Advertisement

Theoretical Chemistry Accounts

, 131:1109 | Cite as

Mayer’s orthogonalization: relation to the Gram-Schmidt and Löwdin’s symmetrical scheme

  • Péter R. Nagy
  • Péter R. Surján
  • Ágnes SzabadosEmail author
Regular Article

Abstract

A method introduced by Mayer (Theor Chem Acc 104:163, 2000) for generating an orthogonal set of basis vectors, perpendicular to an arbitrary start vector, is examined. The procedure provides the complementary vectors in closed form, expressed with the components of the start vector. Mayer’s method belongs to the family of orthogonalization schemes, which keep an arbitrary vector intact without introducing any non-physical sequence-dependence. It is shown that Mayer’s orthogonalization is recovered by performing a two-step combination of the Gram-Schmidt and Löwdin’s symmetrical orthogonalization. Processor time requirement of constructing Mayer’s orthonormal set is proportional to ∼N 2, in contrast to the rough ∼N 3 CPU requirement of performing either a full Gram-Schmidt or Löwdin’s symmetrical orthogonalization. Utility of Mayer’s orthogonalization is demonstrated on an electronic structure application using perturbation theory to improve multiconfigurational wavefunctions.

Keywords

Orthogonalization Mayer vectors 

Notes

Acknowledgments

The authors are indebted to Professor I. Mayer (Budapest) for a detailed inspection and instructive critical remarks on the present study. This work has been supported by the Hungarian National Research Fund (OTKA), grant numbers K-81588 and K-81590. The European Union and the European Social Fund have also provided financial support to the project under the grant agreements TÁMOP 4.2.1./B-09/1/KMR-2010-0003 and 4.2.2./B-10/1-2010-0030.

References

  1. 1.
    Löwdin PO (1950) J Chem Phys 18:365CrossRefGoogle Scholar
  2. 2.
    Löwdin PO (1992) Adv Quantum Chem 23:83CrossRefGoogle Scholar
  3. 3.
    Wannier GH (1937) Phys Rev 52(3):0191CrossRefGoogle Scholar
  4. 4.
    Löwdin PO (1956) Adv Phys 5(17):1CrossRefGoogle Scholar
  5. 5.
    Schweinl HC, Wigner EP (1970) J Math Phys 11(5):1693CrossRefGoogle Scholar
  6. 6.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipes in Fortran 90 (2nd ed.): the art of parallel scientific computing. Cambridge University Press, New York, NYGoogle Scholar
  7. 7.
    Chaturvedi S, Kapoor AK, Srinivasan V (1998) J Phys A-Math Gen 31(19):L367CrossRefGoogle Scholar
  8. 8.
    Löwdin PO (1970) Adv Quantum Chem 5:185CrossRefGoogle Scholar
  9. 9.
    Mayer I (2002) Int J Quantum Chem 90(1):63. doi: 10.1002/qua.981 CrossRefGoogle Scholar
  10. 10.
    Mayer I (2000) Theor Chem Acc 104:163CrossRefGoogle Scholar
  11. 11.
    Mayer I (2003) Simple theorems, proofs, and derivations in quantum chemistry. Kluwer, New YorkGoogle Scholar
  12. 12.
    Beebe NH, Linderberg J (1977) Int J Quantum Chem 12:683CrossRefGoogle Scholar
  13. 13.
    Pedersen TB, Aquilante F, Lindh R (2009) Theor Chem Acc 124:1CrossRefGoogle Scholar
  14. 14.
    Durand P, Malrieu JP (1987) Adv Chem Phys 67:1CrossRefGoogle Scholar
  15. 15.
    Roos B, Andersson K, Fülscher M, Malmqvist PÅ, Serrano-Andrés L, Pierloot K, Merchán M (1996) Adv Chem Phys 93:219CrossRefGoogle Scholar
  16. 16.
    Hoffmann MR, Datta D, Das S, Mukherjee D, Szabados Á, Rolik Z, Surján PR (2009) J Chem Phys 131:204104CrossRefGoogle Scholar
  17. 17.
    Pulay P (2011) Int J Quantum Chem 111:3273CrossRefGoogle Scholar
  18. 18.
    Rolik Z, Szabados Á, Surján PR (2003) J Chem Phys 119:1922CrossRefGoogle Scholar
  19. 19.
    Kobayashi M, Szabados Á, Nakai H, Surján PR (2010) J Chem Theory Comput 6:2024CrossRefGoogle Scholar
  20. 20.
    Surján P, Rolik Z, Szabados Á, Kőhalmi D (2004) Ann Phys (Leipzig) 13:223CrossRefGoogle Scholar
  21. 21.
    Surján PR (1999) Topics Curr Chem 203:63CrossRefGoogle Scholar
  22. 22.
    Dunning TH Jr. (1989) J Chem Phys 90:1007CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Péter R. Nagy
    • 1
  • Péter R. Surján
    • 1
  • Ágnes Szabados
    • 1
    Email author
  1. 1.Laboratory of Theoretical ChemistryEötvös UniversityBudapestHungary

Personalised recommendations