Search

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. [ABSTRACT FROM AUTHOR]

We examine the phase space structures that govern reaction dynamics in the absence of critical points on the potential energy surface. We show that in the vicinity of hyperbolic invariant tori, it is possible to define phase space dividing surfaces that are analogous to the dividing surfaces governing transition from reactants to products near a critical point of the potential energy surface. We investigate the problem of capture of an atom by a diatomic molecule and show that a normally hyperbolic invariant manifold exists at large atom-diatom distances, away from any critical points on the potential. This normally hyperbolic invariant manifold is the anchor for the construction of a dividing surface in phase space, which defines the outer or loose transition state governing capture dynamics. We present an algorithm for sampling an approximate capture dividing surface, and apply our methods to the recombination of the ozone molecule. We treat both 2 and 3 degrees of freedom models with zero total angular momentum. We have located the normally hyperbolic invariant manifold from which the orbiting (outer) transition state is constructed. This forms the basis for our analysis of trajectories for ozone in general, but with particular emphasis on the roaming trajectories. [ABSTRACT FROM AUTHOR]

A model Hamiltonian for the reaction CH4+ → CH3+ + H, parametrized to exhibit either early or late inner transition states, is employed to investigate the dynamical characteristics of the roaming mechanism. Tight/loose transition states and conventional/roaming reaction pathways are identified in terms of time-invariant objects in phase space. These are dividing surfaces associated with normally hyperbolic invariant manifolds (NHIMs). For systems with two degrees of freedom NHIMS are unstable periodic orbits which, in conjunction with their stable and unstable manifolds, unambiguously define the (locally) non-recrossing dividing surfaces assumed in statistical theories of reaction rates. By constructing periodic orbit continuation/bifurcation diagrams for two values of the potential function parameter corresponding to late and early transition states, respectively, and using the total energy as another parameter, we dynamically assign different regions of phase space to reactants and products as well as to conventional and roaming reaction pathways. The classical dynamics of the system are investigated by uniformly sampling trajectory initial conditions on the dividing surfaces. Trajectories are classified into four different categories: direct reactive and non-reactive trajectories, which lead to the formation of molecular and radical products respectively, and roaming reactive and non-reactive orbiting trajectories, which represent alternative pathways to form molecular and radical products. By analysing gap time distributions at several energies, we demonstrate that the phase space structure of the roaming region, which is strongly influenced by nonlinear resonances between the two degrees of freedom, results in nonexponential (nonstatistical) decay. [ABSTRACT FROM AUTHOR]

The Hamiltonian theory of thermodynamics for reversible and irreversible processes is presented in detail. Particularly, it is demonstrated that non-linear molecular dynamics and thermodynamics of equilibrium and non-equilibrium processes can merge in a single dynamical theory by constructing a homogeneous of first degree in momenta Hamiltonian function on the extended thermodynamic state space and in the entropy representation. The geometrical properties of the physical equilibrium state manifold are discussed and results of numerical experiments with the Hénon–Heiles model that dissipates in a simple thermodynamic system with multiple friction parameters are presented. It is well known that this classical paradigm carries most of the non-linear dynamical characteristics of generic molecular systems. The construction of homogeneous Hamiltonians for composite thermodynamic systems is also outlined in the appendixes, where the theory of Hamiltonian thermodynamics is developed in current differential geometry terms. [ABSTRACT FROM AUTHOR]

The diffuse vibrational bands, observed in the ultraviolet photodissociation spectrum of nitrous oxide by exciting the molecule in the first 1A′ state, have recently been attributed to resonances localized mainly in the NN stretch and bend degrees of freedom. To further investigate the origin of this localization, fundamental families of periodic orbits emanating from several stationary points of the 1A′ potential energy surface and bifurcations of them are computed. We demonstrate that center-saddle bifurcations of periodic orbits are the main mechanism for creating stable regions in phase space that can support the partial trapping of the wave packet, and thus they explain the observed spectra. A non-linear mechanical methodology, which involves the calculation of equilibria, periodic orbits, and transition states in normal form coordinates, is applied for an in detail exploration of phase space. The fingerprints of the phase space structures in the quantum world are identified by solving the time dependent Schrödinger equation and calculating autocorrelation functions. This demonstrates that different reaction channels could be controlled if exact knowledge of the phase space structure is available to guide the initial excitation of the molecule. [ABSTRACT FROM AUTHOR]

Despite using potential energy surfaces, multivariable functions on molecular configuration space, to comprehend chemical dynamics for decades, the real happenings in molecules occur in phase space, in which the states of a classical dynamical system are completely determined by the coordinates and their conjugate momenta. Theoretical and numerical results are presented, employing alanine dipeptide as a model system, to support the view that geometrical structures in phase space dictate the dynamics of molecules, the fingerprints of which are traced by following the Hamiltonian flow above saddles. By properly selecting initial conditions in alanine dipeptide, we have found internally free rotor trajectories the existence of which can only be justified in a phase space perspective. [ABSTRACT FROM AUTHOR]

Classical thermodynamics have recently been formulated as Hamiltonian systems elevating the theory of heat to the same level of description as analytical mechanics. Particularly, mechanical and thermodynamical systems can merge in a single dynamical theory to describe equilibrium and non-equilibrium processes by constructing homogeneous of first degree in momenta Hamiltonian functions on the extended thermodynamic phase space and in a variety of representations. By employing geometric Hamiltonian theory we show that classical chemical kinetics can also be put in a Hamiltonian framework with Massieu–Gibbs function as the generating Hamiltonian at constant temperature and pressure. A metric in the physical state manifold as well as the entropy production in irreversible chemical reactions are defined. This way we establish a common computational platform for chemical kinetics and chemical dynamics. Numerical results are presented from the study of consecutive first-order elementary reactions and the non-linear kinetic equations of a model for autocatalytic symmetry breaking chiral reactions. For the latter example, we have examined two cases; that of a closed system with fixed initial concentrations and the steady-state case. • Chemical reaction kinetics as Hamiltonian systems. • Linear consecutive first-order elementary reactions. • Non-linear kinetics of autocatalytic symmetry breaking chiral reactions. [ABSTRACT FROM AUTHOR]

Highlights: [•] We give a dynamical interpretation of the ‘Roaming reaction mechanism’. [•] We give a clear definition of the notions of loose and tight transition states. [•] Periodic orbits as a means to discover new reaction pathways in phase space. [Copyright &y& Elsevier]