T. Uzer

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.

Time-frequency analysis of chaotic systems

We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time–frequency decomposition of a trajectory using wavelets. The ridges of the time–frequency landscape of a trajectory, also called instantaneous frequencies, enable us to analyze the phase space structures. In particular, this method detects resonance trappings and transitions and allows a characterization of the notion of weak and strong chaos. We illustrate the method with the trajectories of the standard map and the hydrogen atom in crossed magnetic and elliptically polarized microwave fields.

Transition State in a Noisy Environment

Transition state theory overestimates reaction rates in solution because conventional dividing surfaces between reagents and products are crossed many times by the same reactive trajectory. We describe a recipe for constructing a time-dependent dividing surface free of such recrossings in the presence of noise. The no-recrossing limit of transition state theory thus becomes generally available for the description of reactions in a fluctuating environment.

Stochastic transition states: Reaction geometry amidst noise

Classical transition state theory (TST) is the cornerstone of reaction-rate theory. It postulates a partition of phase space into reactant and product regions, which are separated by a dividing surface that reactive trajectories must cross. In order not to overestimate the reaction rate, the dynamics must be free of recrossings of the dividing surface. This no-recrossing rule is difficult (and sometimes impossible) to enforce, however, when a chemical reaction takes place in a fluctuating environment such as a liquid. High-accuracy approximations to the rate are well known when the solvent forces are treated using stochastic representations, though again, exact no-recrossing surfaces have not been available. To generalize the exact limit of TST to reactive systems driven by noise, we introduce a time-dependent dividing surface that is stochastically moving in phase space, such that it is crossed once and only once by each transition path.

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.

Identifying reactive trajectories using a moving transition state.

A time-dependent no-recrossing dividing surface is shown to lead to a new criterion for identifying reactive trajectories well before they are evolved to infinite time. Numerical dynamics simulations of a dissipative anharmonic two-dimensional system confirm the efficiency of this approach. The results are compared to the standard fixed transition state dividing surface that is well-known to suffer from recrossings and therefore requires trajectories to be evolved over a long time interval before they can reliably be classified as reactive or nonreactive. The moving dividing surface can be used to identify reactive trajectories in harmonic or moderately anharmonic systems with considerably lower numerical effort or even without any simulation at all.

Celestial Mechanics on a Microscopic Scale

Classical and semiclassical methods are unrivaled in providing an intuitive and computationally tractable approach to the study of atomic, molecular, and nuclear dynamics. An important advantage of such methods is their ability to uncover in a single picture underlying structures that may be hard to extract from the profusion of data supplied by detailed quantum calculations. Modern trends in semiclassical mechanics are described, particularly the combination of group theoretical methods with techniques of nonlinear dynamics. Application is made to intramolecular energy transfer and to the electronic structure of atomic Rydberg states in external electric and magnetic fields.

Semiclassical quantization of slightly nonresonant systems: Avoided crossings, dynamical tunneling, and molecular spectra

An important problem in the spectroscopy of isolated molecules is the passage of a system through an isolated internal nonlinear resonance as the total energy is changed. The spectral signature of such a passage may involve an avoided crossing together with pre‐resonant attraction and/or post‐resonant repulsion of levels. While most previous semiclassical studies have made a clear distinction between resonant and nonresonant problems, we develop an approach which allows for a quantitative description of the passage through an avoided crossing which levels may experience as the total energy of the molecule (as opposed to an external perturbation parameter) is varied. Stimulated by the quantum mechanical and time dependent classical studies of Heller and co‐workers we have developed a time independent semiclassical formalism which is found to provide excellent quantitative agreement with quantum energy levels and splittings as the total energy is changed. The time independent semiclassical approach provides...

A theoretical investigation of microscopic dynamics and time scales of predesorption processes

When an internal mode of an absorbate is excited with enough energy to break the absorbate–surface bond, the adsorbate may desorb in a manner similar to predissociation (or unimolecular reaction) in isolated molecules. In this article, we set up a simple, microscopic model for this ‘‘predesorption’’ process and investigate the influence of various aspects of the model on the time scale of predesorption. These comparisons include quantum mechanical vs classical treatment, and the inclusion vs neglect of surface vibrations. For this simple analog of a ‘‘unimolecular reaction,’’ we find that classical mechanics predicts shorter lifetimes; and that the effect of surface vibrations is far less pronounced in quantum mechanics.

Circularly polarized molecular high harmonic generation using a bicircular laser

We investigate the process of circularly polarized high harmonic generation in molecules using a bicircular laser field. In this context, we show that molecules offer a very robust framework for the production of circularly polarized harmonics, provided their symmetry is compatible with that of the laser field. Using a discrete time-dependent symmetry analysis, we show how all the features (harmonic order and polarization) of spectra can be explained and predicted. The symmetry analysis is generic and can easily be applied to other target and/or field configurations.