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Sparse Matrices in Self-Consistent Field Methods
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
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 [en]
sparse matrix, self-consistent field, Hartree-Fock, Density Functional Theory, Density Matrix Purification
National Category
Theoretical Chemistry
Identifiers
URN: urn:nbn:se:kth:diva-4219ISBN: 978-91-7178-534-3 (print)ISBN: 978-91-7178-534-5 OAI: oai:DiVA.org:kth-4219DiVA: diva2:11291
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
List of papers
1. Systematic sparse matrix error control for linear scaling electronic structure calculations
Open this publication in new window or tab >>Systematic sparse matrix error control for linear scaling electronic structure calculations
2005 (English)In: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 26, no 15, 1628-1637 p.Article in journal (Refereed) Published
Abstract [en]

 Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.

Keyword
matrix error control, linear scaling, density purification
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-6544 (URN)10.1002/jcc.20315 (DOI)000232570300008 ()2-s2.0-27844477884 (Scopus ID)
Note
QC 20101123Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2017-12-14Bibliographically approved
2. Determination of the chemical potential and HOMO/LUMO orbitals in density purification methods
Open this publication in new window or tab >>Determination of the chemical potential and HOMO/LUMO orbitals in density purification methods
2006 (English)In: Chemical Physics Letters, ISSN 0009-2614, E-ISSN 1873-4448, Vol. 432, no 4-6, 591-594 p.Article in journal (Refereed) Published
Abstract [en]

Several density purification methods have been proposed to achieve linear scaling in Hartree-Fock and Kohn-Sham calculations. However, only the density is found, while in the traditional diagonalization method the orbitals are also obtained. This could be seen as a drawback as in many cases one would like at least the HOMO and LUMO orbitals and their orbital energies. In this letter, we show how a value for the chemical potential can be obtained as a by-product of density purification methods at negligible cost. Once the chemical potential is known, MO's around the HOMO-LUMO gap can be calculated with the Spectral Transformation Lanczos method.

Keyword
Chemical analysis, Cost effectiveness, Orbits, Purification
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-6545 (URN)10.1016/j.cplett.2006.10.090 (DOI)000243067000039 ()2-s2.0-34547868145 (Scopus ID)
Note
QC 20101123. Uppdaterad från In press till Published (20101123).Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2017-12-14Bibliographically approved
3. A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations
Open this publication in new window or tab >>A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations
2007 (English)In: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 28, no 16, 2531-2537 p.Article in journal (Refereed) Published
Abstract [en]

A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in HartreeFock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.

Keyword
sparse matrix, C plus plus templates, Hartree-Fock, Density Functional Theory, inverse Cholesky decomposition, symmetric matrix square, electronic-structure calculations, approximate inverse preconditioners, conjugate-gradient method, consistent-field theory, density-matrix, diagonalization, search, purification, atoms
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-16288 (URN)10.1002/jcc.20691 (DOI)000250972500003 ()2-s2.0-35948953260 (Scopus ID)
Note
QC 20100817Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
4. Sparse matrix algebra for quantum modeling of large systems
Open this publication in new window or tab >>Sparse matrix algebra for quantum modeling of large systems
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
Series
Lecture Notes In Computer Science, ISSN 0302-9743 ; 4699
Keyword
Algebra, Mathematical models, Purification, Quantum theory
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-6547 (URN)000250904900011 ()2-s2.0-38049013979 (Scopus ID)978-3-540-75754-2 (ISBN)
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
5. Near-Idempotent Matrices
Open this publication in new window or tab >>Near-Idempotent Matrices
(English)Manuscript (Other academic)
Identifiers
urn:nbn:se:kth:diva-6548 (URN)
Note
QC 20101123Available from: 2006-12-11 Created: 2006-12-11 Last updated: 2010-11-23Bibliographically approved

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