Tensor decomposition in electronic structure calculations on 3D Cartesian grids

In this paper, we investigate 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 applicability of tensor approximations for the numerical solution of Hartree–Fock and Kohn–Sham equations on 3D Cartesian grids. We show that the orthogonal Tucker-type tensor approximation of electron density and Hartree potential of simple molecules leads to low tensor rank representations. 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 × n × n grid. Combined with the Richardson extrapolation, our approach exhibits O ( h 3 ) convergence in the grid-size h = O ( n - 1 ) . Moreover, this requires O ( 3 rn + r 3 ) storage, where r denotes the Tucker rank of the electron density with r = O ( log n ) , almost uniformly in n. For example, calculations of the Coulomb matrix and the Hartree–Fock energy for the CH 4 molecule, with a pseudopotential on the C atom, achieved accuracies of the order of 10 - 6 hartree with a grid-size n of several hundreds. Since the tensor-product convolution in 3D is performed via 1D convolution transforms, our scheme markedly outperforms the 3D-FFT in both the computing time and storage requirements.