# On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals

## Abstract

In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.

- Authors:

- Duke Univ., Durham, NC (United States). Dept. of Chemistry
- Duke Univ., Durham, NC (United States). Dept. of Chemistry; South China Normal Univ., Guangzhou (China). Key Lab. of Theoretical Chemistry of Environment. School of Chemistry and Environment

- Publication Date:

- Research Org.:
- Duke Univ., Durham, NC (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); National Science Foundation (NSF)

- OSTI Identifier:
- 1465680

- Alternate Identifier(s):
- OSTI ID: 1361757

- Grant/Contract Number:
- SC0012575; CHE-13-62927

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 146; Journal Issue: 7; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; protons; Hilbert space; electron affinities; band gap; ground states; local density approximations; electron densities of states; many electron systems

### Citation Formats

```
Li, Chen, and Yang, Weitao.
```*On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals*. United States: N. p., 2017.
Web. doi:10.1063/1.4974988.

```
Li, Chen, & Yang, Weitao.
```*On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals*. United States. doi:10.1063/1.4974988.

```
Li, Chen, and Yang, Weitao. Tue .
"On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals". United States.
doi:10.1063/1.4974988. https://www.osti.gov/servlets/purl/1465680.
```

```
@article{osti_1465680,
```

title = {On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals},

author = {Li, Chen and Yang, Weitao},

abstractNote = {In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.},

doi = {10.1063/1.4974988},

journal = {Journal of Chemical Physics},

number = 7,

volume = 146,

place = {United States},

year = {Tue Feb 21 00:00:00 EST 2017},

month = {Tue Feb 21 00:00:00 EST 2017}

}

Works referenced in this record:

##
Density?functional thermochemistry. III. The role of exact exchange

journal, April 1993

- Becke, Axel D.
- The Journal of Chemical Physics, Vol. 98, Issue 7, p. 5648-5652