Article, “Influence of mass and potential energy surface geometry on roaming in Chesnavich’s CH4+ model,” was published 07 September 2018, in The Journal of Chemical Physics (Vol.149, Issue 9).  Vladimir Krajnak and Stephen Wiggins

It may be accessed via the link below:
https://doi.org/10.1063/1.5044532
DOI: 10.1063/1.5044532

Chesnavich’s model Hamiltonian for the reaction CH⁺₄ → CH⁺₃ + H is known to exhibit a range of interesting dynamical phenomena including roaming. The model system consists of two parts: a rigid, symmetric top representing the CH⁺₃ ion and a free H atom. We study roaming in this model with focus on the evolution of geometrical features of the invariant manifolds in phase space that govern roaming under variations of the mass of the free atom m and a parameter a that couples radial and angular motion. In addition, we establish an upper bound on the prominence of roaming in Chesnavich’s model. The bound highlights the intricacy of roaming as a type of dynamics on the verge between isomerisation and nonreactivity as it relies on generous access to the potential wells to allow reactions as well as a prominent area of high potential that aids sufficient transfer of energy between the degrees of freedom to prevent isomerisation.

In July 2018, three of the CHAMPS PDRA’s: Rafael Garcia Meseguer, Matthaios Katsanikas and Vladimir Krajnak, attended the workshop ‘Geometry of Chemical Reaction Dynamics in Gas and Condensed Phases’ at the Telluride Science Research Center in Telluride, CO, USA. The meeting focussed on theoretical aspects of chemical reaction dynamics and forms a platform for discussions about extensions of theory from lower to higher dimensional systems and from gas to condensed phases. Apart from meeting some of the leading researchers in the field and discussing ideas for future work, the PDRA’s also shared their contributions to the field. Rafael gave a talk on the applications of Lagrangian descriptors to finding dividing surfaces and invariant structures, Matthaios shared his findings on phase space transport in the Caldera model and Vladimir reported on the phase space mechanism behind Roaming in Chesnavich’s model and its evolution under parameter variations.

CHAMPS held a research day at Imperial College London on Friday 13th July 2018. This provided an  opportunity for all of the investigators and PDRAs to discuss their research and obtain feedback from the entire CHAMPS team.

A copy of the schedule for the day can be found here: Research Day 13.07.18 Schedule

Professor Stephen Wiggins – Champs Principle Investigator

Professor Dmitry Shalashilin (left) Dr Dmitry Makhov (right)

Dr Matthaios Katsanikas – Champs PDRA

Dr Vladimir Krajnak – Champs PDRA

Professor Dmitry Shalashilin Co-Investigator (left) Professor Darryl Holm – Co-Investigator (right)

Dr Lars Bratholm – Champs PDRA

Professor Dmitry Shalashilin – Champs Co-Investigator

Professor Barry Carpenter – Champs Co-Investigator

Dr Shibabrat Naik – Champs PDRA

Dr Dmitry Makhov – Champs PDRA

Champs PDRAs – Makhov, Matthaios, Krajnak, Garcia-Meseguer, Naik, Bratholm (left to right)

Geometry of nonadiabatic quantum hydrodynamics

Michael. S. Foskett, Darryl. D. Holm, Cesare Tronci

(Submitted on 3 Jul 2018)

By using standard momentum maps from geometric mechanics, we collectivize the mean-field (MF) and exact factorization (EF) models of molecular quantum dynamics into two different quantum fluid models. After deriving the corresponding quantum fluid models, we regularize each of their Hamiltonians for finite ℏ by introducing spatial smoothing. The ℏ≠0 dynamics of the Lagrangian paths of the classical nuclear fluid flows for both MF and EF can be written and contrasted as finite dimensional canonical Hamiltonian systems for the evolution in phase space of singular solutions called Bohmions, in which each nucleus follows a Lagrangian path in configuration space. Comparison is also made with the variational dynamics of a new type of Bohmian trajectories, which arise from Hamilton’s principle with spatially smoothed quantum potential with finite ℏ.

https://arxiv.org/abs/1807.01031

The CHAMPS Leeds group just published in the Journal of Chemical Physics two papers related to CHAMPS Work Projects 4 and 5 on quantum dynamics and nonadiabatic dynamics. In the first paper “The effect of sampling techniques used in the multiconfigurational Ehrenfest method” by C.Symonds, J.Kattitzi and D.Shalashilin (https://aip.scitation.org/doi/10.1063/1.5020567) Spin-Boson model was used to assess the samplings of Canonical Coherent States basis sets in the phase space quantum mechanics. The paper validated the sampling techniques used in our simulations of ultrafast photochemical reactions. It has been demonstrated that the techniques really work and for the Spin-Boson model the calculations converge to the exact quantum result.  The Figure below shows quantum wave functions in phase space.

In the second paper “Zombie states for description of structure and dynamics of multi-electron systems” by D.Shalashilin (https://aip.scitation.org/doi/10.1063/1.5023209) a new type of Fermionic Coherent State has been introduced, which potentially can be used in simulations of nonadiabatic dynamics in chemistry and photochemistry. Fermionic Coherent States are well known in mathematics.  However, they require complicated algebra of Grassmann numbers not well suited for numerical simulations in computational chemistry.  The paper introduces a coherent antisymmetrised superposition of “dead” and “alive” electronic states called Zombie State (ZS), which do not need Grassmann algebra.  Instead it is replaced by a very simple sign-changing rule in the definition of creation and annihilation operators.  Zombie States can be used as basis functions for quantum propagation just like Canonical Coherent States for distinguishable particles.  As it is shown at the Figure in standard electronic structure and dynamics theories (left frame) some spin-orbitals are occupied by electrons and some are empty. In Zombie States (right frame) all orbitals are occupied, but some electrons are more “alive” than “dead” or more “dead” than “alive”.  The term “Zombie” for simultaneously “dead” and “alive” electrons was proposed by Liz Clark, Bristol School of Mathematics manager, who is acknowledged in the paper!

Robert G. Bergman Lecture: What is a transition state, and why should I care?

13th March 2018. University of California, Berkeley.

Featured Speaker:  Prof. Barry Carpenter, School of Chemistry, University of Bristol

Since the early days of the development of Transition State Theory, there have been two descriptions of what a transition state (TS) is. The one that most chemists use identifies the TS with a saddle point on the potential energy surface (PES). The other is that it is a dividing surface (DS) in phase space, which reactive trajectories cross only once on their transit from reactant to product. Under limited circumstances, the two descriptors can be shown to be equivalent, but in most practical circumstances they are not. The DS description is the more rigorous, and this talk will focus on cases in which the location of the DS in configuration space is far from any PES saddle point. A common example occurs for reactions in solution that involve substantial changes in shape of the solute. If there are parallel paths to competing products after such an event, then incorrect identification of the true location of the TS can lead to a misunderstanding of what controls the product ratio. That, in turn, has obvious consequences for efforts to control product ratios through change of conditions or design of catalysts.

A goal of CHAMPS is to bring together developments in mathematics with problems in chemistry that reveal a new and unique insight. The method of Lagrangian descriptors is an excellent example of such a synergy.

Lagrangian descriptors are a trajectory diagnostic for revealing phase space structures in dynamical systems. The method was originally developed in the context of Lagrangian transport studies in fluid dynamics (Madrid and Mancho 2009) but the wide applicability of the method has recently been recognized in chemistry, see (Craven and Hernandez 2015, Junginger and Hernandez 2016, Feldmaier, Junginger et al. 2017, Junginger, Duvenbeck et al. 2017, Junginger, Main et al. 2017).

The method is very compelling since it is straightforward to implement computationally, the interpretation is evident, and it provides a “high resolution” method for exploring high dimensional phase space with low dimensional slices. It also applies to both Hamiltonian and non-Hamiltonian systems (Lopesino, Balibrea-Iniesta et al. 2017) as well as to systems with arbitrary, even stochastic, time-dependence (Balibrea-Iniesta, Lopesino et al. 2016). Moreover, Lagrangian descriptors can be applied directly to data sets, without the need of an explicit dynamical system (Mendoza, Mancho et al. 2014).

Briefly, Lagrangian descriptors are implemented as follows. Each point in a chosen grid of initial conditions for trajectories in phase space is assigned a value according to the arclength of the trajectory starting at that initial condition after integration for a fixed, finite time, both backward and forward in time (all initial conditions in the grid are integrated for the same time). The idea is that the influence of phase space structures on trajectories will result in differences in arclength of nearby trajectories near a phase space structure. This has been quantified in terms of the notion of “singular structures’ in the Lagrangian descriptor plots, which are easy to recognize visually (Mancho, Wiggins et al. 2013, Lopesino, Balibrea-Iniesta et al. 2017).

Trajectories are the “primitive objects” that are used to explore phase space structure. In fact, phase space structure is “built” from trajectories. For high dimensional phase space this approach is problematic and prone to issues of interpretation since a tightly grouped set of initial conditions may result in trajectories that become “lost” with respect to each other in phase space. The method of Lagrangian descriptors turns this problem on its head by emphasizing the initial conditions of trajectories, rather than the precise location of their futures and pasts, after a specified amount of time.  A low dimensional “slice” of phase space can be selected and sampled with a grid of initial conditions of high resolution. Since the phase space structure is encoded in the initial conditions of the trajectories, no resolution is lost as the trajectories evolve in time.

We remark that the original Lagrangian trajectory diagnostic is the arclength. This has been modified in (Lopesino, Balibrea et al. 2015, Lopesino, Balibrea-Iniesta et al. 2017) where, effectively, a different type of norm is used as a trajectory diagnostic. This has been shown to have many advantages over the arclength. For example, it allows for a rigorous analysis of the notion of “singular structures” in certain cases and the relation of this notion to invariant manifolds. It also allows a natural decomposition of the Lagrangian descriptor in a way that isolates distinct dynamical effects. This was utilized in (Demian and Wiggins 2017) in order to show that a Lagrangian descriptor could be used to detect the Lyapunov periodic orbits in the two degree-of-freedom Henon-Heiles Hamiltonian system.

This figure is from (Mendoza, Mancho et al. 2014) and shows the results of the  use of the Lagrangian descriptor for revealing flow structures in the Gulf Stream using a geophysical  fluid dynamics data set.

The use and further development of Lagrangian descriptors is a topic underlying many of the themes of CHAMPS: from understanding the role of phase space structure in high dimensional systems to dimensional reduction and as well as for machine learning from data sets.

References

Balibrea-Iniesta, F., et al. (2016). “Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems.” International Journal of Bifurcation and Chaos 26(13): 1630036.

Craven, G. T. and R. Hernandez (2015). “Lagrangian descriptors of thermalized transition states on time-varying energy surfaces.” Physical review letters 115(14): 148301.

Demian, A. S. and S. Wiggins (2017). “Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors.” International Journal of Bifurcation and Chaos 27(14): 1750225.

Feldmaier, M., et al. (2017). “Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors.” Chemical Physics Letters 687: 194-199.

Junginger, A., et al. (2017). “Chemical dynamics between wells across a time-dependent barrier: Self-similarity in the Lagrangian descriptor and reactive basins.” The Journal of chemical physics 147(6): 064101.

Junginger, A. and R. Hernandez (2016). “Lagrangian descriptors in dissipative systems.” Physical Chemistry Chemical Physics 18(44): 30282-30287.

Junginger, A., et al. (2017). “Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points.” Physical Review A 95(3): 032130.

Lopesino, C., et al. (2015). “Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps.” Communications in Nonlinear Science and Numerical Simulation 27(1): 40-51.

Lopesino, C., et al. (2017). “A theoretical framework for lagrangian descriptors.” International Journal of Bifurcation and Chaos 27(01): 1730001.

Madrid, J. J. and A. M. Mancho (2009). “Distinguished trajectories in time dependent vector fields.” Chaos: An Interdisciplinary Journal of Nonlinear Science 19(1): 013111.

Mancho, A. M., et al. (2013). “Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems.” Communications in Nonlinear Science and Numerical Simulation 18(12): 3530-3557.

Mendoza, C., et al. (2014). “Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets.” Nonlinear Processes in Geophysics 21(3): 677-689.