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.     

Barrier scattering with complex-valued quantum trajectories: taxonomy and analysis of isochrones

To facilitate the search for isochrones when using complex-valued trajectory methods for quantum barrier scattering calculations, the structure and shape of isochrones in the complex plane were studied. Isochrone segments were categorized based on their distinguishing features, which are shared by each situation studied: High and low energy wave packets, scattering from both thick and thin Gaussian and Eckart barriers of varying height. The characteristic shape of the isochrone is a trifurcated system: Trajectories that transmit the barrier are launched from the lower branch (T), while the middle and upper branches form the segments for reflected trajectories (F and B). In addition, a model is presented for the curved section of the lower branch (from which transmitted trajectories are launched), and important features of the complex extension of the initial wave packet are identified.     

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.     

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.     

Quantum trajectories for resonant scattering

Time‐dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double‐barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi‐bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001     

Bohmian mechanics with complex action: a new trajectory-based formulation of quantum mechanics

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared-it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification-a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wave packet dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. In fact, the current method is shown to be a rigorous extension of generalized Gaussian wave packet dynamics to give exact quantum mechanics. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10(-7) calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.     

Quantum trajectories in complex space: one-dimensional stationary scattering problems

One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.     

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.     

Kinetics and dynamics of the NH3 + H → NH2 + H2 reaction using transition state methods, quasi-classical trajectories, and quantum-mechanical scattering

On a recent analytical potential energy surface developed by two of the authors, an exhaustive kinetics study, using variational transition state theory with multidimensional tunneling effect, and dynamics study, using both quasi-classical trajectory and full-dimensional quantum scattering methods, was carried out to understand the reactivity of the NH(3) + H → NH(2) + H(2) gas-phase reaction. Initial state-selected time-dependent wave packet calculations using a full-dimensional model were performed, where the total reaction probabilities were calculated for the initial ground vibrational state and for four excited vibrational states of ammonia. Thermal rate constants were calculated for the temperature range 200-2000 K using the three methods and compared with available experimental data. We found that (a) the total reaction probabilities are very small, (b) the symmetric and asymmetric N-H stretch excitations enhance the reactivity, (c) the quantum-mechanical calculated thermal rate constants are about one order of magnitude smaller than the transition state theory results, which reproduce the experimental evidence, and (d) quasi-classical trajectory calculations, which were performed with the main goal of analyzing the influence of the zero-point energy problem on the final dynamics results, reproduce the quantum scattering calculations on the same surface.     

Tunneling time in driven double-well system using entangled molecular dynamics method

We calculate quantum tunneling time in the double-well system without perturbation and with symmetry-breaking driven force using the entangled trajectory molecular dynamics method in the present article. Without perturbation, quantum tunneling time decreased with the increase of the energy, and the different contributions of the barrier traversal time and the intrinsic decay time have been shown. The tunneling time dependence on the amplitude and frequency of the symmetry-breaking driven force are present. In the case of weak driven force, tunneling time has a minimum value in the resonant frequency. For strong driven force, chaos brings a huge change to quantum dynamics, tunneling time significantly becomes short. Finally, we directly show the enhancement of quantum tunneling process by chaotic behavior of entangled trajectories and indicate that it was caused by quantum effect.     

Theoretical study of the complex-forming CH + H2 --> CH2 + H reaction

The complex-forming CH + H2 --> CH2 + H reaction is studied employing a recently developed global potential energy function. The reaction probability in the total angular momentum J = 0 limit is estimated with a four-atom quantum wave packet method and compared with classical trajectory and statistical theory results. The formation of complexes from different reactant internal states is also determined with wave packet calculations. While there is no barrier to reaction along the minimum energy path, we find that there are angular constraints to complex formation. Trajectory-based estimates of the low-pressure rate constants are made and compared with experimental results. We find that zero-point energy violation in the trajectories is a particularly severe problem for this reaction.     

Communication: quantum mechanics without wavefunctions

We present a self-contained formulation of spin-free non-relativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications-theoretical, computational, and interpretational-are discussed.     

Tunneling and quantum localization in chaos‐driven symmetric triple well potential: An approach using quantum theory of motion

For a symmetric triple well potential, driven by the forces associated with the bifurcation diagram of a logistic map, the tunneling and quantum localization are studied using quantum theory of motion and time‐dependent Fourier grid Hamiltonian methods. Detailed analysis reveals that application of only asymmetric or symmetric perturbation results into either quantum localization or over‐barrier transition and no tunneling while application of mixed symmetry perturbation gives either tunneling or over‐barrier transition, depending on temporal nature and initial position of the particle. For bifurcative and chaotic symmetric‐asymmetric perturbation, with truncation of mixed symmetry perturbation, a sudden jump in energy causes a transition from the tunneling phenomenon to the over‐barrier transition. With particle located initially near to either of the minima of the unperturbed well, quantum localization, or over‐barrier transition is observed, depending on types of perturbation used.     

Differential and integral cross sections of the N(2D)+H2-->NH+H reaction from exact quantum and quasi-classical trajectory calculations

Exact quantum mechanical state-to-state differential and integral cross sections and their energy dependence have been calculated on an accurate NH2 potential energy surface (PES), using a newly proposed Chebyshev wave packet method. The NH product is found to have a monotonically decaying vibrational distribution and an inverted rotational distribution. The product angular distributions peak in both forward and backward directions, but with a backward bias. This backward bias is more pronounced than that observed previously on a less accurate PES. Both the differential and integral cross sections oscillate mildly with collision energy, indicating the dominance of short-lived resonances. The quantum mechanical results are compared with those obtained from quasi-classical trajectories. The agreement is generally reasonable and the discrepancies can be attributed to the neglect of quantum effects such as tunneling. Detailed analysis of the trajectories revealed that the backward bias in the differential cross section stems overwhelmingly from the fast insertion component of the reaction, augmented with some flux from the abstraction channel, particularly at high collision energies.     

Exact quantum dynamics of N(2D) + H2 --> NH + H reaction: cross-sections, rate constants, and dependence on reactant rotation

Using an exact Chebyshev wave packet method, initial state-specified (upsilon(i)=0, j(i)=0,2) integral cross-sections and rate constants are obtained for the title reaction on the latest ab initio potential energy surface. Reaction probabilities up to J=29 are dependent on the reactant rotation and show mild oscillations superimposed on a broad background. Due to a barrier in the entrance channel, the cross sections increase with energy with clear thresholds and the rate constants vary with temperature in the Arrhenius form. The calculated canonical rate constant is in good agreement with the experimental measurements. Our results also indicate that the quasiclassical trajectory method underestimates the rate due to the neglect of tunneling, while the quantum statistical approach overestimates because of the short lifetime of the reaction intermediate.     

Volterra inverse scattering series method for one‐dimensional quantum barrier scattering

The Volterra inverse scattering series method is developed to obtain the interaction potential for one‐dimensional quantum barrier scattering problems. The Lippmann–Schwinger equation describing quantum barrier scattering is renormalized from a Fredholm to a Volterra integral equation. Employing the Born–Neumann series solution of the Lippmann–Schwinger Volterra equation and a related expansion of the interaction potential in orders of the data, we derive the Volterra inverse scattering series for the reflection and transmission amplitudes. Each term of the interaction potential is computed using the scattering amplitude and the Volterra Green's function. We do not consider the separate issue of extracting scattering amplitudes from quantum cross sections. The triangular nature of the Volterra Green's function significantly reduces computational effort. The Volterra series is then applied to several one‐dimensional quantum barrier scattering problems. Computational results show that the first few terms in the Volterra series can yield accurate interaction barriers. In addition, the potential barriers are calculated using the Born inverse scattering series based on the Lippmann–Schwinger Fredholm equation with the reflection amplitude. The comparison between the Born and Volterra results demonstrates that the Volterra inverse scattering series can provide a more accurate and more efficient method for determining the interaction potential.     

A semiclassical study of wave packet dynamics in anharmonic potentials

Classical and semiclassical methods are developed to calculate and invert the wave packet motion measured in pump-probe experiments. With classical propagation of the Wigner distribution of the initial wave packet created by the pump pulse, we predict the approximate probe signal with slightly displaced recurrence peaks, and derive a set of first-order canonical perturbation expressions to relate the temporal features of the signal to the characteristics of the potential surface. A reduced dynamics scheme based on the Gaussian assumption leads to the correct center of mass motion but does not describe the evolution of the shape of the wave packet accurately. To incorporate the quantum interference into classical trajectories, we propose a final-value representation semiclassical method, specifically designed for the purpose of computing pump-probe signals, and demonstrate its efficiency and accuracy with a Morse oscillator and two kinetically coupled Morse oscillators. For the case of one-color pump probe, a simple phase-space quantization scheme is devised to reproduce the temporal profile at the left-turning point without actual wave packet propagation, revealing a quantum mechanical perspective of the nearly classical pump-probe signal.     

Barrier Height Effect on Cl+H2(D2) Reaction

Three-dimensional time-dependent quantum wave packet calculation was performed to study the reaction dynamics of Cl+H2(D2) on two potential energy surfaces (CW PESs). The first CW PES is with spin-orbit correction; the second is without spin-orbit correction. The integral cross-section and reaction probability as a function of collision energy are calculated in the collision energy range of 0.1 eV to 1.4 eV. For reaction of Cl with D2, the reaction section with spin-orbit correction has a shift toward the high energy because the barrier height increases. As for the reaction of Cl with H2 at low collision energy, it is more reactive on the PES with spin-orbit correction than on the low barrier height PES without spin-orbit correction, due to the tunnel effect for the reaction of the Cl with H2. When the collision energy is higher than 0.7 eV, the reactivity on the low barrier height PES is larger than that on the high barrier height PES. It is believed that the barrier height plays a very important role in the reactivity of Cl with (H2, D2). For the Cl+H2 reaction the barrier width is also very important because of the tunneling effect.