Computational Investigation of Wave Packet Scattering in the Complex Plane: Propagation on a Grid.

The time-dependent scattering of a wave packet from a Gaussian barrier is investigated computationally in the complex z-plane. The initial wave packet and the potential energy are obtained through analytic continuation from functions specified on the real-axis. The wave packet is then propagated on the two-dimensional grid. For a low initial wave packet energy, the time evolution is followed by plotting the following functions: |ψ(z,t)|, real(ψ(z,t)), and the quantum momentum function (QMF), p(z,t). In the reflected packet, an important role is played by ripples (quasi-nodes) forming above the real axis. As these quasi-nodes move down across the real axis, they are 'detected' as 'interference oscillations' in the density. In contrast, the component of the packet below the real axis makes a significant contribution to the transmitted packet. Vector maps of the QMF show hyperbolic flow around quasi-nodes and counterclockwise circular flow around transient stagnation points, where the QMF vanishes. However, when the Pólya vector field (defined by P(z,t) = p*(z,t)) is plotted, circular counterclockwise flow is obtained near the quasi-nodes. The real and imaginary parts of the quantum action function S(z,t) are plotted and the vorticity, defined by the curl of the Pólya field, is used to pinpoint regions of nonanalyticity in the QMF.

Computational Investigation of Wave Packet Scattering in the Complex Plane: Dynamics of Exact Quantum Trajectories.

In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.

Complex trajectories sans isochrones: quantum barrier scattering with rectilinear constant velocity trajectories.

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.

Quantum trajectories in complex phase space: multidimensional barrier transmission.

The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an "equal arrival time surface," defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.

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.

Degradation kinetics of VX on concrete by secondary ion mass spectrometry.

At trace coverages on concrete surfaces, the nerve agent VX (O-ethyl S-2-diisopropylaminoethyl methyl phosphonothiolate) degrades by cleavage of the P-S and S-C bonds, as revealed by periodic secondary ion mass spectrometry (SIMS). The observed kinetics were (pseudo-) first-order, with a half-life of 2-3 h at room temperature. The rate increased with surface pH and temperature, with an apparent second-order constant of k(OH) = 0.64 M(-1) min(-1) at 25 degrees C and an activation energy of 50-60 kJ mol(-1). These values are consistent with a degradation mechanism of alkaline hydrolysis within the adventitious water film on the concrete surface. Degradation of bulk VX on concrete would proceed more slowly.

Hydrolysis of VX on concrete: rate of degradation by direct surface interrogation using an ion trap secondary ion mass spectrometer.

The nerve agent VX (O-ethyl S-2-diisopropylaminoethyl methylphosphonothiolate) is lethal at very low levels of exposure, which can occur by dermal contact with contaminated surfaces. Hence, behavior of VX in contact with common urban or industrial surfaces is a subject of acute interest. In the present study, VX was found to undergo complete degradation when in contact with concrete surfaces. The degradation was directly interrogated at submonolayer concentrations by periodically performing secondary ion mass spectrometry (SIMS) analyses after exposure of the concrete to VX. The abundance of the [VX + H]+ ion in the SIMS spectra was observed to decrease in an exponential fashion, consistent with first-order or pseudo-first-order behavior. This phenomenon enabled the rate constant to be determined at 0.005 min(-1) at 25 degrees C, which corresponds to a half-life of about 3 h on the concrete surface. The decrease in [VX + H]+ was accompanied by an increase in the abundance of the principal degradation product diisopropylaminoethanethiol (DESH), which arises by cleavage of the P-S bond. Degradation to form DESH is accompanied by the formation of ethyl methylphosphonic acid, which is observable only in the negative ion spectrum. A second degradation product was also implicated, which corresponded to a diisopropylvinylamine isomer (perhaps N,N-diisopropyl aziridinium) that arose via cleavage of the S-C bond. No evidence was observed for the formation of the toxic S-2-diisopropylaminoethyl methylphosphonothioic acid. The degradation rate constants were measured at four different temperatures (24-50 degrees C), which resulted in a linear Arrhenius relationship and an activation energy of 52 kJ mol(-1). This value agrees with previous values observed for VX hydrolysis in alkaline solutions, which suggests that the degradation of submonolayer VX is dominated by alkaline hydrolysis within the adventitious water film on the concrete surface.