Charles Jaffé

The geometry of reaction dynamics

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom (DOFs) form the subject of this paper. Our treatment focuses on the (2n − 3)-dimensional normally hyperbolic invariant manifold (NHIM) in nDOF systems associated with a centre × · ·· ×centre × saddle in the phase space flow in the (2n − 1)dimensional energy surface. The NHIM bounds a (2n − 2)-dimensional surface, called a ‘transition state’ (TS) in chemical reaction dynamics, which partitions the energy surface into volumes characterized as ‘before’ and ‘after’ the transformation. This surface is the long-sought momentum-dependent TS beyond two DOFs. The (2n − 2)-dimensional stable and unstable manifolds associated with the (2n − 3)-dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase flow interior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flow exterior to these spherical cylinders is directed away from the TS and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic fields.

Transition state theory for laser-driven reactions

Recent developments in transition state theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, the authors construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and nonreactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. They illustrate the power of our approach on a driven Henon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of transition state theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.

Localized chaos and partial assignability of dynamical constants of motion in the transition to molecular chaos

The implications of approximate dynamical constants of motion for statistical analysis of highly excited vibrational spectra are investigated. The existence of approximate dynamical constants is related to localized chaos and partial assignability of a ‘‘chaotic spectrum.’’ Approximate dynamical constants are discussed in a dynamical symmetry breaking formulation of the transition from periodic to quasiperiodic motion, and from quasiperiodic to chaotic motion. Level repulsion, leading to a Wigner distribution in the case of a strongly chaotic system, is shown to originate in dynamical symmetry breaking via the noncrossing rule that states of the same symmetry do not cross. It is argued that quantum numbers for dynamical constants must be correctly assigned to detect localized chaos in statistical spectroscopy. Two possible kinds of approximate constants, for a ‘‘total polyad number’’ and a bend normal mode, are discussed in relation to two coupling schemes that could govern the transition to chaos in H2O.

Time independent methods in semiclassical mechanics: Adiabatic switching

A time independent algorithm based upon Ehrenfest’s adiabatic hypothesis for the construction of the transformation from the physical variables to a set of action‐angle variables is presented. At the heart of this algorithm is the recognition that the switching parameter need not be treated as a function of time. The calculations of semiclassical energy levels and dipole matrix elements for the simple quartic oscillator are presented to illustrate this approach.

Classical atom-diatom scattering : self-similarity, scaling laws, and renormalization

We have examined the structure of the chattering region of the initial angle–final action plots of He scattered by I2. We demonstrate the existence of asymptotic self‐similarity, determine the asymptotic scaling laws, and outline a renormalization treatment of the structures observed.

Time‐independent adiabatic switching in quantum mechanics

A time‐independent quantum mechanical adiabatic switching algorithm is presented. This algorithm, which is based upon Born’s quantum mechanical adiabatic theorem, is used to calculate the energy levels and eigenstates of the Henon–Heiles Hamiltonian system. The relationship between this algorithm and similar classical algorithms are discussed with particular attention focused upon the correspondence between avoided crossings and classical nonlinear resonances.

Chaotic scattering: An invariant fractal tiling of phase space

The existence of an invariant fractal tiling of phase space for unbound Hamiltonian systems is demonstrated. The fractal properties of this partitioning of phase space is intimately related to the redistribution of energy among the various modes of the system. The existence of this tiling enables one to express the expectation values of physical observables as infinite sums over all of the tiles. Furthermore, knowledge of the scaling laws associated with the tiling then enables one to evaluate these sums.

Classical S‐matrix theory for chaotic atom–diatom collisions

The extension of classical S‐matrix theory to chaotic scattering systems is considered. It is shown that if the fractal structure of the chattering region is understood then the contribution to the S‐matrix elements and the transition probabilities can be expressed as a sum over the infinite number of contributing trajectories and that by using the scaling laws of the fractal that this sum can be evaluated. It is shown that if the transition is classically forbidden then the contribution from the chattering region is significant.

The correspondence between classical nonlinear resonances and quantum mechanical Fermi resonances

The correspondence between classical nonlinear resonances and quantum mechanical Fermi resonances is discussed. The principle result is the recognition that the classical resonant behavior does not contribute to the diagonal matrix elements, but rather contributes to the off‐diagonal matrix elements. As a direct consequence, it is not necessary for the classical dynamics corresponding to the quantum mechanical states involved in a Fermi resonance to exhibit resonant behavior. Instead, it is the classical dynamics associated with the matrix element which connects the quantum mechanical states involved in the Fermi resonance which must exhibit resonant behavior. These results are illustrated numerically using a very simple model of two kinetically coupled Morse oscillators.

Transition state geometry near higher-rank saddles in phase space

We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.