Eric J. Heller

Cellular dynamics : a new semiclassical approach to time-dependent quantum mechanics

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.

Generalized Gaussian wave packet dynamics

We propose an extension of the semiclassical Gaussian wave packet dynamics to eliminate the three main restrictions of this method. The first restriction is that the wave packet is forced to remain Gaussian. This is correct only for quadratic, linear, or constant potentials. The second restriction is that the method is, in general, not able to treat most classically forbidden processes. The third restriction is that the norm is conserved only for Gaussian wave packets. For a superposition of Gaussians this is no longer true. We can eliminate these restrictions by an extension of the method into complex phase space, keeping time real.

CH2I2 photodissociation: Dynamical modeling

A full quantum mechanical calculation is carried out on the first excited state of CH2I2 to model the absorption and emission spectra and examine the photodissociation dynamics from a time dependent point of view. The dissociation at 355 nm is direct in the sense that the wave packet does not revisit the Franck–Condon region. The initial motion of the excited molecule is mainly along the CI2 symmetric stretch coordinate while simultaneously spreading in the antisymmetric stretch coordinate. The molecule then dissociates along a C–I ‘‘local’’ mode; no I2 can be formed in this energy region. Vibrationally hot CH2I radical in the C–I mode is predicted. The model is in good agreement with available experimental results. A simple and intuitive method is presented to construct model potential energy surfaces for two chromophore systems from the potential energy surface and information known for the corresponding single chromophore. CH3I and CH2I2 are used as numerical illustrations.A full quantum mechanical calculation is carried out on the first excited state of CH2I2 to model the absorption and emission spectra and examine the photodissociation dynamics from a time dependent point of view. The dissociation at 355 nm is direct in the sense that the wave packet does not revisit the Franck–Condon region. The initial motion of the excited molecule is mainly along the CI2 symmetric stretch coordinate while simultaneously spreading in the antisymmetric stretch coordinate. The molecule then dissociates along a C–I ‘‘local’’ mode; no I2 can be formed in this energy region. Vibrationally hot CH2I radical in the C–I mode is predicted. The model is in good agreement with available experimental results. A simple and intuitive method is presented to construct model potential energy surfaces for two chromophore systems from the potential energy surface and information known for the corresponding single chromophore. CH3I and CH2I2 are used as numerical illustrations.

Hybrid mechanics: A combination of classical and quantum mechanics

Because classical mechanics is so much easier to handle than quantum mechanics, the time evolution of wave functions for molecular dynamics is often calculated using semiclassical methods. The errors of such methods grow, in general, faster than linearly with time, although they may be quite small for small, but finite times. We therefore propose to use a semiclassical method to calculate the quantum mechanical time propagator for a finite time step (say 1/10 of a vibrational period) and to use this propagator and quantum mechanics for longer times. To describe the quantum time propagator we use a basis set that can describe regions in phase space that are not necessarily rectangular, but can have any shape, that will become important in applications to higher dimensions. We give numerical examples to demonstrate the accuracy of the method.

Ring torsional dynamics and spectroscopy of benzophenone: A new twist

The low energy portion of the high resolution S1←S0 fluorescence excitation spectrum of benzophenone recently reported by Holtzclaw and Pratt [J. Chem. Phys. 84, 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 S1 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 S1 surface 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 S1 surface and the spectroscopy of benzophenone is revealed by comparing a...

Hybrid mechanics. II

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).

Quantum dynamics for vibrational and rotational degrees of freedom using Gaussian wave packets: Application to the three-dimensional photodissociation dynamics of ICN

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 one of these orientations. An application to the 3D photodissociation dynamics 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 absorption spectrum out of vibrational–rotational eigenstates of ICN as well as fully resolved final product dist...

Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier

We investigate the behavior of the quantum bakers 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 Plancks 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.

The eigenfunctions of classically chaotic systems

We study the quantum mechanics of a Hamiltonian system which is classically chaotic: the stadium billiard. We have examined many of the eigenstates of the stadium, up to about the 10 000th. Complex periodic orbits play an active role in shaping the eigenstates.

Gaussian wave packet dynamics and scattering in the interaction picture

Abstract The present work demonstrates that the number of scattering problems which can be handled accurately by the semiclassical (thawed) Gaussian method can be increased substantially if the calculations are done within the framework of the interaction picture. Examples for photodissociation of three atomic molecules are presented.