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A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations
KTH, School of Biotechnology (BIO), Theoretical Chemistry (closed 20110512).
KTH, School of Biotechnology (BIO), Theoretical Chemistry (closed 20110512).
KTH, School of Biotechnology (BIO), Theoretical Chemistry (closed 20110512).
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.

Place, publisher, year, edition, pages
2007. Vol. 28, no 16, 2531-2537 p.
Keyword [en]
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: urn:nbn:se:kth:diva-16288DOI: 10.1002/jcc.20691ISI: 000250972500003Scopus ID: 2-s2.0-35948953260OAI: oai:DiVA.org:kth-16288DiVA: diva2:334330
Note
QC 20100817Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Matrix Algebra for Quantum Chemistry
Open this publication in new window or tab >>Matrix Algebra for Quantum Chemistry
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns methods of reduced complexity for electronic structure calculations.  When quantum chemistry methods are applied to large systems, it is important to optimally use computer resources and only store data and perform operations that contribute to the overall accuracy. At the same time, precarious approximations could jeopardize the reliability of the whole calculation.  In this thesis, the self-consistent field method is seen as a sequence of rotations of the occupied subspace. Errors coming from computational approximations are characterized as erroneous rotations of this subspace. This viewpoint is optimal in the sense that the occupied subspace uniquely defines the electron density. Errors should be measured by their impact on the overall accuracy instead of by their constituent parts. With this point of view, a mathematical framework for control of errors in Hartree-Fock/Kohn-Sham calculations is proposed.  A unifying framework is of particular importance when computational approximations are introduced to efficiently handle large systems.

An important operation in Hartree-Fock/Kohn-Sham calculations is the calculation of the density matrix for a given Fock/Kohn-Sham matrix. In this thesis, density matrix purification is used to compute the density matrix with time and memory usage increasing only linearly with system size. The forward error of purification is analyzed and schemes to control the forward error are proposed. The presented purification methods are coupled with effective methods to compute interior eigenvalues of the Fock/Kohn-Sham matrix also proposed in this thesis.New methods for inverse factorizations of Hermitian positive definite matrices that can be used for congruence transformations of the Fock/Kohn-Sham and density matrices are suggested as well.

Most of the methods above have been implemented in the Ergo quantum chemistry program. This program uses a hierarchic sparse matrix library, also presented in this thesis, which is parallelized for shared memory computer architectures. It is demonstrated that the Ergo program is able to perform linear scaling Hartree-Fock calculations.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. ix, 49 p.
Series
Trita-BIO-Report, ISSN 1654-2312 ; 2008:23
Keyword
linear scaling, reduced complexity, electronic structure, density functional theory, Hartree-Fock, density matrix purification, congruence transformation, inverse factorization, eigenvalues, eigenvectors, numerical linear algebra, occupied subspace, canonical angles, invariant subspace
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-9447 (URN)978-91-7415-160-2 (ISBN)
Public defence
2008-11-27, FB52, Roslagstullsbacken 21, AlbaNova, 13:15 (English)
Opponent
Supervisors
Note
QC 20100908Available from: 2008-11-06 Created: 2008-11-04 Last updated: 2010-09-08Bibliographically approved
2. Quantum Chemistry for Large Systems
Open this publication in new window or tab >>Quantum Chemistry for Large Systems
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis deals with quantum chemistry methods for large systems. In particular, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree-Fock calculations. Density matrix purification, which is a method used to construct the density matrix for a given Fock matrix, is also discussed.

The methods described are not only applicable in the Hartree-Fock case, but also in Kohn-Sham Density Functional Theory calculations, where the Coulomb and exchange matrices are parts of the Kohn-Sham matrix. Screening techniques for reducing the computational complexity of both Coulomb and exchange computations are discussed, including the fast multipole method, used for efficient computation of the Coulomb matrix.

The thesis also discusses how sparsity in the matrices occurring in Hartree-Fock and Kohn-Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. xi, 56 p.
Series
Trita-BIO-Report, ISSN 1654-2312 ; 2007:13
Keyword
quantum chemistry, fast multipole method, density matrix purification, sparse matrices
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-4561 (URN)978-91-7178-797-2 (ISBN)
Public defence
2007-12-12, FA32, Albanova, Roslagstullsbacken 21, 106 91 Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100817Available from: 2007-12-04 Created: 2007-12-04 Last updated: 2010-08-17Bibliographically approved
3. 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
4. Fock Matrix Construction for Large Systems
Open this publication in new window or tab >>Fock Matrix Construction for Large Systems
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This licentiate thesis deals with quantum chemistry methods for large systems. In particular, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree--Fock calculations.The methods described are also applicable in Kohn--Sham Density FunctionalTheory calculations, where the Coulomb and exchange matrices areparts of the Kohn--Sham matrix. Screening techniques for reducing the computational complexity of bot Coulomb and exchange computations are discussed, as well as the fast multipole method, used for efficient computation of the Coulomb matrix.

The thesis also discusses how sparsity in the matrices occurring in Hartree--Fock and Kohn--Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them.

As an example of a possible type of application, the thesis includes a theoretical study of Heisenberg exchange constants, using unrestricted Kohn--Sham Density Functional Theory calculations.

Place, publisher, year, edition, pages
Stockholm: KTH, 2006. viii, 28 p.
Keyword
quantum chemistry
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-4247 (URN)978-91-7178-535-0 (ISBN)
Presentation
2006-12-15, FD41, Albanova, Roslagstullsbacken 21, Stockholm, 11:15
Opponent
Supervisors
Note
QC 20101123Available from: 2006-12-19 Created: 2006-12-19 Last updated: 2010-11-23Bibliographically approved

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