# On extending Kohn-Sham density functionals to systems with fractional number of electrons

## Abstract

Here, we analyze four ways of formulating the Kohn-Sham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the Kohn-Sham and the generalized Kohn-Sham (GKS) non-interacting $$\mathcal{v}$$-representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting *N*-representable densities, (II) ensemble non-interacting *N*-representable densities, (III) non-interacting densities by the Janak construction, and (IV) non-interacting densities whose composing orbitals satisfy the Aufbau occupation principle. By proving the equivalence of the underlying first order reduced density matrices associated with these densities, we show that sets (I), (II), and (III) are equivalent, and all reduce to the Janak construction. Moreover, for functionals with the ensemble v-representable assumption at the minimizer, (III) reduces to (IV) and thus justifies the previous use of the Aufbau protocol within the (G)KS framework in the study of the ground state of fractional electron systems, as defined in the grand canonical ensemble at zero temperature. By further analyzing the Aufbau solution for different density functional approximations (DFAs) in the (G)KS scheme, we rigorously prove that there can be one and only one fractional occupation for the Hartree Fock functional, while there can be multiple fractional occupations for general DFAs in the presence of degeneracy. This has been confirmed by numerical calculations using the local density approximation as a representative of general DFAs. This work thus clarifies important issues on density functional theory calculations for fractional electron systems.

- Authors:

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

- Publication Date:

- Research Org.:
- Temple Univ., Philadelphia, PA (United States); 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:
- 1474043

- Alternate Identifier(s):
- OSTI ID: 1365425

- Grant/Contract Number:
- SC0012575; CHE-13-62927; DMS-14-54939

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 146; Journal Issue: 21; 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; 74 ATOMIC AND MOLECULAR PHYSICS; density-matrix; Kohn-Sham equation; Slater determinant; local density approximations; Hilbert space; density functional theory; Aufbau principle; correlation-consistent basis sets; self consistent field methods; statistical thermodynamics

### Citation Formats

```
Li, Chen, Lu, Jianfeng, and Yang, Weitao.
```*On extending Kohn-Sham density functionals to systems with fractional number of electrons*. United States: N. p., 2017.
Web. doi:10.1063/1.4982951.

```
Li, Chen, Lu, Jianfeng, & Yang, Weitao.
```*On extending Kohn-Sham density functionals to systems with fractional number of electrons*. United States. doi:10.1063/1.4982951.

```
Li, Chen, Lu, Jianfeng, and Yang, Weitao. Mon .
"On extending Kohn-Sham density functionals to systems with fractional number of electrons". United States.
doi:10.1063/1.4982951. https://www.osti.gov/servlets/purl/1474043.
```

```
@article{osti_1474043,
```

title = {On extending Kohn-Sham density functionals to systems with fractional number of electrons},

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

abstractNote = {Here, we analyze four ways of formulating the Kohn-Sham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the Kohn-Sham and the generalized Kohn-Sham (GKS) non-interacting $\mathcal{v}$-representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting N-representable densities, (II) ensemble non-interacting N-representable densities, (III) non-interacting densities by the Janak construction, and (IV) non-interacting densities whose composing orbitals satisfy the Aufbau occupation principle. By proving the equivalence of the underlying first order reduced density matrices associated with these densities, we show that sets (I), (II), and (III) are equivalent, and all reduce to the Janak construction. Moreover, for functionals with the ensemble v-representable assumption at the minimizer, (III) reduces to (IV) and thus justifies the previous use of the Aufbau protocol within the (G)KS framework in the study of the ground state of fractional electron systems, as defined in the grand canonical ensemble at zero temperature. By further analyzing the Aufbau solution for different density functional approximations (DFAs) in the (G)KS scheme, we rigorously prove that there can be one and only one fractional occupation for the Hartree Fock functional, while there can be multiple fractional occupations for general DFAs in the presence of degeneracy. This has been confirmed by numerical calculations using the local density approximation as a representative of general DFAs. This work thus clarifies important issues on density functional theory calculations for fractional electron systems.},

doi = {10.1063/1.4982951},

journal = {Journal of Chemical Physics},

number = 21,

volume = 146,

place = {United States},

year = {Mon Jun 05 00:00:00 EDT 2017},

month = {Mon Jun 05 00:00:00 EDT 2017}

}