A new method for solving the quantum hydrodynamic equations of motion: application to two-dimensional reactive scattering

The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. 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. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method.     

Time-dependent wave packet propagation using quantum hydrodynamics

A new approach for propagating time-dependent quantum wave packets is presented based on the direct numerical solution of the quantum hydrodynamic equations of motion associated with the de Broglie–Bohm formulation of quantum mechanics. A generalized iterative finite difference method (IFDM) is used to solve the resulting set of non-linear coupled equations. The IFDM is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a “smart” Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is generalized to treat higher-dimensional problems and anharmonic potentials. The method is applied to a one-dimensional Gaussian wave packet scattering from an Eckart barrier, a one-dimensional Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. The 2D scattering results represent the first successful application of an accurate direct numerical solution of the quantum hydrodynamic equations to an anharmonic potential energy surface.     

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.     

Reconstructing interference fringes in slit experiments by complex quantum trajectories

There are external and internal representations for a quantum state Ψ. External representation is commonly adopted in the standard quantum mechanics by exploiting probability density function Ψ*Ψ to explain the observed interference fringes in slit experiments. On the other hand, in quantum Hamilton mechanics, the quantum state Ψ has a dynamical representation that reveals the internal mechanism underlying the externally observed interference fringes. The internal representation of Ψ is described by a set of Hamilton equations of motion, by which quantum trajectories of a particle moving in Ψ can be solved. In this article, millions of complex quantum trajectory connecting slits to a screen are solved from the Hamilton equations, and the statistical distribution of their arrivals on the screen is shown to reproduce the observed interference fringes. This appears to be the first quantitative verification of the equivalence between the trajectory‐based statistics and the wavefunction‐based statistics on slit experiments. © 2012 Wiley Periodicals, Inc.     

Comparison of dielectric theories that explicitly include viscoelastic parameters

In this work we compare four dielectric relaxation models that explicitly include the viscoelastic properties of the environment to define the dielectric relaxation process. These models are the Debye, Gamant, DiMarzio-Bishop, and the Havriliak-Havriliak models. Debye's model assumes a Newtonian viscosity and solves the hydrodynamic problem exactly leading to the venerable Debye function. Gamant's used heuristic arguments to include a time-dependent viscosity. These results lead to a splitting of the relaxation process. DiMarzio-Bishop solved the hydrodynamic equations which included a time-dependent viscosity exactly and applied the results to poly(n-octyl methacrylate). The Havriliak-Havriliak approach is based on statistical mechanics arguments that take into account strain energy which is associated with the polarization process. This model also requires a knowledge of the viscoelastic properties of the system. The results of the four models are compared. © 1995 John Wiley & Sons, Inc.     

Forward-backward semiclassical and quantum trajectory methods for time correlation functions

Forward-backward trajectory formulations of time correlation functions are reviewed. Combination of the forward and reverse time evolution operators within the time-dependent semiclassical approximation minimizes phase cancellation, giving rise to an efficient methodology for simulating the dynamics of low-temperature fluids. A quantum mechanical version of the forward-backward formulation, based on the hydrodynamic formulation of time-dependent quantum mechanics, is also available but is practical only for small systems.     

Extended hydrodynamic approach to quantum-classical nonequilibrium evolution. II. Application to nonpolar solvation

The mixed quantum-classical formulation derived in our companion paper [D. Bousquet, K. H. Hughes, D. Micha, and I. Burghardt, J. Chem. Phys. 134, 064116 (2011)], which is based upon a hydrodynamic representation of the classical sector, is applied to nonequilibrium nonpolar solvation dynamics as exemplified by the solvation of the electronically excited NO molecule in a rare gas environment. Derived from a partition of the Hamiltonian into a primary (quantum) part and a secondary (classical) part the hydrodynamic equations are formulated for multi-quantum states and result in explicit equations of motion for populations and coherences. The hierarchy of hydrodynamic equations is truncated by the following approximate closure schemes: Gauss-Hermite closure, dynamical density functional theory approximation, and a generalized Maxwellian closure. A comparison of the dynamics using these three closure methods showed that the suitability of a particular closure scheme was dependent on the initial conditions and the nonequilibrium character of the dynamics.     

Coping with the node problem in quantum hydrodynamics: the covering function method

A conceptually simple approach, the covering function method (CFM), is developed to cope with the node problem in the hydrodynamic formulation of quantum mechanics. As nodes begin to form in a scattering wave packet (detected by a monitor function), a nodeless covering wave function is added to it yielding a total function that is also nodeless. Both local and global choices for the covering function are described. The total and covering functions are then propagated separately in the hydrodynamic picture. At a later time, the actual wave function is recovered from the two propagated functions. The results obtained for Eckart barrier scattering in one dimension are in excellent agreement with exact results, even for very long propagation times t=1.2 ps. The capability of the CFM is also demonstrated for multidimensional propagation of a vibrationally excited wave packet.     

Non-adiabatic quantum molecular dynamics: basic formalism and case study

A general formalism is presented that treats selfconsistently and simultaneously classical atomic motion and quantum electronic excitations in dynamical processes of atomic many-body systems (non-adiabatic quantum molecular dynamics). On the basis of time-dependent density functional theory, coupled highly non-linear equations of motion are derived for arbitrary basis sets for the time-dependent Kohn-Sham orbitals. Possible approximations to make the approach practical for large atomic cluster systems are discussed. As a first application of the still exact equations of motion, non-adiabatic effects in the scattering of H⁺+H, as a case study, are investigated.     

Quantum trajectories in elastic atom-surface scattering: threshold and selective adsorption resonances

The elastic resonant scattering of He atoms off the Cu(117) surface is fully described with the formalism of quantum trajectories provided by Bohmian mechanics. Within this theory of quantum motion, the concept of trapping is widely studied and discussed. Classically, atoms undergo impulsive collisions with the surface, and then the trapped motion takes place covering at least two consecutive unit cells. However, from a Bohmian viewpoint, atom trajectories can smoothly adjust to the equipotential energy surface profile in a sort of sliding motion; thus the trapping process could eventually occur within one single unit cell. In particular, both threshold and selective adsorption resonances are explained by means of this quantum trapping considering different space and time scales. Furthermore, a mapping between each region of the (initial) incoming plane wave and the different parts of the diffraction and resonance patterns can be easily established, an important issue only provided by a quantum trajectory formalism.     

Evaluation of translational friction coefficients of micro-sized spherical probes in nematic liquid crystals

We study the translational friction coefficients of a spherical micrometric probe moving in nematic liquid crystalline fluids, by solving numerically the constitutive hydrodynamic equations of uncompressible isothermal nematic fluids (Leslie–Ericksen equations). The nematic medium is described by a vector field, which specifies the director orientation at each point and by the velocity vector field. Simulations of director dynamics surrounding the moving probe are presented, and the dependence of translational diffusion upon liquid crystal viscoelastic parameters is discussed. The time evolution of director field is studied in the presence of an orienting magnetic field in two characteristic situations, i.e. direction of motion parallel and perpendicular to field. In particular, a detailed analysis is given for the case of a spherical probe in rectilinear motion in nematic MBBA (4-methoxibenzylidene-4′-n-butylaniline), together with a comparison with other nematogens.     

Everyman's derivation of the theory of atoms in molecules

This paper demonstrates that it is straightforward to develop the theory of an atom in a molecule--the extension of quantum mechanics to an open system--by deriving the necessary equations of motion from Schr?dinger's equation, followed by a comparison of the predicted properties with experiment to determine the correct boundary condition. Although less fundamental than the variational derivation of the quantum theory of atoms in molecules, this heuristic approach makes the quantum mechanics of an atom in a molecule accessible to "everyman" possessing a knowledge of Schr?dinger's equation, aiding its general acceptance by experimental chemists.     

A hybrid hydrodynamic-Liouvillian approach to mixed quantum-classical dynamics: application to tunneling in a double well

The hybrid quantum-classical approach of Burghardt and Parlant [Burghardt, I.; Parlant, G. J. Chem. Phys. 2004, 120, 3055], referred to here as the quantum-classical moment (QCM) approach, is demonstrated for the dynamics of a quantum double well coupled to a classical harmonic coordinate. The approach combines the quantum hydrodynamic and classical Liouvillian representations by the construction of a particular type of moments (that is, partial hydrodynamic moments) whose evolution is determined by a hierarchy of coupled equations. For pure states, which are at the center of the present study, this hierarchy terminates at the first order. In the Lagrangian picture, the deterministic trajectories result in dynamics which is Hamiltonian in the classical subspace, while the projection onto the quantum subspace evolves under a generalized hydrodynamic force. Importantly, this force also depends upon the classical (Q, P) variables. The present application demonstrates the tunneling dynamics in both the Eulerian and Lagrangian representations. The method is exact if the classical subspace is harmonic, as is the case for the systems studied here.     

Incorporation of quantum effects for selected degrees of freedom into the trajectory-based dynamics using spatial domains

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.     

Statistical mechanical theory for steady state systems. V. Nonequilibrium probability density

The phase space probability density for steady heat flow is given. This generalizes the Boltzmann distribution to a nonequilibrium system. The expression includes the nonequilibrium partition function, which is a generating function for statistical averages and which can be related to a nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent mechanical work, and for nonequilibrium quantum statistical mechanics.     

Environmental effects on molecules immersed in liquids

A methodology to study environmental effects is thoroughly discussed. It combines molecular quantum mechanics and classical statistical mechanics of molecular fluids. Pair distribution functions collecting statistical information appear quite naturally in the quantum equations describing a single molecule. As well as allowing the computation of any individual molecular property in a liquid phase, this approach satisfies a number of theoretical requirements (dependence on density and temperature, validity in the thermodynamic limit). In a sense, it can be regarded as a useful alternative to the well-known Monte Carlo averaging processes for calculating molecular properties. Numerical applications studying liquid carbon disulphide and liquid carbon tetrachloride at several state points are given. Results cover typical RHF information (CNDO/2) on molecules, and show the sensitivity of the presented methodology to structural changes in liquids.     

Quantized Hamilton Dynamics

The paper describes the quantized Hamilton dynamics (QHD) approach that extends classical Hamiltonian dynamics and captures quantum effects, such as zero point energy, tunneling, decoherence, branching, and state-specific dynamics. The approximations are made by closures of the hierarchy of Heisenberg equations for quantum observables with the higher order observables decomposed into products of the lower order ones. The technique is applied to the vibrational energy exchange in a water molecule, the tunneling escape from a metastable state, the double-slit interference, the population transfer, dephasing and vibrational coherence transfer in a two-level system coupled to a phonon, and the scattering of a light particle off a surface phonon, where QHD is coupled to quantum mechanics in the Schrödinger representation. Generation of thermal ensembles in the extended space of QHD variables is discussed. QHD reduces to classical mechanics at the first order, closely resembles classical mechanics at the higher orders, and requires little computational effort, providing an efficient tool for treatment of the quantum effects in large systems.     

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.