Il Nuovo Cimento D

, Volume 15, Issue 2–3, pp 269–277 | Cite as

A new method for determining excited states of quantum systems

  • G. Grosso
  • L. Martinelli
  • G. Pastori Parravicini
Article

Summary

A new method for computing excited states of a given operatorH is here presented. Our procedure is of particular value when its representation requires an orthonormal basis set of large dimension. In order to obtain the excited state ofH nearest in energy to any chosen trial energyE t, we consider the auxiliary operatorA=(H−E t)2. We show that a reasonable number of relaxations on appropriately generated low-order Krylov subspaces forA is sufficient to produce better and better approximations of its ground state; a high-accuracy final refinement of the ground state ofA is then possible through the standard Lanczos procedure. An important feature of our method is that storage memory limitations, encountered in the conventional determination of all eigenvalues of large systems, are here overcome. As an illustration of the method two significant examples are discussed.

PACS. 71.10

General theories and computational techniques (including many-body perturbation theory, density-functional theory, atomic sphere approximation methods, Fourier decomposition methods, etc.) 

PACS. 31.50

Excited states 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    See, for instanceR. D. Graft, G. Grosso, D. J. Lohrmann, L. Martinelli, S. Moroni, G. Pastori Parravicini andL. Resca: inProgress in Electron Properties of Solids, edited byE. Doni, R. Girlanda, G. Pastori Parravicini andA. Quattropani (Kluwer, Dordrecht, 1989), p. 409.Google Scholar
  2. [2]
    Yu. E. Perlin andM. Wagner (Editors):The Dynamical Jahn-Teller Effect in Localized Systems (North-Holland, Amsterdam, 1984).Google Scholar
  3. [3]
    N. Sakamoto andS. Muramatsu:Phys. Rev. B,17, 868 (1978);J. Phys. Soc. Jpn.,46, 1273 (1979);M. C. M. O'Brien andS. N. Evangelou:J. Phys. C,13, 611 (1980);M. C. M. O'Brien:J. Phys. C,16, 85, 6345 (1983);18, 4963 (1985).CrossRefADSGoogle Scholar
  4. [4]
    L. Martinelli, M. Passaro andG. Pastori Parravicini:Phys. Rev. B,39, 13343 (1989);43, 8395 (1991).CrossRefADSGoogle Scholar
  5. [5]
    F. Bassani, G. Jadonisi andB. Preziosi:Rep. Prog. Phys.,37, 1099 (1974);S. T. Pantelides:Rev. Mod. Phys.,50, 797 (1978);D. J. Lohrman, L. Resca, G. Pastori Parravicini andR. D. Graft:Phys. Rev. B,40, 8404, 8410 (1989).CrossRefADSGoogle Scholar
  6. [6]
    See, for instance,K. W. H. Stevens:Physica B,176, 1 (1992).CrossRefADSGoogle Scholar
  7. [7]
    E. Dagotto andJ. R. Schrieffer:Phys. Rev. B,43, 8705 (1991);H. J. Cannon, R. T. Scalettar andE. Fradkin:Phys. Rev. B,44, 5995 (1991);H. J. Schulz andT. A. L. Ziman:Europhys. Lett.,18, 355, (1992);J. Song andJ. F. Annett:Europhys. Lett.,18, 549 (1992).CrossRefADSGoogle Scholar
  8. [8]
    J. L. Martins andM. L. Cohen:Phys. Rev. B,37, 6134 (1988);J. L. Martins, N. Troullier andS. H. Wei:Phys. Rev. B,43, 2213 (1991).CrossRefADSGoogle Scholar
  9. [9]
    F. Bassani, F. Fumi andM. P. Tosi (Editors):Highlights of Condensed Matter Theory, Proceedings of the International School of Physics «E. Fermi», Course LXXXIX (North-Holland, Amsterdam, 1983).Google Scholar
  10. [10]
    C. Lanczos:J. Res. Nat. Bur. Standards,45, 255 (1950);49, 33 (1952);Applied Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1956).MathSciNetGoogle Scholar
  11. [11]
    R. Haydock, V. Heine andM. J. Kelly:J. Phys. C,5, 2845 (1972);8, 2591 (1975); see alsoD. W. Bullet, R. Haydock, V. Heine andM. J. Kelly: inSolid State Physics, edited byH. Erhenreich, F. Seitz andD. Turnbull, Vol.35 (Academic Press, New York, N.Y., 1980).CrossRefADSGoogle Scholar
  12. [12]
    G. Grosso andG. Pastori Parravicini:Adv. Chem. Phys.,62, 81, 133 (1985) and references quoted therein. See alsoP. Giannozzi, G. Grosso, S. Moroni andG. Pastori Parravicini:Appl. Num. Math.,4, 273 (1988).MathSciNetGoogle Scholar
  13. [13]
    J. C. Cullum andR. A. Willoughby:Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I and II (Birkhauser, Boston, Mass., 1985).Google Scholar
  14. [14]
    B. N. Parlett:The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J., 1980).Google Scholar
  15. [15]
    J. Alberty, J. Greensite andA. Patkos:Phys. Lett. B,138, 405 (1984);E. Dagotto andA. Moreo:Phys. Rev. D,31, 865 (1985).CrossRefADSGoogle Scholar
  16. [16]
    E. R. Gagliano, E. Dagotto, A. Moreo andF. C. Alcaraz:Phys. Rev.,34, 1677 (1986);E. R. Gagliano andC. A. Balseiro:Phys. Rev. Lett.,59, 2999 (1987);E. R. Gagliano andS. Bacci:Phys. Rev. D,36, 546 (1987).MathSciNetADSGoogle Scholar
  17. [17]
    B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema andC. B. Moler: EISPAKGuide, Lecture Notes in Computer Science (Springer, New York, N.Y., 1976).Google Scholar
  18. [18]
    J. H. Wilkinson:The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).Google Scholar
  19. [19]
    D. M. Wood andA. Zunger:J. Phys. A,18, 1343 (1985).MATHMathSciNetCrossRefADSGoogle Scholar
  20. [20]
    J. E. Hirsch andJ. E. Schrieffer:Phys. Rev. B,28, 5353 (1983);D. M. Ceperley andB. Bernu:J. Chem. Phys.,89, 6316 (1988).MathSciNetCrossRefADSGoogle Scholar
  21. [21]
    G. Grosso, L. Martinelli andG. Pastori Parravicini: to be published.Google Scholar
  22. [22]
    P. Giannozzi, G. Grosso andG. Pastori Parravicini:Riv. Nuovo Cimento,13, 1 (1990).Google Scholar

Copyright information

© Società Italiana di Fisica 1993

Authors and Affiliations

  • G. Grosso
    • 1
  • L. Martinelli
    • 1
  • G. Pastori Parravicini
    • 2
  1. 1.Dipartimento di Fisica dell'UniversitàINFM, CISM and GNSMPisaItalia
  2. 2.Dipartimento di Fisica «A. Volta» dell'UniversitàPaviaItalia

Personalised recommendations