Publications

1999
Forrey, R.C., Balakrishnan, N., Dalgarno, A., Haggerty, M.R. & Heller, E.J. Quasiresonant energy transfer in ultracold atom-diatom collisions. Physical review letters 82, 2657 (1999). Publisher's VersionAbstract

Quantum and classical quasiresonant vibration-rotation energy transfer is investigated for ultracold HeH2 collisions. Classical trajectory computations show that extremely strong correlations between Δj and Δv persist at low energies, though the changes themselves are less than one quantum. Quantum computations show that quasiresonant transitions occur in the limit of zero collision energy but that threshold effects become important and that some quasiresonant channels close. The qualitative similarity between classical and quantum results suggests that they share a common mechanism.

1995
Grossmann, F. & Heller, E.J. A semiclassical correlation function approach to barrier tunneling. Chemical physics letters 241, 45–50 (1995). Publisher's VersionAbstract

A time domain approach employing the semiclassical approximation to the quantum mechanical propagator, as applied to Gaussian wavepackets, is used to study the barrier penetration problem. We have observed that qualitative agreement with the exact quantum calculations for the correlation function and the transmission probability can be achieved by considering only classically allowed trajectories. The results slowly tend to the classical step function at the barrier top as a function of the wavepacket center parameter, however. We suggest that a full semiclassical calculation, including nonclassical trajectories with complex energies, would improve the results.

1994
Rost, J.M. & Heller, E.J. Ionization of hydrogen by positron impact near the fragmentation threshold. Physical Review A 49, R4289 (1994). Publisher's VersionAbstract

The semiclassical approximation of Feynman’s path integral is used to calculate the S matrix for the positron-impact ionization of hydrogen. The formulation provides a full scattering amplitude, and more importantly does not require knowledge of the asymptotic three-body Coulomb state in the continuum. In the limit of vanishing excess energy, the results confirm Wannier’s classical model for fragmentation [Phys. Rev. 90, 817 (1953)]. The experimentally observable ratio of fragmentation versus total ionization (including positronium formation) is predicted.

Sepúlveda, M.A. & Heller, E.J. Semiclassical analysis of hierarchical spectra. The Journal of chemical physics 101, 8016–8027 (1994). Publisher's VersionAbstract

The semiclassical techniques developed in the previous paper are applied to the understanding of the hierarchical structure underlying the spectra. This organization, as analyzed by Davis with statistical models, is revealed by continuously changing the energy resolution of the spectra and noting the branching pattern of the peaks. We argue that the greater part of this hierarchical organization can be understood with classical events in the time domain.

Sepúlveda, M.A. & Heller, E.J. Semiclassical calculation and analysis of dynamical systems with mixed phase space. The Journal of chemical physics 101, 8004–8015 (1994). Publisher's VersionAbstract

A semiclassical method for the propagation of arbitrary wave packets in a multidimensional Hamiltonian is presented. The method is shown to be valid for treating Hamiltonian systems whose classical phase space is a combination of chaotic and quasiperiodic motion (mixed dynamics). The propagation can be carried out long enough for the nonlinearities of the system to be important. The nonlinear dynamics is reflected in spectra and correlation functions. We suggest this new semiclassical method can be a tool for analyzing the nonlinear aspects of the vibrational spectra.

1993
Tomsovic, S. & Heller, E.J. Long-time semiclassical dynamics of chaos: The stadium billiard. Physical Review E 47, 282 (1993). Publisher's VersionAbstract

In a recent Letter [Phys. Rev. Lett. 67, 664 (1991)] we found semiclassical propagation to be remarkably accurate in the chaotic stadium billiard long after classical fine structure had developed on a scale much smaller than ħ. We give a complete account of that work and derive an approximate time scale for the validity of the semiclassical approximation as a function of ħ.

Tomsovic, S. & Heller, E.J. Semiclassical construction of chaotic eigenstates. Physical review letters 70, 1405 (1993). Publisher's VersionAbstract

It has been a long-standing problem to understand the eigenfunctions of a system whose classical analog is strongly chaotic. We show that in some cases the eigenfunctions can be constructed by purely semiclassical calculations.

1992
O'Connor, P.W., Tomsovic, S. & Heller, E.J. Accuracy of semiclassical dynamics in the presence of chaos. Journal of statistical physics 68, 131–152 (1992). Publisher's VersionAbstract

We review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.

O'Connor, P.W., Tomsovic, S. & Heller, E.J. Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier. Physica D: Nonlinear Phenomena 55, 340–357 (1992). Publisher's VersionAbstract

We investigate the behavior of the quantum baker's transformation, a system whose classical analogue is completely chaotic, for time scales where the classical mechanics generates phase space structures on a scale smaller than Planck's constant (i.e., past the log time t ≈ ln ħ-1). Surprisingly, we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime.

Heller, E.J. & Tomsovic, S. Time domain approach to semiclassical dynamics: Breaking the log time barrier. Chaos 2, 105 (1992). Publisher's VersionAbstract
Outlines two methods that have been developed for performing the evaluation and demonstrate the quantitative agreement between the quantum and semiclassical dynamics for systems with nonintegrable underlying dynamics. Method for calculating the semiclassical and semiclassical propagation of initial states and correlation functions for nonlinear and chaotic dynamics.
1991
Heller, E.J. Wavepacket dynamics and quantum chaology. Session LII-Chaos and quantum physics (Les Houches, France, 1989) (1991).
Heller, E.J. Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics. The Journal of chemical physics 94, 2723–2729 (1991). Publisher's VersionAbstract

A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time‐dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.

Tomsovic, S. & Heller, E.J. Semiclassical dynamics of chaotic motion: unexpected long-time accuracy. Physical review letters 67, 664 (1991). Publisher's VersionAbstract

Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green’s function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell (h). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.

Gauss, J. & Heller, E.J. A time-dependent semiclassical wavepacket method using a fast Fourier transform (FFT) algorithm. Computer Physics Communications 63, 375–388 (1991). Publisher's VersionAbstract

A new semiclassical propagator based on a local expansion of the potential up the wavepacket is proposed. Formulas for the propagator are derived and the implementation using grid and fast Fourier transform (FFT) methods is discussed. The semiclassical propagator can be improved up to the exact quantum mechanical limit by including anharmonic corrections using a split operator approach. Preliminary applications to the CH3I photodissociation problem show the applicability and accuracy of the proposed method.

1990
Heller, E.J. Theoretical studies of time-of-flight and atom and molecular surface collision. (1990).
Heller, E.J. Mode mixing and chaos induced by potential surface crossings. The Journal of chemical physics 92, 1718–1727 (1990). Publisher's VersionAbstract

The effect of Born–Oppenheimer potential energy surface crossings on energy transfer in polyatomic molecules is investigated, classically and quantum mechanically. The hopping from one energy surface to another is enough to cause classical chaos, and strong mixing of the levels quantum mechanically. The r a t e of classical mixing determines the e x t e n t of quantum mixing, even though classical mixing is complete at long times.

1989
Huber, D., Ling, S., Imre, D.G. & Heller, E.J. Hybrid mechanics. II. J. Chem. Phys. 90, 7317 (1989). Publisher's VersionAbstract

We recently published a new method for the calculation of the time evolution of a wave function. We used an accurate approximate method to calculate the time propagator for a finite time Δt. Numerical calculations showed that this scheme works quite accurately, but that it is not more efficient than conventional methods. In this paper we propose to use a very fast and simple, but less accurate semiclassical method for the calculation of the time propagator. The approximation consists in the replacement of the Hamiltonian by a quadratic approximation around the center of the evolving wave packet called thawed Gaussian dynamics. We show by numerical examples in one and two dimensions that, despite this crude approximation, we achieve nearly the same accuracy as in the foregoing paper, but with an efficiency that is typically more than an order of magnitude better. We further show that the method is able to describe tunneling and long time dynamics (e.g., 1000 vibrational periods).

Henriksen, N.E. & Heller, E.J. Quantum dynamics for vibrational and rotational degrees of freedom using Gaussian wave packets: Application to the three-dimensional photodissociation dynamics of ICN. J. Chem. Phys. 91, 4700 (1989). Publisher's VersionAbstract

We present an approach to quantum dynamics, based entirely on Cartesian coordinates, which covers vibrational as well as rotational motion. The initial state is represented in terms of multidimensional Gaussian wave packets. Rotational adaptation to angular momentum eigenstates is done by using angular momentum projection operators. This gives an initial state represented as a weighted superposition of Gaussians with different average orientation in space. It is shown that the subsequent dynamics can be determined from the dynamics of Gaussians corresponding to just o n e of these orientations. An application to the 3Dphotodissociationdynamics of ICN is presented. All six degrees of freedom which describe the internal motion of the triatomic are included, the only approximation introduced in the present calculation being the thawed Gaussian wave packet approximation for the dynamics. The total absorptionspectrum out of vibrational–rotational eigenstates of ICN as well as fully resolved final product distributions are calculated.

1988
Huber, D., Heller, E.J. & Littlejohn, R.G. Generalized Gaussian wave packet dynamics, Schro¨dinger equation, and stationary phase approximation. J. Chem. Phys. 89, 2003 (1988). Publisher's VersionAbstract

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time‐dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.

Frederick, J.H., Heller, E.J., Ozment, J.L. & Pratt, D.W. Ring torsional dynamics and spectroscopy of benzophenone: A new twist. J. Chem. Phys. 88, 2169 (1988). Publisher's VersionAbstract

The low energy portion of the high resolution S 1S 0fluorescence excitation spectrum of benzophenone recently reported by Holtzclaw and Pratt [J. Chem. Phys. 8 4, 4713 (1986)] is modeled here using a simple two‐degree‐of‐freedom vibrational Hamiltonian. The Hamiltonian features a 1:1 nonlinear resonance between the two low frequency ring torsional modes of the molecule in its S 1 state. Line positions and intensities of the two major spectral progressions are well reproduced using parameters similar to those derived from earlier matrix diagonalizations. The comparison of the theory and experiment results in a determination of the displacement of the S 1surface relative to the ground electronic state along the symmetric torsional coordinate and permits a calculation of the excitation spectra of various isotopically substituted molecules not yet measured in the laboratory. A clear picture of the relationship between the dynamics on the S 1surface and the spectroscopy of benzophenone is revealed by comparing a time domain analysis of the experimental data with wave packet dynamics on the model S 1surface. This comparison provides new insight into energy flow in the isolated molecule and permits a qualitative simulation of the effects of collisional quenching on the fluorescencespectrum. We also discuss, using a classical trajectory analysis, the resonance dynamics of the torsional modes and note the existence of heretofore undetected local modes in the high resolution spectrum.

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