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# Energy window stochastic density functional theory

## Abstract

We report that linear scaling density functional theory is important for understanding electronic structure properties of nanometer scale systems. Recently developed stochastic density functional theory can achieve linear or even sublinear scaling for various electronic properties without relying on the sparsity of the density matrix. The basic idea relies on projecting stochastic orbitals onto the occupied space by expanding the Fermi-Dirac operator and repeating this for N _{χ} stochastic orbitals. Often, a large number of stochastic orbitals are required to reduce the statistical fluctuations (which scale as N$$-1/2\atop{χ}$$) below a tolerable threshold. In this work, we introduce a new stochastic density functional theory that can efficiently reduce the statistical fluctuations for certain observable and can also be integrated with an embedded fragmentation scheme. The approach is based on dividing the occupied space into energy windows and projecting the stochastic orbitals with a single expansion onto all windows simultaneously. This allows for a significant reduction of the noise as illustrated for bulk silicon with a large supercell. Finally, we also provide theoretical analysis to rationalize why the noise can be reduced only for a certain class of ground state properties, such as the forces and electron density.

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Hebrew Univ. of Jerusalem (Israel). Fritz Haber Center of Molecular Dynamics and Institute of Chemistry
- Univ. of California, Los Angeles, CA (United States)
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Tel Aviv Univ., Ramat Aviv (Israel). The Raymond and Beverly Sackler Center of Computational Molecular and Materials Science

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Univ. of California, Oakland, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division

- OSTI Identifier:
- 1577611

- Alternate Identifier(s):
- OSTI ID: 1563022

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 151; Journal Issue: 11; 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; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Chen, Ming, Baer, Roi, Neuhauser, Daniel, and Rabani, Eran. Energy window stochastic density functional theory. United States: N. p., 2019.
Web. doi:10.1063/1.5114984.
```

```
Chen, Ming, Baer, Roi, Neuhauser, Daniel, & Rabani, Eran. Energy window stochastic density functional theory. United States. doi:10.1063/1.5114984.
```

```
Chen, Ming, Baer, Roi, Neuhauser, Daniel, and Rabani, Eran. Fri .
"Energy window stochastic density functional theory". United States. doi:10.1063/1.5114984.
```

```
@article{osti_1577611,
```

title = {Energy window stochastic density functional theory},

author = {Chen, Ming and Baer, Roi and Neuhauser, Daniel and Rabani, Eran},

abstractNote = {We report that linear scaling density functional theory is important for understanding electronic structure properties of nanometer scale systems. Recently developed stochastic density functional theory can achieve linear or even sublinear scaling for various electronic properties without relying on the sparsity of the density matrix. The basic idea relies on projecting stochastic orbitals onto the occupied space by expanding the Fermi-Dirac operator and repeating this for Nχ stochastic orbitals. Often, a large number of stochastic orbitals are required to reduce the statistical fluctuations (which scale as N$-1/2\atop{χ}$) below a tolerable threshold. In this work, we introduce a new stochastic density functional theory that can efficiently reduce the statistical fluctuations for certain observable and can also be integrated with an embedded fragmentation scheme. The approach is based on dividing the occupied space into energy windows and projecting the stochastic orbitals with a single expansion onto all windows simultaneously. This allows for a significant reduction of the noise as illustrated for bulk silicon with a large supercell. Finally, we also provide theoretical analysis to rationalize why the noise can be reduced only for a certain class of ground state properties, such as the forces and electron density.},

doi = {10.1063/1.5114984},

journal = {Journal of Chemical Physics},

number = 11,

volume = 151,

place = {United States},

year = {2019},

month = {9}

}

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