# Locality of correlation in density functional theory

## Abstract

The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi (TF) approximation in the non-relativistic semiclassical (or large-Z) limit for all matter, i.e., the kinetic energy becomes local. Exchange also becomes local in this limit. Numerical data on the correlation energy of atoms support the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around a large particle number are equivalent to local density approximations and their strong relevance to density functional approximations. Analyzing highly accurate atomic correlation energies, we show that E{sub C} → −A{sub C} ZlnZ + B{sub C}Z as Z → ∞, where Z is the atomic number, A{sub C} is known, and we estimate B{sub C} to be about 37 mhartree. The local density approximation yields A{sub C} exactly, but a very incorrect value for B{sub C}, showing that the local approximation is less relevant for the correlation alone. This limit is a benchmark for the non-empirical construction of density functional approximations. We conjecture that, beyond atoms, the leading correction to the local density approximation in the large-Z limit generally takes this form, but with B{sub C} a functional of the TF density for themore »

- Authors:

- Department of Chemistry, University of California, Irvine, California 92697 (United States)
- Department of Physics and Astronomy, Ball State University, Muncie, Indiana 47306 (United States)
- Qld Micro- and Nanotechnology Centre, Griffith University, Nathan, Qld 4111 (Australia)
- CNR-Istituto di Nanoscienze, Via Campi 213A, I-41125 Modena (Italy)

- Publication Date:

- OSTI Identifier:
- 22679023

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 145; Journal Issue: 5; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; CATALYST SUPPORTS; DENSITY FUNCTIONAL METHOD; ELECTRON CORRELATION; KINETIC ENERGY; LOCALITY; NUMERICAL DATA; SEMICLASSICAL APPROXIMATION; THOMAS-FERMI MODEL

### Citation Formats

```
Burke, Kieron, Cancio, Antonio, Gould, Tim, and Pittalis, Stefano.
```*Locality of correlation in density functional theory*. United States: N. p., 2016.
Web. doi:10.1063/1.4959126.

```
Burke, Kieron, Cancio, Antonio, Gould, Tim, & Pittalis, Stefano.
```*Locality of correlation in density functional theory*. United States. doi:10.1063/1.4959126.

```
Burke, Kieron, Cancio, Antonio, Gould, Tim, and Pittalis, Stefano. Sun .
"Locality of correlation in density functional theory". United States.
doi:10.1063/1.4959126.
```

```
@article{osti_22679023,
```

title = {Locality of correlation in density functional theory},

author = {Burke, Kieron and Cancio, Antonio and Gould, Tim and Pittalis, Stefano},

abstractNote = {The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi (TF) approximation in the non-relativistic semiclassical (or large-Z) limit for all matter, i.e., the kinetic energy becomes local. Exchange also becomes local in this limit. Numerical data on the correlation energy of atoms support the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around a large particle number are equivalent to local density approximations and their strong relevance to density functional approximations. Analyzing highly accurate atomic correlation energies, we show that E{sub C} → −A{sub C} ZlnZ + B{sub C}Z as Z → ∞, where Z is the atomic number, A{sub C} is known, and we estimate B{sub C} to be about 37 mhartree. The local density approximation yields A{sub C} exactly, but a very incorrect value for B{sub C}, showing that the local approximation is less relevant for the correlation alone. This limit is a benchmark for the non-empirical construction of density functional approximations. We conjecture that, beyond atoms, the leading correction to the local density approximation in the large-Z limit generally takes this form, but with B{sub C} a functional of the TF density for the system. The implications for the construction of approximate density functionals are discussed.},

doi = {10.1063/1.4959126},

journal = {Journal of Chemical Physics},

number = 5,

volume = 145,

place = {United States},

year = {Sun Aug 07 00:00:00 EDT 2016},

month = {Sun Aug 07 00:00:00 EDT 2016}

}