Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT
We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response timedependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original nonHermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is selfadjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a lowmemory alternative to the Lanczos approach, but the algorithm may require more matrixvector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and mediumsized molecules are also presented.
 Authors:

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 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Univ. of Minnesota, Minneapolis, MN (United States)
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Chemical Theory and Computation
 Additional Journal Information:
 Journal Volume: 11; Journal Issue: 11; Journal ID: ISSN 15499618
 Publisher:
 American Chemical Society
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1378640
Brabec, Jiri, Lin, Lin, Shao, Meiyue, Govind, Niranjan, Yang, Chao, Saad, Yousef, and Ng, Esmond G. Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT. United States: N. p.,
Web. doi:10.1021/acs.jctc.5b00887.
Brabec, Jiri, Lin, Lin, Shao, Meiyue, Govind, Niranjan, Yang, Chao, Saad, Yousef, & Ng, Esmond G. Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT. United States. doi:10.1021/acs.jctc.5b00887.
Brabec, Jiri, Lin, Lin, Shao, Meiyue, Govind, Niranjan, Yang, Chao, Saad, Yousef, and Ng, Esmond G. 2015.
"Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT". United States.
doi:10.1021/acs.jctc.5b00887. https://www.osti.gov/servlets/purl/1378640.
@article{osti_1378640,
title = {Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT},
author = {Brabec, Jiri and Lin, Lin and Shao, Meiyue and Govind, Niranjan and Yang, Chao and Saad, Yousef and Ng, Esmond G.},
abstractNote = {We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response timedependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original nonHermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is selfadjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a lowmemory alternative to the Lanczos approach, but the algorithm may require more matrixvector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and mediumsized molecules are also presented.},
doi = {10.1021/acs.jctc.5b00887},
journal = {Journal of Chemical Theory and Computation},
number = 11,
volume = 11,
place = {United States},
year = {2015},
month = {10}
}