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Density matrix purification with rigorous error control
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
KTH, Skolan för bioteknologi (BIO), Teoretisk kemi.
2008 (Engelska)Ingår i: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, nr 7Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
2008. Vol. 128, nr 7
Nyckelord [en]
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
Nationell ämneskategori
Kemi
Identifikatorer
URN: urn:nbn:se:kth:diva-14325DOI: 10.1063/1.2826343ISI: 000253336800008PubMedID: 18298139Scopus ID: 2-s2.0-39749201822OAI: oai:DiVA.org:kth-14325DiVA, id: diva2:332226
Anmärkning
QC 20100803Tillgänglig från: 2010-08-03 Skapad: 2010-08-03 Senast uppdaterad: 2020-03-09Bibliografiskt granskad
Ingår i avhandling
1. Matrix Algebra for Quantum Chemistry
Öppna denna publikation i ny flik eller fönster >>Matrix Algebra for Quantum Chemistry
2008 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2008. s. ix, 49
Serie
Trita-BIO-Report, ISSN 1654-2312 ; 2008:23
Nyckelord
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
Nationell ämneskategori
Teoretisk kemi
Identifikatorer
urn:nbn:se:kth:diva-9447 (URN)978-91-7415-160-2 (ISBN)
Disputation
2008-11-27, FB52, Roslagstullsbacken 21, AlbaNova, 13:15 (Engelska)
Opponent
Handledare
Anmärkning
QC 20100908Tillgänglig från: 2008-11-06 Skapad: 2008-11-04 Senast uppdaterad: 2010-09-08Bibliografiskt granskad
2. Quantum Chemistry for Large Systems
Öppna denna publikation i ny flik eller fönster >>Quantum Chemistry for Large Systems
2007 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH, 2007. s. xi, 56
Serie
Trita-BIO-Report, ISSN 1654-2312 ; 2007:13
Nyckelord
quantum chemistry, fast multipole method, density matrix purification, sparse matrices
Nationell ämneskategori
Teoretisk kemi
Identifikatorer
urn:nbn:se:kth:diva-4561 (URN)978-91-7178-797-2 (ISBN)
Disputation
2007-12-12, FA32, Albanova, Roslagstullsbacken 21, 106 91 Stockholm, 13:00
Opponent
Handledare
Anmärkning
QC 20100817Tillgänglig från: 2007-12-04 Skapad: 2007-12-04 Senast uppdaterad: 2010-08-17Bibliografiskt granskad

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