Bijoy K. Dey

Hamilton-Jacobi equation for the least-action/least-time dynamical path based on fast marching method.

Classical dynamics can be described with Newtons equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethians fast marching method. Using this method, we can--starting from an arbitrary initial conformation--find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H(2)-->H(2)+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales.

A Hamilton–Jacobi type equation for computing minimum potential energy paths

A new method for computing minimum-energy reaction paths is presented. Unlike existing approaches (e.g. intrinsic reaction coordinate methods), our approach works for any reactant configuration: the structure of the transition state, reactive intermediates and product will be determined by the algorithm, and so need not be known beforehand. The method we have developed is based on solving a Hamilton–Jacobi type equation. Specifically, we introduce a speed function so that the ‘first arrival times’ from the Hamilton–Jacobi equation correspond to least-potentials. Then, adopting a back-tracing method, we can use the first arrival times to determine the minimum-energy path between any classically allowed molecular conformation and the initial (reactant) conformation. The method is illustrated by applying it to six different systems: (1) a model system with four different minima in the potential energy surface, (2) a model Muller–Brown potential, (3) the isomerization reaction of malonaldehyde using a fitting potential energy surface, (4) a model Minyaev–Quapp potential representative of con- and dis-rotations of two BH2 groups in the BH2–CH2–BH2 molecule, (5) the F + H2→FH + H reaction and (6) the H + FH → HF + H reaction. Our results demonstrate that the proposed method represents a robust alternative to existing techniques for finding chemical reaction paths.

Coherently controlled nanoscale molecular deposition.

Quantum interference effects are shown to provide a means of controlling and enhancing the focusing of a collimated neutral molecular beam onto a surface. The nature of the aperiodic pattern formed can be altered by varying laser field characteristics and the system geometry.

Direct ab initio calculation of ground-state electronic energies and densities for atoms and molecules through a time-dependent single hydrodynamical equation

By using an imaginary-time evolution technique, coupled with the minimization of an expectation value, ground-state electron densities and energies have been directly calculated for six atomic and molecular systems (He, Be++, Ne, H2, HeH+, He2++), from a single time-dependent (TD) quantum fluid dynamical equation of motion whose real-time solution yields the TD electron density. For all the systems, a local Wigner-type correlation functional has been employed. For Ne, a local exchange functional is used while, for all the other systems, the exchange energy is calculated exactly. The static (ground-state) results are of beyond-Hartree–Fock quality for all the species.

Direct calculation of ground-state electronic densities and properties of noble gas atoms through a single time-dependent hydrodynamical equation

Ground-state electronic densities and properties of noble gas atoms (He, Ne, Ar, Kr and Xe) have been calculated through a single time-dependent quantum fluid dynamical equation of motion. The equation has been transformed through imaginary time into a diffusion equation which is then numerically solved in order to reach a global minimum. The present results compare favourably with other available values.

Computing tunneling paths with the Hamilton–Jacobi equation and the fast marching method

We present a new method for computing the most probable tunneling paths based on the minimum imaginary action principle. Unlike many conventional methods, the paths are calculated without resorting to an optimization (minimization) scheme. Instead, a fast marching method coupled with a back-propagation scheme is used to efficiently compute the tunneling paths. The fast marching method solves a Hamilton–Jacobi equation for the imaginary action on a discrete grid where the action value at an initial point (usually the reactant state configuration) is known in the beginning. Subsequently, a back-propagation scheme uses a steepest descent method on the imaginary action surface to compute a path connecting an arbitrary point on the potential energy surface (usually a state in the product valley) to the initial state. The proposed method is demonstrated for the tunneling paths of two different systems: a model 2D potential surface and the collinear reaction. Unlike existing methods, where the tunneling path is based on a presumed reaction coordinate and a correction is made with respect to the reaction coordinate within an ‘adiabatic’ approximation, the proposed method is very general and makes no assumptions about the relationship between the reaction coordinate and tunneling path.

STRIPPED ION-HELIUM ATOM COLLISION DYNAMICS WITHIN A TIME-DEPENDENT QUANTUM FLUID DENSITY FUNCTIONAL THEORY

A nonperturbative, time-dependent (TD) quantum mechanical approach is described for studying the collision dynamics between the He atom and a fully stripped ion. The method combines quantum fluid dynamics and density functional theory to solve two coupled equations: one for the trajectory of the projectile nucleus and the other for the electronic charge distribution of the target atom. The computed TD and frequency-dependent properties provide detailed features of the collision process. Inelastic and ionization cross sections are also reported.

Optimal reduced dimensional representation of classical molecular dynamics

An optimal reduced space method for capturing the low-frequency motion in classical molecular dynamics calculations is presented. This technique provides a systematic means for carrying out reduced-dimensional calculations in an effective set of reduced coordinates. The method prescribes an optimal reduced subspace linear transformation for the low frequency motion. The method is illustrated with a dynamics calculation for a model potential, where the original six-dimensional space is reduced to two (three) dimensions, depending on the desired frequency cutoff value.

ALTERNATING DIRECTION IMPLICIT TECHNIQUE AND QUANTUM EVOLUTION WITHIN THE HYDRODYNAMICAL FORMULATION OF SCHRODINGER'S EQUATION

Abstract An alternative method of quantum dynamics is presented. The method is based on the hydrodynamical formulation of the time-dependent Schrodinger equation originally given by David Bohm in his quest for establishing a hidden variable alternative to the quantum mechanics. A new alternating direction implicit technique has been employed to decouple many-dimensional hydrodynamical equations into a set of one-dimensional equations which have been solved numerically by adopting a recently developed flux corrected transport algorithm. We apply the method to describe the dynamics of a quantum particle in three spatial dimensions where analytical solutions are known.

Helium atom in intense and superintense laser fields: A new theoretical approach

A quantum hydrodynamical study is made of the dynamical changes of a helium atom interacting with lasers of two different intensities, but having the same frequency. Under the intense laser field, electron density oozes out of the helium atom by absorbing laser photons and getting promoted to higher excited states including the continuum. Under the superintense field, electron density partly moves away from the helium nucleus but remains in the “quasi-bound” dressed states along with the laser field, thus suppressing ionization.