skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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:
ORCiD logo [1]; ORCiD logo [2];  [3]
  1. Duke Univ., Durham, NC (United States). Dept. of Chemistry
  2. Duke Univ., Durham, NC (United States). Dept. of Chemistry, Dept. of Mathematics and Dept. of Physics
  3. 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:
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 = {2017},
month = {6}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals
journal, November 2006

  • Ruzsinszky, Adrienn; Perdew, John P.; Csonka, Gábor I.
  • The Journal of Chemical Physics, Vol. 125, Issue 19
  • DOI: 10.1063/1.2387954

Communication: Two types of flat-planes conditions in density functional theory
journal, July 2016

  • Yang, Xiaotian Derrick; Patel, Anand H. G.; Miranda-Quintana, Ramón Alain
  • The Journal of Chemical Physics, Vol. 145, Issue 3
  • DOI: 10.1063/1.4958636

Challenges for Density Functional Theory
journal, December 2011

  • Cohen, Aron J.; Mori-Sánchez, Paula; Yang, Weitao
  • Chemical Reviews, Vol. 112, Issue 1
  • DOI: 10.1021/cr200107z

Density functionals for coulomb systems
journal, September 1983

  • Lieb, Elliott H.
  • International Journal of Quantum Chemistry, Vol. 24, Issue 3
  • DOI: 10.1002/qua.560240302

Variational Principle for Many-Fermion Systems
journal, February 1981


Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems
journal, February 2009


Many-electron self-interaction error in approximate density functionals
journal, November 2006

  • Mori-Sánchez, Paula; Cohen, Aron J.; Yang, Weitao
  • The Journal of Chemical Physics, Vol. 125, Issue 20
  • DOI: 10.1063/1.2403848

The Hartree-Fock theory for Coulomb systems
journal, February 1977

  • Lieb, Elliott H.; Simon, Barry
  • Communications in Mathematical Physics, Vol. 53, Issue 3
  • DOI: 10.1007/bf01609845

Fractional charge perspective on the band gap in density-functional theory
journal, March 2008


Assessment of Gaussian-2 and density functional theories for the computation of enthalpies of formation
journal, January 1997

  • Curtiss, Larry A.; Raghavachari, Krishnan; Redfern, Paul C.
  • The Journal of Chemical Physics, Vol. 106, Issue 3
  • DOI: 10.1063/1.473182

Fractional spins and static correlation error in density functional theory
journal, September 2008

  • Cohen, Aron J.; Mori-Sánchez, Paula; Yang, Weitao
  • The Journal of Chemical Physics, Vol. 129, Issue 12
  • DOI: 10.1063/1.2987202

Error bound for the Hartree-Fock energy of atoms and molecules
journal, July 1992

  • Bach, Volker
  • Communications in Mathematical Physics, Vol. 147, Issue 3
  • DOI: 10.1007/bf02097241

Insights into Current Limitations of Density Functional Theory
journal, August 2008


Consequences of extending 1‐matrix energy functionals from pure–state representable to all ensemble representable 1 matrices
journal, August 1980

  • Valone, Steven M.
  • The Journal of Chemical Physics, Vol. 73, Issue 3
  • DOI: 10.1063/1.440249

Fractional Electron Loss in Approximate DFT and Hartree–Fock Theory
journal, October 2015

  • Peach, Michael J. G.; Teale, Andrew M.; Helgaker, Trygve
  • Journal of Chemical Theory and Computation, Vol. 11, Issue 11
  • DOI: 10.1021/acs.jctc.5b00804