The Chesnavich Model for Ion-Molecule Reactions: A Rigid Body Coupled to a Particle

The Chesnavich Model for Ion-Molecule Reactions: A Rigid Body Coupled to a Particle

Gregory S. Ezra and Stephen Wiggins

International Journal of Bifurcation and Chaos, Vol. 29, No. 2 (2019) 1950025

DOI: 10.1142/S0218127419500251

 In this paper, we present a derivation of Chesnavich’s Hamiltonian for a model ion-molecule reaction. The model system has the basic structure of a rigid body coupled to a structureless particle. Using the form of the potential energy of interaction given by Chesnavich, we derive the equilibria, determine their stability, and construct two, two-dimensional invariant manifolds and determine their stability.

ijbc_29_issue-02_cover

 

Upcoming CHAMPS workshop on Exact Factorization and Bohmian Mechanics

CHAMPS – Workshop on Exact Factorization and Bohmian Mechanics

Engineers House The Promenade Clifton Down, Bristol BS8 3NB

Monday 22nd April 2019

Methodologies for separating nuclear and electronic motions are fundamental to chemistry.

In recent years the notion of “Exact Factorization” has emerged as a compelling approach for analysing and interpreting the complete wave function for a system of nuclei and electrons evolving in a time-dependent external potential.

Another related approach is Bohmian mechanics, which is known for many years, but recently has received much of attention.

The purpose of this workshop is to bring together experts in this area to discuss the current “state-of-the-art” and the prospects for future development.

To register your interest in this event please email the Champs Project Manager for further information – champs-project@bristol.ac.uk

 

Event schedule 22.04.19

 

Upcoming CHAMPS Workshop on “Discovering Phase Space Structure and Reaction Mechanisms from Trajectory Data Sets”

CHAMPS – Workshop on “Discovering Phase Space Structure and Reaction Mechanisms from Trajectory Data Sets” 

Engineers House The Promenade Clifton Down, Bristol BS8 3NB

Tuesday 19th March 2019

Trajectories generated by Hamilton’s equations are a fundamental quantity for understanding reaction mechanisms in chemistry. Trajectories are an inherently phase space object, i.e. they describe the change in configuration space coordinates and momentum coordinates. Consequently, their behaviour is governed by phase space structures. Theoretical and computational advances now allow the generation of large sets of trajectories. Consequently, there has been significant activity in recent years in developing theoretical and computational strategies for discovering “structure” in these data sets. This is a “first of its kind”  workshop that will bring together  people in the applied mathematics and chemistry communities that are working on these issues.

To register your interest in this event please email the Champs Project Manager for further information – champs-project@bristol.ac.uk

 

19.03.19 Workshop Schedule of Talks

Congratulation to CHAMPS PDRA Lars Bratholm for being awarded a Alan Turing Institute Fellowship

Congratulation to CHAMPS PDRA Lars Bratholm for being awarded a Alan Turing Institute Fellowship

We are please that Dr. Lars Bratholm is in the first group of Alan Turing Institute fellows at the University of Bristol. This is a notable accomplishment and we anticipate that this will serve to  further develop links between CHAMPS and the Alan Turing Institute.

https://www.turing.ac.uk/people/researchers/lars-andersen-bratholm

Information about the Alan Turing Institute can be found here:

https://www.turing.ac.uk/about-us

 

Lars Bratholm - photo for website
L. Bratholm | Bristol

Phase Space Structure and Transport in a Caldera Potential Energy Surface

Phase Space Structure and Transport in a Caldera Potential Energy Surface

Matthaios Katsanikas and Stephen Wiggins

International Journal of Bifurcation and Chaos, Vol. 28, No. 13 (2018) 1830042

https://doi.org/10.1142/S0218127418300422

ijbc.28.issue-13.cover

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors

V. J. Garcia-Garrido, F. Balibrea-Iniesta, S. Wiggins, A. M. Mancho, and C. Lopesino,  Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors, Regular and Chaotic Dynamics, 23(6), 751-766 (2018).

https://link.springer.com/article/10.1134/S1560354718060096

In this paper we show that Lagrangian Descriptors (LDs), a technique based on Dynamical Systems Theory (DST), are able to reveal the phase space structures present in the well known Arnold’s cat map. This discrete dynamical system, which represents a classical example of an Anosov diffeomorphism that is strongly mixing,  provides us with a benchmark model to test the performance of LDs and their capability to detect fixed points, periodic orbits and their stable and unstable manifolds present in chaotic maps. In this work we show, both from a theoretical and a numerical perspective, how LDs reveal the invariant manifolds of the periodic orbits of the cat map. The application of this methodology in this setting clearly illustrates the chaotic behaviour of the cat map and highlights some technical numerical difficulties that arise in the identification of its phase space structures.