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Cholesky Decomposition Techniques in Electronic Structure Theory

  • Francesco AquilanteEmail author
  • Linus Boman
  • Jonas Boström
  • Henrik Koch
  • Roland Lindh
  • Alfredo Sánchez de Merás
  • Thomas Bondo Pedersen
Chapter
Part of the Challenges and Advances in Computational Chemistry and Physics book series (COCH, volume 13)

Abstract

We review recently developed methods to efficiently utilize the Cholesky decomposition technique in electronic structure calculations. The review starts with a brief introduction to the basics of the Cholesky decomposition technique. Subsequently, examples of applications of the technique to ab inito procedures are presented. The technique is demonstrated to be a special type of a resolution-of-identity or density-fitting scheme. This is followed by explicit examples of the Cholesky techniques used in orbital localization, computation of the exchange contribution to the Fock matrix, in MP2, gradient calculations, and so-called method specific Cholesky decomposition. Subsequently, examples of calibration of the method with respect to computed total energies, excitation energies, and auxiliary basis set pruning are presented. In particular, it is demonstrated that the Cholesky method is an unbiased method to derive auxiliary basis sets. Furthermore, details of the implementational considerations are put forward and examples from a parallel Cholesky decomposition scheme is presented. Finally, an outlook and perspectives are presented, followed by a summary and conclusions section. We are of the opinion that the Cholesky decomposition method is a technique that has been overlooked for too long. We have just recently started to understand how to efficiently incorporate the method in existing ab initio programs. The full potential of the Cholesky technique has not yet been fully explored.

Keywords

Cholesky decomposition Reduced scaling Low-rank tensor approximation Inner projection Density fitting Multi-configurational methods Orbital localization Linear-scaling correlation methods 

Notes

Acknowledgments

ASM acknowledges financial suport from Spanish FEDER+MEC project CTQ2007-67143-C02-01/BQU. RL and JB would like to thank the Swedish Research Council directly and through the Linnaeus Center of Excellence on Organizing Molecular Matter at Lund University, Sweden, for financial support. TBP acknowledges financial support from the CoE Centre for Theoretical and Computational Chemistry (179568/V30). FA would like to acknowledge funding from the Swiss National Science Foundation (SNF), Grant No. 200020-120007. HK and LB would like to thank the Norwegian Research Council for support (Grant No. 154011/420). The authors thank Eirik Hjertenæs at the Norwegian University of Science and Technology for proof-reading the manuscript.

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Francesco Aquilante
    • 1
    Email author
  • Linus Boman
    • 2
  • Jonas Boström
    • 3
  • Henrik Koch
    • 2
  • Roland Lindh
    • 4
  • Alfredo Sánchez de Merás
    • 5
  • Thomas Bondo Pedersen
    • 6
  1. 1.Department of Physical Chemistry, Sciences IIUniversity of GenevaGeneva 4Switzerland
  2. 2.Department of ChemistryNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Department of Theoretical Chemistry, Chemical CenterUniversity of LundLundSweden
  4. 4.Department of Quantum ChemistryUppsala UniversityUppsalaSweden
  5. 5.Instituto de Ciencia MolecularUniversitat de ValènciaValenciaSpain
  6. 6.Department of Chemistry, Centre for Theoretical and Computational ChemistryUniversity of OsloOsloNorway

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