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BIT Numerical Mathematics

, Volume 36, Issue 3, pp 563–578 | Cite as

Solution of large eigenvalue problems in electronic structure calculations

  • Y. Saad
  • A. Stathopoulos
  • J. Chelikowsky
  • K. Wu
  • S. Öğüt
Article

Abstract

Predicting the structural and electronic properties of complex systems is one of the outstanding problems in condensed matter physics. Central to most methods used in molecular dynamics is the repeated solution of large eigenvalue problems. This paper reviews the source of these eigenvalue problems, describes some techniques for solving them, and addresses the difficulties and challenges which are faced. Parallel implementations are also discussed.

Key words

Eigenvalue problem electronic structure calculation paralell implementation 

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References

  1. 1.
    S. Baroni and P. Giannozzi,Towards very large-scale electronic-structure calculations, Europhys. Lett. 17 (1992), pp. 547–552.Google Scholar
  2. 2.
    E. J. Bylaska, S. Kohn, S. Baden, A. Edelman, R. Kawai, M. E. Ong, and J. Weare,Scalable parallel numerical methods and software tools for material design, in Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, San Francisco, CA, 1995, pp. 219–224.Google Scholar
  3. 3.
    J. R. Chelikowsky, N. Troullier, and Y. Saad,Finite-difference-pseudopotential method: electronic structure calculations without a basis, Phys. Rev. Lett., 72 (1994), pp. 1240–1243.Google Scholar
  4. 4.
    J. R. Chelikowsky and M. L. Cohen,Handbook on Semiconductors Vol. 1, P. T. Landsberg ed., Elsevier, Amsterdam, 1992.Google Scholar
  5. 5.
    J. R. Chelikowsky, N. Troullier, K. Wu, and Y. Saad,Higher order finite difference pseudopotential method: An application to diatomic molecules, Phys. Rev. B 50 (1994), pp. 11355–11364.Google Scholar
  6. 6.
    J. R. Chelikowsky, N. R. Troullier, X. Jing, D. Dean, N. Binggeli, K. Wu, and Y. Saad,Algorithms for the structural properties of clusters, Computer Physics Communications, 85 (1995), pp. 325–335.Google Scholar
  7. 7.
    D. T. Colbert and W. H. Miller,A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method, J. Comput. Phys., 96 (1992), pp. 1982–1991.Google Scholar
  8. 8.
    M. Crouzeix, B. Philippe, and M. Sadkane,The Davidson method, SIAM J. Sci. Comput., 15 (1994), pp. 62–76.Google Scholar
  9. 9.
    J. Cullum and R. A. Willoughby,Lanczos algorithms for large symmetric eigenvalue computations 2: Programs, Progress in Scientific Computing, v. 4, Birkhauser, Boston, 1985.Google Scholar
  10. 10.
    J. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart,Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp., 30 (1976), pp. 772–795.Google Scholar
  11. 11.
    E. R. Davidson,The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Comput. Phys., 17 (1975), 87–94.Google Scholar
  12. 12.
    D. E. Ellis and G. S. Painter, Phys. Rev. B, 2 (1970), pp. 2887.Google Scholar
  13. 13.
    B. Fornberg and D. M. Sloan,A review of pseudospectral methods for solving partial differential equations, Acta Numerica, (1994), pp. 203–267.Google Scholar
  14. 14.
    A. Geist, A. Beguelin, J. Dongarra, W. Jiang, R. Manchek, and V. Sunderam,PVM3 users's guide and reference manual, Tech. Report TM-12187, Oak Ridge National Laboratory, TN, 1994.Google Scholar
  15. 15.
    X. Jing, N. R. Troullier, J. R. Chelikowsky, K. Wu, and Y. Saad,Vibrational modes of silicon nanostructures, Solid State Communication, 96:4 (1995), pp. 231–235.Google Scholar
  16. 16.
    X. Jing, N. R. Troullier, D. Dean, N. Binggeli, J. R. Chelikowsky, K. Wu, and Y. Saad,Ab initio molecular dynamics simulations of Si clusters using the higher-order finite-difference-pseudopotential method, Phys. Rev. B, 50 (1994), pp. 12234–12237.Google Scholar
  17. 17.
    G. E. Kimball and G. H. Shortley, Phys. Rev., 45 (1934), pp. 815.Google Scholar
  18. 18.
    C. M. Kirkpatrick and D. S. Marynick,Localized molecular orbital studies of transition metal complexes. II. Simple π-accepting ligands, J. Comput. Chem., 6 (1985), pp. 142–147.Google Scholar
  19. 19.
    S. R. Kohn and S. Baden,The parallelization of an adaptive multigrid eigenvalue solver with LPARX, in Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, San Francisco, CA, 1995, pp. 552–557.Google Scholar
  20. 20.
    W. Kohn and L. J. Sham, Phys. Rev. A, 140 (1965), pp. 1133.Google Scholar
  21. 21.
    X.-P. Li, R. W. Nunes, and D. Vanderbilt,Density-matrix electronic-structure method with linear system-size scaling, Phys. Rev. B, 47 (1993), pp. 10891–10894.Google Scholar
  22. 22.
    B. Liu, in: Numerical Algorithms in Chemistry: Algebraic Methods, C. Moler and I. Shavitt eds., LBL-8158 Lawrence Berkeley Laboratory, 1978.Google Scholar
  23. 23.
    R. B. Morgan and D. S. Scott,Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 817–825.Google Scholar
  24. 24.
    B. N. Parlett and D. S. Scott,The Lanczos algorithm with selective orthogonalization, Math. Comp., 33 (1979), pp. 217–238.Google Scholar
  25. 25.
    Y. Saad,Numerical Methods for Large Eigenvalue Problems, Manchester Univ. Press, Manchester, 1992.Google Scholar
  26. 26.
    Y. Saad and A. V. Malevsky,P-SPARSLIB: A portable library of distributed memory sparse iterative solvers, Research Report 95/180, University of Minnesota Supercomputer Institute, Minneapolis, MN, September 1995.Google Scholar
  27. 27.
    Y. Saad and K. Wu,Design of an iterative solution module for a parallel matrix library (P-SPARSLIB), Applied Numerical Mathematics, 19 (1995), pp. 343–357.Google Scholar
  28. 28.
    A. Stathopoulos and C. F. Fischer,A Davidson program for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix, Comput. Phys. Commun., 79 (1994), pp. 268–290.Google Scholar
  29. 29.
    A. Stathopoulos, A. Ynnerman, and C. F. Fischer,A PVM implementation of the MCHF atomic structure package, The International Journal of Supercomputer Applications and High Performance Computing, 10 (1996), pp. 41–61,Google Scholar
  30. 30.
    A. P. Seitsonen, M. J. Puska, and R. M. Nieminen,Real-space electronic-structure calculations: Combination of the finite-difference and conjugate-gradient methods, Phys. Rev. B, 51 (1995), pp. 14057–14061.Google Scholar
  31. 31.
    C. H. Tong, T. F. Chan, and C. C. J. Kuo,Multilevel filtering preconditioners: Extensions to more general elliptic problems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 227–242.Google Scholar
  32. 32.
    N. R Troullier and J. L. Martins,Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B, 43 (1991), pp. 1993–1997.Google Scholar

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • Y. Saad
    • 1
  • A. Stathopoulos
    • 1
  • J. Chelikowsky
    • 2
  • K. Wu
    • 1
  • S. Öğüt
    • 2
  1. 1.Department of Computer ScienceUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Chemical EngineeringUniversity of MinnesotaMinneapolisUSA

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