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We analyze randomly selected trajectories for cesium ions recombining with fluoride (iodide) ions in the presence of argon (xenon) atoms.

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus)., 34 pages. The material of Section 4 (included to achieve a self-contained presentation) almost coincides with that of Section 4 in arXiv:1612.07653

The direct three-body recombination reactions Cs+ + X− + R → CsX + R (X = F, I and R = Ar, Xe) are studied within the quasiclassical trajectory method using diabatic semiempirical potential energy surfaces, the encounters of the ions being non-central. The collision energies range between 1 and 10 eV (values typical for low temperature plasma), while the so-called delay parameter, which characterizes the delay in the arrival of the neutral atom R in relation to the time instant when the distance between the ions attains its minimum, is equal to 0 or 20%. The calculation results include the recombination excitation functions, the opacity functions, and the vibrational and rotational energy distributions of the recombination products. All the four reactions considered exhibit similar overall statistical dynamics, but each process has its own features. On the whole, for both the recombining pairs Cs+ + F− and Cs+ + I−, xenon is more effective than argon as an acceptor of excess energy from the ion pair. The rotational energy distributions of the salt molecules CsF and CsI are almost equilibrium, whereas the vibrational energy distributions are strongly non-equilibrium.

The design of a source of a seeded supersonic beam of alkali halide molecules or other vaporized solids is described. At a carrier gas (hydrogen or helium) stagnation pressure of 0.5–5 atm, a stagnation temperature of 1000 K, and a partial seed species pressure of 10–2 Torr, the intensity of the salt beam was 1014 sr–1 s–1 and the energy of molecules in the beam varied from 5.5 to 7.5 eV for a mixture with helium and from 7.5 to 15.5 eV for a mixture with hydrogen. The modular design of the source provides the easy replacement of its functional units.

We propose a general impulsive model for scattering of molecules from a flat solid surface. It is assumed within the framework of this model that an encounter of an atom (or ion) with the surface is a series of elastic (in the direction normal to the surface) hits of the atom against surface pseudoparticles, the hits instantly following each other. To each atom, one assigns two infinite sequences of masses of pseudoparticles. The model is a far-reaching generalization of the well-known hard cube model. Criteria for both finiteness and infinity of series of hits are formulated, based on the masses of pseudoparticles and the mass of the atom. It is shown that in virtually all the cases, any number of hits in a series occurs with a positive probability. The proposed model does not satisfy the reciprocity condition., 18 pages and 1 figure, submitted to Rendiconti Lincei. Scienze Fisiche e Naturali

We compare the dynamics of two elementary reactions of recombination of atomic ions, namely, of the termolecular process of two ions and the third body (1) and of the bimolecular interaction of an ion with the diatomic cluster (2) consisting of the second ion and the neutral atom. It is shown that, despite the fact that both processes yield the same products, the main dynamical characteristics of these processes differ. First of all this concerns the excitation function which confines recombination for (2) by a collision energy up to 2 eV. The opacity functions of both processes have different structures. The distributions of the product vibrational energy in both the processes exhibit a non-equilibrium character, while the rotational energy distributions are equilibrium.

An impulsive model of dissociation of diatomic molecules with an ionic bond (for instance, alkali halide molecules) at a graphite surface is proposed. Within the framework of this model, a single encounter of an ion with the surface can consist of several elastic hits of the ion against graphite pseudoparticles of various masses. These hits instantly follow each other and are described in general in the same way as in Logan’s and Stickney’s well-known “hard-cube model” of scattering of atoms from a solid surface. The calculation results for dissociation of KI molecules with a translational energy from 4 to 14 eV are presented.

We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question., Comment: 22 pages

Using the method of classical trajectories, the motion of a pair of ions Cs+ and Cl– in three non-convex closed cavities of complicated geometry containing neutral cylindrical “crosspieces” and charged spherical “nuclei” is simulated. The collisions of the ions with the cavity boundary, crosspieces, and nuclei are supposed to be inelastic. The signs of the charges of the nuclei are varied. In each cavity, in one of the calculation series, the nuclei are assumed to be shielded (to carry charges smaller than 1 a.u. in absolute value), whereas in another calculation series, the possibility of the mutual neutralization of an ion and an oppositely charged nucleus at their encounter is taken into account. The statistics of various events, such as the “stickings” of ions to nuclei, neutralizations of ions and nuclei, collisions of ions with obstacles, recombinations of ions, and dissociations of the CsCl ionic-bond molecule are analyzed.

We describe a hard sphere model of direct three-body recombination of the Cs+ and Br− ions in the presence of neutral atoms Hg, Xe, Kr, or Ar as the third bodies. Calculations are carried out for the ion approach energy and the third body energy in the range from 1 to 10 eV under the assumption of non-central approach of the ions. The calculation results include the dependences of the total recombination probability on these energies as well as the opacity functions for two impact parameters and the dependences of the recombination probability on the angles determining the mutual orientation of the velocities of the reagents. The classification of the three-body collisions according to the sequences of pairwise encounters of the particles is considered. The most widespread mechanism of energy removal from the ionic pair is a single impact of the third body with the Br− ion.

V. I. Arnold (12 June 1937–3 June 2010) published several papers where he described, in the form of recollections, his two earliest research problems (superpositions of continuous functions and quasi-periodic motions in dynamical systems), the main results and their interrelations: [A1], then [A2] (reprinted as [A4, A6]), and [A3] (translated into English by the author as [A5]). The first exposition [A1] has never been translated into English; however, it contains many details absent in the subsequent articles. It seems therefore that publishing the English translation of the paper [A1] would not be superfluous. What follows is this translation. In many cases, the translator gives complete bibliographic descriptions of various papers mentioned briefly in the original Russian text. The English translations of papers in Russian are also pointed out where possible. A related material is contained also in Arnold’s recollections “On A.N. Kolmogorov”. Slightly different versions of these reminiscences were published several times in Russian and English [A7–A12]. The early history of KAM theory is also discussed in detail in the recent brilliant semi-popular book [A13].

Using the method of classical trajectories, we have simulated the motion of a pair of ions Cs+ and Cl− in ellipsoidal cavities containing neutral cylindrical “crosspieces” and charged spherical “nuclei” (both the signs and the absolute values of the charges of the nuclei were varied). Both elastic and inelastic encounters of the ions with the cavity boundary, crosspieces, and nuclei are considered. Most attention is paid to the statistics of the “captures” of trajectories by the nuclei (for elastic encounters of the ions with the obstacles) and to that of the “stickings” of the ions to the nuclei (for inelastic encounters), as well as to the statistics of the ion recombination events and the dissociation events of the corresponding molecule with an ionic bond. The dynamical consequences of nuclei shielding (i.e., a decrease in the absolute values of the nuclei charges) have been determined.

Single-stage three-body recombination of ions and recombination via intermediate complex formation can be considered as two limiting cases of a general mechanism involving two stages separated by some time interval T , namely, energy removal by the third body and closest approach of the ions. Any of these can occur first. Particular dynamics of nascent molecule stabilization is determined by various kinematic parameters including two collision energies and the lag T . In this paper, recombination dynamics of the Cs + and Br − ions in the presence of the Xe atom is studied by quasiclassical trajectories. If T vanishes (one-stage process), stabilization proceeds chiefly via energy transfer in encounters of Xe with both ions approaching each other. If the energy removal stage is late, dynamics changes drastically, stabilization occurring via interaction of Xe mainly with only one ion. This three-body recombination mechanism may be of considerable importance at low pressures in gas medium.

The reversible context 2 in KAM theory refers to the situation where dim FixG < 1/2 codim T, here FixG is the fixed point manifold of the reversing involution G and T is the invariant torus one deals with. Up to now, this context has been entirely unexplored. We obtain a first result on the persistence of invariant tori in the reversible context 2 (for the particular case where dim Fix G = 0) using J. Moser’s modifying terms theorem of 1967.

Using the method of classical trajectories, we have simulated the motion of a pair of oppositely charged ions in spherical and ellipsoidal cavities, including cavities that contain cylindrical “crosspieces” and positively charged “nuclei.” We supposed that each of the ions reflected off the cavity wall, crosspieces, and nuclei according to the elastic impact law. Most of attention is focused on the statistics of the events of ion recombination and those of dissociation of the corresponding molecule with an ionic bond. When nuclei are absent, recombination and dissociation events are possible only on collisions of the ions with the cavity wall or crosspieces. In the presence of nuclei, on the other hand, the major part of the events occur in time intervals between the collisions of the ions with the cavity wall, crosspieces, or nuclei.

The influence of interaction potential parameters of likely charged ions on cross sections of various channels of a reaction of a pair of diatomic molecules with ionic bonds was studied in terms of quasi-classical trajectory simulation with the use of linear mean-square regressions. In the regression approach, the dependence of the cross section of a given reaction channel on potential parameters at each fixed collision energy is approximated by a linear function. We determined the region of softness parameters of the Cs+-Rb+ and Cl−-I− interaction potentials. This region was optimum for the reproduction of experimental excitation functions of atomic and complex positive ions for the CsCl + RbI → products reaction.

By means of quantum mechanical simulation of the reaction F + H2(v = 0; j = 0, 1, 2) → H + HF(v′, j′) on the Stark-Werner ground state potential energy surface at collision energies of 1.84, 2.74, and 3.42 kcal/mol, we have analyzed interference of the “partial waves” corresponding to different values of the total angular momentum J. As the vibrational quantum number v′ of the HF(v′, j′) product increases, the interference for the HF forward scattering becomes noticeably more constructive. This is probably the reason for the maximum in the angular distributions of the HF(v′= 3) molecules at small scattering angles that was discovered experimentally by D.M. Neumark, A.M. Wodtke, G.N. Robinson, C.C. Hayden, and Y.T. Lee, J. Chem. Phys. 82 (7), 3045 (1985) at the same collision energies. We have also determined the intervals of J values most effective for forward scattering of the HF(v′, j′) molecules.

In Aquilanti, Lombardi, and Sevryuk, J. Chem. Phys. 2004, V. 121, No. 12, P. 5579 and Sevryuk, Lombardi, and Aquilanti, Phys. Rev. A. 2005, V. 72, No. 3, P. 033201, we defined several partitions of the total kinetic energy of a system of classical particles into terms corresponding to various motion modes. In this work, we study the statistics of these terms for clusters with the number of particles N from 3 to 100 (at randomly selected particle coordinates and velocities). Some new kinetic energy components are defined and studied. Two limiting situations are considered, those of particles of equal masses and particles whose masses vary randomly. With equal masses, the mean values of almost all cluster kinetic energy components are expressed in terms of N with the use of very simple equations.

Ten open problems in Kolmogorov–Arnold–Moser theory for finite-dimensional dynamical systems are presented and discussed. These problems concern the preservation of frequencies and Floquet exponents of invariant tori of partially integrable systems under small perturbations in the presence of external parameters.

We studied the nature and collision energy dependence of the maximum that appears in the angular distributions of the HF (v′ = 3) product of the F + H2 (v = 0; j = 0, 1, 2) → H + HF (v′, j′) reaction at small scattering angles θ in the center-of-mass frame. This maximum and its increase as the collision energy increased were discovered in the well-known experiment described by D.M. Neumark, A.M. Wodtke, G.N. Robinson, C.C. Hayden, and Y.T. Lee, J. Chem. Phys. 82 (7), 3045 (1985). In order to determine the nature of the maximum, we performed quantum-mechanical simulation of the reaction on the Stark-Werner ground state potential energy surface at collision energies of 1.84, 2.74, and 3.42 kcal/mol corresponding to the above-mentioned experiment and calculated the vibrationally and rotationally resolved differential cross sections dσv′j′/dΩ of the reaction. The maximum under consideration was found to be due to a superposition of two effects, namely, the absence of HF (v′ = 3; j′) products with large j′ because of energy restrictions and an increase in the relative amplitude of quantum-mechanical oscillations on dσv′j′/dΩ cross sections at small j′ and θ as v′ increased. Oscillations on dσ3j′/dΩ cross sections with small j′ are responsible for the maximum observed.

We consider quasi-periodic (with $N$ basic frequencies) non-autonomous perturbations of Hamiltonian, reversible, volume preserving, and dissipative systems. The unperturbed systems possess analytic families of invariant $n$-tori carrying conditionally periodic motions, are allowed to depend on external parameters, and are assumed to satisfy just very weak nondegeneracy conditions. We construct invariant $(n+N)$-tori in perturbed systems following M.R. Herman's approach: additional external parameters are introduced to remove degeneracies and then are eliminated via an appropriate number-theoretical lemma concerning Diophantine approximations of dependent quantities.

The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on “classical” definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three- and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.

We consider perturbations of moderately degenerate integrable or partially integrable Hamiltonian systems, so that unperturbed invariant n-tori with prescribed frequencies or frequency ratios do not persist, but there is preservation of, say, the first d < n frequencies or their ratios. Lagrangian and lower dimensional tori are treated in a unified way. The proofs are very simple and follow Herman's idea of 1990: we introduce external parameters to remove degeneracies and then eliminate these parameters making use of a suitable number-theoretical lemma concerning Diophantine approximations of dependent quantities. Parallel results for reversible, volume preserving and dissipative systems are also presented.

Rigorous definitions are presented for the kinematic angular momentum K of a system of classical particles (a concept dual to the conventional angular momentum J), the angular momentum L(xi) associated with the moments of inertia, and the contributions to the total kinetic energy of the system from various modes of the motion of the particles. Some key properties of these quantities are described-in particular, their invariance under any orthogonal coordinate transformation and the inequalities they are subject to. The main mathematical tool exploited is the singular value decomposition of rectangular matrices and its differentiation with respect to a parameter. The quantities introduced employ as ingredients particle coordinates and momenta, commonly available in classical trajectory studies of chemical reactions and in molecular dynamics simulations, and thus are of prospective use as sensitive and immediately calculated indicators of phase transitions, isomerizations, onsets of chaotic behavior, and other dynamical critical phenomena in classical microaggregates, such as nanoscale clusters.

The first-order post-adiabatic representations in the sense of Klar and Fano (Klar, H.; Fano, U. Phys. Rev. Lett. 1976, 37, 1132−1134) are obtained for two-channel stationary Schrodinger equations describing the interaction of the fluorine and chlorine (in the 2P state) and oxygen and sulfur (in the 3P state) atoms with a number of closed-shell particles, in particular, with rare gases, hydrogen, deuterium, and methane. Information on the adiabatic potentials and the nonadiabatic coupling comes mostly from scattering experiments, although some input from ab initio quantum mechanical calculations is also exploited. Various trends in the behavior of the first-order post-adiabatic coupling are analyzed, and the optimal ways to estimate the smallness of this coupling are discussed. The best measure of the strength of the post-adiabatic coupling of order s is found to be the differences between the respective post-adiabatic potentials of orders s and s + 1. A rigorous proof is given of the fact that post-adiab...

The connection between the salient features of the potential energy surface (PES) and the dynamical characteristics of the elementary collision process is studied using a correlation approach based on quasiclassical trajectory simulations. The method is demonstrated for the reaction F + D2 --D + DF for which the scattering characteristics were calculated on a model family of PES's based on a London-Eyring-Polanyi-Sato-type five-parameter equation. The correlations between the reactive cross section and the vibrational and rotational quantum numbers and the scattering angle of the DF product, and the various parameters of the collinear and noncollinear PES's, such as the location and height of the minimal barrier and the Sato coefficients, are reported. Although usually correlations between two variables suffice, in some cases coefficients of correlation among three variables are required. The role of nonlinear parameter dependencies in computing the correlation coefficients is also considered. The correlation approach makes it possible to examine a large set of potential surfaces without intermediate human control and obtain quantitative information.

The most adequate description of the dynamical behavior of F+H 2 →H+HF, one of the most important model elementary chemical reactions, can be achieved on potential energy surfaces (PES) with a non-collinear transition state configuration. Each such surface is characterized by the particle interaction angle where the minimum of the barrier height is attained. In the present paper, an extended LEPS method where the Sato coefficients are given by a five-parameter expression is proposed and used to examine how the dependence of the barrier height E 0 on the interaction angle γ affects the chemical scattering dynamics. We show that in this model system, the minimum of the dependence E 0 ( γ ) and its shape determine, to a great extent, the total reaction cross-section, the HF( v f ) product branching ratio and the angular distribution of the product scattering.

We propose an impulsive model for the interaction between two alkali halide molecules. The excitation functions for various channels of the reactions RbBr+RbBr, CsBr+CsBr, RbCl+CsI, and RbI+CsCl obtained within this model exhibit a qualitative agreement with the results of exact quasiclassical trajectory calculations.

We compute the rotationally resolved differential cross sections for F(2P3/2)+D2(v=0,j) inelastic scattering as well as opacity functions for D2 rotational excitation and the reaction F+D2→D+DF. Two values of the collision energy (89.7 and 187 meV) and two initial D2 rotational states (j=0 and j=1) are probed. Four calculation techniques have been compared: the quasiclassical trajectory approach and the Wigner method on the ground state (12A′) surface, wave packet propagation (with the D2 vibrational degree of freedom treated quantum mechanically) on the 12A′ surface, and wave packet propagation on the two coupled surfaces 12A′ and 22A′. The effect of the nonadiabatic spin–orbit coupling on the nonreactive F+D2 scattering is almost negligible, whereas the reaction cross sections in the two‐surface wave packet propagation treatment are considerably smaller than those in the calculations taking into account the ground state surface only.

We compute the vibrationally resolved integral cross sections σ R (ν′), differential cross sections, and opacity functions for the reaction F + H 2 ( ν = 0, j = 0, 1) → H + HF( ν ′, j ′) on the 6SEC potential energy surface by the quasiclassical trajectory technique as well as the Wigner method. In both cases, the Langer correction of the initial H 2 rotational energy is implemented. The results are compared with the recent quasiclassical trajectory calculations on the same surface without the Langer correction and quantum mechanical calculations. It is shown that use of the Wigner phase space functions to describe the H 2 and HF vibrational degrees of freedom reduces the total integral cross sections, smooths the selective HF( ν ′ = 3) forward peak in the HF center-of-mass angular distributions, and dramatically increases the σ R ( ν ′ = 3)/ σ R ( ν ′ = 2) vibrational branching ratio. These results indicate that the disagreements between the quasiclassical trajectory cross sections and the quantum mechanical ones observed in the previous works cannot be minimized by just a proper choice and weighting of the initial conditions in the quasiclassical trajectory calculations.

Various properties of post-adiabatic representations of multichannel Schrodinger equations are described in the general context of adiabatic and classical path approximations as used in atomic and molecular physics. The van der Waals interactions of fluorine, chlorine, and oxygen atoms with rare gases, hydrogen, methane, and hydrogen halides are considered: it is found that in some of these systems, the first-order post-adiabatic scheme exhibits a smaller coupling than the adiabatic representation, thus providing an appropriate choice of the basis functions for a decoupling approximation.

We calculate the 0→1, 1→0, 1→2, 2→1, and 2→3 vibrational transition probabilities of the CO molecule trapped in the center of icosahedral argon clusters with 12, 54, and 146 Ar atoms at various cluster temperatures. The semiclassical scheme has been employed, where the impurity vibrations are quantized while all the other degrees of freedom are treated classically. The transition probabilities increase with the cluster size but exhibit an inverse temperature dependence.

Despite the ubiquity of recombination processes in nature and various technologies, presently little is known about the dynamics of these processes. This article reports on a quasi-classical trajectory study of the dynamics of the direct three-body recombination of Cs(+) and Br(-) ions in the presence of a Xe atom at energies of the ion encounter and that of the third body ranging from 0.2 to 10 eV. Several dynamical mechanisms of stabilization of the recombining ion pair have been found. Head-on ion encounters are characterized by two mechanisms of removing the energy from the recombining pair. In the case of nonzero impact parameters of ion encounters, the dynamics leading to the stabilization of the nascent CsBr molecule becomes much more complicated and three new mechanisms appear. They depend mainly on the energy of the ion encounter, on the energy of the collision of the ion pair with the third body, and on the impact parameter of the ion encounter and the impact parameter of the third body. The common feature of all the three mechanisms is the transfer of a fraction of the rotational energy of the recombining pair to the third body. This transfer plays a key role in the stabilization of the molecule. The dynamical peculiarities observed are expected to be common for the recombination of the charged and neutral particles.

Within the impulsive framework, the energy transfer processes in collisions of atoms with diatomic molecules are considered. In the case of noncollinear collisions involving multiple impacts between the particles, analytic expressions for the amount of the collision energy transferred to the internal degrees of freedom of the molecule have been derived. The limiting cases of these expressions are the well-known Mahan (a single impact) and Mahan-Shin (collinear collisions) formulas. The efficiency of energy transfer in collisions of cesium halide molecules with xenon atoms has been computed as an example; the results obtained agree well with the data of accurate trajectory calculations.

The asymptotic equivalence of systems of two ordinary first‐order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points. The set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed. By persistently using the determinant properties of the transformation matrices, the number of integration operations at each step of this procedure is halved compared with the algorithms known before. The theory is specialized to the case of time‐independent one‐dimensional Schrodinger equations with singular potentials. Some generalizations to multichannel Schrodinger equations are also presented.

We present Kolmogorov–Arnold–Moser (or KAM) Theory regarding typicality of quasi-periodic invariant tori, partly from a historical and partly from a pedagogical point of view. At the same time we aim at a unified approach of the theory in various dynamical settings: the ‘classical’ Hamiltonian setting of Lagrangean tori, the Hamiltonian lower dimensional isotropic tori, the dissipative case of quasi-periodic attractors, etc. Also we sketch the theory of quasi-periodic bifurcations, where resonances cause Cantorization and fraying of the bifurcation sets known from the cases of equilibrium points and periodic orbits. Here the concept of Whitney differentiability plays a central role, which locally organizes the nowhere dense union of persistent quasi-periodic invariant tori, of positive measure. At the level of torus bundles this Cantorization is observed as well, where the geometry of the torus bundles turns out to be persistent. In the meantime we briefly deal with the natural affine structure of quasi-periodic tori, with uniqueness of most of the KAM tori, and with the mechanisms of the destruction of resonant unperturbed tori. Other parts of the theory, such as the Hamiltonian higher

Relative differential and integral cross sections for scattering of fluorine atoms by hydrogen bromide and hydrogen iodide have been measured at collision energies from 5 to 175 meV. Treating the experimental data in terms of a single potential of the Lennard-Jones type, we have evaluated the well depths and equilibrium distances of the long-range interaction potentials.

A phase-space approach is proposed for molecular dynamics simulations, which serve as a bridge between detailed descriptions of microscopic world and macroscopic properties of matter. The introduction-aside from the angular momentum of spatial rotations-of other "hyperangular" momenta (the overall grand angular momentum of a cluster of particles and those describing the deformation and rearrangement modes) permits one to analyze different degrees of freedom and to extract, from simulation data, a kinetic energy partition in terms of phase-space invariants. Model calculations illustrate how these provide specific signatures of critical behavior, such as energy thresholds for openings of chaotic pathways in small clusters and for phase transitions in nanoaggregates., European Union through the COST D9 Action and the Human Potential Research Network Theoretical Studies of Electronic and Molecular Processes in Molecules and Clusters; MIUR; ENEA; ASI; Turkish Academy of Sciences (TÜBA)

This is an introduction to the contributions by the lecturers at the mini-symposium on symplectic and contact geometry. We present a very general and brief account of the prehistory of the field and give references to some seminal papers and important survey works.

Consider a mechanical system of s particles with the interaction forces which are independent of velocities or even functions in velocities. Such a system ruled by the Newton equation d 2 r/dt 2 = F(r,v), r ∈ ℝ3s, v = dr/dt, F(r, -v) = F(r,v), is time reversible, i.e., reversing all the velocities v reverses all the trajectories in the configuration space ℝ3s. More generally, an autonomous differential equation du/dt = V(u) and the corresponding vector field V are said to be reversible if there exists a phase space involution G (a mapping for which G 2 = Id) that reverses the direction of time: TG o V = -V o G, where TG is the differential of G, i.e., G transforms the field V into the opposite field -V. This means that with u(t), also G(u(-t)) is a solution of the equation. In the above example, G: (r,v) ↦ (r, -v).

High resolution HF product time-of-flight spectra measured for the reactive scattering of F atoms from n-H2(p-H2) molecules at collision energies between 69 and 81 meV are compared with exact coupled-channel quantum mechanical calculations based on the Stark-Werner ab initio ground state potential energy surface. Excellent agreement between the experimental and computed rotational distributions is found for the HF product vibrational states v'=1 and v'=2. For the v'=3 vibrational state the agreement, however, is less satisfactory, especially for the reaction with p-H2. The results for v'=1 and v'=2 confirm that the reaction dynamics for these product states is accurately described by the ground electronic state 1 (2)A' potential energy surface. The deviations for HF(v'=3, j'or =2) are attributed to an enhancement of the reaction resulting from the 25% fraction of excited ((2)P(12)) fluorine atoms in the reactant beam.

It is shown that in hard sphere (impulsive) collisions of atoms with diatomic molecules, complete conversion of the collision energy into the internal energy of the diatomic partner is possible for any number of impacts between the elastic balls representing the particles. The corresponding collision geometries and relations between the masses of the particles are described in detail.

The excitation functions for the formation of cations Cs+, Rb+, CsRbCl+ and CsRbI+ in collisions of alkali-metal halide molecules CsCl and RbI have been measured in crossed molecular beams for the collision energy ranging from 3 to 10 eV. Trajectory simulation of dissociative processes in the CsCl + RbI system has been performed on a potential-energy surface chosen to be the sum of six pairwise interaction potentials. The calculated excitation functions agree well with the experimental data. Simulation of the CsCl + RbI reaction by a hard-sphere model has also been carried out.