Gaussian Wave Packet Dynamics Research Articles

A wavepacket — path integral method for curve-crossing problems: application to resonance Raman spectra and photodissociation cross sections

A multidimensional Gaussian wavepacket dynamics — path integral method (GWD-PI) is utilized to study time evolution on two nonadiabatically coupled anharmonic potential energy surfaces. The proposed technique relies on single surface propagations in which the nuclear coordinate wavepacket hops from one potential surface to the other at selected times. Appropriate sequences of hopping times are considered and the output wavepackets associated with each sequence are summed coherently. Good agreement is obtained between GWD-PI and exact quantum dynamics for a two-dimensional model of photodissociation, even in the case of strong and spatially dependent nonadiabatic coupling.

Semiclassical Dynamics and Quantum Control in Condensed Phases: Application to I2in a Solid Argon Matrix

A novel scheme is developed which allows for the combination of classical sampling techniques and quantum wave packet dynamics to study both the inhomogeneous structural effects and the homogeneous dynamical effects in condensed phases. We utilize this methodology to theoretically investigate quantum control of the vibrational dynamics of a chromophore embedded in a condensed-phase environment. We consider control of the vibrational dynamics on an excited electronic state of I 2 that has been embedded in a low-temperature argon matrix, to compare with the work of Apkarian, Zadoyan, Martens, and co-workers. The high dimensionality of such systems precludes the possibility of an exact quantum treatment. To overcome this difficulty we take a semiclassical approach using Gaussian wave packet dynamics in the weak response regime. We compare the numerical simulation with experimental pump-probe measurements of Zadoyan and Apkarian, and we find reasonable agreement over the short time interval within which we will attempt to control the vibrational dynamics of the system in this work. Our calculations predict that coherent quantum control is indeed possible in this condensed-phase system at sufficiently short times and provide a measure of how its effectiveness falls off with time in comparison with the parallel gas-phase case. Finally, we summarize some of the conclusions about quantum control which may be drawn from this work and our other theoretical studies of quantum control in condensed-phase environments.

A Wavepacket−Path Integral Method for Curve-Crossing Dynamics

A combined wavepacket−path integral method is proposed for computing the time evolution of nuclear coordinate wavepackets on several nonradiatively coupled potential energy surfaces. In order to make the technique useful for multidimensional problems, the method uses the Gaussian wavepacket dynamics (GWD) algorithm (Heller, E. J.; J. Chem. Phys. 1975, 62, 1544) to perform nuclear coordinate wavepacket evolution on successive individual surfaces as prescribed by the path integral formalism. Since the GWD algorithm is not exact for anharmonically coupled potential energy surfaces, this introduces some error into the calculation. However, as long as the GWD algorithm is sufficiently accurate to treat single-surface propagation on all of the relevant potential energy surfaces, the proposed method is expected to yield useful results. A simple illustration is given using a one-dimensional model of a direct photodissociation process involving two nonradiatively coupled electronic states. Good agreement with numerically exact results is obtained, even in the case of strong, and position-dependent, nonradiative coupling.

Spectroscopy of potential barriers: An analytic line-shape formula for broad resonances.

Using a time-dependent formulation of Franck-Condon spectra, we analyze the spectra associated with the transitions into a saddle-point region of an unbound Born-Oppenheimer potential. We find a spectral peak at an energy near the barrier top, which we associate with the existence of a quasi-bound-state. We refer to this state as ``barrier resonance.'' Employing a quadratic approximation to the potential and the quantum dynamics of Gaussian wave packets, we develop a line-shape formula that accurately fits this spectral feature. This line-shape formula corresponds to an infinite set of complex energy poles rather than the usual single-pole approach used in the Breit-Wigner theory. We apply the line-shape formula to barrier resonances on the Eckart potential and in the H+${\mathrm{H}}_{2}$ chemical reaction.

A semi-classical implementation of quantum control using Gaussian wave packet dynamics

Quantum control has previously been implemented using the exact time dependent Schrödinger equation (TDSE). Such calculations are limited by computational difficulty to systems with only a few degrees of freedom. In order to implement quantum control in many-atom systems, with sufficient accuracy and computational speed, we explore here a semi-classical implementation in the weak response regime whereby the TDSE is solved approximately using Gaussian wave packet (GWP) dynamics. We demonstrate by example that, for suitable systems and targets, the GWP representation of the evolving quantum system can be used to accurately calculate globally optimal light fields.

Chaos and low-order corrections to classical mechanics or geometrical optics.

Based on simple first-order quantum corrections to classical equations of motion, which we show to be closely related to Gaussian wave-packet dynamics (GWD) and a time-dependent variational principle (TDVP), we deduce that quantum corrections to classical dynamics should typically become most pronounced when the classical system becomes chaotic. The time duration over which classical dynamics, GWD, or TDVP may provide good approximations is much shorter when the classical dynamics are chaotic. However, for certain situations involving very short laser pulses, these approximations can be very accurate. The same concepts are applicable to paraxial wave optics, which may offer simpler experimental studies of ``quantum chaos'': the distinction between classical and ``quantum'' chaos is in large part the distinction between ray versus wave behavior.

Hamilton‐Jacobi dynamics for the solution of time dependent quantum problems II. Wave packet propagation in two dimensional, nonlinearly coupled oscillators – exact and time‐dependent SCF‐solutions–

AbstractMethods for the approximate numerical integration of the time dependent Schrödinger equation with given initial conditions (the initial wave packet) are presented. The methods are based on the Schrödinger representation of the quantum dynamic system. The quantum dynamic equations are transformed into Hamilton‐Jacobi type equations of motion as they occur in multi particle classical dynamics, i.e. standard techniques in molecular dynamics can be applied for the integration. The dynamics of minimum uncertainty Gaussian wave packets in strongly non‐harmonic, nonlinearly coupled oscillators are studied as examples. The numerically exact solutions are compared to time dependent SCF approximations of the wave packet.

Gaussian wavepacket dynamics in an anharmonic system

An exact quantum solution is obtained for the problem of Gaussian wavepacket motion in an anharmonic potential. Wavepacket evolution is described by means of an autocorrelation function. The solution is expressed via the theta function with time-dependent argument and parameter. A simple physical analogy is proposed for the wavepacket motion. The problem of calculating the width of the wavepacket “revival zones” is considered.

Gaussian wavepacket dynamics and the Landau—Zener model for non-adiabatic transitions

The Landau—Zener model for transitions between two linear diabatic potentials is examined. We derive, in the weak-coupling limit, an expression for the transition probability where the classical trajectory and the constant velocity approximations are abandoned and replaced by quantum dynamics described by a Gaussian wavepacket. A remarkable agreement with the results from the simple Landau—Zener formula is observed.

Multidimensional variational Gaussian wave packet dynamics with application to photodissociation spectroscopy

The McLachlan variational principle for the time-dependent Schrödinger equation is utilized in conjunction with extant localized Guassian wave packet technology to deduce equations of motion for general multidimensional Gaussians. These equations of motion are characterized by the same simplicity as the local quadratic expansion results of Heller [J. Chem. Phys. 62, 1544 (1975)]. However, the resultant variational wave packet evolution is shown to be an improvement over its local quadratic analog as a tool for computing certain photodissociation spectra. Numerical examples drawn from the Beswick–Jortner model of ICN photodissociation [Chem. Phys. 24, 1 (1977)] are presented.

Time-of-flight spectra of a particle scattering from a collinear harmonic lattice at finite temperature

A new formalism is developed for computing the time-of-flight spectrum of a particle scattering from a collinear harmonic lattice prepared at finite temperature. We use a time-domain transcription to construct an S-matrix formalism that can be easily implemented via Gaussian wave packet dynamics. Numerical results are presented for a particle scattered from a lattice containing 100 oscillators at several temperature values.

Resonance enhanced electron stimulated desorption

A theory is presented which accounts for ‘‘giant’’ enhancements in electron stimulated desorption (ESD) yields from adsorbate-covered surfaces if the incident electrons become trapped in a shape or Feshbach resonance associated with the adsorbate. The resulting temporary negative ion is displaced inwards towards the surface as a result of the force provided by the image screening charge. Upon reneutralization, the ‘‘desorbate’’ can be returned high on the dissociative repulsive wall of the neutral-surface potential curve. This process has been modeled within the context of semiclassical Gaussian wave packet dynamics. Recent observations of such giant enhancements in the ESD yields for the system O(a)/Pd(111) are explained in terms of this model, and an atomic physics basis for the resonance in atomic oxygen is proposed.

The hamilton formalism of gaussian wave-packet dynamics

Gaussian wave packets have been used extensively to describe short-time localised processes in small molecular systems. We present a formalism to compute approximately the quantal time evolution of Gaussian wave packets for N-body systems. The wave-packet dynamics are obtained by invoking the time-dependent variational principle in Lagrangian form. It is then shown that the parametrisation of a Gaussian wave packet can be chosen so as to yield a Hamilton-like set of evolution equations for the wave-packet parameters. The resulting formalism proves to be energy conserving. For those parts of the system for which a classical description is appropriate, a localisation limit recovers the classical Hamilton equations. Also a mixed classical-quantal description of an N-body system is feasible.

Semiclassical Gaussian wave packet dynamics for collinear reactive scattering

The dynamics of collinear exchange reactions is treated within a semiclassical Gaussian wave packet (GWP) description. Two extensions to the previous methodology of GWP dynamics are presented. The first involves the evaluation of the interaction picture wave function propagators directly via the GWP solution to the time-dependent Schrödinger equation. The second involves use of a sum of GWPs to represent the initial translational plane wave on a finite interval. As usual, the vibrational wave function is also represented as a sum of GWPs, where the expansion coefficients are chosen to yield the desired initial vibrational state. The limitations of the previous GWP dynamical methods as applied to the reactive scattering problem are pointed out and the solutions provided by the present extensions are discussed. Results for the H+H2 collinear exchange reaction using the Porter–Karplus II potential energy surface are shown.

Generalized Gaussian wave packet dynamics, Schrödinger equation, and stationary phase approximation

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.

Gaussian wave packet dynamics and scattering in the interaction picture

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.

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.

The exact eigenfunctions and eigenvalues of a particle in a box obtained using Gaussian wavepacket dynamics

Exact eigenfunctions for a particle in a box are obtained using Gaussian wavepacket dynamics. The eigenfunctions are obtained by propagating, without approximation, an infinite set of Gaussian wavepackets that collectively satisfy the boundary conditions of the problem, being coherent states appropriate to this problem. The method of images is applied to enforce these boundary conditions. This technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space, regardless of the dimension of the region. Also, it is shown that the use of frozen Gaussians along with the De Leon-Heller spectral quantisation method gives the exact solution for the one-dimensional problem as well as for the above multi-dimensional problems, provided the components of the momentum of the wavepackets are chosen at random.

The adiabatic switching of semiclassical wave functions

The behavior of semiclassical wave functions during adiabatic deformations of the Hamiltonian is studied. A practical technique based on Gaussian wave packet dynamics is presented which allows a semiclassical eigenstate for one Hamiltonian to be adiabatically switched into an eigenstate for a different Hamiltonian. The validity of this technique is established by studying the properties of the Hamilton–Jacobi equation during adiabatic processes. The method is shown to yield good results for several one- and two-dimensional model problems. It is shown that converged semiclassical eigenstates can be obtained even if the trajectories have become chaotic by the end of the switching process.

Gaussian wave-packet dynamics

The time-dependent variational principle is applied to the manifold of Gaussian wave packets. The translational and spreading degrees of freedom are treated equally. Comparison is made with frozen Gaussian dynamics and with the exact Schrodinger dynamics.