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Sparse matrix algebra for quantum modeling of large systems
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
2007 (English)In: Applied Parallel Computing - STATE OF THE ART IN SCIENTIFIC COMPUTING     / [ed] Kagstrom B, Elmroth E, Dongarra J, Wasniewski J, Berlin, Germany: Springer-Verlag , 2007, 90-99 p.Conference paper, Published paper (Refereed)
Abstract [en]

Matrices appearing in Hartree-Fock or density functional theory coming from discretization with help of atom-centered local basis sets become sparse when the separation between atoms exceeds some system-dependent threshold value. Efficient implementation of sparse matrix algebra is therefore essential in large-scale quantum calculations. We describe a unique combination of algorithms and data representation that provides high performance and strict error control in blocked sparse matrix algebra. This has applications to matrix-rnatrix multiplication, the Trace-Correcting Purification algorithm and the entire self-consistent field calculation.

Place, publisher, year, edition, pages
Berlin, Germany: Springer-Verlag , 2007. 90-99 p.
Series
Lecture Notes In Computer Science, ISSN 0302-9743 ; 4699
Keyword [en]
Algebra, Mathematical models, Purification, Quantum theory
National Category
Theoretical Chemistry
Identifiers
URN: urn:nbn:se:kth:diva-6547ISI: 000250904900011Scopus ID: 2-s2.0-38049013979ISBN: 978-3-540-75754-2 (print)OAI: oai:DiVA.org:kth-6547DiVA: diva2:11289
Conference
8th International Workshop on Applied Parallel Computing (PARA 2006), Umea, SWEDEN, JUN 18-21, 2006
Note
QC 20101123. Uppdaterad från Manuskript till Konferensbidrag (20101123).Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2010-11-23Bibliographically approved
In thesis
1. Sparse Matrices in Self-Consistent Field Methods
Open this publication in new window or tab >>Sparse Matrices in Self-Consistent Field Methods
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis is part of an effort to enable large-scale Hartree-Fock/Kohn-Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn-Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is σ(n3). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix-matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix-matrix multiplication can be performed with complexity σ(n).

A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO-LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method.

Place, publisher, year, edition, pages
Stockholm: Bioteknologi, 2006. x, 38 p.
Keyword
sparse matrix, self-consistent field, Hartree-Fock, Density Functional Theory, Density Matrix Purification
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-4219 (URN)978-91-7178-534-3 (ISBN)978-91-7178-534-5 (ISBN)
Presentation
2006-12-15, FD41, AlbaNova, Roslagstullsbacken 21, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20101123Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2010-11-23Bibliographically approved

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