Using nonproduct quadrature grids to solve the vibrational Schrödinger equation in 12D

Avila, Gustavo; Carrington, Tucker

doi:10.1063/1.3549817

Cited by 123 publications

(197 citation statements)

References 102 publications

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“…When quadrature is used, all potential terms are treated together, and there is no need to evaluate matrix-vector products term by term. There is also no need to approximate the potential as a sum of terms depending on one, two, three, etc coordinates [52,53,54]. When one uses the full potential, and therefore multidimensional quadrature, the size of the quadrature grid is an important problem.…”

Section: Mathematical Methods For High-dimensional Problemsmentioning

confidence: 99%

Mathematical Methods in Quantum Molecular Dynamics

2016

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Section: Mathematical Methods For High-dimensional Problemsmentioning

confidence: 99%

Mathematical Methods in Quantum Molecular Dynamics

2016

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“…At each step, the n c and k c indices are constrained among themselves. Everything is clearly explained in [90]. The Smolyak grid is a sum of smaller direct product grids and therefore has structure that makes it possible to evaluate MVPs by doing sums sequentially.…”

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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“…Differences between the energies of ref. 50 and experiment are due to the potential. Differences between the energies of ref.…”

Section: Application To a 12-d Model For Acetonitrile (Ch3cn)mentioning

confidence: 99%

Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices

Leclerc

Carrington

2014

The Journal of Chemical Physics

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120

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We propose an iterative method for computing vibrational spectra that significantly reduces the memory cost of calculations. It uses a direct product primitive basis, but does not require storing vectors with as many components as there are product basis functions. Wavefunctions are represented in a basis each of whose functions is a sum of products (SOP) and the factorizable structure of the Hamiltonian is exploited. If the factors of the SOP basis functions are properly chosen, wavefunctions are linear combinations of a small number of SOP basis functions. The SOP basis functions are generated using a shifted block power method. The factors are refined with a rank reduction algorithm to cap the number of terms in a SOP basis function. The ideas are tested on a 20-D model Hamiltonian and a realistic CH 3 CN (12 dimensional) potential. For the 20-D problem, to use a standard direct product iterative approach one would need to store vectors with about 10 20 components and would hence require about 8 × 10 11 GB. With the approach of this paper only 1 GB of memory is necessary. Results for CH 3 CN agree well with those of a previous calculation on the same potential.

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Using nonproduct quadrature grids to solve the vibrational Schrödinger equation in 12D

Cited by 123 publications

References 102 publications

Mathematical Methods in Quantum Molecular Dynamics

Mathematical Methods in Quantum Molecular Dynamics

Iterative Methods for Computing Vibrational Spectra

Calculating vibrational spectra with sum of product basis functions without storing full-dimensional vectors or matrices

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