CHAMPS (Chemistry and Mathematics in Phase Space) Workshop 19th March 2019 – “Discovering Phase Space Structure and Reaction Mechanisms from Trajectory Data Sets”

Report on the CHAMPS (Chemistry and Mathematics in Phase Space) Workshop

19th March 2019 – “Discovering Phase Space Structure and Reaction Mechanisms from Trajectory Data Sets” held at Engineers House, Clifton Bristol.

Workshop Photo - 19.03.19 group IMG_2183

 This one day workshop was very much in the spirit of the CHAMPS project in that it brought together a group of chemists, mathematicians, and physicists all concerned with different aspects of the fundamental problem of revealing the phase space structures governing reaction dynamics using trajectory based diagnostics. The trajectory based diagnostic used by most of this group was the method of Lagrangian descriptors (a brief description of this approach was given in https://champsproject.com/2018/03/01/lagrangian-descriptors-from-fluid-dynamics-to-mathematics-to-chemistry/). This method has emerged as a flexible and “easy to code” method that can be used in a wide variety of settings relevant to chemical reaction dynamics.

Ana Mancho gave the first talk and discussed the basic ideas behind the method of Lagrangian descriptors and the different settings in which they can be applied. In particular, it was demonstrated how they could be applied to complex data sets in  geophysical fluid dynamics settings.  She was followed by Victor Garcia Garrido how the method could be applied in higher dimensional settings in order to detect periodic orbits and normally hyperbolic invariant manifolds (NHIMs) in general. Next Florentino Borondo showed how the method of Lagrangian descriptors could be applied to a study of  lithium cyanide isomerization. He showed how the method revealed the stable and unstable manifolds of a hyperbolic periodic orbit that mediated the isomerization reaction. Intriguingly, he demonstrated the existence of a parabolic periodic orbit in the reaction region and argued that it played an important role in the isomerization reaction. The full implications of this observation required further study. Rigoberto Hernandez  showed that the method of Lagrangian descriptors could be extended to completely new situations for reaction dynamics. In particular, he considered dissipative systems where the time dependence is stochastic and driven systems where the  dividing surfaces vary in time. This theme was extended by Fabio Revuelta who considered the situation where the environment exhibits memory effects and is modelled by colored noise. Thomas Bartsch considered the reaction dynamics of a system with three reactive channels that is known to be strongly chaotic at all energies—the monkey saddle.  He discussed the phase space structures controlling reaction dynamics in this situation. Joerg Main  concluded the main talks by discussing  how neural networks could be used to reveal the phase space structures, such as NHIMs,  that  govern the reaction dynamics. He showed how this approach could be used to locate time dependent NHIMs in driven systems and, from this, compute rate constants in multidimensional systems.

The day was concluded by a lively series of “lightning talks” (20 slides, 15 seconds per slide, for a total of 5 minutes) presented by the CHAMPS postdocs.

Workshop Photo - 19.03.19 PDRAs IMG_2183

The small size of the workshop encouraged much interaction amongst the participants and encouraged further collaborations amongst the participants.  We hope to hear more about the fruits of these interactions in a follow-on workshop in the near future.

 

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.