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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
S. Baroni and P. Giannozzi,Towards very large-scale electronic-structure calculations, Europhys. Lett. 17 (1992), pp. 547–552.
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.
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.
J. R. Chelikowsky and M. L. Cohen,Handbook on Semiconductors Vol. 1, P. T. Landsberg ed., Elsevier, Amsterdam, 1992.
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.
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.
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.
M. Crouzeix, B. Philippe, and M. Sadkane,The Davidson method, SIAM J. Sci. Comput., 15 (1994), pp. 62–76.
J. Cullum and R. A. Willoughby,Lanczos algorithms for large symmetric eigenvalue computations 2: Programs, Progress in Scientific Computing, v. 4, Birkhauser, Boston, 1985.
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.
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.
D. E. Ellis and G. S. Painter, Phys. Rev. B, 2 (1970), pp. 2887.
B. Fornberg and D. M. Sloan,A review of pseudospectral methods for solving partial differential equations, Acta Numerica, (1994), pp. 203–267.
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.
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.
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.
G. E. Kimball and G. H. Shortley, Phys. Rev., 45 (1934), pp. 815.
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.
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.
W. Kohn and L. J. Sham, Phys. Rev. A, 140 (1965), pp. 1133.
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.
B. Liu, in: Numerical Algorithms in Chemistry: Algebraic Methods, C. Moler and I. Shavitt eds., LBL-8158 Lawrence Berkeley Laboratory, 1978.
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.
B. N. Parlett and D. S. Scott,The Lanczos algorithm with selective orthogonalization, Math. Comp., 33 (1979), pp. 217–238.
Y. Saad,Numerical Methods for Large Eigenvalue Problems, Manchester Univ. Press, Manchester, 1992.
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.
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.
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.
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,
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.
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.
N. R Troullier and J. L. Martins,Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B, 43 (1991), pp. 1993–1997.
Work supported by NSF grants DMR-9217287 and ASC 95-04038, and by the Minnesota Supercomputer Institute
About this article
Cite this article
Saad, Y., Stathopoulos, A., Chelikowsky, J. et al. Solution of large eigenvalue problems in electronic structure calculations. Bit Numer Math 36, 563–578 (1996). https://doi.org/10.1007/BF01731934
- Eigenvalue problem
- electronic structure calculation
- paralell implementation