The Bending Dynamics of Acetylene
C. Jung* and H.S. Taylor**
*UNAM Instituto de Matematicas
Unidad Cuernavaca
Av. Universidad s/n,
62191 Cuernavaca, Mexico
**Department of Chemistry
University of Southern California,
Los Angeles, CA 90089-0482, USA
The effective four degree freedom Hamilitonian fitted to
15000 cm-1 by successive diagonalization and parameter adjustment
to absorption and dispersed fluorescence spectra,(1) is converted
to a four degree of freedom classical Hamiltonian in action angle
variables. A specific canonical transform is then used to convert
to an effective Hamiltonian in which for the first time the two
effective constants of the motion are seen explicitly and which
now is a function of two abstract action variables and two abstract
angle variables. The constants of the motion are the polyad quantum
number and the total bending angular momentum (here eventually
set to zero as it is in the experiment); the former measures
the total occupation of the cis and trans bend modes here considered
as basis states. The 2D abstract phase space Hamiltonian has
symmetries which suggest the existence of two key periodic orbits
that will be shown to organize the phase space dynamics of the
Hamiltonian over its fitted energy range. These periodic orbits
are numerically followed in energy. Additionally, surface of
sections are generated to observe the surrounding tori or chaos
as the case may be. The generic motions of these periodic orbits
which for a given P and changing energy, are librations and rotations
in abstract angle configuration space, show how the organized
abstract motions change with P and within a polyad.
To obtain chemically recognizable motions, we start with
the general solution of a four degree of freedom adopted model
that envisions the "bending" acetylene as two opposite
outward facing, bottom to bottom attached cups. Each cup contains
a ball representing the Hydrogen atom. This is a two 2D spherical
harmonic oscillator model. The constants of the general solution,
whose values specify the motions of the H atoms in parallel face
to face planes on each end of the acetene, are shown to be determined
and correlated by choosing a trajectories in the abstract space.
When the organizing periodic orbits or their nearby tori based
2D trajectories are each separately plugged into the constants
in the general solution, 4D trajectories are obtained that specify
the various ways the hydrogens move in their respective planes.
Below P=9, the motion is regular with cis and trans motions
dominating the top and bottom of the polyads, respectively. In
the middle a beat motion (linear combinations of cis and trans)
appear. Above P=9, cis and trans no longer exist and now the
bottom and top of the polyads are regular with local and counter
rotating motions, respectively. The middle of the polyads have
a mixed phase space. The regular regions embedded in the chaos,
show beautiful correlated motions whose pictures will be shown.
The fraction of phase space with chaotic motion increases up
to about P=16 and then decreases, in agreement with the regularity
of the energy level patterns seen when comparing level diagrams
of polyads. In general in regular regions the levels are separated
by the frequency of the organizing periodic orbit taken halfway
between the levels. This alone shows a quantum connection. We
have quantized (by diagonalization) the 2D effective semi-classical
actions-angle Hamiltonian. We get the same eigenvalues as when
the spectra was fitted using a diagonalization in 4D space. We
see that the 2D wavefunctions in the two angle variables lie
on top of the abstract stable periodic orbits, allowing us to
count nodes and complete the assignment in the regular region.
"Chaotic" states, scarred by one or more unstable periodic
orbits, also appear and explain the mixings that occur at the
given energy in the polyad.
(1). M.P. Jacobson, J.P. O'Brien, R.J. Silby and R. Field,
Preprint. 1998
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