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

Venera Khoromskaia, Boris N. Khoromskij, and Reinhold Schneider

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Submission date: 22. Jun. 2011 (revised version: September 2011)
Pages: 19
published in: Computational methods in applied mathematics, 11 (2011) 3, p. 327-341 
DOI number (of the published article): 10.2478/cmam-2011-0018
Bibtex
MSC-Numbers: 65F30, 65N35, 65F50, 65F10
Keywords and phrases: tensor-structured methods, QTT format, Hartree-Fock equation, Electronic structure calculations, Coloumb and exchange matrices
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Abstract:
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 × n × 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.