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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]

A trajectory method of calculating tunneling probabilities from phase integrals along straight line tunneling paths, originally suggested by Makri and Miller [J. Chem. Phys. 91, 4026 (1989)] and recently implemented by Truhlar and co-workers [Chem. Sci. 5, 2091 (2014)], is tested for oneand two-dimensional ab initio based potentials describing hydrogen dissociation in the 1B1 excited electronic state of pyrrole. The primary observables are the tunneling rates in a progression of bending vibrational states lying below the dissociation barrier and their isotope dependences. Several initial ensembles of classical trajectories have been considered, corresponding to the quasiclassical and the quantum mechanical samplings of the initial conditions. It is found that the sampling based on the fixed energy Wigner density gives the best agreement with the quantum mechanical dissociation rates. [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]

The interaction of the glycine radical on the side walls of both armchair and zigzag single walled carbon nanotubes is investigated by density functional theory. It is found that the interaction potential of the N-centered glycine radical with the tubes has a minimum of 16.9 (armchair) and 20.2 (zigzag) kcal/mol with respect to the dissociation products.

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 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]

Small and large molecules may localize their energy in specific bonds or generally in vibrational modes for extended periods of time, an effect which may have dramatic consequences in reaction dynamics. Periodic orbits offer the means to identify phase space regions with localized motions. The author demonstrate that techniques to locate periodic orbits developed for small molecules can be applied to large molecules such as alanine dipeptide. The widely used empirical force fields are employed and principal families of periodic orbits associated with local-type motions and emanated from the lowest energy minima and saddle points are investigated. Continuation of these families at high energies unravels the stable and unstable regions of phase space as well as elementary bifurcations such as saddle nodes. [ABSTRACT FROM AUTHOR]

Low-lying resonances of the CH2(ã 1A1) system (J=0) in an accurate ab initio potential energy surface are studied using a filter-diagonalization method. The width of these resonances fluctuates by more than two orders of magnitude and on average increases with the energy. Analysis of the resonance states concludes that the unimolecular decay of the excited molecular system near the dissociation threshold is neither mode specific nor statistical state specific. This is apparently due to remnant regularity embedded in the largely chaotic classical phase space, as evidenced by periodic orbit analysis. As a result, the Rice–Ramsperger–Kassel–Marcus and statistical adiabatic channel models overestimate the average unimolecular decay rate. The implications of the resonances for the bimolecular C(1D)+H2 reaction are also discussed. [ABSTRACT FROM AUTHOR]

The Huggins band of ozone is investigated by means of dynamics calculations using a new (diabatic) potential energy surface for the 3 1A′(1B2) state. The good overall agreement of the calculated spectrum of vibrational energies and intensities with the experimental spectrum, especially at low to intermediate excitation energies, is considered as evidence that the Huggins band is due to the two Cs potential wells of the 1B2 state rather than the single C2v well of the 2 1A′(1A1) state. The vibrational assignment of the “cold bands,” based on the nodal structure of wave functions, on the whole supports the most recent experimental assignment [J. Chem. Phys. 115, 9311 (2001)]. The quantum mechanical spectrum is analyzed in terms of classical periodic orbits and the structure of the classical phase space.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The Huggins band of ozone is investigated by means of exact dynamics calculations using a new (diabatic) potential energy surface for the 1B2 state. The remarkable agreement with the measured spectrum strongly suggests that the Huggins band is due to the two Cs potential wells of the 1B2 state. The vibrational assignment, based on the nodal structure of wave functions, supports the most recent experimental assignment. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Based on an accurate potential energy surface [J. Chem. Phys. 113, 4598 (2000)] we calculated ca. 700 bound state energies and wave functions of nonrotating HOBr using the filter-diagonalization method. Similar to HOCl, a 1:2 anharmonic resonance between the HOBr bending and the OBr stretching mode determines the general structure of the level spectrum. One of the results of this resonance is a saddle-node bifurcation at which a new class of states (''dissociation states'') comes into existence, which advance along the HO-Br dissociation path. Because the resonance condition at low energies is better fulfilled for HOBr, the bifurcation occurs at considerably lower energies than for HOCl. The results of the quantum mechanical calculations are interpreted in terms of classical periodic orbits (continuation/bifurcation diagram) and a semiclassical analysis based on a spectroscopic Hamiltonian, which is fitted to the exact energy levels and also taking into account the wave functions. [ABSTRACT FROM AUTHOR]

Transitions from normal-mode to local-mode states in the two bending oscillations of acetylene observed in spectroscopic studies, as well as the existence of localized vibrational states in vinylidene above the isomerization barrier to acetylene, are examined by constructing continuation/bifurcation diagrams of periodic orbits on a global potential-energy surface. The principal families of periodic orbits, which provide a faithful representation of the overtone vibrational states, show pitchfork bifurcations for the asymmetric stretch and the two bending families of acetylene, the fingerprints of the normal- to local-mode transitions, whereas for vinylidene most of the principal families of periodic orbits remain stable above the isomerization potential barrier, thus supporting the existence of localized vinylidene states. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A systematic approach has employed a nonlinear mechanics to study the vibrational spectra such as those of [C.sub.c]O protein. The studies have shown that proton movement adjacent to active site has led to significant perturbations of the [Fe.sup.IV][double bond]O isotopic difference vibrational spectra in cytochrome c oxidase, without a change in oxidation state of the metal sites.

The complete vibrational spectrum of the H[O.sub.2]([X.sup.2]A') radical is obtained quantum mechanically on an accurate potential energy surface (PES). Fermi resonances are identified and the vibrational dynamics is analyzed by using a periodic orbit approach.

Discusses the use of high order finite difference algorithms for solving the Schrodinger equation in molecular dynamics. Background on pseudospectral methods; Finite difference methods; Relations between finite difference and pseudospectral methods; Application to iodine molecule.

We present exact quantum mechanical bound-state calculations for HCP using an ab initio potential energy surface. The main result is the existence of two distinct families of bending states: one with normal-mode-type behavior and wave functions confined to small bending angles and the other one sampling the isomerization path all the way from H–CP to CP–H. Stable periodic orbits provide a clear-cut assignment. Possible relations to recent spectroscopic observations are discussed. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Families of periodic orbits and the classical survival probability function are calculated for ozone in the excited 1B2 state. It is shown that some of the observed oscillations in the experimentally extracted and quantum mechanically calculated time autocorrelation functions can be explained by certain types of periodic orbits. Disagreements between the calculated and experimental correlation function are attributed to inaccuracies of the potential function used. [ABSTRACT FROM AUTHOR]

Quantum mechanical calculations on KCN, LiCN, and ArHCl with two-dimensional (2D) potentials show that the Born–Oppenheimer separation scheme with the slow coordinate as the adiabatic variable is reliable for regular states but not for chaotic states. Three-dimensional (3D) calculations on KCN and ArHCl, where the coupling of CN or HCl motion with the other coordinates is introduced through the kinetic part of the Hamiltonian, demonstrate the existence of an extra constant of motion even for high excitations of the diatom. In 3D, cuts through the wave function show regular nodal patterns across the regular and chaotic coordinates. However, the number of nodes displaced by the chaotic coordinate varies for different values of the third constant coordinate. It is suggested that these plots provide a useful diagnostic aid for multidimensional irregular states. The adiabatic separation of regular motion gives results in excellent agreement with fully coupled 3D calculations. [ABSTRACT FROM AUTHOR]

Classical and quantum mechanical calculations on the vibrational motions of LiCN using a realistic potential are presented. These, together with recent results for KCN, are analyzed using indicators that have been proposed for identifying quantum chaos. These are nodal structure, dominant coefficient, overlapping avoided crossings, second differences, and spectral distribution. We find an early onset of chaos in LiCN and KCN with good agreement between the indicators. Localized quasiperiodic trajectories and regular states are found well into the chaotic region for both isomers of LiCN. LiCN is a weakly coupled system in contrast to strongly coupled KCN. The utility of the indicators is discussed in the light of these results. The link between floppy molecules and chaos suggests that floppy systems are suitable for the experimental investigation of vibrational chaos. [ABSTRACT FROM AUTHOR]

Presents quantum mechanical bound-state calculations for highly excited vibrational states of HCP using an ab initio potential energy surface. Wave functions of the states; Fermi resonance governing the energy spectrum; Changes in the bending character of the states; Large-amplitude motion of the states.

In this review we discuss the recendy described roaming mechanism for chemical reactions from the point of view of nonlinear dynamical systems in phase space. The recognition of the roaming phenomenon shows the need for further developments in our fundamental understanding of basic reaction dynamics, as is made clear by considering some questions that cut across most studies of roaming: Is the dynamics statistical? Can :ransition state theory be applied to estimate roaming reaction rates? What role do saddle points on the potential energy surface play in explaining the behavior of roaming trajectories? How do we construct a dividing surface that is appropriate for describing the transformation from reactants to products for roaming trajectories? How should we define the roaming region? We snow that the phase space perspective on reaction dynamics provides the setting in which these questions can be properly framed and answered. We illustrate these ideas by considering photodissociation of formaldehyde. Tne phasespace formulation allows an unambiguous description of all possible reactive events, which also allows us to uncover the phase space mechanism that explains which trajectories roam, as opposed to evolving toward a different reactive event. [ABSTRACT FROM AUTHOR]