Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization

Poirier, Bill; Salam, A.

doi:10.1063/1.1767511

Cited by 43 publications

(42 citation statements)

References 48 publications

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“…[30][31][32][33][34][35][36][37][38] Wavelets are localized in both the position and momentum representations, and are commonly used for spectral analysis, but have also found application to quantum dynamics. 33,34,[36][37][38] Here we use a different approach to generating PSL functions, one that has recently been introduced by Dawes and Carrington. 3,39 They use the method of simultaneous diagonalization ͑SD͒, which seeks a single set of eigenfunctions that diagonalize the position and momentum operator matrices.…”

Section: Introductionmentioning

confidence: 99%

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

McCormack

2006

The Journal of Chemical Physics

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We investigate the viability of dynamical pruning of localized basis sets in time-dependent quantum wave packet methods. Basis functions that have a very small population at any given time are removed from the active set. The basis functions themselves are time independent, but the set of active functions changes in time. Two different types of localized basis functions are tested: discrete variable representation ͑DVR͒ functions, which are localized in position space, and phase-space localized ͑PSL͒ functions, which are localized in both position and momentum. The number of functions active at each point in time can be as much as an order of magnitude less for dynamical pruning than for static pruning, in reactive scattering calculations of H 2 on the Pt͑211͒ stepped surface. Scaling of the dynamically pruned PSL ͑DP-PSL͒ bases with dimension is considerably more favorable than for either the primitive ͑direct product͒ or DVR bases, and the DP-PSL basis set is predicted to be three orders of magnitude smaller than the primitive basis set in the current state-of-the-art six-dimensional reactive scattering calculations.

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Section: Introductionmentioning

confidence: 99%

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

McCormack

2006

The Journal of Chemical Physics

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“…Many authors have implemented basis pruning strategies [16,18,19,53,62,63,[70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. Although pruning has the obvious advantage that it decreases the size of the vectors one must store and the spectral range of the Hamiltonian matrix, if one uses an iterative method, it complicates the evaluation of MVPs.…”

Section: Using Pruning To Reduce Both Basis and Grid Sizementioning

confidence: 99%

Iterative Methods for Computing Vibrational Spectra

Carrington

2018

Mathematics

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I review some computational methods for calculating vibrational spectra. They all use iterative eigensolvers to compute eigenvalues of a Hamiltonian matrix by evaluating matrix-vector products (MVPs). A direct-product basis can be used for molecules with five or fewer atoms. This is done by exploiting the structure of the basis and the structure of a direct product quadrature grid. I outline three methods that can be used for molecules with more than five atoms. The first uses contracted basis functions and an intermediate (F) matrix. The second uses Smolyak quadrature and a pruned basis. The third uses a tensor rank reduction scheme.

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“…On the other hand, in the von Neumann basis set [11,12] each basis function is localized on a unit cell of size h in phase space. However, despite the formal completeness of the vN basis set [13], attempts to utilize this basis in quantum numerical calculations have been plagued with numerical errors [4,14].…”

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confidence: 99%

Phase-Space Approach to Solving the Time-Independent Schrödinger Equation

Shimshovitz

Tannor

2012

Phys. Rev. Lett.

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We propose a method for solving the time independent Schrödinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)] we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method for a challenging two-dimensional potential where the FGH method breaks down.

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Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization

Cited by 43 publications

References 48 publications

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

Dynamical pruning of static localized basis sets in time-dependent quantum dynamics

Iterative Methods for Computing Vibrational Spectra

Phase-Space Approach to Solving the Time-Independent Schrödinger Equation

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