The Chebyshev propagator for quantum systems

A detailed comparison of the time-dependent wave packet method using the split operator propagator and recently introduced Chebyshev real wave packet approach for calculating reactive scattering processes is reported. As examples, the state-to-state differential cross sections of the H + H(')D(v(0) = 0,j(0) = 1) --> H(')D + H/H(')H + D reaction, the state-to-state reaction probabilities of the (16)O + (35)O(2) (v(0) = 0,j(0) = 0) --> (17)O + (16)O(18)O/(18)O + (16)O(17)O reaction, the H + O(2) --> O + HO reaction, and the F + HD --> HF + D reaction are calculated, using an efficient reactant-coordinate-based method on an L-shape grid which allows the extraction of the state-to-state information of the two product channels simultaneously. These four reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these two propagation methods for describing reactive scattering dynamics. The results indicate that the Chebyshev real wave packet method is typically more accurate, particularly for reactions dominated by long-lived resonances. However, the split operator approach is often more cost effective, making it a method of choice for fast reactions. In addition, our results demonstrate accuracy of the reactant-coordinate-based method for extracting state-to-state information.

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“…27,40 In the reactant coordinates, the Hamiltonian for a given total angular momentum J can be written as…”

Section: A Hamiltonian and Basis Functions In Reactant Coordinatesmentioning

confidence: 99%

Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations

Sun

Lee

Guo

et al. 2009

The Journal of Chemical Physics

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“…Both limitations can be overcome by an approach where we expand the time evolution operator U (t, t 0 ) = U (t − t 0 ) = U ( t) into a finite series of first-kind Chebyshev polynomials of order k: T k (x) = cos(k arccos(x)). We then obtain [4,20,21] 1]. In practice, we use α = 0.01.…”

Section: Iterative Methodsmentioning

confidence: 99%

Numerical approaches to time evolution of complex quantum systems

et al. 2009

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We examine several numerical techniques for the calculation of the dynamics of quantum systems. In particular, we single out an iterative method which is based on expanding the time evolution operator into a finite series of Chebyshev polynomials. The Chebyshev approach benefits from two advantages over the standard time-integration Crank-Nicholson scheme: speedup and efficiency. Potential competitors are semiclassical methods such as the Wigner-Moyal or quantum tomographic approaches. We outline the basic concepts of these techniques and benchmark their performance against the Chebyshev approach by monitoring the time evolution of a Gaussian wave packet in restricted one-dimensional (1D) geometries. Thereby the focus is on tunnelling processes and the motion in anharmonic potentials. Finally we apply the prominent Chebyshev technique to two highly non-trivial problems of current interest: (i) the injection of a particle in a disordered 2D graphene nanoribbon and (ii) the spatiotemporal evolution of polaron states in finite quantum systems. Here, depending on the disorder/electron-phonon coupling strength and the device dimensions, we observe transmission or localisation of the matter wave.Comment: 8 pages, 3 figure

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“…In the FOM case, meeting the former requirement demands basically to get rid of the (time-consuming) step by step computational procedures traditionally employed in molecular dynamics. 21,22 FEM is an alternative approach [23][24][25] endowed with right prerogatives for dealing efficiently with the long-time dynamics involved in FOM, i.e.. to yield directly the solution of eq. (4) for the dynamical state of the mapped system after imposition of the external periodic force.…”

Section: Fom Speedup By Fast-time Evaluation Methodsmentioning

confidence: 99%