On extending KohnSham density functionals to systems with fractional number of electrons
Here, we analyze four ways of formulating the KohnSham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the KohnSham and the generalized KohnSham (GKS) noninteracting $$\mathcal{v}$$representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting Nrepresentable densities, (II) ensemble noninteracting Nrepresentable densities, (III) noninteracting densities by the Janak construction, and (IV) noninteracting 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 vrepresentable 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:

^{[1]}
;
^{[2]}
;
^{[3]}
 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:
 Grant/Contract Number:
 SC0012575; CHE1362927; DMS1454939
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Chemical Physics
 Additional Journal Information:
 Journal Volume: 146; Journal Issue: 21; Journal ID: ISSN 00219606
 Publisher:
 American Institute of Physics (AIP)
 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) (SC22); National Science Foundation (NSF)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 74 ATOMIC AND MOLECULAR PHYSICS; densitymatrix; KohnSham equation; Slater determinant; local density approximations; Hilbert space; density functional theory; Aufbau principle; correlationconsistent basis sets; self consistent field methods; statistical thermodynamics
 OSTI Identifier:
 1474043
 Alternate Identifier(s):
 OSTI ID: 1365425
Li, Chen, Lu, Jianfeng, and Yang, Weitao. On extending KohnSham density functionals to systems with fractional number of electrons. United States: N. p.,
Web. doi:10.1063/1.4982951.
Li, Chen, Lu, Jianfeng, & Yang, Weitao. On extending KohnSham density functionals to systems with fractional number of electrons. United States. doi:10.1063/1.4982951.
Li, Chen, Lu, Jianfeng, and Yang, Weitao. 2017.
"On extending KohnSham 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 KohnSham 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 KohnSham (KS) density functionals with a fractional number of electrons, through extending the constrained search space from the KohnSham and the generalized KohnSham (GKS) noninteracting $\mathcal{v}$representable density domain for integer systems to four different sets of densities for fractional systems. In particular, these density sets are (I) ensemble interacting Nrepresentable densities, (II) ensemble noninteracting Nrepresentable densities, (III) noninteracting densities by the Janak construction, and (IV) noninteracting 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 vrepresentable 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 = {2017},
month = {6}
}