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Efficient Density Matrix Methods for Large Scale Electronic Structure Calculations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Efficient and accurate methods for computing the density matrix are necessary to be able to perform large scale electronic structure calculations. For sufficiently sparse matrices, the computational cost of recursive polynomial expansions to construct the density matrix scales linearly with increasing system size. In this work, parameterless stopping criteria for recursive polynomial expansions are developed. The proposed stopping criteria automatically adapt to a change in the requested accuracy, perform at almost no additional cost and do not require any user-defined tolerances.

Compared to the traditional diagonalization approach, in linear scaling methods molecular orbitals are not readily available. In this work, the interior eigenvalue problem for the Fock/Kohn-Sham matrix is coupled to the recursive polynomial expansions. The idea is to view the polynomial, obtained in the recursive expansion, as an eigenvalue filter, giving large separation between eigenvalues of interest. An efficient method for computation of homo and lumo eigenvectors is developed. Moreover, a method for computation of multiple eigenvectors around the homo-lumo gap is implemented and evaluated.

An original method for inverse factorization of Hermitian positive definite matrices is developed in this work. Novel theoretical tools for analysis of the decay properties of matrix element magnitude in electronic structure calculations are proposed. Of particular interest is an inverse factor of the basis set overlap matrix required for the density matrix construction. It is shown that the proposed inverse factorization algorithm drastically reduces the communication cost compared to state-of-the-art methods.

To perform large scale numerical tests, most of the proposed methods are implemented in the quantum chemistry program Ergo, also presented in this thesis. The recursive polynomial expansion in Ergo is parallelized using the Chunks and Tasks matrix library. It is shown that the communication cost per process of the recursive polynomial expansion implementation tends to a constant in a weak scaling setting.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2019. , p. 51
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1858
Keywords [en]
electronic structure; linear scaling; density matrix; density functional theory; Hartree-Fock; recursive polynomial expansion; density matrix purification; eigenvectors; molecular orbitals; stopping criteria; inverse factorization; matrix sparsity; parallelization
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-393387ISBN: 978-91-513-0756-5 (print)OAI: oai:DiVA.org:uu-393387DiVA, id: diva2:1353081
Public defence
2019-11-08, 2446, ITC (Informationsteknologiskt centrum), Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2019-10-16 Created: 2019-09-20 Last updated: 2019-11-12
List of papers
1. Parameterless stopping criteria for recursive density matrix expansions
Open this publication in new window or tab >>Parameterless stopping criteria for recursive density matrix expansions
2016 (English)In: Journal of Chemical Theory and Computation, ISSN 1549-9618, E-ISSN 1549-9626, Vol. 12, p. 5788-5802Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-310710 (URN)10.1021/acs.jctc.6b00626 (DOI)000389866500011 ()
Projects
eSSENCE
Available from: 2016-10-26 Created: 2016-12-19 Last updated: 2019-09-20Bibliographically approved
2. On-the-fly computation of frontal orbitals in density matrix expansions
Open this publication in new window or tab >>On-the-fly computation of frontal orbitals in density matrix expansions
2018 (English)In: Journal of Chemical Theory and Computation, ISSN 1549-9618, E-ISSN 1549-9626, Vol. 14, p. 139-153Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-338750 (URN)10.1021/acs.jctc.7b00968 (DOI)000419998300013 ()29193971 (PubMedID)
Projects
eSSENCE
Available from: 2017-12-01 Created: 2018-01-12 Last updated: 2019-09-20Bibliographically approved
3. Ergo: An open-source program for linear-scaling electronic structure calculations
Open this publication in new window or tab >>Ergo: An open-source program for linear-scaling electronic structure calculations
2018 (English)In: SoftwareX, E-ISSN 2352-7110, Vol. 7, p. 107-111Article in journal (Refereed) Published
National Category
Software Engineering
Identifiers
urn:nbn:se:uu:diva-348090 (URN)10.1016/j.softx.2018.03.005 (DOI)000457139300020 ()
Projects
eSSENCE
Available from: 2018-04-05 Created: 2018-04-10 Last updated: 2019-09-20Bibliographically approved
4. Localized inverse factorization
Open this publication in new window or tab >>Localized inverse factorization
2018 (English)In: Computing Research Repository, no 1812.04919Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-381327 (URN)
Projects
eSSENCE
Available from: 2018-12-12 Created: 2019-04-08 Last updated: 2019-09-20Bibliographically approved
5. Efficient computation of the density matrix with error control on distributed computer systems
Open this publication in new window or tab >>Efficient computation of the density matrix with error control on distributed computer systems
2019 (English)In: Computing Research Repository, no 1909.12533Article in journal (Other academic) Submitted
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-393386 (URN)
Projects
eSSENCE
Available from: 2019-09-27 Created: 2019-09-20 Last updated: 2019-09-30Bibliographically approved
6. Multiple eigenvectors around the homo–lumo gap as a cheap by-product in linear scaling electronic structure calculations
Open this publication in new window or tab >>Multiple eigenvectors around the homo–lumo gap as a cheap by-product in linear scaling electronic structure calculations
2019 (English)In: Computing Research Repository, no 1909.11662Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-393385 (URN)
Projects
eSSENCE
Available from: 2019-09-25 Created: 2019-09-20 Last updated: 2019-09-30Bibliographically approved

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