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.

 

 

A.S. Demian and S. Wiggins, Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors, International Journal of Bifurcation and Chaos, 27 (4), 1750225 (2017).

http://www.worldscientific.com/doi/abs/10.1142/S021812741750225X

 The purpose of this paper is to apply Lagrangian Descriptors, a concept used to describe phase space structure, to autonomous Hamiltonian systems with two degrees of freedom in order to detect periodic solutions. We propose a method for Hamiltonian systems with saddle-center equilibrium and apply this approach to the classical Hénon–Heiles system. The method was successful in locating the unstable Lyapunov orbits in phase space.

B.K. Carpenter, G.S. Ezra, S. C. Farantos, Z. C. Kramer, and S. Wiggins. Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points, Regular and Chaotic Dynamics, 23(1), 60-79 (2018).

https://link.springer.com/article/10.1134/S1560354718010069

 

In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the “roaming mechanism” whose reaction dynamics are of current interest in the chemistry community.

B.K. Carpenter, G.S. Ezra,  S. C. Farantos, Z. C. Kramer, and S. Wiggins. Empirical Classification of Trajectory Data: An Opportunity for the Use of Machine Learning in Molecular Dynamics. J. Phys. Chem. B. DOI: 10.1021/acs.jpcb.7b08707

Publication Date (Web): October 2, 2017.
https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.7b08707

This paper uses trajectory data and machine learning approaches to “learn” phase space structures. Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degree of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent.  Below the energy of the plane, the dynamics are demonstrably chaotic.  However, most of the work in this paper involves trajectories at a fixed energy that is 1% above that of the plane, in which regime the dynamics exhibit behavior characteristic of chaotic scattering.  The trajectories are analyzed without reference to the potential, as if they had been generated in a typical direct molecular dynamics simulation.  The questions addressed are whether one can recover useful information about the structures controlling the dynamics in phase space from the trajectory data alone, and whether, despite the at least partially chaotic nature of the dynamics, one can make statistically meaningful predictions of trajectory outcomes from initial conditions.  It is found that key unstable periodic orbits, which can be identified on the analytical potential, appear by simple classification of the trajectories, and that the specific roles of these periodic orbits in controlling the dynamics are also readily discerned from the trajectory data alone.  Two different approaches to predicting trajectory outcomes from initial conditions are evaluated, and it is shown that the more successful of them has ~90% success.  The results are compared with those from a simple neural network, which has higher predictive success (97%) but requires the information obtained from the “by-hand” analysis to achieve that level.  Finally, the dynamics, which occur partly on the very flat region of the potential, show characteristics of the much-studied phenomenon called “roaming.” On this potential, it is found that roaming trajectories are effectively “failed” periodic orbits, and that angular momentum can be identified as a key controlling factor, despite the fact that it is not a strictly conserved quantity.  It is also noteworthy that roaming on this potential occurs in the absence of a “roaming saddle,” which has previously been hypothesized to be a necessary feature for roaming to occur.