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

By Venera Khoromskaia. 2010

18 citations recorded by the source corpus.

Abstract We propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator. Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the “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 point of our method is a low-rank tensor representation of arising functions and operators on an n×n×n Cartesian grid and the 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 evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Computational Theory and Mathematics, Numerical Methods & Scientific Computing, Computational Chemistry & Drug Discovery

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In Atomic and Molecular Physics, and Optics

In Computational Mathematics

In Computational Theory and Mathematics

In Numerical Methods & Scientific Computing

In Computational Chemistry & Drug Discovery

Citations in the corpus

DMRG Approach to Fast Linear Algebra in the TT-Format (2011, unknown): “…were ﬁrst obtained in the series of papers by D.V. Savostyanov and others [14, 15, 16, 17, 18] and in the works of V. Khoromskaia and B. Khoromskij [19, 20, 21, 22, 23] for the Tucker format, that also has nice stability properties, but can not be used in high dimensions due to the exponential…”
Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis (2013, methodology): “The presented grid-based approach benefits from the fast O(n log n) tensor-product convolution with the 6D Newton kernel over a large n(3)×n(3) grid, [25], which has already proved the numerical efficiency in the evaluation of the Hartree and exchange integrals [22, 19, 23].”