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Matrix Algebra for Quantum ChemistryPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: KTH , 2008. , p. ix, 49
##### Series

Trita-BIO-Report, ISSN 1654-2312 ; 2008:23
##### Keywords [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, id: diva2:114034
##### Public defence

2008-11-27, FB52, Roslagstullsbacken 21, AlbaNova, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

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##### Note

QC 20100908Available from: 2008-11-06 Created: 2008-11-04 Last updated: 2010-09-08Bibliographically approved
##### List of papers

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.

1. Rotations of occupied invariant subspaces in self-consistent field calculations$(function(){PrimeFaces.cw("OverlayPanel","overlay332182",{id:"formSmash:j_idt495:0:j_idt499",widgetVar:"overlay332182",target:"formSmash:j_idt495:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Density matrix purification with rigorous error control$(function(){PrimeFaces.cw("OverlayPanel","overlay332226",{id:"formSmash:j_idt495:1:j_idt499",widgetVar:"overlay332226",target:"formSmash:j_idt495:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification$(function(){PrimeFaces.cw("OverlayPanel","overlay114024",{id:"formSmash:j_idt495:2:j_idt499",widgetVar:"overlay114024",target:"formSmash:j_idt495:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Recursive inverse factorization$(function(){PrimeFaces.cw("OverlayPanel","overlay114025",{id:"formSmash:j_idt495:3:j_idt499",widgetVar:"overlay114025",target:"formSmash:j_idt495:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Truncation of Small Matrix Elements Based on the Euclidean Norm for Blocked Data Structures$(function(){PrimeFaces.cw("OverlayPanel","overlay336347",{id:"formSmash:j_idt495:4:j_idt499",widgetVar:"overlay336347",target:"formSmash:j_idt495:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. A hierarchic sparse matrix data structure for large-scale Hartree-Fock/Kohn-Sham calculations$(function(){PrimeFaces.cw("OverlayPanel","overlay334330",{id:"formSmash:j_idt495:5:j_idt499",widgetVar:"overlay334330",target:"formSmash:j_idt495:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. Hartree-Fock calculations with linearly scaling memory usage$(function(){PrimeFaces.cw("OverlayPanel","overlay332243",{id:"formSmash:j_idt495:6:j_idt499",widgetVar:"overlay332243",target:"formSmash:j_idt495:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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