetric scattering properties will differ in detail but will still be asymmetric.
The procedure described here can be applied to any choice of initial phase of the
radiation field. The symbolic dynamics of the scattering process, and the resulting
transfer matrix, can provide an important tool for disentangling reflection from
transmission in such scattering processes.
4.5 Chaos in the HOCl Molecular System
One of the most active regions of a molecule involves the vibrational dynamics
of its atomic constituents just above the dissociation energy of the molecule. The
vibrational dynamics determine the pathways for dissociation and recombination
of the constituents of the molecule. In most molecules, little is known about these
dynamical processes because they have so many degrees of freedom. The nonlinear
dynamics of systems with more than two degrees of freedom is not well understood.
Recent studies of the classical nonlinear vibrational dynamics of small
molecules, above the dissociation energy, have revealed bifurcations and dynamical
structures that can have significant influence on the dissociation dynamics of the
molecule (Farantos et al. 2009; Schinke et al. 2010; Schinke 2011; Mauguiere
et al. 2011; Joyeux et al. 2005). Therefore, small molecules provide an important
laboratory for studying the classical-quantum correspondence in systems with more
than two degrees of freedom.
One molecule that has provided some insight into molecular dissociation dynamics is HOCl. The HOCl molecule consists of three atoms, H, O, and Cl, with masses
m H , m O and m Cl , respectively, that lie in a plane. If we assume that the total
angular momentum of the system L tot = 0, then the plane remains stationary
and all motion occurs in the plane of the molecule. We can introduce lab-frame
coordinates (x , y , z ) and body-frame coordinates (x, y, z). The origin of the bodyframe coordinates is the center-of-mass of the molecule. When L tot ≡0, we can
assume that all of the dynamics occurs in the (x , z ) and (x, z) planes and that any
angular momentum vectors generated by internal rotations of the molecule (such
vectors must add to zero because L tot = 0) lie along the y and y axes in the lab and
body-fixed frames, respectively.
In the body-frame, t 1 is a vector that connects the center of mass of HO to Cl
and t 2 is a vector that connects H to O (see Fig. 4.17). The vector t 1 has length
R and the vector t 2 has length r. The vibrational motion of HO does not contribute
significantly to the HOCl dynamics at lower energies (Skokov et al. 1999; Jost 1999;
Weiss et al. 2000; Joyeux et al. 2005). Therefore, we set r = r o , where r o is the
ground state bond length of HO. This constraint, together with the condition that
123
(2011), it is determined that, for the line of initial conditions shown in Fig. 4.16, of
those crossing the tendril, 44.24% are transmitted and 55.76% are reflected.
The PSS used here to analyze the scattering dynamics (we plot p and x each
time φ = π/2 + n2π , with n = 0, 1, 2, . . .) corresponds to a particular phase of the
radiation field at time t = 0. If the phase of the field is different at time t = 0, then
the asymmetric scattering properties will differ in detail but will still be asymmetric.
The procedure described here can be applied to any choice of initial phase of the
radiation field. The symbolic dynamics of the scattering process, and the resulting
transfer matrix, can provide an important tool for disentangling reflection from
transmission in such scattering processes.
4.5 Chaos in the HOCl Molecular System
One of the most active regions of a molecule involves the vibrational dynamics
of its atomic constituents just above the dissociation energy of the molecule. The
vibrational dynamics determine the pathways for dissociation and recombination
of the constituents of the molecule. In most molecules, little is known about these
dynamical processes because they have so many degrees of freedom. The nonlinear
dynamics of systems with more than two degrees of freedom is not well understood.
Recent studies of the classical nonlinear vibrational dynamics of small
molecules, above the dissociation energy, have revealed bifurcations and dynamical
structures that can have significant influence on the dissociation dynamics of the
molecule (Farantos et al. 2009; Schinke et al. 2010; Schinke 2011; Mauguiere
et al. 2011; Joyeux et al. 2005). Therefore, small molecules provide an important
laboratory for studying the classical-quantum correspondence in systems with more
than two degrees of freedom.
One molecule that has provided some insight into molecular dissociation dynamics is HOCl. The HOCl molecule consists of three atoms, H, O, and Cl, with masses
m H , m O and m Cl , respectively, that lie in a plane. If we assume that the total
angular momentum of the system L tot = 0, then the plane remains stationary
and all motion occurs in the plane of the molecule. We can introduce lab-frame
coordinates (x , y , z ) and body-frame coordinates (x, y, z). The origin of the bodyframe coordinates is the center-of-mass of the molecule. When L tot ≡0, we can
assume that all of the dynamics occurs in the (x , z ) and (x, z) planes and that any
angular momentum vectors generated by internal rotations of the molecule (such
vectors must add to zero because L tot = 0) lie along the y and y axes in the lab and
body-fixed frames, respectively.
In the body-frame, t 1 is a vector that connects the center of mass of HO to Cl
and t 2 is a vector that connects H to O (see Fig. 4.17). The vector t 1 has length
R and the vector t 2 has length r. The vibrational motion of HO does not contribute
significantly to the HOCl dynamics at lower energies (Skokov et al. 1999; Jost 1999;
Weiss et al. 2000; Joyeux et al. 2005). Therefore, we set r = r o , where r o is the
ground state bond length of HO. This constraint, together with the condition that