NDLI: Numerical Solution of the HartreeFock Equation in Multilevel Tensor-Structured Format

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Khoromskaia, V. Khoromskij, B. N. Flad, H.-J.
Copyright Year	2011
Abstract	In this paper, we describe a novel method for a robust and accurate iterative solution of the self-consistent HartreeFock 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 discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three-dimensional Cartesian grid. The approximation error is estimated by $O(h^3)$, where $h=O(n^{-1})$ is the mesh size of an $n\times n\times n$ 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 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 the 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.
Starting Page	45
Ending Page	65
Page Count	21
File Format	PDF
ISSN	10648275
DOI	10.1137/090777372
e-ISSN	10957197
Issue Number	1
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-01-06
Access Restriction	Subscribed
Subject Keyword	discrete convolution Coloumb and exchange matrices tensor-truncated methods Spectral, collocation and related methods Other matrix algorithms Iterative methods for linear systems orthogonal Tucker tensor decomposition canonical model HartreeFock equation multilevel self-consistent field iteration Sparse matrices
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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