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Rotations of occupied invariant subspaces in self-consistent field calculations
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
KTH, School of Biotechnology (BIO), Theoretical Chemistry.
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.

Place, publisher, year, edition, pages
2008. Vol. 49, no 3, 032103- p.
Keyword [en]
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: urn:nbn:se:kth:diva-14313DOI: 10.1063/1.2884588ISI: 000254537500003Scopus ID: 2-s2.0-41549132179OAI: oai:DiVA.org:kth-14313DiVA: diva2:332182
Note
QC 20100803Available from: 2010-08-03 Created: 2010-08-03 Last updated: 2010-09-08Bibliographically 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

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