Ronnie Kosloff

An accurate and efficient scheme for propagating the time dependent Schrödinger equation

A new propagation scheme for the time dependent Schrodinger equation is based on a Chebychev polynomial expansion of the evolution operator U=exp(−iHt). Combined with the Fourier method for calculating the Hamiltonian operation the scheme is not only extremely accurate but is up to six times more efficient than the presently used second order differencing propagation scheme.

A fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics

A new method is presented for the solution of the time dependent SchrBdinger equation in its application to physical and chemical molecular phenomena. The method is based on discretizing space and time on a grid, and using the Fourier method to produce both spatial derivatives, and second order differencing for time derivatives. The method conserves norm and energy, and preserves quantum mechanical commutation relations. One- and twodimensional examples, where a comparison to analytic results is possible, are investigated. Many phenomena of physics and chemistry have a common dynamical evolution pattern. This pattern begins with an initial state which under the influence of the potentials, evolves through time to produce a final asymptotic state. Among such phenomena are chemical reactions, photodissociation, unimolecular breakdown, surface scattering, and desorption. The present study presents a numerical solution based on quantum mechanics for the common pattern, or for the time dependent evolution of the state of the system. The importance of exact numerical solutions is twofold, first these solutions give a quantitative description of physical problems and also are of use as a bench mark to check approximation methods and to define their range of validity.

A nonreflecting boundary condition for discrete acoustic and elastic wave equations

One of the nagging problems which arises in application of discrete solution methods for wave‐propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually override the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting boundary conditions were introduced for the finite‐difference method (Clayton and Enquist, 1977; Reynolds, 1978). These boundary conditions are based on replacing the wave equation in the boundary region by one‐way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. This approach has been relatively successful, except that its effectiveness degrades for event...

Absorbing boundaries for wave propagation problems

Abstract Reflections or wraparound from boundaries of numerical grids have always presented a difficulty in applying discrete methods to simulate physical phenomena. This study presents a systematic derivation of absorbing boundary conditions which can be used in a wide class of wave equations. The derivation is applied to the Schrodinger equation and to the acoustic equation in one and two dimensions. The effectiveness of the absorbing boundary conditions can be evaluated apriori on the basis of analytic solutions.

Coherent pulse sequence induced control of selectivity of reactions: Exact quantum mechanical calculations

We present a novel approach to the control of selectivity of reaction products. The central idea is that in a two‐photon or multiphoton process that is resonant with an excited electronic state, the resonant excited state potential energy surface can be used to assist chemistry on the ground state potential energy surface. By controlling the delay between a pair of ultrashort (femtosecond) laser pulses, it is possible to control the propagation time on the excited state potential energy surface. Different propagation times, in turn, can be used to generate different chemical products. There are many cases for which selectivity of product formation should be possible using this scheme. We illustrate the methodology with numerical application to a variety of model two degree of freedom systems with two inequivalent exit channels. Branching ratios obtained using a swarm of classical trajectories are in good qualitative agreement with full quantum mechanical calculations.

A direct relaxation method for calculating eigenfunctions and eigenvalues of the schrödinger equation on a grid

Abstract Eigenfunctions and eigenvalues of the Schrodinger equation are determined by propagating the Schrodinger equation in imaginary time. The method is based on representing the Hamiltonian operation on a grid. The kinetic energy is calculated by the Fourier method. The propagation operator is expanded in a Chebychev series. Excited states are obtained by filtering out the lower states. Comparative examples include: eigenfunctions and eigenvalues of the Morse oscillator, the Henon-Heiles system and weakly bound states of He on a Pt surface.

WAVEPACKET DANCING: ACHIEVING CHEMICAL SELECTIVITY BY SHAPING LIGHT PULSES

Abstract The Tannor-Rice pump-dump scheme for controlling the selectivity of product formation in a chemical reaction is improved by development of a method for optimizing the field of a particular product with respect to the shapes of the pump and dump pulses. Numerical studies of the optimization of product yield in a model system of the same type as studied by Tannor and Rice illustrate the enhancement possible with pulse shaping.

A Fourier method solution for the time dependent Schrödinger equation: A study of the reaction H++H2, D++HD, and D++H2

A new quantum mechanical time dependent integrator was used in the study of wave packet dynamics on potentials which include a deep well. The purpose of the study was to find the conditions, if any, for complex formation. The integrator, which is stable, conserves energy and norm and was used on the H++H2 system whose classical trajectory had been previously worked out. Almost no complex formation is found for the H++H2 system and its isotopic analogs. For high translational energies there was a good correspondence with the classical trajectory results, while for low translational energies where the classical trajectories become complex, the quantum calculations still show nonstatistical behavior. For even lower energies, a quantum effect took place leading to zero reactivity.

IMPULSIVE EXCITATION OF COHERENT VIBRATIONAL MOTION GROUND SURFACE DYNAMICS INDUCED BY INTENSE SHORT PULSES

A framework for understanding impulsively photoinduced vibrational coherent motion on the ground electronic surface is presented. In particular strong resonant excitation to a directly dissociative electronic surface is considered. Three distinct approaches are employed. A two surface Fourier wavepacket method explicitly including the field explores this process in isolated molecules. A coordinate dependent two‐level system is employed to develop a novel analytical approximation to the photoinduced quantum dynamics. The negligible computational requirements make it a powerful interactive tool for reconstructing the impulsive photoexcitation stage. Its analytical nature provides closed form expressions for the photoinduced changes in the material. Finally the full simulation of the process including the solvent effects is carried out by a numerical propagation of the density operator. In all three techniques the excitation field is treated to all orders, allowing an analysis of current experiments using st...

Quantum computing by an optimal control algorithm for unitary transformations.

Quantum computation is based on implementing selected unitary transformations representing algorithms. A generalized optimal control theory is used to find the driving field that generates a prespecified unitary transformation. The approach is independent of the physical implementation of the quantum computer and it is illustrated for one and two qubit gates in model molecular systems, where only part of the Hilbert space is used for computation.