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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
25/2009

Computation of the Hartree-Fock Exchange by the Tensor-structured Methods

Venera Khoromskaia

Abstract

We introduce the novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is the (nonlocal) integral operator in $\mathbb{R}^3 \times \mathbb{R}^3$. Usually, this challenging computational problem is solved by laborious analytical evaluation of the two-electron integrals using "analytically separable" Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated "grey-box" numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set.

The core of our method is the low-rank tensor representation of arising functions and operators on $n\times n\times n$ Cartesian grid, and implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size $n$ enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for computation of the exchange operator and a recent tensor method of the Coulomb matrix evaluation are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide the new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

Received:
Jun 23, 2009
Published:
Jun 25, 2009
MSC Codes:
65F30, 65F50, 65N35
Keywords:
Hartree-Fock operator, exchange matrix, discrete tensor operations

Related publications

inJournal
2010 Journal Open Access
Venera Khoromskaia

Computation of the Hartree-Fock exchange by the tensor-structured methods

In: Computational methods in applied mathematics, 10 (2010) 2, pp. 204-218