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A spectrum slicing method for the Kohn-Sham problem. (English) Zbl 1264.82014
Summary: Solving the Kohn-Sham equation, which arises in density functional theory, is a standard procedure to determine the electronic structure of atoms, molecules, and condensed matter systems. The solution of this nonlinear eigenproblem is used to predict the spatial and energetic distribution of electronic states. However, obtaining a solution for large systems is computationally intensive because the problem scales super-linearly with the number of atoms.
Here, we demonstrate a divide and conquer method that partitions the necessary eigenvalue spectrum into slices and computes each partial spectrum on an independent group of processors in parallel. We focus on the elements of the spectrum slicing method that are essential to its correctness and robustness such as the choice of filter polynomial, the stopping criterion for a vector iteration, and the detection of duplicate eigenpairs computed in adjacent spectral slices. Some of the more prominent aspects of developing an optimized implementation are discussed.

82B10 Quantum equilibrium statistical mechanics (general)
82-08 Computational methods (statistical mechanics) (MSC2010)
81V45 Atomic physics
35Q55 NLS equations (nonlinear Schrödinger equations)
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F50 Computational methods for sparse matrices
Full Text: DOI
[1] Bekas, C.; Kokiopoulou, E.; Saad, Y., Polynomial filtered Lanczos iterations with applications in density functional theory, SIAM journal on matrix analysis and applications, 30, 1, 397-418, (2008) · Zbl 1159.65319
[2] Bjorck, Ake; Golub, Gene H., Numerical methods for computing angles between linear subspaces, Mathematics of computation, 27, 123, 579-594, (1973) · Zbl 0282.65031
[3] Chelikowsky, J.R.; Chan, T.-L.; Alemany, M.M.G.; Dalpian, G., Rep. prog. phys., 74, 046501, (2011)
[4] Chelikowsky, J.R., The pseudopotential-density functional method applied to nanostructures, Journal of physics D: applied physics, 33, 8, R33-R50, (2000)
[5] Daniel, J.W.; Gragg, W.B.; Kaufman, L.; Stewart, G.W., Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Mathematics of computation, 30, 772-795, (1976) · Zbl 0345.65021
[6] Guerra, C. Fonseca; Snijders, J.G.; te Velde, G.; Baerends, E.J., Theor. chem. acc., 99, 391, (1998)
[7] Hernandez, E.; Gillan, M.J., Phys. rev. B, 51, 10157, (1995)
[8] Jackson, Dunham, The theory of approximation, (1930), American Mathematical Society New York · Zbl 0008.05803
[9] Jay, Laurent O.; Hanchul, Kim; Saad, Yousef; Chelikowsky, James R., Electronic structure calculations in plane-wave codes without diagonalization, Computer physics communications, 118, 21-30, (1999) · Zbl 1001.65038
[10] Kohn, W.; Sham, L.J., Self-consistent equations including exchange and correlation effects, Phys. rev., 140, A1133, (1965)
[11] Lanczos, C., An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, Journal of research of the national bureau of standards, 45, 255-282, (1950)
[12] Parlett, B.N., The symmetric eigenvalue problem, (1980), Prentice-Hall Englewood Cliffs · Zbl 0431.65016
[13] Rivlin, Theodore J., An introduction to the approximation of functions, (1981), Dover New York · Zbl 0489.41001
[14] Rocca, Dario; Gebauer, Ralph; Saad, Yousef; Baroni, Stefano, Turbo charging time-dependent density-functional theory with Lanczos chains, J. chem. phys., 128, 154105, (2008)
[15] Zhou, Y.; Saad, Y.; Tiago, M.L.; Chelikowsky, J.R., Self-consistent-field calculations using Chebyshev-filtered subspace iteration, Journal of computational physics, 219, 172-184, (2006) · Zbl 1105.65111
[16] Yunkai, Zhou; Saad, Yousef, A Chebyshev-Davidson algorithm for large symmetric eigenproblems, SIAM journal on matrix analysis and applications, 29, 3, 954-971, (2007) · Zbl 1151.65321
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