## Minimax rational approximation of the Fermi-Dirac distribution

## Abstract

Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ ^{–1})) poles to achieve an error tolerance ϵ at temperature β ^{–1} over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δ _{occ}, the occupied energy interval. Furthermore, this is particularly beneficial when Δ >> Δ _{occ}, such as in electronic structure calculations that use a large basis set.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1335130

- Report Number(s):
- SAND-2016-3915J

Journal ID: ISSN 0021-9606; JCPSA6; 638848

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 16; 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

```
Moussa, Jonathan E. Minimax rational approximation of the Fermi-Dirac distribution. United States: N. p., 2016.
Web. doi:10.1063/1.4965886.
```

```
Moussa, Jonathan E. Minimax rational approximation of the Fermi-Dirac distribution. United States. doi:10.1063/1.4965886.
```

```
Moussa, Jonathan E. Thu .
"Minimax rational approximation of the Fermi-Dirac distribution". United States. doi:10.1063/1.4965886. https://www.osti.gov/servlets/purl/1335130.
```

```
@article{osti_1335130,
```

title = {Minimax rational approximation of the Fermi-Dirac distribution},

author = {Moussa, Jonathan E.},

abstractNote = {Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use O(log(βΔ)log(ϵ–1)) poles to achieve an error tolerance ϵ at temperature β–1 over an energy interval Δ. We apply minimax approximation to reduce the number of poles by a factor of four and replace Δ with Δocc, the occupied energy interval. Furthermore, this is particularly beneficial when Δ >> Δocc, such as in electronic structure calculations that use a large basis set.},

doi = {10.1063/1.4965886},

journal = {Journal of Chemical Physics},

number = 16,

volume = 145,

place = {United States},

year = {2016},

month = {10}

}

Other availability

Cited by: 1 work

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