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

By Venera Khoromskaia, Boris N. Khoromskij, Reinhold Schneider. 2011

27 citations recorded by the source corpus.

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 quantized-TT (QTT) formats. It leads to O(log n) complexity at computationally extensive stages in the rank-structured calculation with the respective 3D Hartree and exchange potentials discretized on large n×n×n Cartesian grids. The numerical examples for some volumetric organic molecules confirm that the QTT ranks of these potentials are nearly independent of the one-dimension grid size n. Thus, paradoxically, the complexity of the grid-based evaluation of the Coulumb and exchange matrices becomes almost independent of the grid size, being regulated only by the structure of a molecular system. As a result, the grid approximation of the Hartree-Fock equation allows to gain the high resolution with a guaranteed accuracy.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Spectroscopy

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “QTT has been applied to the solution of PDEs and eigenvalue problems [144, 154, 162], evaluation of boundary integrals in BEM [164], convolution [107] and the FFT [63, 108, 230].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “It is worth to note, that the QTT-format was already successfully applied to solving highdimensional elliptic equations [28], multiparametric SPDEs [27], and in electronic structure calculation [18] and the log-volume complexity was con rmed numerically and theoretically [9], so it is a natural idea”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “To use all the benefits from tensor decompositions for low-dimensional data, we use the QTT format [26, 27, 28, 29, 44, 30, 45, 40].”
Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis (2013, background): “theoretical concepts of the rank-structured representation to operators and functions as well as the novel tensor-structured numerical methods designed for solving multi-dimensional PDEs have been developed and successfully applied [26, 7, 23, 20, 21, 9].”