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.