MiS Preprint Repository

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 ($n\ge 128$), the tensor-product convolution scheme markedly outperforms the $3D-$ FFT in both the computing time and storage requirements.