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Matrix Algebra for Quantum Chemistry
KTH, School of Biotechnology (BIO), Theoretical 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 [en]
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: urn:nbn:se:kth:diva-9447ISBN: 978-91-7415-160-2 (print)OAI: oai:DiVA.org:kth-9447DiVA: diva2:114034
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
List of papers
1. Rotations of occupied invariant subspaces in self-consistent field calculations
Open this publication in new window or tab >>Rotations of occupied invariant subspaces in self-consistent field calculations
2008 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, no 3, 032103- p.Article in journal (Refereed) Published
Abstract [en]

In this article, the self-consistent field (SCF) procedure as used in Hartree-Fock and Kohn-Sham calculations is viewed as a sequence of rotations of the so-called occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence. (C) 2008 American Institute of Physics.

Keyword
Electronic-structure calculations; density-matrix search; fast multipole method; convergence acceleration; expansion methods; large systems; hartree-fock; diagonalization; optimization; purification
National Category
Chemical Sciences
Identifiers
urn:nbn:se:kth:diva-14313 (URN)10.1063/1.2884588 (DOI)000254537500003 ()2-s2.0-41549132179 (Scopus ID)
Note
QC 20100803Available from: 2010-08-03 Created: 2010-08-03 Last updated: 2017-12-12Bibliographically approved
2. Density matrix purification with rigorous error control
Open this publication in new window or tab >>Density matrix purification with rigorous error control
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, no 7Article in journal (Refereed) Published
Abstract [en]

Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification. (C) 2008 American Institute of Physics.

Keyword
Electronic-structure calculations; linear scaling computation; consistent-field theory; fast multipole method; chebyshev expansion methods; binding molecular-dynamics; conjugate-gradient method; hartree-fock; tight-binding; parallel computation
National Category
Chemical Sciences
Identifiers
urn:nbn:se:kth:diva-14325 (URN)10.1063/1.2826343 (DOI)000253336800008 ()2-s2.0-39749201822 (Scopus ID)
Note
QC 20100803Available from: 2010-08-03 Created: 2010-08-03 Last updated: 2017-12-12Bibliographically approved
3. Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification
Open this publication in new window or tab >>Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, no 17, 176101- p.Article in journal (Refereed) Published
Abstract [en]

Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.

Keyword
purification, Lanczos, spectral filter, density functional theory, eigenvalue, eigenvector
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-9445 (URN)10.1063/1.2913072 (DOI)000256232400044 ()2-s2.0-43149088044 (Scopus ID)
Note
QC 20100908Available from: 2008-11-04 Created: 2008-11-04 Last updated: 2017-12-14Bibliographically approved
4. Recursive inverse factorization
Open this publication in new window or tab >>Recursive inverse factorization
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, no 10, 104105- p.Article in journal (Refereed) Published
Abstract [en]

A recursive algorithm for the inverse factorization S−1=ZZ* of Hermitian positive definite matrices S is proposed. The inverse factorization is based on iterative refinement [A.M.N. Niklasson, Phys. Rev. B 70, 193102 (2004)] combined with a recursive decomposition of S. As the computational kernel is matrix-matrix multiplication, the algorithm can be parallelized and the computational effort increases linearly with system size for systems with sufficiently sparse matrices. Recent advances in network theory are used to find appropriate recursive decompositions. We show that optimization of the so-called network modularity results in an improved partitioning compared to other approaches. In particular, when the recursive inverse factorization is applied to overlap matrices of irregularly structured three-dimensional molecules.

Keyword
Hermitian matrices, matrix decomposition, recursion method, sparse matrices, congruence transformation, inverse factorization, iterative refinement
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-9446 (URN)10.1063/1.2884921 (DOI)000254025300007 ()2-s2.0-40849134177 (Scopus ID)
Note
QC 20100908Available from: 2008-11-04 Created: 2008-11-04 Last updated: 2017-12-14Bibliographically approved
5. Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures
Open this publication in new window or tab >>Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures
2009 (English)In: Journal of Computational Chemistry, ISSN 0192-8651, E-ISSN 1096-987X, Vol. 30, no 6, 974-977 p.Article in journal (Refereed) Published
Abstract [en]

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.

Keyword
sparsity, linear scaling, Hartree-Fock, DFT, density functional theory, blocked data structure, Euclidean norms, Lanczos, sparse matrix, Frobenius norm, electronic-structure calculations, consistent-field theory, density-matrix, expansion methods, diagonalization, minimization, purification, search
National Category
Theoretical Chemistry
Identifiers
urn:nbn:se:kth:diva-18301 (URN)10.1002/jcc.21120 (DOI)000264651200015 ()2-s2.0-65449174900 (Scopus ID)
Note
QC 20100817Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
6. 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
7. Hartree-Fock calculations with linearly scaling memory usage
Open this publication in new window or tab >>Hartree-Fock calculations with linearly scaling memory usage
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, no 18Article in journal (Refereed) Published
Abstract [en]

We present an implementation of a set of algorithms for performing Hartree-Fock calculations with resource requirements in terms of both time and memory directly proportional to the system size. In particular, a way of directly computing the Hartree-Fock exchange matrix in sparse form is described which gives only small addressing overhead. Linear scaling in both time and memory is demonstrated in benchmark calculations for system sizes up to 11 650 atoms and 67 204 Gaussian basis functions on a single computer with 32 Gbytes of memory. The sparsity of overlap, Fock, and density matrices as well as band gaps are also shown for a wide range of system sizes, for both linear and three-dimensional systems. (C) 2008 American Institute of Physics.

Keyword
Electronic-structure calculations; fast multipole method; consistent-field theory; density-matrix search; convergence acceleration; parallel computation; expansion methods; exchange matrix; coulomb matrix; large systems
National Category
Chemical Sciences
Identifiers
urn:nbn:se:kth:diva-14334 (URN)10.1063/1.2918357 (DOI)000255983500013 ()2-s2.0-43949128760 (Scopus ID)
Note
QC 20100803Available from: 2010-08-03 Created: 2010-08-03 Last updated: 2017-12-12Bibliographically approved

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