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Efficient and accurate algorithms for solving the Bethe-Salpeter eigenvalue problem for crystalline systems. (English) Zbl 1473.65044

Summary: Optical properties of materials related to light absorption and scattering are explained by the excitation of electrons. The Bethe-Salpeter equation is the state-of-the-art approach to describe these processes from first principles (ab initio), i.e. without the need for empirical data in the model. To harness the predictive power of the equation, it is mapped to an eigenvalue problem via an appropriate discretization scheme. The eigenpairs of the resulting large, dense, structured matrix can be used to compute dielectric properties of the considered crystalline or molecular system. The matrix always shows a \(2\times 2\) block structure. Depending on exact circumstances and discretization schemes, one ends up with a matrix structure such as \begin{align*} H_1=\begin{bmatrix} A & B \\ - B & - A \end{bmatrix}\in\mathbb{C}^{2n\times 2 n},\quad & A=A^{\mathsf{H}},B= B^{\mathsf{H}},\\ \text{or }\qquad H_2=\begin{bmatrix} A & B \\ - B^{\mathsf{H}} & - A^{\mathsf{T}}\end{bmatrix}\in\mathbb{C}^{2n\times 2 n}\text{ or } \mathbb{R}^{2n\times 2n},\quad & A=A^{\mathsf{H}},B=B^{\mathsf{T}}. \end{align*} \(H_1\) can be acquired for crystalline systems (see [T. Sander et al., “Beyond the Tamm-Dancoff approximation for extended systems using exact diagonalization”, Phys. Rev. B, 92, Article ID 045209 (2015)]), \(H_2\) is a more general form found e.g. in [M. Shao et al., SIAM J. Matrix Anal. Appl. 39, No. 2, 683–711 (2018; Zbl 1391.65089)] and [C. Penke et al., “High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem”, Parallel Comput. 96, Article ID 102639 (2020)], which can for example be used to study molecules. Additionally, certain definiteness properties may hold. In this work, we compile theoretical results characterizing the structure of \(H_1\) and \(H_2\) in the language of non-standard scalar products. These results enable us to develop a generalized perspective on the currently used direct solution approach for matrices of form \(H_1\). This new viewpoint is used to develop two alternative methods for solving the eigenvalue problem. Both have advantages over the method currently in use and are well suited for high performance environments and only rely on basic numerical linear algebra building blocks. The results are extended to hold even without the mentioned definiteness property, showing the usefulness of our new perspective.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Citations:

Zbl 1391.65089
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References:

[1] Sander, T.; Maggio, E.; Kresse, G., Beyond the Tamm-Dancoff approximation for extended systems using exact diagonalization, Phys. Rev. B, 92, Article 045209 pp. (2015)
[2] Vorwerk, C.; Aurich, B.; Cocchi, C.; Draxl, C., Bethe-Salpeter equation for absorption and scattering spectroscopy: implementation in the exciting code, Electron. Struct., 1, 3, Article 037001 pp. (2019)
[3] Sagmeister, S.; Ambrosch-Draxl, C., Time-dependent density functional theory versus Bethe-Salpeter equation: an all-electron study, Phys. Chem. Chem. Phys., 11, 4451-4457 (2009)
[4] Penke, C.; Marek, A.; Vorwerk, C.; Draxl, C.; Benner, P., High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter eigenvalue problem, Parallel Comput., 96, Article 102639 pp. (2020)
[5] The Top500 list, Available at http://www.top500.org.
[6] Hedin, L.; Lundqvist, S., Effects of electron-electron and electron-phonon interactions on the one-electron states of solids, (Seitz, F.; Turnbull, D.; Ehrenreich, H., Solid State Physics, Vol. 23 (1970), Academic Press), 1-181
[7] Bai, Z.; Li, R.-C., Minimization principles for the linear response eigenvalue problem I: Theory, SIAM J. Matrix Anal. Appl., 33, 4, 1075-1100 (2012) · Zbl 1263.65078
[8] Bai, Z.; Li, R.-C., Minimization principles for the linear response eigenvalue problem II: Computation, SIAM J. Matrix Anal. Appl., 34, 2, 392-416 (2013) · Zbl 1311.65102
[9] Bai, Z.; Li, R.-C., Minimization principles and computation for the generalized linear response eigenvalue problem, BIT, 54, 1, 31-54 (2014) · Zbl 1293.65053
[10] Bai, Z.; Li, R.-C.; Lin, W.-W., Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods, Sci. China Math., 59, 8, 1443-1460 (2016) · Zbl 1358.65021
[11] Bai, Z.; Li, R.-C., Recent progress in linear response eigenvalue problems, (Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, Lect. Notes Comput. Sci. Eng., vol. 117 (2017), Springer, Cham), 287-304
[12] Casida, M., Time-dependent density functional response theory for molecules, (Recent Advances in Density Functional Methods (1995), World Scientific), 155-192
[13] Papakonstantinou, P., Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices, Europhys. Lett., 78, 1, 12001 (2007) · Zbl 1244.65054
[14] Shao, M.; da Jornada, F. H.; Lin, L.; Yang, C.; Deslippe, J.; Louie, S. G., A structure preserving Lanczos algorithm for computing the optical absorption spectrum, SIAM J. Matrix Anal. Appl., 39, 2, 683-711 (2018) · Zbl 1391.65089
[15] Henneke, F.; Lin, L.; Vorwerk, C.; Draxl, C.; Klein, R.; Yang, C., Fast optical absorption spectra calculations for periodic solid state systems, Commun. Appl. Math. Comput. Sci., 15, 1, 89-113 (2020) · Zbl 1465.35346
[16] Mackey, D. S.; Mackey, N.; Tisseur, F., Structured factorizations in scalar product spaces, SIAM J. Matrix Anal. Appl., 27, 3, 821-850 (2005) · Zbl 1098.15005
[17] Benner, P.; Kressner, D.; Mehrmann, V., Skew-Hamiltonian and Hamiltonian eigenvalue problems: Theory, algorithms and applications, (Proc. Conf. Appl Math. Scientific Comp. (2005), Springer-Verlag), 3-39 · Zbl 1069.65034
[18] Benner, P.; Faßbender, H.; Yang, C., Some remarks on the complex J-symmetric eigenproblem, Linear Algebra Appl., 544, 407-442 (2018) · Zbl 1392.65086
[19] Onida, G.; Reining, L.; Rubio, A., Electronic excitations: density-functional versus many-body Green’s-function approaches, Rev. Modern Phys., 74, 601-659 (2002)
[20] Kressner, D., (Numerical Methods for General and Structured Eigenvalue Problems. Numerical Methods for General and Structured Eigenvalue Problems, Lecture Notes in Computational Science and Engineering, vol. 46 (2005), Springer-Verlag, Berlin Heidelberg), xiv+258 · Zbl 1079.65041
[21] Benner, P.; Mehrmann, V.; Xu, H., A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils, Numer. Math., 78, 3, 329-358 (1998) · Zbl 0889.65036
[22] Higham, N. J., (Functions of Matrices: Theory and Computation. Functions of Matrices: Theory and Computation, Applied Mathematics (2008), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA), xx+425 · Zbl 1167.15001
[23] Chi, B. E., The eigenvalue problem for collective motion in the random phase approximation, Nuclear Phys. A, 146, 2, 449-456 (1970)
[24] Van Loan, C. F., A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl., 61, 233-251 (1984) · Zbl 0565.65018
[25] Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Oxford University Press: Oxford University Press Oxford · Zbl 0258.65037
[26] Benner, P.; Byers, R.; Barth, E., Algorithm 800. Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices I: The square-reduced method, ACM Trans. Math. Softw., 26, 1, 49-77 (2000) · Zbl 1137.65338
[27] Shao, M.; da Jornada, F. H.; Yang, C.; Deslippe, J.; Louie, S. G., Structure preserving parallel algorithms for solving the Bethe-Salpeter eigenvalue problem, Linear Algebra Appl., 488, 148-167 (2016) · Zbl 1330.65059
[28] Stewart, G. W.; Sun, J.-G., Matrix Perturbation Theory (1990), Academic Press: Academic Press New York · Zbl 0706.65013
[29] Gulans, A.; Kontur, S.; Meisenbichler, C.; Nabok, D.; Pavone, P.; Rigamonti, S.; Sagmeister, S.; Werner, U.; Draxl, C., Exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory, J. Phys.: Condens. Matter, 26, 36, Article 363202 pp. (2014)
[30] Deslippe, J.; Samsonidze, G.; Strubbe, D. A.; Jain, M.; Cohen, M. L.; Louie, S. G., BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures, Comput. Phys. Comm., 183, 6, 1269-1289 (2012)
[31] Sangalli, D.; Ferretti, A.; Miranda, H.; Attaccalite, C.; Marri, I.; Cannuccia, E.; Melo, P.; Marsili, M.; Paleari, F.; Marrazzo, A.; Prandini, G.; Bonfà, P.; Atambo, M. O.; Affinito, F.; Palummo, M.; Molina-Sánchez, A.; Hogan, C.; Grüning, M.; Varsano, D.; Marini, A., Many-body perturbation theory calculations using the yambo code, J. Phys.: Condens. Matter, 31, 32, Article 325902 pp. (2019)
[32] Golub, G. H.; Van Loan, C. F., (Matrix Computations. Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences (2013), Johns Hopkins University Press: Johns Hopkins University Press Baltimore) · Zbl 1268.65037
[33] Benner, P.; Mehrmann, V.; Xu, H., A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems, Electron. Trans. Numer. Anal., 8, 115-126 (1999) · Zbl 0936.65045
[34] Benner, P.; Kressner, D.; Sima, V.; Varga, A., Die SLICOT-toolboxen für MATLAB, at-Automatisierungstechnik, 58, 1, 15-25 (2010)
[35] Mehl, C., Finite-dimensional indefinite inner product spaces and applications in numerical analysis, (Operator Theory (2015), Springer Basel: Springer Basel Basel), 1-17
[36] Benner, P.; Faßbender, H.; Watkins, D. S., Two connections between the SR and HR eigenvalue algorithms, Linear Algebra Appl., 272, 17-32 (1997) · Zbl 0899.65018
[37] Watkins, D., The Matrix Eigenvalue Problem (2007), Society for Industrial and Applied Mathematics · Zbl 1142.65038
[38] Bunse-Gerstner, A., An analysis of the HR algorithm for computing the eigenvalues of a matrix, Linear Algebra Appl., 35, 155-173 (1981) · Zbl 0462.65021
[39] Bunse-Gerstner, A., An algorithm for the symmetric generalized eigenvalue problem, Linear Algebra Appl., 58, 43-68 (1984) · Zbl 0575.65028
[40] Brebner, M.; Grad, J., Eigenvalues of \(A x = \lambda B x\) for real symmetric matrices \(A\) and \(B\) computed by reduction to a pseudosymmetric form and the HR process, Linear Algebra Appl., 43, 99-118 (1982) · Zbl 0485.65029
[41] Mehl, C.; Mehrmann, V.; Xu, H., Structured decompositions for matrix triples: SVD-like concepts for structured matrices, Oper. Matrices, 3, 3, 303-356 (2009) · Zbl 1188.15012
[42] Parlett, B. N.; Chen, H. C., Use of indefinite pencils for computing damped natural modes, Linear Algebra Appl., 140, 53-88 (1990) · Zbl 0725.65055
[43] Day, D., An efficient implementation of the nonsymmetric Lanczos algorithm, SIAM J. Matrix Anal. Appl., 18, 3, 566-589 (1997) · Zbl 0873.65037
[44] Campos, C.; Roman, J. E., Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems, BIT, 56, 4, 1213-1236 (2016) · Zbl 1355.65057
[45] Nakatsukasa, Y.; Higham, N. J., Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD, SIAM J. Sci. Comput., 35, 3, A1325-A1349 (2013) · Zbl 1326.65049
[46] Auckenthaler, T.; Blum, V.; Bungartz, H.-J.; Huckle, T.; Johanni, R.; Krämer, L.; Lang, B.; Lederer, H.; Willems, P., Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations, Parallel Comput., 37, 12, 783-794 (2011)
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