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MiS Preprint

Tensor decomposition in electronic structure calculations on 3D Cartesian grids

Sambasiva Rao Chinnamsetty, Heinz-Jürgen Flad, Venera Khoromskaia and Boris N. Khoromskij


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 $n\times n\times n $ grid. Combined with Richardson extrapolation, our approach exhibits $O(h^{3})$ convergence with $h=O(n^{-1})$, and requires $O(3rn + r^3)$ storage, where $r$ denotes the Tucker rank of the electron density with $r\ll n$ almost uniformly in $n$ (specifically, $r=O(\log n)$). For example, Hartree-Fock calculations for the CH$_4$ molecule, with a pseudopotential on the C atom, achieved accuracies of the order of $10^{-5}$ hartree with a grid-size $n$ of several hundreds. For large $3D$ grids ($n\geq 128$), the tensor-product convolution scheme markedly outperforms the $3D-$ FFT in both the computing time and storage requirements.

Jul 26, 2007
Jul 26, 2007
Tucker-type tensor decomposition, Hartree-Fock equation, discrete convolution, orthogonal adaptive tensor-product basis

Related publications

2009 Repository Open Access
Boris N. Khoromskij, Venera Khoromskaia, Sambasiva Rao Chinnamsetty and Heinz-Jürgen Flad

Tensor decomposition in electronic structure calculations on 3D Cartesian grids

In: Journal of computational physics, 228 (2009) 16, pp. 5749-5762