A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Huang, Yilun; Zhu, Wei; Kouri, Donald J.; Hoffman, David K.

doi:10.1016/0009-2614(93)85523-q

Cited by 119 publications

(60 citation statements)

References 11 publications

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“…The partitioning of the problem into a calculation of direct scattering and of resonance decay makes the hybrid approach also more efficient than a TI approach in which the coefficients of the Chebyshev sum are integrated analytically over time [63][64][65] and where convergence is ensured by the use of a modified Chebyshev recursion. 65 This is shown in Fig.…”

Section: A the 2d Problemmentioning

confidence: 99%

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

Kroes

Wiesenekker

Baerends

et al. 1996

The Journal of Chemical Physics

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We investigate the usefulness of a hybrid method for scattering with resonances. Wave packet propagation is used to obtain the time-dependent wave function ⌿(t) up to some time T at which direct scattering is over. Next, ⌿(t) is extrapolated beyond T employing resonance eigenvalues and eigenfunctions obtained in a Lanczos procedure, using ⌿(T) as starting vector to achieve faster convergence. The method is tested on one two-dimensional ͑2D͒ and one four-dimensional ͑4D͒ reactive scattering problem, affected by resonances of widths 0.1-5 meV. Compared to long time wave packet propagation, the hybrid method allows large reductions in the number of Hamiltonian operations N H required for obtaining converged reaction probabilities: A reduction factor of 24 was achieved for the 2D problem, and a factor of 6 for the 4D problem.

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Section: A the 2d Problemmentioning

confidence: 99%

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

Kroes

Wiesenekker

Baerends

et al. 1996

The Journal of Chemical Physics

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“…It is based on the well known approximation of an analytical function by the Faber polynomial series [3] (see also the textbooks [4]). The Faber approximation method has been applied to quantum scattering problems [5] to compute the causal Green's function for the Schrödinger equation. The Faber polynomial approximation of the exponential of a non-Hermitian operator has also been used to solve the initial value problem for the Liouville -von Neumann equation that describes the time evolution of the density matrix in statistical systems [6,7].…”

Section: Introductionmentioning

confidence: 99%

Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Borisov¹,

Shabanov²

2006

Journal of Computational Physics

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Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schrödinger equation i∂Ψ/∂t = HΨ, where H is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector Ψ composed of the electromagnetic fields and auxiliary matter fields describing the medium response. In this representation, the initial value problem is solved by applying the fundamental solution exp(−itH) to the initial field configuration. The Faber polynomial approximation of the fundamental solution is used to develop a numerical algorithm for propagation of broad band wave packets in passive media. The action of the Hamiltonian on the wave function Ψ is approximated by the Fourier grid pseudospectral method. The algorithm is global in time, meaning that the entire propagation can be carried out in just a few time steps. A typical time step is much larger than that in finite differencing schemes, ∆t F ≫ H −1 . The accuracy and stability of the algorithm is analyzed. The Faber propagation method is compared with the Lanczos-Arnoldi propagation method with an example of scattering of broad band laser pulses on a periodic grating made of a dielectric whose dispersive properties are described by the Rocard-Powels-Debye model. The Faber algorithm is shown to be more efficient. The Courant limit for time stepping, ∆t C ∼ H −1 , is exceeded at least in 3000 times in the Faber propagation scheme.

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“…In form, the above expression for the expansion coefficient, d k (E), is different from the ones given by Neuhauser et al, 33 Mandelshtam et al, 35 and Kouri et al 36 This has prompted us to check its validity by comparing it numerically with the other three. We have found it to give similar results to the formulas of Neuhauser et al Like the time-independent LHFD method, we can arrive at an expression for product state distributions after obtaining the resonance wave functions…”

Section: ͑23͒mentioning

confidence: 84%

Calculation of product state distributions from resonance decay via Lanczos subspace filter diagonalization: Application to HO2

Zhang

Smith

2001

The Journal of Chemical Physics

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Articles you may be interested inResonance phenomena associated with the unimolecular dissociation of HO 2 have been investigated quantum-mechanically by the Lanczos homogeneous filter diagonalization ͑LHFD͒ method. The calculated resonance energies, rates ͑widths͒, and product state distributions are compared to results from an autocorrelation function-based filter diagonalization ͑ACFFD͒ method. For calculating resonance wave functions via ACFFD, an analytical expression for the expansion coefficients of the modified Chebyshev polynomials is introduced. Both dissociation rates and product state distributions of O 2 show strong fluctuations, indicating the dissociation of HO 2 is essentially irregular.

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A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Cited by 119 publications

References 11 publications

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

Dissociative chemisorption of H2 on Cu(100): A four-dimensional study of the effect of parallel translational motion on the reaction dynamics

Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Calculation of product state distributions from resonance decay via Lanczos subspace filter diagonalization: Application to HO2

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