Close Coupling Wave Packet Method Research Articles

Sticking and desorption of hydrogen on graphite: A comparative study of different models

We study the physisorption of atomic hydrogen on graphitic surfaces with four different quantum mechanical methods: perturbation and effective Hamiltonian theories, close coupling wavepacket, and reduced density matrix propagation methods. Corrugation is included in the modeling of the surface. Sticking is a fast process which is well described by all methods. Sticking probabilities are of the order of a few percent in the collision energy range 0-25 meV, but are enhanced for collision energies close to those of diffraction resonances. Sticking also increases with surface temperature. Desorption is a slow process which involves multiphonon processes. We show, however, how to correct the close coupling wavepacket method to account for such phenomena and obtain correct time constants for initial state decay. Desorption time constants are in the range of 20-50 ps for a surface temperature of 300 K.

Generalized Pseudospectral Method for Solving the Time-Dependent Schrödinger Equation Involving the Coulomb Potential

We present an accurate and efficient generalized pseudospectral method for solving the time-dependent Schrödinger equation for atomic systems interacting with intense laser fields. In this method, the time propagation of the wave function is calculated using the well-known second-order split-operator method implemented by the numerically exact, fast transform between the grid and spectral representations. In the grid representation, the radial coordinate is discretized using the Coulomb wave discrete variable representation (CWDVR), and the angular dependence of the wave function is expanded in the Gauss–Legendre–Fourier grid. In the spectral representation, the wave function is expanded in terms of the eigenfunctions of the field-free zero-order Hamiltonian. Calculations on the high order harmonic generation and ionization dynamics of hydrogen atom in strong laser pulses are presented to demonstrate the accuracy and efficiency of the present method. This new algorithm will be found more computationally attractive than the close-coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.

Numerical solution of the three-dimensional time-dependent Schr?dinger equation and its application

We present an accurate and effective pseudospectral method for solving the three-dimensional time-dependent Schrdinger equation involving the Coulomb potential. In this method, the Hamiltonian is evaluated by exploiting the two representations of the wave function. One is a grid representation, in which the angular dependence of the wave function is expanded in a two-dimensional Gauss-Legendre-Fourier grid in the coordinate space of polar and azimuthal angles. The radial coordinate is discretized using a discrete variable representation constructed from the Coulomb wave function （CWDVR）. The other is a spectral representation, in which the wave function is expanded in a set of square integrable functions chosen as the eigenfunctions of a zero-order Hamiltonian. The time of propagation of the wave function is calculated using the well-known second-order split-operator method implemented through the transform between the grid and spectral representations. Calculations on the photo-absorption strength of hydrogen atom are presented to demonstrate the accuracy of present method in low energy limit by the time-dependent wave-packet propagation method. As another example, the present method is applied to multiphoton ionization of H atom. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate numerical calculations. The new algorithm will be found more efficient than the close coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.

Non-adiabatic wavepacket dynamics for charge-exchange processes in ion-atom collisions: application to

The collisional system is studied as a test case to assess the efficiency of a full quantum time-dependent method to calculate charge-exchange cross sections. The molecular diabatic electronic channels are calculated at the CASSCF level from the adiabatic potential energy curves and the radial coupling matrix. The elements of the collision matrix are extracted, for each value of the total angular momentum K, by Fourier transforming the post-collisional amplitudes of the wavepackets which are prepared in the entry channel and propagated on the coupled effective channels for each value of K. The results are compared with previous calculations of cross sections obtained by solving the stationary scattering equations and with recent experimental results. The close-coupling wavepacket method also provides a time-dependent picture of the collision dynamics for any K value. The kinematic isotopic effect is also examined.

Time-dependent reactive scattering in hyperspherical coordinates: A close coupled wave packet approach

We present a close coupled wave packet method to treat reactive collisions in hyperspherical coordinates. Within this approach, the information needed for the time-dependent propagation is the same as that required for a time-independent close coupling treatment. The method is tested on the simple collinear D+H2 reaction. We compare the time-dependent technique with the more established time-independent close coupling treatment. Finally, we show that time-dependent properties are useful to gain insight into the detailed mechanism of the reaction.

Performance of close-coupled wave packet methods for molecule-corrugated surface scattering

The H2+LiF(001) system was used to investigate the performance of the hybrid close-coupling wave packet (CCWP) method and of a symmetry adapted, fully close-coupled wave packet (SAWP) method for a molecule–surface problem characterized by fairly high corrugation. In the calculations, a realistic, φ-dependent model potential was used. The calculations were performed for a collision energy of 0.2 eV, with H2 initially in its j=0 rotational state at normal incidence to the surface. Large increases in the computational efficiencies of both wave packet methods were achieved by taking advantage of the potential coupling matrices associated with both methods becoming sparser with increasing molecule–surface distance. For the present model problem and employing this increased sparseness at longer range, the SAWP method is faster than the CCWP method by a factor of 2. The potential usefulness of the SAWP method for dissociative chemisorption problems is discussed.

Prediction of large differences in diffraction of cold beams of para-H 2 and normal-H 2 by LiF(001)

The close-coupling wave packet method has been used for performing calculations on rotationally and diffractionally inelastic scattering of H 2 from LiF(001), using a model potential. Inclusion of the quadropole-ionic lattice interaction in the model leads to large differences between the diffraction of H 2 in its j = 0 rotational state and diffraction of j = 1 H 2. For j = 1 1 H 2 large probabilities are obtained for changes of the magnetic rotational quantum number m j . It should be possible to verify some of our results by diffraction experiments employing cold beams of para-H 2 and normal-H 2.

Dissociative adsorption of H2 using the close-coupling wave packet method

A time-dependent quantum mechanical wave packet method was used to study the dynamics of the dissociative adsorption of H2 and D2 on a flat, static surface. A three-dimensional (3D) model was used in which the molecular rotational, vibrational, and center-of-mass translational motion normal to the surface are treated. In the close-coupling wave packet method the wave function is represented using a combination of a basis set expansion for the rotational degrees of freedom and a 2D L-shaped grid for the vibrational and translational coordinates. The time propagation is carried out by expanding the time-evolution operator in a series of Chebyshev polynomials. The molecule–surface interaction is described using a modified London–Eyring–Polanyi–Sato (LEPS) potential with parameters chosen to represent the H2/Ni(100) system. The dissociation probability was calculated for different incident energies and initial rotational and vibrational states and compared to the results of other theoretical calculations. Higher incident energies are required for D2 dissociation than for H2. The barrier height and zero-point energies at the saddle point can be determined from the energy dependence of the dissociation probabilities for H2 and D2.

Partial cross sections in the predissociation of quasibound states of NeHF. An adiabatic channel interpretation

We have computed partial cross sections for the internal rotational predissociation of certain quasibound states of the NeHF van der Waals complex, using the time-dependent close-coupled wave packet method. These cross sections show strong selectivity in the orbital angular momentum of the outgoing fragments. We provide an explanation of these propensities based on an adiabatic channel interpretation, and discuss the observable physical consequences of the partial cross sections we calculate.

Pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordinates

In this paper we describe a numerically efficient pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordinates. In this method the translational kinetic energy operator is evaluated with a Fourier transform. The angular dependence of the wave function is expanded on a two-dimensional grid in coordinate space and the angular part of the Laplacian is evaluated by a Gauss–Legendre–Fourier transform between the coordinate and conjugate angular momentum representations. The potential energy operator is diagonal. Calculations performed for a model system representing H2 scattering from a static corrugated surface yield transition probabilities identical to those obtained with the close coupled wave packet (CCWP) method. The new algorithm will be more efficient than the CCWP method for problems in which a large number of rotational states are coupled.

CLOSE COUPLING WAVE PACKET THEORY OF ATOM- DIATOM SCATTERING

In the space-fixed(SF)coordinate frame and body-fixed(BF) coordinate frame, we cal-culete the state to state transitional probabilities for the He-H2 system by using the rencently developed close coupling wave packet method. The results shows that (1) the two theories formulated in the SF frame and BF frame are equivalent, and the calculated results of the two method coincide with that of close coupling method, (2) the calculational efficience in the SF, frame is higher than that in the BF frame as the number of coupled terms increases.

Inelastic molecule-corrugated surface scattering using the close-coupling wave-packet method

A computer program for calculating transition probabilities for rotationally and translationally inelastic scattering of homonuclear diatomic molecules from static, corrugated surfaces is described. The program uses an exact quantum-mechanical method which combines a close-coupling expansion of the internal states with a time-dependent wave-packet description of the center-of-mass motion. The wave function is propagated in time using an expansion of the time-evolution operator in a series of Chebyshev polynomials.

Close-coupled wave-packet calculations of the direct inelastic scattering of NO(X 2Π) from Ag(111)

In this article we report an application of the time-dependent close-coupled wave-packet (CCWP) method to the rotationally inelastic scattering of NO(X 2Π) molecules from a rigid, flat Ag(111) surface. Previous applications of the CCWP method have been restricted to either direct scattering off purely repulsive potentials requiring short propagation times, or weakly physisorbed systems in which only a few internal states are coupled. The calculations reported here were performed for a molecule in an initial state with a momentum distribution peaked around E=6700 cm−1 scattering off a strongly anisotropic potential with a well depth of 4400 cm−1 and a long-range tail. Numerical procedures were introduced which enhance the efficiency of the CCWP method whenever a large number of internal states or a large number of grid points are needed to simulate the collision. For the current application to NO–Ag these techniques reduced the required CPU time by more than an order of magnitude. The resulting state–to–state transition probabilities are compared with previous time-independent close-coupled calculations, and with the semiclassical self-consistent eikonal method (SCEM). The agreement between the two quantum-mechanical methods is well within the accuracy of both numerical procedures. A comparison of the instantaneous transition probabilities calculated throughout the propagation shows good agreement between the CCWP and the SCEM calculations at high collision energies.

A numerically exact full wave packet approach to molecule–surface scattering

A numerically exact spectral method for solving the time-dependent Schrödinger equation in spherical coordinates is described. The angular dependence of the wave function is represented on a two-dimensional grid of evenly spaced points. The fast Fourier transform algorithm is used to transform between the angle space representation of the wave function and its conjugate representation in momentum space. The time propagation of the wave function is evaluated using an expansion of the time evolution operator as a series of Chebyshev polynomials. Calculations performed for a model system representing H2 scattering from a rectangular corrugated surface yield transition probabilities that are in excellent agreement with those obtained using the close-coupling wave packet (CCWP) method. However, the new method is found to require substantially more computation time than the CCWP method because of the large number of grid points needed to represent the angular dependence of the wave function and the variation in the number of terms required in the Chebyshev representation of the time evolution operator.

Body frame close coupling wave packet approach to gas phase atom–rigid rotor inelastic collisions

The close coupling wave packet (CCWP) method is formulated in a body-fixed representation for atom–rigid rotor inelastic scattering. For total J>jmax the computational cost of propagating the coupled channel wave packets in the body frame scales approximately as N3/2 where N is the total number of channels. For large numbers of channels, this will be much more efficient than the space frame CCWP method previously developed which scales approximately as N2 under the same conditions. Timing results are reported for a model system for 25, 64, 144, 256, and 969 channels. The calculations were run with total J=30 and with parameters corresponding to a heavy diatom. The results for some representative transitions are compared to the identical transitions obtained using the space frame formalism. For all values of N, the body frame computations ran faster than the corresponding ones using the space frame, with the ratio increasing to a value of 4.5 for 969 channels. This is a significant improvement which will continue to increase with N, encouraging us to believe that the body frame CCWP method will prove practical for calculating scattering information for more realistic inelastic and reactive systems.

Application of the close coupling wave packet method to long lived resonance states in molecule–surface scattering

In this paper we describe the application of the close coupling wave packet (CCWP) method to the study of HD and H2 scattering from a Ag(111) surface. The presence of rotationally mediated selective adsorption (RMSA) resonances at low collision energies for the HD/Ag(111) system requires that the scattering wave function be propagated for extremely long times to allow the resonant states to decay. A new procedure is presented for obtaining both the magnitude and phase of the S-matrix elements from the scattered wave packet which drastically reduces the size of the spatial grid required. Both the transition probabilities and collision lifetimes obtained from CCWP calculations are in excellent agreement with those predicted by the CC method, demonstrating that both the moduli and phases of S-matrix elements can be obtained over a range of collision energies from the propagation of a single wave packet.

Molecule–corrugated surface scattering calculations using the close coupling wave packet method

Numerically exact quantum calculations for scattering of N2 off a model corrugated rigid lattice are reported. The computations were done using the recently developed close coupling-wave packet method for treating quantum scattering. Results for all energies in the range 0.010–0.040 eV are obtained from a single wave packet propagation. The calculations included even N2 rotor j states j=0–12, all mj’s, and sufficient grid points to describe diffraction states −8≤m≤8, −4≤n≤4, for a total of 13 923 channels. The computations, which required a total of 230 minutes of computer time, were carried out on the CRAY2 supercomputer at the University of Minnesota Supercomputer Center under National Science Foundation support.

Close-coupling wave packet approach to numerically exact molecule–surface scattering calculations

In this paper we describe the theory and application of the recently developed close-coupling wave packet (CCWP) method to the study of transition probabilities of H2 scattered from flat and corrugated surfaces. We present an improved method of analyzing the final wave function which permits S matrices and transition probabilities to be obtained over a wide range of energies from the propagation of a single wave packet. Transition probabilities obtained using the CCWP method are in excellent agreement with those obtained from time-independent close-coupling (CC) calculations. For the present three-dimensional H2-corrugated surface scattering study, the CCWP calculations require roughly one-tenth of the computation time of the CC method. Our results indicate that the CCWP method should be a very efficient method of obtaining highly accurate scattering results both for standard collision problems and for collision problems which cannot be readily treated using standard CC methods.