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MiS Preprint

Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format

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


In this paper, we describe a novel method for robust and accurate iterative solution of the self-consistent Hartree-Fock equation in $\mathbb{R}^3$ based on the idea of tensor-structured computation of the electron density and the nonlinear Hartree and (nonlocal) exchange operators at all steps of the iterative process. We apply the self-consistent field (SCF) iteration to the Galerkin discretisation in a set of low separation rank basis functions that are solely specified by the respective values on a 3D Cartesian grid. The approximation error is estimated by $O(h^3)$, where $h=O(n^{-1})$ is the mesh size of $n\times n \times n$ tensor grid, while the numerical complexity to compute the Galerkin matrices scales linearly in $n \log n$.

We propose the tensor-truncated version of the SCF iteration using the traditional direct inversion in the iterative subspace (DIIS) scheme enhanced by the multilevel acceleration with the grid dependent termination criteria at each discretization level. This implies that the overall computational cost scales almost linearly in the univariate problem size $n$.

Numerical illustrations are presented for the all electron case of H$_2$O, and pseudopotential case of CH$_4$ and CH$_3$OH molecules. The proposed scheme is not restricted to a priori given rank-1 basis sets allowing analytically integrable convolution transform with the Newton kernel, that opens further perspectives for promotion of the tensor-structured methods in computational quantum chemistry.

Jul 24, 2009
Jul 28, 2009
MSC Codes:
65F30, 65F50, 65N35, 65F10
Hartree-Fock equation, Tucker/canonical models, discrete multivariate convolution, Tensor-truncated methods, multilevel SCF iteration

Related publications

2011 Repository Open Access
Boris N. Khoromskij, Venera Khoromskaia and Heinz-Jürgen Flad

Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format

In: SIAM journal on scientific computing, 33 (2011) 1, pp. 45-65