Tensor Numerical Methods in Quantum Chemistry

Venera Khoromskaia; Boris N. Khoromskij

Published by De Gruyter 2018

Tensor Numerical Methods in Quantum Chemistry

Venera Khoromskaia and Boris N. Khoromskij

https://doi.org/10.1515/9783110365832

Overview
Contents

About this book

The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results　 from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations.

Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 10¹⁵.
Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules.
Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing.

This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.

Author / Editor information

Venera Khoromskaia and Boris N. Khoromskij, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany.

Topics

Frontmatter Publicly Available Download PDF	I
Contents Publicly Available Download PDF	V
1. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
2. Rank-structured formats for multidimensional tensors Requires Authentication Unlicensed Licensed Download PDF	9
3. Rank-structured grid-based representations of functions in ℝ^d Requires Authentication Unlicensed Licensed Download PDF	37
4. Multiplicative tensor formats in ℝ^d Requires Authentication Unlicensed Licensed Download PDF	79
5. Multidimensional tensor-product convolution Requires Authentication Unlicensed Licensed Download PDF	87
6. Tensor decomposition for analytic potentials Requires Authentication Unlicensed Licensed Download PDF	99
7. The Hartree–Fock equation Requires Authentication Unlicensed Licensed Download PDF	105
8. Multilevel grid-based tensor-structured HF solver Requires Authentication Unlicensed Licensed Download PDF	111
9. Grid-based core Hamiltonian Requires Authentication Unlicensed Licensed Download PDF	129
10. Tensor factorization of grid-based two-electron integrals Requires Authentication Unlicensed Licensed Download PDF	141
11. Fast grid-based Hartree–Fock solver by factorized TEI Requires Authentication Unlicensed Licensed Download PDF	157
12. Calculation of excitation energies of molecules Requires Authentication Unlicensed Licensed Download PDF	179
13. Density of states for a class of rank-structured matrices Requires Authentication Unlicensed Licensed Download PDF	201
14. Tensor-based summation of long-range potentials on finite 3D lattices Requires Authentication Unlicensed Licensed Download PDF	215
15. Range-separated tensor format for many-particle systems Requires Authentication Unlicensed Licensed Download PDF	241
Bibliography Requires Authentication Unlicensed Licensed Download PDF	271
Index Requires Authentication Unlicensed Licensed Download PDF	287

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate students and researchers in numerical analysis, scientific computing, and materials science.
Pages
- Front matter: 8
- Main content: 289
Keywords: Tensornumerische Methoden

eBook

Published: June 11, 2018
ISBN: 9783110365832

Hardcover

Published: June 11, 2018
ISBN: 9783110370157