

Preprint 44/2009
Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format
Boris N. Khoromskij, Venera Khoromskaia, and Heinz-Jürgen Flad
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Submission date: 24. Jul. 2009 (revised version: October 2010)
Pages: 26
published in: SIAM journal on scientific computing, 33 (2011) 1, p. 45-65
DOI number (of the published article): 10.1137/090777372
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: Hartree-Fock equation, Tucker/canonical models, discrete multivariate convolution, Tensor-truncated methods, multilevel SCF iteration
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Abstract:
In this paper, we describe a novel
method for robust and accurate iterative solution of the self-consistent
Hartree-Fock equation in 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
, where
is the mesh size of
tensor grid,
while the numerical complexity to compute the Galerkin matrices
scales linearly in
.
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
O, and pseudopotential case of CH
and CH
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.