Trajectory approach to dissipative quantum phase space dynamics: Application to barrier scattering

The Caldeira-Leggett master equation, expressed in Lindblad form, has been used in the numerical study of the effect of a thermal environment on the dynamics of the scattering of a wave packet from a repulsive Eckart barrier. The dynamics are studied in terms of phase space trajectories associated with the distribution function, W(q,p,t). The equations of motion for the trajectories include quantum terms that introduce nonlocality into the motion, which imply that an ensemble of correlated trajectories needs to be propagated. However, use of the derivative propagation method (DPM) allows each trajectory to be propagated individually. This is achieved by deriving equations of motion for the partial derivatives of W(q,p,t) that appear in the master equation. The effects of dissipation on the trajectories are studied and results are shown for the transmission probability. On short time scales, decoherence is demonstrated by a swelling of trajectories into momentum space. For a nondissipative system, a comparison is made of the DPM with the "exact" transmission probability calculated from a fixed grid calculation.     

Quantum hydrodynamics: capturing a reactive scattering resonance

The hydrodynamic equations of motion associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a meshless method based upon a moving least-squares approach. An arbitrary Lagrangian-Eulerian frame of reference and a regridding algorithm which adds and deletes computational points are used to maintain a uniform and nearly constant interparticle spacing. The methodology also uses averaged fields to maintain unitary time evolution. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. A new and more robust artificial viscosity algorithm is presented which gives accurate scattering results and is capable of capturing quantum resonances. The methodology is applied to a one-dimensional model chemical reaction that is known to exhibit a quantum resonance. The correlation function approach is used to compute the reactive scattering matrix, reaction probability, and time delay as a function of energy. Excellent agreement is obtained between the scattering results based upon the quantum hydrodynamic approach and those based upon standard quantum mechanics. This is the first clear demonstration of the ability of moving grid approaches to accurately and robustly reproduce resonance structures in a scattering system.     

Multidimensional quantum trajectories: applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.     

Higher-order split operator schemes for solving the Schrödinger equation in the time-dependent wave packet method: applications to triatomic reactive scattering calculations

The efficiency of the numerical propagators for solving the time-dependent Schr?dinger equation in the wave packet approach to reactive scattering is of vital importance. In this Perspective, we first briefly review the propagators used in quantum reactive scattering calculations and their applications to triatomic reactions. Then we present a detailed comparison of about thirty higher-order split operator propagators for solving the Schr?dinger equation with their applications to the wave packet evolution within a one-dimensional Morse potential, and the total reaction probability calculations for the H + HD, H + NH, H + O(2), and F + HD reactions. These four triatomic reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these higher-order propagation methods for describing reactive scattering dynamics. Our calculations reveal that the most often used second-order split operator method is typically more efficient for a direct reaction, particularly for those involving flat potential energy surfaces. However, the optimal higher-order split operator methods are more suitable for a reaction with resonances and intermediate complexes or a reaction experiencing potential energy surface with fluctuations of considerable amplitude. Three 4th-order and one 6th-order split operator methods, which are most efficient for solving reactive scattering in various conditions among the tested ones, are recommended for general applications. In addition, a brief discussion on the relative performance between the Chebyshev real wave packet method and the split operator method is given. The results in this Perspective are expected to stimulate more applications of (high-order) split operators to the quantum reactive scattering calculation and other related problems.     

Direct quantum mechanical calculation of the F + H2 → HF + H thermal rate constant

Accurate full-dimensional quantum mechanical thermal rate constant values have been calculated for the F+H₂→HF+H reaction on the Stark–Werner ab initio potential energy surface. These calculations are based on a flux correlation functions and employ a rigorous statistical sampling scheme to account for the overall rotation and the MCTDH scheme for the wave packet propagation. Our results shed some light on discrepancies on the thermal rate found for previous flux correlation based calculations with respect to accurate reactive scattering results. The resonance pattern of the all-J cumulative reaction probability is analyzed in terms of the partial wave contributions.     

Semiclassical wave packet study of ozone forming reaction

We have applied the semiclassical wave packet method (SWP) to calculate energies and lifetimes of the metastable states (scattering resonances) in a simplified model of the ozone forming reaction. All values of the total angular momentum up to J=50 were analyzed. The results are compared with numerically exact quantum mechanical wave packet propagation and with results of the time-independent WKB method. The wave functions for the metastable states in the region over the well are reproduced very accurately by the SWP; in the classically forbidden region and outside of the centrifugal barrier, the SWP wave functions are qualitatively correct. Prony's method was used to extract energies and lifetimes from the autocorrelation functions. Energies of the metastable states obtained using the SWP method are accurate to within 0.1 and 2 cm(-1) for under-the-barrier and over-the-barrier states, respectively. The SWP lifetimes in the range of 0.5相似文献   

Three-dimensional time-dependent wave-packet calculations of OBrO absorption spectra

The absorption spectra of the C(2A2) <-- X(2B1) transition of the OBrO molecule are calculated using three-dimensional time-dependent wave-packet method in Radau coordinates for a total angular momentum J=0. The wave packet is propagated using the split operator technique associated with fast Fourier transform. Employing the basis functions obtained by one-dimensional Fourier grid Hamiltonian method, the initial wave packet is calculated directly on the three-dimensional Fourier grid. The numerical model is characterized by simplicity and efficiency. The ab initio potential surfaces for the C(2A2) and X(2B1) states are used in the calculation. The calculated absorption spectra of the C(2A2) <-- X(2B1) transition of OBrO molecule agree well with the experimental results.     

Semiclassical wave packet study of anomalous isotope effect in ozone formation

We applied the semiclassical initial value representation method to calculate energies, lifetimes, and wave functions of scattering resonances in a two-dimensional potential for O+O2 collision. Such scattering states represent the metastable O3* species and play a central role in the process of ozone formation. Autocorrelation functions for scattering states were computed and then analyzed using the Prony method, which permits one to extract accurate energies and widths of the resonances. We found that the results of the semiclassical wave packet propagation agree well with fully quantum results. The focus was on the 16O16O18O isotopomer and the anomalous isotope effect associated with formation of this molecule, either through the 16O16O+18O or the 16O+16O18O channels. An interesting correlation between the local vibration mode character of the metastable states and their lifetimes was observed and explained. New insight is obtained into the mechanism by which the long-lived resonances in the delta zero-point energy part of spectrum produce the anomalously large isotope effect.     

Quantum wave packet study of the insertion reaction S+H2

S(¹D)+H₂→SH+H reaction at zero total angular momentum has been studied by using a time-dependent quantum real wave packet method. State-to-state and state-to-all reactive scattering probabilities for a broad range of energy are calculated. The probabilities show many sharp peaks that ascribed to reactive scattering resonances. The density plots of the wave function shows that the reaction presents a pure insertion pathway.     

Scattered wave packet formalism for the energy-resolved reaction probability

The scattered wave packet formalism developed for a quantum subsystem interacting with reservoirs through open boundaries is utilized to calculate the energy-resolved transmission probability. The total wave function is split into incident and scattered components. Markovian outgoing wave boundary conditions are imposed on the scattered or total wave function by the polynomial method. The wave packet correlation function approach is employed to compute the energy-resolved transmission probability for a one-dimensional potential barrier and a one-dimensional model chemical reaction exhibiting a quantum resonance. Accurate results demonstrate that this formalism can significantly reduce the number of grid points required in a dynamical calculation for the reaction probability.     

Enhanced creation of dispersive monolayer phonons in XePt(111) by inelastic helium atom scattering at low energies

Conditions likely to lead to enhanced inelastic atomic scattering that creates shear horizontal (SH) and longitudinal acoustic (LA) monolayer phonons are identified, specifically examining the inelastic scattering of (4)He atoms by a monolayer solid of XePt(111) at incident energies of 2-25 meV. There is strong inelastic scattering for both dispersive phonon branches (SH and LA) of the monolayer at incident energies below 8 meV. Several improvements enable more complete wave packet calculations of the inelastic scattering than in previous work. Long propagation times are made feasible by adding an absorbing potential at large distance. The times now extend to beyond 100 ps and enable a clarification of processes involving transient trapping of the He atoms. The wave packet is made more monochromatic by significantly increasing the spatial width of the initial Gaussian shape. The narrower energy distribution in the incident beam then enables a demonstration of strong energy dependence of the scattering over a scale of less than 0.3 meV.     

Wave Packet Approach to Adiabatic and Nonadiabatic Dynamics of Cold Inelastic Scatterings

Due to the extremely large de Broglie wavelength of cold molecules, cold inelastic scattering is always characterized by the time-independent close-coupling (TICC) method. However, the TICC method is difficult to apply to collisions of large molecular systems. Here, we present a new strategy for characterizing cold inelastic scattering using wave packet (WP) method. In order to deal with the long de Broglie wavelength of cold molecules, the total wave function is divided into interaction, asymptotic and long-range regions (IALR). The three regions use different numbers of ro-vibrational basis functions, especially the long-range region, which uses only one function corresponding to the initial ro-vibrational state. Thus, a very large grid range can be used to characterize long de Broglie wavelengths in scattering coordinates. Due to its better numerical scaling law, the IALR-WP method has great potential in studying the inelastic scatterings of larger collision systems at cold and ultracold regimes.     

Angle dependent total cross sections and the optical theorem

Cross sections are either represented by generalized asymptotical partial wave expansions or obtained as a spherical average of an appropriate differential cross section. In these cases it is usually assumed that the total scattering cross section, as a property of a scattering object, does not depend on the incident angles. This viewpoint is supported by common knowledge in connection with low energy scattering. However this unconscious belief is not always correct. In the present paper we will show that a non-spherical scatterer may exhibit strong dependence on the incident direction. To do this we will represent the scattering data of the most general potential, separable in ellipsoidal coordinates, in perturbed ellipsoidal (Lamé) wave functions. These functions arise when variables in the Schr?dinger equation are separated in an ellipsoidal coordinate system. The Lamé wave functions are analogous to spherical- and Bessel functions in the case of spherical symmetry. We will expand the total scattering cross section and derive the optical theorem explicitly demonstrating the incident angle dependence for such a class of potentials. As an illustration we will present and display some calculations of the total cross section versus incident direction. Unexpected behavior will be discussed and explained. We also use results from classical acoustic scattering by a triaxial ellipsoid. The general character of the ellipsoidal coordinate system is emphasized.     

Moving boundary truncated grid method: Multidimensional quantum dynamics

The moving boundary truncated grid (TG) method is used to study wave packet dynamics of multidimensional quantum systems. As time evolves, appropriate Eulerian grid points required for propagating a wave packet are activated and deactivated with no advance information about the dynamics. This method is applied to the Henon-Heiles potential and wave packet barrier scattering in two, three, and four dimensions. Computational results demonstrate that the TG method not only leads to a great reduction in the number of grid points needed to perform accurate calculations but also is computationally more efficient than the full grid calculations.     

Quantum hydrodynamics: application to N-dimensional reactive scattering

The quantum hydrodynamic equations associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a new methodology which gives an accurate, unitary, and stable propagation of a time dependent quantum wave packet [B. K. Kendrick, J. Chem. Phys. 119, 5805 (2003)]. The methodology is applied to an N-dimensional model chemical reaction with an activation barrier. A parallel version of the methodology is presented which is designed to run on massively parallel supercomputers. The computational scaling properties of the parallel code are investigated both as a function of the number of processors and the dimension N. A decoupling scheme is introduced which decouples the multidimensional quantum hydrodynamic equations into a set of uncoupled one-dimensional problems. The decoupling scheme dramatically reduces the computation time and is highly parallelizable. Furthermore, the computation time is shown to scale linearly with respect to the dimension N=2,...,100.     

Analysis of barrier scattering with real and complex quantum trajectories

In this study, an analysis of the one-dimensional Eckart and Gaussian barrier scattering problems is undertaken using approximate quantum trajectories. Individual quantum trajectories are computed using the derivative propagation method (DPM). Both real-valued and complex-valued DPM quantum trajectories are employed. Of interest are the deep tunneling and the higher energy barrier scattering problems in cases in which the scattering barrier is "thick" by comparison to the width of the initial wave packet. For higher energy scattering problems, it is found that real-valued DPM trajectories very accurately reproduce the transmitted probability densities at low orders when compared to large fixed-grid calculations. However, higher orders must be introduced to obtain good probabilities for deep tunneling problems. Complex-valued DPM is found to accurately reproduce transmitted probability densities at low order for both the deep tunneling and the higher energy scattering problems. Of particular note, complex-classical trajectories are found to very nearly give the exact result for the deep barrier tunneling scattering problem, and the complex DPM converges well at high orders for these thick barrier scattering problems. A variety of analyses are performed to elucidate the dynamics of complex-valued DPM trajectories. The complex-extended barrier potentials are examined in detail, including an analysis of the complex force. Of particular interest are initial conditions for complex-valued DPM trajectories known as isochrones. All trajectories launched from an isochrone arrive on the real axis on the transmitted side of the barrier at the same time. The computation and properties of isochrones as well as the behavior of the initial wave packet in the complex plane are also examined.     

Evanescent Wave Light Scattering A Fusion of the Evanescent Wave and Light Scattering Techniques to the Study of Colloids and Polymers Near the Interface

The evanescent wave light scattering technique, which is produced by a fusion of the evanescent wave technique and light scattering technique, is a very powerful and useful tool for investigation of colloidal particles and polymers near the surface and interfaces. We have developed two kinds of evanescent wave light scattering apparatuses. One is the evanescent wave dynamic light scattering (EVDLS) technique and the other is the evanescent wave light scattering microscope (EVLSM). By EVDLS, the diffusion behavior of a colloidal particle near the interface can be extracted quantitatively as a function of the distance from the interface. The diffusion coefficient was smaller than those for particles in bulk, reflecting electrostatic and hydrodynamic interactions. By EVLSM, the interaction potential profile between a colloidal particle and the surface in dispersion can be evaluated directly. EVLSM will play an important role in colloidal interaction studies, especially at a low ionic strength. It is also pointed out that a particle dynamics study is also possible by the EVLSM technique. A new field will be developed in colloid science and polymer science by application of the evanescent wave light scattering technique, i. e. a fusion of the evanescent light and a light scattering techniques.     

Colloidal dynamics near a wall studied by evanescent wave light scattering: experimental and theoretical improvements and methodological limitations

The dynamics of colloidal spheres near to a wall is studied with an evanescent wave scattering setup that allows for an independent variation of the components of the scattering wave vector normal and parallel to the wall. The correlation functions obtained with this novel instrumentation are interpreted on the basis of an expression for their short time behavior that includes hydrodynamic interactions between the colloidal spheres and the wall. The combination of the evanescent wave scattering setup and the exact expression for the short time behavior of correlation functions allows for an unambiguous measurement of the particle mobility parallel and normal to the wall by means of light scattering. It is possible to measure the viscous wall drag effect on the dynamics of particles with radii as small as 27 nm, where, however, the method reaches its limits due to the low scattering intensities of such small particles.