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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.