Henon Heiles Research Articles

The zero-point energy problem in classical trajectory simulations at dissociation threshold

Quasiclassical trajectory calculations offer a cost-effective means of investigating the dynamics of chemical reactions. However, they suffer from the zero-point energy (ZPE) problem, whereby the (quantum) ZPE motion can contribute to an overestimation of the rate coefficient. This paper reports on some dynamics of the Hénon–Heiles system. Dynamics of the water molecule at energies just below the (quantum) dissociation threshold, are also reported. The TRAPZ method [Lim and McCormack, J. Chem. Phys. 102, 1705 (1995)] leads to a definite improvement over unconstrained classical mechanics.

Integrability properties of a generalized Lamé equation: Applications to the Henon–Heiles system

Integrability properties of a generalized Lamé equation: Applications to the Henon–Heiles system

A general and efficient filter-diagonalization method without time propagation

A general and efficient filter-diagonalization method is proposed to solve the quantum mechanical eigenvalue/eigenstate problem in a pre-specified energy range. This method is in similar spirit to the original filter-diagonalization approach, but it eliminates the time propagation by expanding the filter operator directly in the energy space in terms of Chebychev polynomials. Our approach is more efficient than the existing methods because neither time propagation nor time-to-energy transformation is needed. It also allows the choice of the most efficient filter operator for the system of interest. This method is tested for a one-dimensional Morse oscillator and for the two-dimensional Henon–Heiles system. The results show excellent convergence and accuracy.

The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions

A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrödinger equation. The method has been used previously for the solution of Fokker–Planck equations. The Fokker–Planck equation can be transformed to a Schrödinger equation with a potential of the form that occurs in supersymmetric quantum mechanics. For this class of potentials, the ground state wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrödinger equation. For the Fokker–Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrödinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O2 and Ar–Xe with realistic pair potentials. For O2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrödinger equation with the Henon–Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrödinger equation is very rapid with this approach.

Symplectic reversible integrators: Predictor–corrector methods

A new fourth order predictor–corrector integration scheme is presented. The unique feature of the new algorithm and what distinguishes it from a Gear predictor–corrector is that the method is derived from the Trotter decomposition of a specially formulated evolution operator and as such, is both symplectic and reversible. In addition, the method retains the useful property of Gear methods that only one force evaluation per time step is required. The new integrator is tested on a harmonic plus quartic oscillator and the Henon–Heiles system. Comparisons are made to the second order velocity Verlet integrator, the true fourth order Yoshida/Suzuki schemes and fourth order Gear. In all cases, the new method works well, giving energy conservation and trajectories of much better quality than velocity Verlet and of comparable quality to the results of the true fourth order schemes for the same computational cost as velocity Verlet.

Normal coordinates–finite elements calculation of 3D vibrational energy levels: Henon—Heiles and Eckart potentials, H molecule

AbstractTheoretical and numerical results related to the calculation of multidimensional vibrational levels are presented. A description of the methodological details of a very general method (normal coordinates–finite elements, NC–FEM) is provided. Several representative three‐dimensional (3D) systems (Henon–Heiles and Eckart potentials, and the H molecule) are studied, and NC–FEM results are compared with those published by other authors. For the H, a vibrational Hamiltonian expressed in terms of the three internuclear distances is integrated, and the results obtained are compared with the experimental ones. © 1994 by John Wiley & Sons, Inc.

A generalized Henon–Heiles system and related integrable Newton equations

A detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q22. Their connection with fifth-order one-component soliton equations is discussed. Lax representations are constructed, and the bi-Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

Elliptic Baker–Akhiezer functions and an application to an integrable dynamical system

An ansatz for elliptic Baker–Akhiezer functions is used to find new solutions for various spectral problems—the Sturm–Liouville problem with four and five-gap Lamé potentials and two third-order problems. One of the third-order problems is the classical Halphen equation with a spectral parameter. The eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized Hènon–Heiles system.

A method to constrain vibrational energy in quasiclassical trajectory calculations

In this paper, we present a general method to constrain the classical energy of a vibrational mode to be greater than a specifled amount. In particular, zero-point energy constraints can be applied with this method to (zero-order) vibrational modes of a polyatomic system or complex. A demonstration of the method is made for a model two-mode Henon–Heiles Hamiltonian.

The quantum Henon–Heiles problem with Coriolis coupling: A comparison of algebraic and exact results

The results of an algebraically computed double Van Vleck perturbation theory are reported for combined anharmonic and Coriolis perturbations to a degenerate harmonic oscillator. The results to sixth order in the anharmonic coupling and comparable Coriolis coupling are in excellent agreement with exact calculations for systems with anharmonic splittings of up to 5% to 10% of the vibrational spacing. Particular care is required in handling Fermi resonance interactions in the algebraic computations.

Classical and quantal pseudoergodic regions of the Henon–Heiles system

This work describes an approximate and temporary form of ergodicity, called pseudoergodicity. This is a classical phenomenon that is of particular interest since it has a straightforward quantum analog and can be expected to influence the dynamics of a quantum system more directly than rigorous ergodicity. A procedure is described for locating phase space regions where classical motion is pseudoergodic (pe) and this technique is applied to the Henon–Heiles system excited to near its escape energy. Certain pe zones containing chaotic trajectories appear to coincide with the vague tori of Reinhardt and co-workers while other such zones have less familiar forms. When the Henon–Heiles system is observed for long times, a single pe region becomes larger than others, thus marking incipient ergodicity on much of the energy shell. The classical pe regions are compared to quantum pe zones, calculated by a slight modification of a technique presented in an earlier paper. It is found that certain classical pe regions have close quantum analogs while others do not, for reasons probably related to the short time scale for quantum-classical correspondence. Among the classical regions that do not have quantum analogs is the aforementioned dominant pe zone. The implication is that the present quantum system does not display behavior similar to the truly large-scale ergodicity that occurs in the classical system.

Characteristics of power spectra for regular and chaotic systems

Power spectra for chaotic and quasiperiodic systems are analyzed in detail. Differences in convergence with increasing T are carefully examined and a number of important results are emphasized. First, we demonstrate that a finite time power spectrum is incapable, in principle, of providing an unambiguous distinction between a grassy power spectrum associated with a chaotic system and a singular power spectrum associated with quasiperiodic motion. By contrast the statistics of power spectra at fixed energy are shown to provide an adequate method for making such a distinction. Second we show two routes to linking the power spectrum to the physically meaningful spectral density. One route results from a correct analysis of the Wiener–Khinchin theorem wherein the power spectrum and spectral density are shown to be related through a weak limit over frequencies. The second route emerges from a study of the asymptotic behavior of averages over chaotic power spectrum leading to a method for extracting, via extrapolation to the long time limit, the smooth spectral density from the grassy power spectra associated with chaotic behavior. Computations on the cat map, and Henon–Heiles and NaClK Hamiltonians are provided throughout.

Local ergodicity as a probe for chaos in quantum systems: Application to the Henon–Heiles system

Classical chaos is usually accompanied by ‘‘local’’ ergodicity—ergodic behavior in regions of phase space that are generally smaller than, but of the same dimensionality as, the energy surface. If there are strong bottlenecks in phase space impeding the relaxation to statistical equilibrium in the full region of ergodicity, it is often possible to define an approximate form of ergodic behavior in a smaller subregion where relaxation occurs temporarily but quickly. Such ‘‘pseudoergodicity’’ is also a symptom of chaos. We use the presence of quantum behavior that mimics local ergodicity and pseudoergodicity as a probe for the influence of classical chaos on quantum dynamics. We show that the quantum analog of a pseudoergodic region in phase space is generally formed by superpositions of energy eigenstates. The superposition states spanning a given pseudoergodic zone have similar expectation values and small off-diagonal elements to other states with similar energy for a certain class of operators. We perform calculations which identify the ergodic regions for two versions of the quantum Henon–Heiles system. For the ‘‘more classical’’ of these systems we find good agreement between the proportion of quantum states in pseudoergodic zones and the proportion of classical phase space occupied by chaotic trajectories. We also find that the quantum pseudoergodic regions can be identified with classical vague tori of the precessing and librating types. Different ergodic regions are separated from one another by what appear to be the quantum analogs of the precessor–librator separatrix and other classical bottlenecks. For the ‘‘less classical’’ of the systems, we find that classical chaos is reflected in the quantum dynamics to a smaller, but still noticeable, degree.

Time-independent adiabatic switching in quantum mechanics

A time-independent quantum mechanical adiabatic switching algorithm is presented. This algorithm, which is based upon Born’s quantum mechanical adiabatic theorem, is used to calculate the energy levels and eigenstates of the Henon–Heiles Hamiltonian system. The relationship between this algorithm and similar classical algorithms are discussed with particular attention focused upon the correspondence between avoided crossings and classical nonlinear resonances.

On understanding the relationship between structure in the potential surface and observables in classical dynamics: A functional sensitivity analysis approach

The relationship between structure in the potential surface and classical mechanical observables is examined by means of functional sensitivity analysis. Functional sensitivities provide maps of the potential surface, highlighting those regions that play the greatest role in determining the behavior of observables. A set of differential equations for the sensitivities of the trajectory components are derived. These are then solved using a Green’s function method. It is found that the sensitivities become singular at the trajectory turning points with the singularities going as η−3/2, with η being the distance from the nearest turning point. The sensitivities are zero outside of the energetically and dynamically allowed region of phase space. A second set of equations is derived from which the sensitivities of observables can be directly calculated. An adjoint Green’s function technique is employed, providing an efficient method for numerically calculating these quantities. Sensitivity maps are presented for a simple collinear atom–diatom inelastic scattering problem and for two Henon–Heiles type Hamiltonians modeling intramolecular processes. It is found that the positions of the trajectory caustics in the bound state problem determine regions of the highest potential surface sensitivities. In the scattering problem (which is impulsive, so that ‘‘sticky’’ collisions did not occur), the positions of the turning points of the individual trajectory components determine the regions of high sensitivity. In both cases, these lines of singularities are superimposed on a rich background structure. Most interesting is the appearance of classical interference effects. The interference features in the sensitivity maps occur most noticeably where two or more lines of turning points cross. The important practical motivation for calculating the sensitivities derives from the fact that the potential is a function, implying that any direct attempt to understand how local potential regions affect the behavior of the observables by repeatedly and systematically altering the potential will be prohibitively expensive. The functional sensitivity method enables one to perform this analysis at a fraction of the computational labor required for the direct method.

Semiclassical ergodic properties of the Henon–Heiles system

This paper investigates the applicability of the concepts of classical ergodicity and the results of semiclassical ergodic theory to quantum mechanical systems that are away from the classical limit and that do not necessarily become ergodic as ℏ → 0. To carry out this study, a new quantity, called the ‘‘constancy’’ FA of classical property A, is introduced. This constancy is defined in terms of the autocorrelation function of A and its statistical equilibrium value. It measures, on an absolute scale, the extent to which the dynamics of A behave statistically. The FA for a set of quadratically integrable properties A can be used to define the ‘‘degree of ergodicity’’ of a classical system with respect to this set. This analysis motivates introduction of the quantum analog of FA as a means of judging the applicability of classical ergodicity concepts to quantum mechanical systems. To illustrate these ideas, calculations for the Henon–Heiles system are carried out which compare the quantum and classical analogs of the constancies and of the ‘‘almost microcanonical’’ autocorrelation functions from which they are formed. The results indicate that the quantum and classical systems exhibit similar forms of partial ergodicity at high energy. This conclusion supports the approach introduced here for identifying the quantum implications of classical ergodicity. As a consequence of the good quantum-classical agreement, the partially ergodic nature of the classical behavior is reflected in the distribution of certain quantum-mechanical matrix elements. At lower energy, the applicability of classical ergodicity concepts to this system is more limited due to differences in the quantum and classical dynamics. The kinds of quantum-classical discrepancies that limit the implications of classical ergodicity for quantum systems are identified.

Nonadiabatic interactions of anharmonically coupled oscillators

The validity of the adiabatic approximation applied to systems of two coupled oscillators–typically represented by the Henon–Heiles’ Hamiltonian–is examined. Through explicit analytical estimates of nonadiabatic couplings, it is shown that the adiabatic approximation is appropriate in all cases, except for the case of 1:1 Fermi resonance between the two oscillators. In this case, nonadiabatic couplings are found to be important already at an excitation energy of about two quanta, i.e., at much lower energies than Ec, the critical energy for the classically observed quasiperiodic–chaotic transition. Nonadiabatic interactions are dominantly horizontal at these energies (E<Ec), however, indicating some regularity in the energy spectrum at low excitation, despite the early breakdown of the adiabatic separation.

The quantum normal form and its equivalents

A quantum analog, called the quantum normal form, of the classical Birkhoff–Gustavson normal form is presented. The algebraic relationship between the quantum and Birkhoff–Gustavson normal forms has been established by developing the latter using Lie transforms. It is shown that the Birkhoff–Gustavson normal form can be obtained from the quantum normal form. Using an anharmonic oscillator and a Henon–Heiles system as test cases, the equivalence between the quantum normal form and the Rayleigh–Schrödinger perturbation method is shown. This equivalence provides an algebraic connection between the Birkhoff–Gustavson normal form and the Rayleigh–Schrödinger perturbation approach. The question of Weyl and torus quantizations of the Birkhoff–Gustavson normal form is discussed in the light of the quantum normal form.

Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems

Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie–Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon–Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three-quarters of the dissociation energy).

Bifurcating families of periodic solutions for the generalized Henon–Heiles potential

Generalizations of the Floquet method are developed for investigating periodic solutions in classical systems of coupled oscillators. A new iterative procedure which converges on exact initial conditions for periodic solutions is applied to the generalized Henon–Heiles potential. A stability map for the symmetric mode is generated and families of periodic solutions bifurcating from the symmetric mode at stability/instability boundaries are investigated using the iterative procedure.