Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; Govind, Niranjan; Yang, Chao

doi:10.1016/j.cpc.2017.07.017

Title: Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Journal Article · Fri Dec 01 00:00:00 EST 2017 · Computer Physics Communications

DOI:https://doi.org/10.1016/j.cpc.2017.07.017· OSTI ID:1425428

Vecharynski, Eugene ^[1]; Brabec, Jiri ^[1]; Shao, Meiyue ^[1]; Govind, Niranjan ^[2]; Yang, Chao ^[1]

Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Lab.

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Ministry of Education, Youth and Sports (Cambodia); USDOE Office of Science (SC), Biological and Environmental Research (BER)

Grant/Contract Number:: AC02-05CH11231; AC02-05CH1123; KC-030106062653; AC05-76RL01830; AC05-76RL1830

OSTI ID:: 1425428

Alternate ID(s):: OSTI ID: 1395271; OSTI ID: 1576835

Report Number(s):: PNNL-SA-114405; ark:/13030/qt8rf6x4kg; TRN: US1802105

Journal Information:: Computer Physics Communications, Vol. 221, Issue C; ISSN 0010-4655

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 8 works

Citation information provided by
Web of Science

References (28)

Stability analysis for solutions of the closed shell Kohn–Sham equation Bauernschmitt, Rüdiger; Ahlrichs, Reinhart The Journal of Chemical Physics, Vol. 104, Issue 22 https://doi.org/10.1063/1.471637	journal	June 1996
Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT Brabec, Jiri; Lin, Lin; Shao, Meiyue Journal of Chemical Theory and Computation, Vol. 11, Issue 11 https://doi.org/10.1021/acs.jctc.5b00887	journal	October 2015
Minimization Principles for the Linear Response Eigenvalue Problem II: Computation Bai, Zhaojun; Li, Ren-Cang SIAM Journal on Matrix Analysis and Applications, Vol. 34, Issue 2 https://doi.org/10.1137/110838972	journal	January 2013
Linear Scaling Solution of the Time-Dependent Self-Consistent-Field Equations Challacombe, Matt Computation, Vol. 2, Issue 1 https://doi.org/10.3390/computation2010001	journal	March 2014
On the Efficiency of Algorithms for Solving Hartree–Fock and Kohn–Sham Response Equations Kauczor, Joanna; Jørgensen, Poul; Norman, Patrick Journal of Chemical Theory and Computation, Vol. 7, Issue 6 https://doi.org/10.1021/ct100729t	journal	April 2011
An indefinite variant of LOBPCG for definite matrix pencils Kressner, Daniel; Pandur, Marija Miloloža; Shao, Meiyue Numerical Algorithms, Vol. 66, Issue 4 https://doi.org/10.1007/s11075-013-9754-3	journal	August 2013
Solution of the large matrix equations which occur in response theory Olsen, Jeppe; Jensen, Hans Jørgen Aa; Jørgensen, Poul Journal of Computational Physics, Vol. 74, Issue 2 https://doi.org/10.1016/0021-9991(88)90081-2	journal	February 1988
Passing the one-billion limit in full configuration-interaction (FCI) calculations Olsen, Jeppe; Jørgensen, Poul; Simons, Jack Chemical Physics Letters, Vol. 169, Issue 6 https://doi.org/10.1016/0009-2614(90)85633-N	journal	June 1990
An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules Stratmann, R. Eric; Scuseria, Gustavo E.; Frisch, Michael J. The Journal of Chemical Physics, Vol. 109, Issue 19 https://doi.org/10.1063/1.477483	journal	November 1998
Advances in molecular quantum chemistry contained in the Q-Chem 4 program package Shao, Yihan; Gan, Zhengting; Epifanovsky, Evgeny Molecular Physics, Vol. 113, Issue 2 https://doi.org/10.1080/00268976.2014.952696	journal	September 2014
NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations Valiev, M.; Bylaska, E. J.; Govind, N. Computer Physics Communications, Vol. 181, Issue 9, p. 1477-1489 https://doi.org/10.1016/j.cpc.2010.04.018	journal	September 2010
Minimization Principles for the Linear Response Eigenvalue Problem I: Theory Bai, Zhaojun; Li, Ren-Cang SIAM Journal on Matrix Analysis and Applications, Vol. 33, Issue 4 https://doi.org/10.1137/110838960	journal	January 2012
A Trace Minimization Algorithm for the Generalized Eigenvalue Problem Sameh, Ahmed H.; Wisniewski, John A. SIAM Journal on Numerical Analysis, Vol. 19, Issue 6 https://doi.org/10.1137/0719089	journal	December 1982
The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices Davidson, Ernest R. Journal of Computational Physics, Vol. 17, Issue 1 https://doi.org/10.1016/0021-9991(75)90065-0	journal	January 1975
A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems G. Sleijpen, Gerard L.; Van der Vorst, Henk A. SIAM Journal on Matrix Analysis and Applications, Vol. 17, Issue 2 https://doi.org/10.1137/S0895479894270427	journal	April 1996
An iterative method for calculating several of the extreme eigensolutions of large real non-symmetric matrices Rettrup, Sten Journal of Computational Physics, Vol. 45, Issue 1 https://doi.org/10.1016/0021-9991(82)90104-8	journal	January 1982
Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method Knyazev, Andrew V. SIAM Journal on Scientific Computing, Vol. 23, Issue 2 https://doi.org/10.1137/S1064827500366124	journal	January 2001
Basis selection in LOBPCG Hetmaniuk, U.; Lehoucq, R. Journal of Computational Physics, Vol. 218, Issue 1 https://doi.org/10.1016/j.jcp.2006.02.007	journal	October 2006
A Block Orthogonalization Procedure with Constant Synchronization Requirements Stathopoulos, Andreas; Wu, Kesheng SIAM Journal on Scientific Computing, Vol. 23, Issue 6 https://doi.org/10.1137/S1064827500370883	journal	January 2002
A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix Vecharynski, Eugene; Yang, Chao; Pask, John E. Journal of Computational Physics, Vol. 290 https://doi.org/10.1016/j.jcp.2015.02.030	journal	June 2015
Density‐functional thermochemistry. III. The role of exact exchange Becke, Axel D. The Journal of Chemical Physics, Vol. 98, Issue 7, p. 5648-5652 https://doi.org/10.1063/1.464913	journal	April 1993
Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules Hehre, W. J.; Ditchfield, R.; Pople, J. A. The Journal of Chemical Physics, Vol. 56, Issue 5, p. 2257-2261 https://doi.org/10.1063/1.1677527	journal	March 1972
The influence of polarization functions on molecular orbital hydrogenation energies Hariharan, P. C.; Pople, J. A. Theoretica Chimica Acta, Vol. 28, Issue 3 https://doi.org/10.1007/BF00533485	journal	January 1973
Optimization of Gaussian-type basis sets for local spin density functional calculations. Part I. Boron through neon, optimization technique and validation Godbout, Nathalie; Salahub, Dennis R.; Andzelm, Jan Canadian Journal of Chemistry, Vol. 70, Issue 2 https://doi.org/10.1139/v92-079	journal	February 1992
Large-scaleRPA calculations of chiroptical properties of organic molecules: Program RPAC Bouman, Thomas D.; Hansen, Aage E.; Voigt, Bj�rn International Journal of Quantum Chemistry, Vol. 23, Issue 2 https://doi.org/10.1002/qua.560230230	journal	February 1983
A classical mechanics technique for quantum linear response Tsiper, E. V. Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 34, Issue 12 https://doi.org/10.1088/0953-4075/34/12/102	journal	June 2001
Vibrational states of nuclei in the random phase approximation Thouless, D. J. Nuclear Physics, Vol. 22, Issue 1 https://doi.org/10.1016/0029-5582(61)90364-9	journal	January 1961
Variational procedure and generalized Lanczos recursion for small-amplitude classical oscillations Tsiper, E. V. Journal of Experimental and Theoretical Physics Letters, Vol. 70, Issue 11 https://doi.org/10.1134/1.568257	journal	December 1999

Cited By (2)

S erenity : A subsystem quantum chemistry program Unsleber, Jan P.; Dresselhaus, Thomas; Klahr, Kevin Journal of Computational Chemistry, Vol. 39, Issue 13 https://doi.org/10.1002/jcc.25162	journal	January 2018
A FEAST Algorithm for the Linear Response Eigenvalue Problem Teng, Zhongming; Lu, Linzhang Algorithms, Vol. 12, Issue 9 https://doi.org/10.3390/a12090181	journal	August 2019

Figures / Tables (10)

Similar Records

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Journal Article · Fri Dec 01 00:00:00 EST 2017 · Computer Physics Communications · OSTI ID:1425428

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; +2 more

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Journal Article · Thu Jun 01 00:00:00 EDT 2017 · IEEE Transactions on Parallel and Distributed Systems · OSTI ID:1425428

Aktulga, Hasan Metin; Afibuzzaman, Md.; Williams, Samuel; +6 more

Parallel eigensolver for H(curl) problems using H1-auxiliary space AMG preconditioning

Technical Report · Wed Nov 15 00:00:00 EST 2006 · OSTI ID:1425428

Kolev, T V; Vassilevski, P S

Related Subjects

97 MATHEMATICS AND COMPUTING
Time dependent density functional theory
Linear response eigenvalue problem
Preconditioned eigensolvers