

Preprint 65/2007
Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Sambasiva Rao Chinnamsetty, Heinz-Jürgen Flad, Venera Khoromskaia, and Boris N. Khoromskij
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Submission date: 26. Jul. 2007 (revised version: November 2007)
Pages: 20
published in: Journal of computational physics, 228 (2009) 16, p. 5749-5762
DOI number (of the published article): 10.1016/j.jcp.2009.04.043
Bibtex
Keywords and phrases: Tucker-type tensor decomposition, Hartree-Fock equation, discrete convolution, orthogonal adaptive tensor-product basis
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Abstract:
In this paper we propose a novel approach based on the combination of
Tucker-type
and canonical tensor decomposition techniques for the efficient numerical
approximation of functions and operators in electronic structure
calculations. In particular, we study potential applications of tensor
approximations for
the numerical solution of Hartree-Fock and Kohn-Sham equations on 3D
Cartesian grids.
Low-rank orthogonal Tucker-type tensor approximations are investigated
for electron densities and Hartree potentials of simple molecules,
where exponential convergence with respect to the Tucker rank is observed.
This enables an efficient tensor-product convolution scheme for the
computation of the Hartree potential
using a collocation-type approximation via piecewise constant basis
functions on a uniform grid.
Combined with Richardson extrapolation,
our approach exhibits
convergence with
,
and requires
storage,
where r denotes the Tucker rank of the electron density with
almost uniformly in n (specifically,
).
For example, Hartree-Fock calculations for the CH
molecule, with a
pseudopotential on the C atom,
achieved accuracies of the order of
hartree with a grid-size n of
several hundreds.
For large 3D grids (
), the tensor-product convolution
scheme markedly outperforms
the 3D- FFT in both the computing time and storage requirements.