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

## Abstract

We present two efficient iterative algorithms for solving the linear response eigen- value 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 a product eigenvalue problem that is self-adjoint with respect to a K-inner product. 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. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

- Authors:

- 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.

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); Ministry of Education, Youth and Sports (Cambodia)

- OSTI Identifier:
- 1406765

- Report Number(s):
- PNNL-SA-114405

Journal ID: ISSN 0010-4655; 48614; KC0301060

- DOE Contract Number:
- AC02-05CH11231; AC02-05CH1123; KC-030106062653; AC05-76RL1830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Computer Physics Communications; Journal Volume: 221; Journal Issue: C

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Eigensolvers; Linear Response Time-dependent Density Functional Theory; Environmental Molecular Sciences Laboratory

### Citation Formats

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao.
```*Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory*. United States: N. p., 2017.
Web. doi:10.1016/j.cpc.2017.07.017.

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, & Yang, Chao.
```*Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory*. United States. doi:10.1016/j.cpc.2017.07.017.

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. Fri .
"Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory". United States.
doi:10.1016/j.cpc.2017.07.017.
```

```
@article{osti_1406765,
```

title = {Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory},

author = {Vecharynski, Eugene and Brabec, Jiri and Shao, Meiyue and Govind, Niranjan and Yang, Chao},

abstractNote = {We present two efficient iterative algorithms for solving the linear response eigen- value 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 a product eigenvalue problem that is self-adjoint with respect to a K-inner product. 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. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.},

doi = {10.1016/j.cpc.2017.07.017},

journal = {Computer Physics Communications},

number = C,

volume = 221,

place = {United States},

year = {Fri Dec 01 00:00:00 EST 2017},

month = {Fri Dec 01 00:00:00 EST 2017}

}