# Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory

## Abstract

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper, we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluatesmore »

- Authors:

- University of California, Berkeley, CA (United States). Dept. of Mathematics
- University of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Laboratory, Berkeley, CA (United States). Computational Research Division

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1497884

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 147; Journal Issue: 14; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

### Citation Formats

```
Jia, Weile, and Lin, Lin. Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory. United States: N. p., 2017.
Web. doi:10.1063/1.5000255.
```

```
Jia, Weile, & Lin, Lin. Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory. United States. doi:10.1063/1.5000255.
```

```
Jia, Weile, and Lin, Lin. Wed .
"Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory". United States. doi:10.1063/1.5000255. https://www.osti.gov/servlets/purl/1497884.
```

```
@article{osti_1497884,
```

title = {Robust determination of the chemical potential in the pole expansion and selected inversion method for solving Kohn-Sham density functional theory},

author = {Jia, Weile and Lin, Lin},

abstractNote = {Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper, we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.},

doi = {10.1063/1.5000255},

journal = {Journal of Chemical Physics},

number = 14,

volume = 147,

place = {United States},

year = {2017},

month = {10}

}

*Citation information provided by*

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#### Figures / Tables:

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