# Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations

## Abstract

Recently, we have proposed the adaptive local basis set for electronic structure calculations based on Kohn–Sham density functional theory in a pseudopotential framework. The adaptive local basis set is efficient and systematically improvable for total energy calculations. In this paper, we present the calculation of atomic forces, which can be used for a range of applications such as geometry optimization and molecular dynamics simulation. We demonstrate that, under mild assumptions, the computation of atomic forces can scale nearly linearly with the number of atoms in the system using the adaptive local basis set. We quantify the accuracy of the Hellmann–Feynman forces for a range of physical systems, benchmarked against converged planewave calculations, and find that the adaptive local basis set is efficient for both force and energy calculations, requiring at most a few tens of basis functions per atom to attain accuracies required in practice. Since the adaptive local basis set has implicit dependence on atomic positions, Pulay forces are in general nonzero. However, we find that the Pulay force is numerically small and systematically decreasing with increasing basis completeness, so that the Hellmann–Feynman force is sufficient for basis sizes of a few tens of basis functions per atom. Wemore »

- Authors:

- Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (United States)
- Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720 (United States)
- (United States)
- Physics Division, Lawrence Livermore National Laboratory, Livermore, CA 94550 (United States)

- Publication Date:

- OSTI Identifier:
- 22622279

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 335; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ACCURACY; ALLOY SYSTEMS; ATOMS; BENCHMARKS; COMPUTERIZED SIMULATION; CONVERGENCE; DENSITY FUNCTIONAL METHOD; ELECTRONIC STRUCTURE; HYDROGEN; LIQUIDS; MOLECULAR DYNAMICS METHOD; OPTIMIZATION; SILICON ALLOYS

### Citation Formats

```
Zhang, Gaigong, Lin, Lin, E-mail: linlin@math.berkeley.edu, Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, Hu, Wei, E-mail: whu@lbl.gov, Yang, Chao, E-mail: cyang@lbl.gov, and Pask, John E., E-mail: pask1@llnl.gov.
```*Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.12.052.

```
Zhang, Gaigong, Lin, Lin, E-mail: linlin@math.berkeley.edu, Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, Hu, Wei, E-mail: whu@lbl.gov, Yang, Chao, E-mail: cyang@lbl.gov, & Pask, John E., E-mail: pask1@llnl.gov.
```*Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations*. United States. doi:10.1016/J.JCP.2016.12.052.

```
Zhang, Gaigong, Lin, Lin, E-mail: linlin@math.berkeley.edu, Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, Hu, Wei, E-mail: whu@lbl.gov, Yang, Chao, E-mail: cyang@lbl.gov, and Pask, John E., E-mail: pask1@llnl.gov. Sat .
"Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations". United States.
doi:10.1016/J.JCP.2016.12.052.
```

```
@article{osti_22622279,
```

title = {Adaptive local basis set for Kohn–Sham density functional theory in a discontinuous Galerkin framework II: Force, vibration, and molecular dynamics calculations},

author = {Zhang, Gaigong and Lin, Lin, E-mail: linlin@math.berkeley.edu and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 and Hu, Wei, E-mail: whu@lbl.gov and Yang, Chao, E-mail: cyang@lbl.gov and Pask, John E., E-mail: pask1@llnl.gov},

abstractNote = {Recently, we have proposed the adaptive local basis set for electronic structure calculations based on Kohn–Sham density functional theory in a pseudopotential framework. The adaptive local basis set is efficient and systematically improvable for total energy calculations. In this paper, we present the calculation of atomic forces, which can be used for a range of applications such as geometry optimization and molecular dynamics simulation. We demonstrate that, under mild assumptions, the computation of atomic forces can scale nearly linearly with the number of atoms in the system using the adaptive local basis set. We quantify the accuracy of the Hellmann–Feynman forces for a range of physical systems, benchmarked against converged planewave calculations, and find that the adaptive local basis set is efficient for both force and energy calculations, requiring at most a few tens of basis functions per atom to attain accuracies required in practice. Since the adaptive local basis set has implicit dependence on atomic positions, Pulay forces are in general nonzero. However, we find that the Pulay force is numerically small and systematically decreasing with increasing basis completeness, so that the Hellmann–Feynman force is sufficient for basis sizes of a few tens of basis functions per atom. We verify the accuracy of the computed forces in static calculations of quasi-1D and 3D disordered Si systems, vibration calculation of a quasi-1D Si system, and molecular dynamics calculations of H{sub 2} and liquid Al–Si alloy systems, where we show systematic convergence to benchmark planewave results and results from the literature.},

doi = {10.1016/J.JCP.2016.12.052},

journal = {Journal of Computational Physics},

number = ,

volume = 335,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2017},

month = {Sat Apr 15 00:00:00 EDT 2017}

}