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MiS Preprint

QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations

Venera Khoromskaia, Boris N. Khoromskij and Reinhold Schneider


In this paper, the tensor-structured numerical evaluation of the Coulomb and exchange operators in the Hartree-Fock equation is supplemented by the usage of recent quantics-TT (QTT) formats.

It leads to $O(\log n)$ complexity at computationally extensive stages in the rank-structured calculation of the respective 3D and 6D integral operators including the Newton convolving kernel, and discretized on the $n\times n\times n$ Cartesian grid. The numerical examples for some volumetric organic molecules show that the QTT ranks of the Coulomb and exchange operators are nearly independent on the one-dimension grid size $n$. Thus, paradoxically, the complexity of the grid-based evaluation of the 3D integral operators becomes almost independent on the grid size, being regulated only by the structure of a molecular system. Hence, the grid-based approximation of the Hartree-Fock equation allows to gain a guaranteed accuracy. In numerical illustrations we present the QTT approximation of the Hartree and exchange operators for some moderate size molecules.

MSC Codes:
65F30, 65N35, 65F50, 65F10
tensor-structured methods, QTT format, Hartree-Fock equation, Electronic structure calculations, Coloumb and exchange matrices

Related publications

2011 Journal Open Access
Venera Khoromskaia, Boris N. Khoromskij and Reinhold Schneider

QTT representation of the Hartree and exchange operators in electronic structure calculations

In: Computational methods in applied mathematics, 11 (2011) 3, pp. 327-341