# Global and local curvature in density functional theory

## Abstract

Piecewise linearity of the energy with respect to fractional electron removal or addition is a requirement of an electronic structure method that necessitates the presence of a derivative discontinuity at integer electron occupation. Semi-local exchange-correlation (xc) approximations within density functional theory (DFT) fail to reproduce this behavior, giving rise to deviations from linearity with a convex global curvature that is evidence of many-electron, self-interaction error and electron delocalization. Popular functional tuning strategies focus on reproducing piecewise linearity, especially to improve predictions of optical properties. In a divergent approach, Hubbard U-augmented DFT (i.e., DFT+U) treats self-interaction errors by reducing the local curvature of the energy with respect to electron removal or addition from one localized subshell to the surrounding system. Although it has been suggested that DFT+U should simultaneously alleviate global and local curvature in the atomic limit, no detailed study on real systems has been carried out to probe the validity of this statement. In this work, we show when DFT+U should minimize deviations from linearity and demonstrate that a “+U” correction will never worsen the deviation from linearity of the underlying xc approximation. However, we explain varying degrees of efficiency of the approach over 27 octahedral transition metal complexesmore »

- Authors:

- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 22678941

- 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; AMMONIA; DENSITY FUNCTIONAL METHOD; ELECTRONIC STRUCTURE; ERRORS; EXPERIMENTAL DATA; OPTICAL PROPERTIES; TRANSITION ELEMENTS

### Citation Formats

```
Zhao, Qing, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Ioannidis, Efthymios I., and Kulik, Heather J., E-mail: hjkulik@mit.edu.
```*Global and local curvature in density functional theory*. United States: N. p., 2016.
Web. doi:10.1063/1.4959882.

```
Zhao, Qing, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Ioannidis, Efthymios I., & Kulik, Heather J., E-mail: hjkulik@mit.edu.
```*Global and local curvature in density functional theory*. United States. doi:10.1063/1.4959882.

```
Zhao, Qing, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Ioannidis, Efthymios I., and Kulik, Heather J., E-mail: hjkulik@mit.edu. Sun .
"Global and local curvature in density functional theory". United States.
doi:10.1063/1.4959882.
```

```
@article{osti_22678941,
```

title = {Global and local curvature in density functional theory},

author = {Zhao, Qing and Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Ioannidis, Efthymios I. and Kulik, Heather J., E-mail: hjkulik@mit.edu},

abstractNote = {Piecewise linearity of the energy with respect to fractional electron removal or addition is a requirement of an electronic structure method that necessitates the presence of a derivative discontinuity at integer electron occupation. Semi-local exchange-correlation (xc) approximations within density functional theory (DFT) fail to reproduce this behavior, giving rise to deviations from linearity with a convex global curvature that is evidence of many-electron, self-interaction error and electron delocalization. Popular functional tuning strategies focus on reproducing piecewise linearity, especially to improve predictions of optical properties. In a divergent approach, Hubbard U-augmented DFT (i.e., DFT+U) treats self-interaction errors by reducing the local curvature of the energy with respect to electron removal or addition from one localized subshell to the surrounding system. Although it has been suggested that DFT+U should simultaneously alleviate global and local curvature in the atomic limit, no detailed study on real systems has been carried out to probe the validity of this statement. In this work, we show when DFT+U should minimize deviations from linearity and demonstrate that a “+U” correction will never worsen the deviation from linearity of the underlying xc approximation. However, we explain varying degrees of efficiency of the approach over 27 octahedral transition metal complexes with respect to transition metal (Sc–Cu) and ligand strength (CO, NH{sub 3}, and H{sub 2}O) and investigate select pathological cases where the delocalization error is invisible to DFT+U within an atomic projection framework. Finally, we demonstrate that the global and local curvatures represent different quantities that show opposing behavior with increasing ligand field strength, and we identify where these two may still coincide.},

doi = {10.1063/1.4959882},

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}

}