Abstract
In this paper, we give a comprehensive error analysis for an approximate solution method for the generalized eigenvalue problems arising for instance in the context of electronic structure computations based on density functional theory. The solution method has been demonstrated to excel as compared to established solvers in both computational effort and scaling for parallelization. Here we estimate the improvement provided by our proposed subspace method starting from the initial approximations for instance provided in the course of the self-consistent field iteration, showing that in general the approximation quality is improved by our method to yield sufficiently accurate eigenvalues.
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Bendtsen, C., Nielsen, O., Hansen, L.: Solving large nonlinear generalized eigenvalue problems from density functional theory calculations in parallel. Appl. Numer. Math. 37, 189–199 (2001)
Blaha, P., Hofstätter, H., Koch, O., Laskowsky, R., Schwarz, K.: Iterative diagonalization in APW based methods in electronic structure calculations. J. Comput. Phys. 229, 453–460 (2010)
Blaha, P., Schwarz, K., Madsen, G.: Electronic structure calculations of solids using the WIEN2k package for material sciences. Comput. Phys. Commun. 147, 71–76 (2002)
Blaha, P., Schwarz, K., Madsen, G., Kvasnicka, D., Luitz, J.: An augmented plane wave plus local orbital program for calculating crystal properties. ISBN 3-9501031-1-2 (2001)
Crouzeix, M., Philippe, B., Sadkane, M.: The Davidson method. SIAM J. Sci. Comput. 15, 62–76 (1994)
Davidson, E.: The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 87–94 (1975)
Garcia-Cervera, C.J., Lu, J., Xuan, Y., Weinan, E.: Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory. Phys. Rev. B 79, 115110 (2009)
Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136(3B), B864 (1964)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1990)
Kohn, W., Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140(4A), A1133 (1965)
Kresse, G., Furthmüller, J.: Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996)
Kresse, G., Furthmüller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996)
Kresse, G., Furthmüller, J.: VASP the GUIDE (2007). Available at http://cms.mpi.univie.ac.at/VASP
Lascoux, A., Pragacz, P.: Jacobians of symmetric polynomials. Ann. Comb. 6, 169–172 (2002)
Laskowski, R., Blaha, P.: Unraveling the structure of the h-BN/Rh(111) nanomesh with ab initio calculations. J. Phys. Condens. Matter 20, 064207 (2008)
Madsen, G.K.H., Blaha, P., Schwarz, K., Sjöstedt, E., Nordström, L.: Efficient linearization of the augmented plane-wave method. Phys. Rev. B 64, 195134 (2001)
Pulay, P.: Convergence acceleration of iterative sequences. The case of SCF iteration. Chem. Phys. Lett. 73, 393 (1980)
Rayson, M.J., Briddon, P.R.: Rapid iterative method for electronic-structure eigenproblems using localized basis functions. Comput. Phys. Commun. 178, 128–134 (2008)
Schwarz, K.: DFT calculations of solids with LAPW and WIEN2k. Solid State Commun. 176, 319–328 (2003)
Schwarz, K., Blaha, P.: Solid state calculations using WIEN2k. Comput. Mater. Sci. 28, 259–273 (2003)
Singh, D.: Simultaneous solution of diagonalization and self-consistency problems for transition-metal systems. Phys. Rev. B 40, 5428–5431 (1989)
Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., Van der Vorst, H.A.: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36(3), 595–633 (1996)
Sleijpen, G.L.G., Van der Vorst, H.A.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17(2), 401–425 (1996)
Teter, M.P., Payne, M., Allan, D.C.: Solution of Schrödinger’s equation for large systems. Phys. Rev. B 40(18), 12255 (1989)
van Dam, H.J.J., van Lenthe, J.H., Sleijpen, G.L.G., van der Vorst, H.A.: An improvement of Davidson’s iteration method. J. Comput. Chem. 17(3), 267–272 (1996)
VandeVondele, J., Hutter, J.: An efficient orbital transformation mathod for electronic structure calculations. J. Chem. Phys. 118(10), 4365–4369 (2003)
Wood, D.M., Zunger, A.: A new method for diagonalising large matrices. J. Phys. A: Math. Gen. 18, 1343–1359 (1985)
Yang, C., Gao, W., Meza, J.C.: On the convergence of the self-consistent field iteration for a class of nonlinear eigenvalue problems. SIAM J. Matrix Anal. Appl. 30, 1773–1788 (2009)
Zhou, Y., Saad, Y., Tiago, M., Chelikowsky, J.: Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration. Phys. Rev. E 74, 066704 (2006)
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Hofstätter, H., Koch, O. An approximate eigensolver for self-consistent field calculations. Numer Algor 66, 609–641 (2014). https://doi.org/10.1007/s11075-013-9751-6
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DOI: https://doi.org/10.1007/s11075-013-9751-6
Keywords
- Electronic structure computations
- Density functional theory
- Generalized eigenvalue problem
- Iterative diagonalization