Gaussian wave packet dynamics and scattering in the interaction picture

We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.

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.

Methods 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 and quantum tunneling in asymmetric double-well systems

This paper concerns an approximation of the expectation values of the position and momentum of the solution to the semiclassical Schrödinger equation with a Gaussian as the initial condition. Of particular interest is the approximation obtained by our symplectic/Hamiltonian formulation of the Gaussian wave packet dynamics that introduces a correction term to the conventional formulation using the classical Hamiltonian system by Hagedorn and others. The main result is a proof that our formulation gives a higher-order approximation than the classical formulation does to the expectation value dynamics under certain conditions on the potential function. Specifically, as the semiclassical parameter \(\varepsilon \) approaches 0, our dynamics gives an \(O(\varepsilon ^{3/2})\) approximation of the expectation value dynamics, whereas the classical one gives an \(O(\varepsilon )\) approximation.

Gaussian wavepacket dynamics in an anharmonic system

The theoretical description of heavy-ion collisions at intermediate beam energies, 10 MeV ≲, E/A ≲ 150 MeV, is still in an unsatisfactory state. Different factors contribute to that situation. Thus, for one, the nucleon excitation energies are low and in that comparable to nucleon localization energies, indicating a likely importance of the quantal effects. With the change in the beam energy in the discussed range, the dynamics changes from that dominated by the mean field to that dominated by collisions (as evidenced in the appearance of the flow balance-energy). As excitation energies grow, they begin to exceed the average binding energies per nucleon and, within the mentioned range of E/A, a massive production of intermediate-mass fragments (IMF) takes place[1]. The production, in terms of IMF multiplicity or total mass that IMF carry, maximizes at E/A ~ 75 MeV. The description of the intermediate and light fragment production is beyond the capability of common single-particle models of collisions[2]. The single-particle models with fluctuating forces[3,4] can describe fragment production, but miss the shell effects and the discreteness of the mass and charge numbers. The involved limitation is recognized once one realizes that, in the very central Au + Au collisions at 100 MeV/nucleon, the probability for a proton to come out from the reaction as a constituent of an a particle is close[5] to 50%. Within the Boltzmann-Langevin model with the fluctuating forces[3,4], the α particle plays no distinguished role. Statistical models[6,7] account for the shell effects, but miss the reaction dynamics. The importance of the dynamics is seen, in particular, in the fact that the collective outward flow energy in the reactions is comparable to the thermal energy.

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.

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

Critical behavior of polar modes in lead phosphate near the ferroelastic phase transition

The resonant charge-exchange process by ion–ion collisions is investigated in nonideal plasmas. The interaction Hamiltonian in nonideal plasmas is given by an pseudopotential model that takes into account the plasma screening effect as well as the nonideal collective effect. Screened atomic states of the target and projectile ions in nonideal plasmas are obtained by the variational principle and perturbational analysis. The semiclassical straight-line trajectory method is used to describe the behavior of the projectile ion. It is found that the resonant charge-exchange probability in nonideal plasmas described by the pseudopotential is smaller than that in ideal plasmas described by the Debye–Hückel potential. The maximum position of the resonant charge-exchange probability, i.e., the position which takes place the resonant change-exchange process, is found to recede from the target ion with an increase of the nonideal effect.

Variational and wave packet perturbation theory methods are implemented to study the effects on photodissociation and scattering cross sections due to nonadiabatic coupling between two electronic potential energy surfaces. The simplest variational method we implemented can handle large coupling strengths between surfaces, but its restriction to one Gaussian wave packet per surface is found to be too severe in some instances. In such situations the wave packet perturbation theory, which coherently superimposes multiple Gaussians on each surface, overcomes the difficulties encountered by the variational method and is seen to give quite accurate photodissociation and Landau–Zener cross sections over a wide energy range, so long as the coupling between surfaces is reasonably small. In this paper we give details of the appropriate computational schemes to adopt for calculating photodissociation and scattering cross sections from propagating wave packets. In addition, we discuss the strengths and weaknesses of Gaussian wave packet methods for studying nonadiabatic effects.

A time dependent wave packet approach to gas–surface scattering is presented. This semiclassical method is based on Gaussian wave functions whose average positions and momenta are those of classical trajectories. The initial states are represented as superpositions of Gaussian wave packets. These wave packets are propagated individually and after the collision, the scattering information is extracted by projecting them onto the final states in a coherent way, according to the initial expansion coefficients. The powerful new approach allows the treatment of surface imperfections as well as the inclusion of more degrees of freedom of the surface or the gas particle. The accuracy of the present semiclassical method is tested by a comparison with exact quantal calculations for He–LiF diffractive scattering and in most cases excellent agreement is found. In addition, the wave packet approach is used to calculate diffraction probabilities at high energies and to examine the interference structure of the scattered particles as a function of the size of the surface from which the scattering occurs.

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.

In this paper we develop a criterion of the applicability for the trajectory‐based semiclassical propagation methods that are exact for Gaussian wavepackets in locally quadratic potentials. The error estimate takes into account a deviation of the wavefunction from the Gaussian form due to the anharmonicity of a potential. The derivation is based on the time‐dependent noncoherent solutions of the Schrödinger equation for a general quadratic potential that form a complete orthonormal time‐dependent basis. The anharmonicity is treated as a time‐dependent cubic perturbation, which is further related to the nonlinearity of a force acting on trajectories. The criterion is expressed in terms of scalar products of the average force, its first moments, and dispersion of the wavefunction and is efficient in the context of dynamics with the Bohmian trajectories. We give an application for the Bohmian dynamics with the approximate quantum potential, for the Gaussian wavepacket dynamics and for the semiclassical initial value representation propagator. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004