Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPhase space structures governing reaction dynamics in rotating molecules
223
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Recently the phase space structures governing reaction dynamics in Hamiltonian systems have been identified and algorithms for their explicit construction have been developed. These phase space structures are induced by saddle type equilibrium points which are characteristic for reaction type dynamics. Their construction is based on a Poincar{e}-Birkhoff normal form. Using tools from the geometric theory of Hamiltonian systems and their reduction we show in this paper how the construction of these phase space structures can be generalized to the case of the relative equilibria of a rotational symmetry reduced $N$-body system. As rotations almost always play an important role in the reaction dynamics of molecules the approach presented in this paper is of great relevance for applications.
rate research
Read More
Hamiltonian dynamical systems possessing equilibria of ${saddle} times {centre} times...times {centre}$ stability type display emph{reaction-type dynamics} for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow emph{bottlenecks} created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a emph{Normally Hyperbolic Invariant Manifold (NHIM)}, whose emph{stable and unstable manifolds} have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) emph{dividing surface} which locally divides an energy surface into two components (`reactants and `products), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in emph{transition state theory} where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface. We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the threedimensional space $R^3$, and two schematic models which capture many of the essential features of the dynamics for $n$-DoF systems. In addition, we elucidate the structure of the NHIM.
The transformation of a system from one state to another is often mediated by a bottleneck in the systems phase space. In chemistry these bottlenecks are known as emph{transition states} through which the system has to pass in order to evolve from reactants to products. The chemical reactions are usually associated with configurational changes where the reactants and products states correspond, e.g., to two different isomers or the undissociated and dissociated state of a molecule or cluster. In this letter we report on a new type of bottleneck which mediates emph{kinetic} rather than configurational changes. The phase space structures associated with such emph{kinetic transition states} and their dynamical implications are discussed for the rotational vibrational motion of a triatomic molecule. An outline of more general related phase space structures with important dynamical implications is given.
The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with $2$ degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood - the reaction rate decreases as a function of energy beyond a critical value. Using lobe dynamics we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive upper and lower bounds of the reaction rate, which are desirable for practical reasons.
We study time evolution of Wigner function of an initially interacting one-dimensional quantum gas following the switch-off of the interactions. For the scenario where at $t=0$ the interactions are suddenly suppressed, we derive a relationship between the dynamical Wigner function and its initial value. A two-particle system initially interacting through two different interactions of Dirac delta type is examined. For a system of particles that is suddenly let to move ballistically (without interactions) in a harmonic trap in d dimensions, and using time evolution of one-body density matrix, we derive a relationship between the time dependent Wigner function and its initial value. Using the inverse Wigner transform we obtain, for an initially harmonically trapped noninteracting particles in $d$ dimensions, the scaling law satisfied by the density matrix at time $t$ after a sudden change of the trapping frequency. Finally, the effects of interactions are analyzed in the dynamical Wigner function.
Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
